Modeling and Suppressing Unwanted Parasitic Interactions in Superconducting Circuits

Xuexin Xu

Jülich Forschungszentrum & RWTH Aachen University

CHAPTER: ABSTRACT

Superconducting qubits are based on collective excitations in Josephson junctions. Their sufficiently high controllability and low-noise interaction with one another make them a viable candidate for implementing large-scale quantum computation. Although there are great improvements in the coherence of these qubits, the progression towards building a fault-tolerant quantum computer is still a major mission. This issue can be reflected in the imperfect gate fidelity. A source of such infidelity is the fundamental parasitic interaction between coupled qubits. This thesis deals with such spurious interaction in two- and three-qubit circuits. The parasitic interaction reflects a bending between computational and noncomputational levels. This bending creates a parasitic ZZ interaction. We first study the possibility of zeroing the ZZ interaction in two qubit combinations: a pair of interacting transmons, as well as a hybrid pair of transmon coupled to capacitively shunted flux qubit (CSFQ). We utilize our theory to accurately simulate experimental results taken by our collaborators from measuring a CSFQ-transmon pair in the absence and presence of cross-resonance (CR) gate.  The impressive agreements between our theory and experiment motivated us to study the characteristics of a CR gate that performs with 99.9% fidelity in the absence of static ZZ interaction. Since the CR pulse produces an additional ZZ component on top of the static part, we propose a new strategy for zeroing total ZZ interaction, and name it dynamical ZZ freedom. This freedom can exist in all-transmon circuits and allow to make perfect entanglement. Based on our findings, we propose a new two-qubit gate, namely the parasitic-free (PF) gate. Moreover, we show that we can take advantage of the ZZ interaction and make it stronger in order to perform a controlled-Z gate.  Finally, we study the impact of a third qubit on the two-qubit gate performance and discuss several examples to illustrate the properties of two-body ZZ and three-body ZZZ interactions in circuits with more than two qubits.

CHAPTER: ZUSAMMENFASSUNG

Supraleitende Qubits basieren auf kollektiven Anregungen in Josephson-Kontakten. Ihre ausreichend hohe Kontrollierbarkeit und rauscharme Wechselwirkung untereinander machen sie zu einem praktikablen Kandidaten für die Implementierung von Quantencomputern in großem Maßstab. Obwohl es große Verbesserungen in der Kohärenz dieser Qubits gibt, ist der Weg zum Bau eines fehlertoleranten Quantencomputers noch eine große Aufgabe. Dieses Problem kann sich in der unvollkommenen Gattertreue widerspiegeln. Eine Quelle solcher Untreue ist die fundamentale parasitäre Wechselwirkung zwischen wechselwirkenden Qubits. Diese Arbeit befasst sich mit dieser parasitären Wechselwirkung in Zwei- und Drei-Qubit-Schaltungen. Die parasitäre Wechselwirkung spiegelt eine Biegung zwischen rechnerischer und nicht rechnerischer Ebene wider. Diese Biegung erzeugt eine parasitäre ZZ-Wechselwirkung. Wir untersuchen zunächst die Möglichkeit, die ZZ-Wechselwirkung in zwei Qubit-Kombinationen auf Null zu setzen: ein Paar von wechselwirkenden Transmonen sowie ein hybrides Paar von Transmonen, die an ein kapazitiv geshuntetes Flux-Qubit (CSFQ) gekoppelt sind. Wir nutzen unsere Theorie, um die experimentellen Ergebnisse unserer Kollaborateure bei der Messung eines CSFQ-Transmon-Paares in Abwesenheit und Anwesenheit eines Cross-Resonance (CR) Gatters genau zu simulieren.  Die beeindruckenden Übereinstimmungen zwischen unserer Theorie und dem Experiment motivierten uns, die Eigenschaften eines CR-Gatters zu untersuchen, das mit 99,9% Genauigkeit in Abwesenheit der statischen ZZ-Wechselwirkung arbeitet. Da der CR-Puls zusätzlich zum statischen Teil eine weitere ZZ-Komponente erzeugt, schlagen wir eine neue Strategie zur Aufhebung der gesamten ZZ-Wechselwirkung vor und bezeichnen diese als dynamische ZZ-Freiheit. Diese Freiheit kann in All-Transmon-Schaltkreisen existieren und erlaubt es perfekte Verschränkung zu erzeugen. Basierend auf unseren Erkenntnissen schlagen wir ein neues Zwei-Qubit-Gatter vor, nämlich das parasitenfreie (PF) Gatter. Außerdem zeigen wir, dass wir die ZZ-Wechselwirkung ausnutzen und verstärken können, um ein kontrolliertes Z-Gatter zu realisieren.  Schließlich untersuchen wir den Einfluss eines dritten Qubits auf das Verhalten des Zwei-Qubit-Gatters und diskutieren mehrere Beispiele, um die Eigenschaften der Zwei-Körper-ZZ- und Drei-Körper-ZZZ-Wechselwirkungen in Schaltungen mit mehr als zwei Qubits zu veranschaulichen.

CHAPTER: INTRODUCTION

§ QUANTUM COMPUTATION

Quantum computers develop computation technology based on quantum mechanical phenomena, e.g. superposition, entanglement and interference. The field of quantum computing can be traced back to 1980's. Paul Benioff first put forward a theoretical demonstration of a quantum mechanical prototype of the Turing machine <cit.>, but it was not computationally powerful than a conventional classical one. Two years later, Richard Feynman proposed the first nontrivial application of quantum physics and believed that it can be only simulated on quantum computers <cit.>.  Subsequently, David Deutsch developed this idea of applying quantum mechanics to solve computational problems in computer science <cit.>.  However, these nice ideas destined to remain on paper until 1994 when Peter Shor found the first potentially useful way to use a hypothetical quantum computer, and invented the polynomial-time quantum computer algorithm for integer factorization <cit.>. Another remarkable algorithm was discovered by Lov Grover for searching an unsorted database <cit.>. Both of them provide a superior speed up over classical counterpart.

The laws of quantum mechanics enable to do more than encoding what can be done with bit. They help to solve problems which no classical computer can solve in any feasible amount of time. This ability is known as quantum supremacy or quantum advantage <cit.>. Then it comes to the question, how to perform quantum computing on a certain physical system? About twenty years ago, David DiVincenzo organized a set of characteristics for the physical realization of a quantum computer, known as DiVincenzo's criteria <cit.>, and they will be briefly reviewed here.

§.§ What is a qubit?

In contrast to classical computers which use bits, quantum computing merges two great scientific revolutions of the 20th century: computer science and quantum physics, and uses the quantum states of an object to produce what's called “quantum bit”, namely qubit. A well-characterized qubit should have a distinguished two-level system from higher energy levels.
The two energy levels of a physical qubit are often defined as ground state $|0\rangle$  and excited state $|1\rangle$, which are two orthogonal vectors in a two-dimensional vector space. The energy difference between the two states is referred to as the qubit frequency by assuming the reduced Planck constant $\hbar\equiv1$.

Similar to classical computers, physical qubits should have the ability to be scaled up to any numbers.  A logical qubit used for programming, consisting of one or more physical qubits, can be in a superposition, partly in $|0\rangle$ and partly in $|1\rangle$ state. There are hundreds of physical systems which have well-defined two level system, and are easily scalable. These quantum systems can be classified in terms of the qubit characteristics. For instance, qubits can be based on atomic physics such as trapped ions and neutral atoms. Trapped ions are confined and suspended in free space using electromagnetic fields <cit.>, while neutral atoms are realized in optical or magnetic traps encoded in hyperfine-Zeeman ground substates <cit.>.

Other popular qubits are based on solid-state systems that are integrated into a solid material with the nanofabrication techniques. Qubits in solid state systems take advantage of spin states or charge states such as semiconductor qubits <cit.>, superconducting qubits <cit.> and topological qubits <cit.>.
Beyond natural and artificial atomic systems, photonics is also a promising platform and can be manipulated to create a quantum processor in semiconductor chips, e.g. Silicon chip <cit.>, making possible photon-photon interactions in a compact chip-integrated device.

§.§ How are qubit states reset?

Registers in classical computers should be initialized to a known value before starting computation. Equivalently in quantum computers qubit states should be initialized usually to ground state. During quantum operations qubits heat up. Therefore in order to be able to run quantum algorithms several times one after the other, one needs a quick cooling method, i.e. resetting. This can be realized by natural cooling, also called passive cooling, with the residual coupling to the environment. Although passive cooling benefits in the energy efficiency and lower financial cost, this protocol is inherently time consuming. The reason is that passive cooling transfers heat from qubit systems to the environment at the speed of relaxation rate, which is always minimized to lower the probability of errors in a coherent quantum computation. Another approach is active cooling which becomes attractive by relying on an external approach to enhance the heat flow. For example,  by applying laser-based Doppler-cooling the effective temperature of trapped ions  was rapidly reduced <cit.>; driving qubits can induce subsequent relaxation into the ground state for in superconducting circuits <cit.>. Other promising tools like quantum-circuit refrigerator (QCR) <cit.> are also demonstrated to be able to speed up qubit initialization.

§.§ How long does a qubit last?

Quantum systems cannot live long due to irreversible heat and quasiparticle exchange with environment, which is characterized by coherence timecoherence time <cit.>. Environment surrounding a qubit cannot be directly controlled or fully isolated, therefore the qubit-environment interaction can produce undesirable quantum state changes. Bloch-Redfield theory describes the time evolution of such qubit-environment coupling system with Markovian approximation and shows that decoherence can be represented on the Bloch sphere using longitudinal relaxation, transverse relaxation and pure dephasingdephasing <cit.>,  as shown in Fig. <ref>.

(-355,110)(a) Longitudinal relaxation
(-220,110)(b) Pure dephasing
(-110,110)(c) Transverse relaxation

(a) Longitudinal relaxation (b) pure dephasing (c) transverse relaxation on the Bloch sphere.

Longitudinal relaxation arises from the energy exchange between qubit and environment with transverse noise coupled to the qubit in the $x-y$ plane,  while pure dephasing results from longitudinal noise along the $z$ axis. Combination of these two losses leads to the transverse relaxation, which loses the orientation of the original trajectory. The two relaxations characterize coherence times $T_1$ and $T_2$ as

\begin{eqnarray}
\Gamma_1&\equiv&\frac{1}{T_1},\\
\Gamma_2&\equiv&\frac{1}{T_2}=\frac{\Gamma_1}{2}+\Gamma_{\varphi}.
\end{eqnarray}

$T_1$ is the time required by a qubit to relax from the first excited state to the ground state. $T_2$ is the average time in which the energy-level splitting remains unchanged. To perform high-fidelity quantum computing, coherence times of logical qubits must be longer than the operation time. Among physical qubits, trapped ions can last longer, e.g. up to a few hundred seconds or even a couple of minutes, by suppressing the limiting factors such as critical current noise <cit.>, magnetic-field fluctuations <cit.>, frequency instability and leakage of the microwave reference-oscillator <cit.>.  In superconducting qubits  the lifetime $T_1$ in the past ten years for a few microseconds, has been improved to around 100 $\rm{\mu s}$ <cit.>, and most recently exceeds 300 $\rm{\mu s}$ by replacing fabricating material niobium with tantalum <cit.>.
In semiconductor spin qubits, the relatively short coherence times of 0.1 to 1 $\rm{\mu s}$  <cit.>, has been improved by five orders of magnitude <cit.>.

§.§ What is a universal qubit gate set?

Quantum gates are basic operations on a small number of qubits. They are the building blocks of quantum circuits, like XOR gates for conventional digital counterpart. Unlike many classical logic gateslogic gate, quantum gates are reversible. Such reversible implementations can be expressed using a special class of logic gates, known as universal sets. In order to perform arbitrary unitary operationsunitary operator, it is sufficient to have the ability of performing all members of a universal gateuniversal quantum gates set. The Solovay-Kitaev theorem <cit.> shows that it is possible to define a universal gate set with a discrete number of elements that can approximate any unitary with arbitrary precision, and the following gate sets can be  easily proved universal: the set of controlled NOT (CNOT) gate and all single qubit gates; the set of CNOT gate, Hadamard gate and suitable phase flips; the set of three-qubit Toffoli gate and Hadamard gate.

In reality, most of the physical systems proposed so far consider only two-qubit unitary transformations. Therefore realizing high-performance single qubit gates and CNOT gate is crucially important for quantum algorithms.

Single qubit gates correspond to rotations of a spin about the Cartesian axis, and can be implemented directly with external driving pulses. Usually quantum gates are written as Pauli matrices, e.g. several commonly used quantum gates are

\begin{equation}
\rm X=\left[\begin{array}{c c}
\end{array}\right],
\rm Y=\left[\begin{array}{c c}
\end{array}\right],
\rm Z=\left[\begin{array}{c c}
\end{array}\right],
\rm S=\left[\begin{array}{c c}
\end{array}\right],
\end{equation}

which change the amplitudes or phase of the states $|0\rangle$ and $|1\rangle$. Any single qubit operation can be decomposed into three Bloch sphereBloch sphere rotations and a global phase factor, so it is easy to perform in experiments. To date single-qubit gate can be realized accurately enough for fault-tolerant quantum computing <cit.>.

A CNOT gate is the controlled version of the X gate and performs a selective negation of the target qubit as

\begin{equation}
\rm CNOT=\left[\begin{array}{cccc}
\end{array}\right].
\end{equation}

Construction of CNOT gates varies in gate decomposition and qubits of physical properties. For instance, in trapped ions the CNOT gate can be realized by using a linear trap and laser beams <cit.>; in silicon quantum dots, it can be completed by efficiently resonant drive <cit.>. The challenge to such  a gate is that two-qubit gates are not accurate enough for quantum error correction although these years some progresses have been made <cit.>. More improvement is needed to suppress error rates such that high-fidelity quantum gates are realized.

§.§ Measurement

Measuring quantum states determines computational results projected on the “classical” world. There are three required subsystems for such measurementmeasurement: (1) Quantum system whose properties are remaining unknown; (2) Measurement apparatus, behaves as a classical system that can interact with qubits; (3) Environment surrounding the apparatus whose presence supplies the decoherence.  One way to describe quantum measurements is to project the eigenstates of measurable quantities. A Hermitian measurement operator carries eigenvalues corresponding to the outcomes of measurement. For a mixed state, the unconditioned final state of the system is a classical mixture of the output states of all possible measurement outcomes. However, such projective measurements are too restrictive and can change or destroy the characteristics of quantum systems.

More generalized quantum measurements are so called Positive Operator-Valued Measure (POVM)POVM, which can be viewed as a collection of positive Hermitian operators. The POVM process is a statistical measurement since each operator corresponds to an outcome of the measurement with a real and non-negative probability. This measurement is not repeatable, meaning that one can find different results when repeating the measurement which is independent of a particular realization. These properties make POVM the most general measurement one can perform on a quantum system. For instance, it can be used to measure phonon fluctuation in the ground state <cit.> and hyperfine qubit levels <cit.> with trapped ions, perform state tomography of an exchange-only spin qubit <cit.>, entangle massive fermionic qubits <cit.> and also study the characteristics of superconducting qubits <cit.>.

§ SUPERCONDUCTING QUANTUM COMPUTING

Superconducting qubits have become one of most promising candidates that satisfy  DiVincenzo's criteria and can implement large-scale quantum computing. This type of qubits is macroscopic in size and lithographically defined with microfabrication technology, making it easy to build and capacitively or inductively coupling to the others. Frequency domain of the superconducting qubits allows that the state of a qubit can be changed by existing commercial microwave devices and equipment.

Superconducting quantum computing is an implementation of quantum computers based on superconducting circuits. Here an arbitrary single qubit gate is achievable by external microwave drive. Two-qubit gates such as CNOT gates are realized by combining cross-resonance gate <cit.> or microwave-activated phase (MAP) gate <cit.> with single qubit rotations. The major milestone was the demonstration of the first quantum supremacy on a 53-qubit chip <cit.>.

However, there are yet outstanding challenges against high-performance quantum computing <cit.>. Aside from the limited coherence timescoherence time, the always-on parasitic interaction among qubits does affect the quality of multi-qubit gates. In such circuits, qubits are coupled to one another via shared couplers. Although the nonlinear circuit element Josephson junctionsJosephson junction can approximate superconducting qubits as a two-level system, transitions between computational and noncomputational subspaces inevitably take place. In some cases, this interaction is wanted and should be strengthened to create desired entanglement. But most of time it is unwanted, and
sets barriers to reach the error correction threshold. More precisely, the parasitic interaction accumulates phase error in the computational states, and eventually destroys the multi-qubit gates. Therefore, it must be carefully suppressed during the gate operations.

Particularly, the parasitic ZZ interaction between a pair of superconducting qubits is a limiting factor for two-qubit gates and quantum error correction. Although the ZZ interaction is always one or two orders of magnitude weaker than the coupling strength, it obviously degrades the performance of many quantum gates <cit.>. Furthermore, driven two-qubit gates suffer from additional ZZ component produced by external pulses <cit.>. Therefore zeroing qubits from the undesirable ZZ interaction is highly demanded to boost up the gate quality, and is an important step toward bringing down the overhead of physical qubits.
Specially, for a few gates such as controlled-Z (CZ) gates, the ZZ interaction is wanted, then we should take advantage of it and make it stronger <cit.>.

The main goal of this book is to improve two-qubit gate performance by either mitigating unwanted parasitic ZZ interaction, or improving wanted ZZ interaction, whenever the qubits are idle or driven by external pulses. We aim to develop a theory that can describe the characteristics of ZZ interaction in different circuits. Beyond that, we will also generalize our theory of parasitic interaction to a multi-qubit scheme, and find out the difference from the two-body interaction.

§ OUTLINE

In chapter <ref> we present generalized formalism of circuit quantization from corresponding Lagrangian and Hamiltonian. In particular, we introduce the basic elements of the superconducting circuit, e.g. the LC resonator and Josephson junctions, and describe several types of qubits in details such as transmon and capacitively shunted flux qubit (CSFQ), as well as their variations. After highlighting the Schrieffer-Wolff transformation, we move to the core part of the Hamiltonian model — circuit quantum electrodynamics (circuit QED) by taking several examples.

In chapter <ref>, we theoretically analyze the origin of static ZZ interaction using different theory models. We explore the possibility of static ZZ freedom in two types of devices:  CSFQ-transmon pair and transmon-transmon pair coupled via a harmonic bus resonator, and derive the condition of zeroing static ZZ interaction for both pairs. We apply our theory to simulate the experimental hybrid CSFQ-transmon circuit, which is fabricated at IBM and measured at Syracuse university by our collaborators. We calculate qubit and resonator frequencies and extract their coupling strengths as well as static ZZ interaction versus flux threading the CSFQ.

Following the experiment, in chapter <ref> we explain what a cross-resonance (CR) gate is and how it is realized. We study the error source to the gate like classical crosstalk beyond the intrinsic noise, and show how the errors impact the CR gate and what we can do to improve the gate fidelity.  Our theory predicts the prerequisite to achieve high-performance CR gate with static ZZ freedom. Inspired by the experiment, we keep studying the inherent symmetry in a driven system, and demonstrate the presence of dynamical ZZ freedom.

Chapter <ref> introduces some novel gates based on the study of the tunable coupler. We first propose a parasitic free (PF) gate by cancelling unwanted ZZ interaction and suppressing external flux noise. Beyond that, we take advantage of the wanted ZZ interaction
to perform a CZ gate, and propose a scheme where the CZ gate can be fully turned off by tuning qubit frequency.

Chapter <ref> presents preliminary results of three-qubit interaction in two setups: triangle geometry and qubit chain. By calculating Pauli coefficients in the three-qubit computational subspace, we study characteristics of two-body ZZ and three-body ZZZ interactions, and take several examples to illustrate them.

CHAPTER: SUPERCONDUCTING QUBITS AND CIRCUIT QED

Superconducting qubits have been proved to be versatile on a macroscopic scale for the study of quantum technology, such as quantum computing <cit.>, quantum simulation <cit.>, quantum sensing <cit.>, and quantum information processing <cit.>.
Due to the amazing properties of the nonlinear Josephson junctionsJosephson junction at sufficiently low temperatures, superconducting qubits act as natural atoms with discrete level structures, and exhibit extraordinary properties on integration, readout and tunability. Over the past decades, this type of qubits has been become a leading candidate for realizing building blocks of a quantum computer.

Generally speaking, there are three classes of superconducting qubits based on the relevant degrees of freedom: charge qubit <cit.>, flux qubit <cit.> and phase  qubitphase qubit <cit.>. However these qubits suffer from either charge or flux noise. To reduce the sensitivity to these noises, transmon <cit.> and capacitively shunted flux qubit (CSFQ) <cit.> have been designed theoretically and tested experimentally by adding a large capacitance shunted to one Josephson junction, and nowadays widely used in the lab.

Moreover, superconducting quantum circuits have led to circuit quantum electrodynamics (circuit QEDcircuit QED), which provides an approach to study the fundamental light-matter interaction in the solid-state platform. In contrast to cavity quantum electrodynamics (cavity QEDcavity QED), quantized electromagnetic fields in the microwave frequency domain are stored in a on-chip resonator and superconducting qubits behave as quantum objects.
The artificial atom–field coupling can be easily increased to hundreds of megahertz, and then stepped into the strong coupling regime <cit.>. This results in faster operation since the transition dipole is mainly determined by the qubit geometry. Circuit QED now is a very active research direction as well as a leading architecture for quantum computation. Beyond superconducting qubits, circuit QED can also deal with hybrid quantum systems at its origin, losing their individual identity and combine into something new <cit.>.

This chapter starts with introduction of the basic concepts of superconducting qubits and circuit QED, including circuit quantization, Josephson effect and several types of superconducting qubits mainly focusing on transmon and CSFQ. By means of the  perturbative approach, we move to the regime where the qubit-resonator detuning is larger than coupling strength, namely dispersive regimedispersive regime, and study the models of one qubit coupled to a harmonic oscillator, and two qubits coupled via a bus resonator.

§ CIRCUIT QUANTIZATION

In this section we briefly introduce how to quantize a non-dissipative electrical circuit, most of the discussion below is from Ref. <cit.>. The dimension of a physical circuit is small enough so that electromagnetic waves propagate across the circuit “almost” instantaneously, this is so called lumped approximation. Within this consideration, a circuit can be described as a network of elements meeting at nodes, and elements between two nodes construct a branch as shown in Fig. <ref>.

An example of electrical circuit consists of several loops, same-species elements meet at active nodes while different species elements meet at passive modes.

For each branch at time $t$ two independent variables voltage $v(t)$ and current $i(t)$ flow through it. Conventionally absorbed energy by an element $E=\int i(t)v(t)dt$ is assumed to be negative, in other words at $t=-\infty$ both voltage and current are zero. Then the two vectors should have opposite sign. To proceed Hamiltonian description of electrical circuits, we generalize the definition of fluxes and charges which are derived from branch voltages and currents by

\begin{eqnarray}\label{eq.fluxcharge}
\end{eqnarray}

A dispersive element for which the voltage $v(t)$ is only a function of the charge $Q(t)$ and not directly of the time $t$ or any other variables, is said to be a capacitive element. Similarly, a dispersive element for which the current $i(t)$ is only a function of the flux $\Phi(t)$  is said to be an inductive element. For linear capacitance and inductance, the relation is given by $v(t)=Q(t)/C$ and $i(t)=\Phi(t)/L$, and energy stored in these basic elements can also  be calculated as  $Q^2(t)/2C$ and $\Phi^2(t)/2L$, respectively. Before finding the degrees of freedom of an arbitrary conservative circuit, one should take into account the build-in constitutive relations at each node and loop

\begin{eqnarray}\label{eq.kirchhoff}
&&\sum_k i_k(t)=0 \hspace{0.03in}\qquad\quad\qquad {\textrm {The Kirchhoff's current law}},\\
&&\sum_k v_k(t)=0 \qquad\quad\quad\quad {\textrm {The Kirchhoff's voltage law}},
\end{eqnarray}

which is the famous Kirchhoff's law. The two equations above correspond to two standard methods to write the classical Hamiltonian: the node method and the loop method. In the next, we only show the node method which is applicable for most practical problems.

With the above background, we summarize the following  steps to write a Hamiltonian from the electrical circuit:

1. Separate the circuit into capacitive sub-network and inductive sub-network. Inductances and capacitances meet at active nodes, while only capacitances or only inductances converge at passive nodes.

2. Name a particular active node `‘ground”, and choose the set of branches that connect the ground via capacitances to every other active or passive node without forming any loops, this is so called spanning tree method.

3. Classify inductive energy into potential energy $U$ and capacitive energy into kinetic energy $T$, here one can rewrite the kinetic energy in terms of flux by noticing that $v=\dot\phi$ with $\phi$ being the node flux.

4. Write the Lagrangian of the network as a function of fluxes at nodes in the form of $\mathcal{L}(\cdots,\phi_i,{\dot\phi}_i,\cdots)=T-U$, one can check that by applying Lagrange’s equations.

5. Find the generalized momentum (node charge) which is canonical conjugate of node fluxes, defined as $q_i=\partial \mathcal{L}/\partial{\dot\phi}_i$;

6. Express the Hamiltonian in terms of the conjugate pair $q_i$ and $\phi_i$ by performing Legendre transform $H=\sum_iq_i\dot{\phi_i}-\mathcal{L}$.

Replace the variables with corresponding operators by straightforward adding a hat
$\phi\rightarrow\hat\phi$ and $q\rightarrow\hat q$ which satisfies the commutator relation $[\hat\phi,\hat q]=i\hbar$, description of electrical circuit is changed from classical to quantum regime. For simplicity, the hats on operators can be dropped.

§ LC RESONATOR

The transmission line resonator (TLR) <cit.> is a narrow one-dimensional device consisting of a full-wave section of superconducting coplanar waveguide. Each mode of the TLR is an independent harmonic oscillator. This 1D transmission line can be expressed by a chain of LC oscillators exhibiting standing wave resonances with less small loss, making it possible to readout and measure a circuit with high fidelity. In the vicinity of fundamental frequencies the line can be used as a LC circuit shown in Fig. <ref>.

Isolated LC resonator with kinetic energy $T$ stored in the inductor and potential $U$ stored in the capacitor.

The LC circuit is an idealized model of zero resistance since it assumes that currents can flow essentially without dissipation. The only one degree of freedom current, flows in the wire of the inductor and accumulates charge on the capacitor. A LC resonator only contains one inductor and one capacitor, so the Lagrangian can be easily obtained following steps described in the circuit quantization

\begin{equation}
\mathcal{L}=T-U=\frac{1}{2}C{\dot\phi}^2-\frac{\phi^2}{2L},
\end{equation}

with $\phi$ being flux threading the inductor. Conjugate term charge $q$ can be derived from $q=\partial \mathcal{L}/\partial\dot\phi$.
By using Legendre transformation, the Hamiltonian of a LC resonator is written as

\begin{equation}\label{eq.LC}
\end{equation}

Let us introduce standard creation and annihilation operators $r$ and $r^{\dagger}$ satisfying the commutation relation $[r,r^{\dagger}]=1$, and express them in the form of

\begin{equation}\label{eq.Lca}
\begin{split}
\phi&=\phi_{\rm{ZPF}}(r+r^{\dagger}),\\
q&=-i q_{\rm{ZPF}}(r-r^{\dagger}),
\end{split}
\end{equation}

where zero point fluctuation is expressed as $\phi_{\rm{ZPF}}=\sqrt{\hbar Z_0/2}$ and $q_{\rm{ZPF}}=\sqrt{\hbar/2Z_0}$ with $Z_0=\sqrt{L/C}$ being the characteristic impedance. The zero point energy is the lowest possible energy described by Heisenberg uncertainty principle such that $\phi_{\rm ZPF}q_{\rm ZPF}=\hbar/2$. Hamiltonian then is quantized as a harmonic oscillator

\begin{equation}\label{eq.QLC}
\end{equation}

whose eigenstates are the Fock states $|n\rangle$, satisfying $r=\sum_n\sqrt{n+1}|n\rangle\langle n+1|$, $r^{\dagger}=\sum_n\sqrt{n+1}|n+1\rangle\langle n|$ and $r^{\dagger}r\left|n\right>=n\left|n\right>$. Qubit frequency is written as $\omega_0=1/\sqrt{LC}$. In the rest of this manuscript, $1/2$ corresponding to zero-point energy is dropped.

§ JOSEPHSON JUNCTION

A Josephson junctionJosephson junction (JJ) is a quantum mechanical device which consists of two narrow superconductors separated by a thin insulator as shown in Fig. <ref>.
In superconductors a pair of electrons are attractively bound and behave more or less like particles, namely Cooper pairs, and nearly all of the pairs will be locked down at the lowest energy in exactly the same state. However,
if the insulator is thin enough there is a probability for electron pairs to tunnel. This effect later became known as Josephson tunneling.

Sketch of a Josephson junction.

Based on Ginzburg–Landau theory, in the superconducting state the wave function $\Psi_L$ is the amplitude to find all electrons on the left side, and $\Psi_R$ is the corresponding function to find it on the right side. By substituting two eigenvectors $\Psi_L=\sqrt{n_L}e^{i\varphi_L}$ and $\Psi_R=\sqrt{n_R}e^{i\varphi_R}$ into the Schrödinger equation across the junction, one can show

\begin{eqnarray}
i\hbar\frac{\partial}{\partial t}\sqrt{n_L}e^{i\varphi_L}&=&U_L\sqrt{n_L}e^{i\varphi_L}+K\sqrt{n_R}e^{i\varphi_R},\\
i\hbar\frac{\partial}{\partial t}\sqrt{n_R}e^{i\varphi_R}&=&U_R\sqrt{n_R}e^{i\varphi_R}+K\sqrt{n_L}e^{i\varphi_L},
\end{eqnarray}

where $\varphi_L$ and $\varphi_R$ are the phases on the two sides of the junction and $n_L$, $n_R$ are the density of electrons at those two points. $K$ is a coupling constant for the wave functions across the barrier. If $K$ were zero, these two equations would describe the lowest energy state with energy $U$ of each superconductor.  The two equations can also be simplified as

\begin{eqnarray}
\frac{\partial n_L}{\partial t}&=&-\frac{\partial n_R}{\partial t}=\frac{2K}{\hbar}\sqrt{n_L n_R}\sin\varphi,\\
\frac{\partial \varphi}{\partial t}&=&-\frac{U_R-U_L}{\hbar},
\end{eqnarray}

with $\varphi=\varphi_R-\varphi_L$ being the phase difference across the junction. The first equation says what $n_L$ and $n_R$ would be if there were no extra electric forces due to an unbalance between the electron fluid and the background of positive ions. It tells how the densities would start to change, and therefore describe the kind of current that would begin to flow. Let us suppose the electric potential difference across the junction is $V$, then the energy difference between the two superconductors is $U_R-U_L=-2eV$. If the two superconductors are made of the same materials, then charge carrier density becomes universal such that $n_L=n_R=n$. Note that supercurrent $I$ is proportional to the time derivative of charge carrier density $n$. Simplifying the solution of Schrödinger equation yields the following relations

\begin{eqnarray}\label{eq.JJ}
I_s&=&I_c\sin\varphi \qquad\qquad \textrm{The first~Josephson~relation,}\\
\frac{\partial\varphi}{\partial t}&=&\frac{2eV}{\hbar} \quad\qquad\qquad  \textrm{The second~Josephson~relation,}
\end{eqnarray}

where $I_s$ is the supercurrent, $I_c$ proportional to $n$ is the critical current.

Magnetic flux quantum is defined as $\Phi_0=h/2e$, therefore the voltage across the junction can be defined as $V=d\Phi/dt$ which is analogous to Faraday's law with $\Phi=(\Phi_0/2\pi)\varphi$ being the flux. The energy stored in the JJ is

\begin{equation}\label{eq.Jenergy}
U=\int I_s V dt=\int I_c \sin\varphi d\Phi=-E_J\cos\varphi,
\end{equation}

with $E_J=I_c\Phi_0/2\pi$ being called the Josephson energy.

§ TRANSMONS

Connecting the JJ in series with a capacitor and a voltage source constitutes an artificial two-level electronic system: a single Cooper-pair box (CPB) also called charge qubitcharge qubit <cit.> as shown in Fig. <ref>.

Circuit scheme of the charge qubit with external voltage source $V_g$.

One can take the node flux and write the circuit Lagrangian as

\begin{equation}
\mathcal{L}=\frac{C_J}{2}\dot\Phi^2+\frac{C_g}{2}(\dot\Phi-V_g)^2+E_J\cos\Phi/\Phi_0,
\end{equation}

with which conjugate variable of the flux is

\begin{equation}
Q=\frac{\partial\mathcal{L}}{\partial \dot\Phi}=(C_J+C_g)\dot\Phi-C_gV_g.
\end{equation}

In this case the Hamiltonian of the system is given by

\begin{equation}\label{eq.HtrQ}
H=\frac{({Q}-C_g V_g)^2}{2C_\Sigma}-E_J\cos{\Phi}/\Phi_0,
\end{equation}

where $Q=2en$ is the charge with $n$ being the Cooper pair numbers and $C_\Sigma=C_J+C_g$. The Hamiltonian can also be rewritten in terms of dimensionless canonical variables $n=Q/2e$ and $\varphi=\Phi/\Phi_0$ with ${{\varphi}}\in [0,2\pi]$

\begin{equation}\label{eq.Htrn}
\end{equation}

where $E_C=e^2/2C_\Sigma$ is the charging energy and  $n_g=C_g V_g/2e$ is the normalized gate induced charge.

§.§ Fixed-frequency transmons

The most popular evolution of the charge qubit is transmon in which CPB is shunted by an additional large capacitance as shown in Fig. <ref>.

Circuit scheme of the transmon qubit with shunting capacitance $C_s$.

This Hamiltonian is exactly the same as charge qubit with a new $C_{\Sigma}=C_s+C_J+C_g$. The difference is that large shunting capacitance can dramatically suppress charging energy. Namely  CPB operates in the limit $E_J/E_C\ll1$ whereas the transmon has $E_J/E_C\gg1$. Later we will show how the ratio makes the transmon exponentially less sensitive to charge noise.

To solve the eigenvalues of the Hamiltonian in Eq. (<ref>), Schrödinger equation in the phase basis is written as <cit.>

\begin{equation}\label{eq.schrn}
\left[4E_C(-i\frac{\partial}{\partial\varphi}-n_g)^2-E_J\cos{\varphi}\right]\psi_k(\varphi)=E_k\psi_k(\varphi),
\end{equation}

where $\psi_k(\varphi)$ is the eigenstate which satisfies the periodic condition $\psi_k(\varphi)=\psi_k(\varphi+2\pi)$. By multiplying both sides of Eq. (<ref>) with $\exp{(-i n_g \varphi)}$ and assuming that $\Psi_k(\varphi)=\exp{(-i n_g \varphi)}\psi_k(\varphi)$, $n_g$ is absorbed into the new eigenstate, the equation above then is simplified to

\begin{equation}\label{eq.ms}
\left[-4E_C \frac{\partial^2}{\partial\varphi^2}-E_J\cos\varphi\right]\Psi_k(\varphi)=E_k\Psi_k(\varphi).
\end{equation}

This equation can be simply recast to the standard Mathieu equation, with the following solutions <cit.>

\begin{align}\label{eq.mathieu}
\end{align}

where ${\mathscr{M}}_A$ is Mathieu's characteristic value, int$(x)$ rounds to the integer closest to $x$, $a$ mod $b$ denotes the usual modulo operation, and $a$ div $b$ gives the integer quotient of $a$ and $b$. Note that eigenvalues only depend on the normalized gate charge $n_g$ and the ratio of $E_J$ and $E_C$.

The lowest three eigenvalues as a function of $n_g$ for several $E_J /E_C$ are shown in Fig. <ref>. One can see the anharmonic levels rely on the ratio of $E_J/E_C$ with anharmonicity defined by $\delta=E_{12}-E_{01}$.  Increasing $E_J /E_C$ definitely suppresses the sensitivity of the system with respect to offset charge $n_g$, finally resulting in a fixed frequency transmon as shown in Fig. <ref>(d). Large ratio and sufficient anharmonicity makes a transmon operationally a suitable qubit, since levels can be individually addressed with different frequencies.

In order to further quantify the dependency of energy levels on the ratio, we define the peak-to-peak charge dispersion with $\epsilon_{ij}=E_{ij}(n_g=1/2)-E_{ij}(n_g=0)$. Both anharmonicity and charge dispersion are plotted in Fig. <ref>. It shows that energy levels become less sensitive to the offset charge $n_g$ as $E_J/E_C$ increases, which is consistent with Fig. <ref>. In fig. <ref>(b) the relative anharmonicity $\delta/E_C$ is negative and can be approximated to $\delta\approx-E_C$, a result that we will derive via other approaches in the next section.

The first three eigenvalues of the Hamiltonian Eq. (<ref>) versus the offset $n_g$, at different $E_J/E_C$. Here $E_{10}$ is evaluated at the degeneracy point $n_g=0.5$.

(a) Charge dispersion (b) relative anharmonicity as a function of  $E_J/E_C$ at $n_g=1/2$.

§.§ Flux tunable transmons

Due to charge-insensitive property, transmon has been employed as a central component of several scalable platforms <cit.>, with applications to a wide range of quantum computation and quantum information processing. The transmon discussed in Fig. <ref> was frequency fixed, here we introduce another type of transmon whose frequency is tunable <cit.>. The circuit scheme of the asymmetric transmonasymmetric transmon is shown in Fig. <ref>. The JJ is replaced by the DC superconducting quantum-interference device (SQUIDSQUID) consisting of two JJ's threading by external flux $\Phi_{\rm ext}$, with which the frequency is tuned within a specific domain.

Circuit scheme of the asymmetric transmon qubit.

Circuit Hamiltonian with more than one junction cannot be directly recast to the Mathieu equation. Therefore it is useful to learn how to approximately simplify it.
In phase basis, potential of the asymmetric transmon is written as $U=-E_{J1}\cos\varphi_1-E_{J2}\cos\varphi_2$ with $\varphi_1$ and $\varphi_2$ being the phases across two junctions. The change in macroscopic phase around a closed loop must be zero modulo $2\pi$, so we have

\begin{equation}
\oint\nabla \varphi \cdot d\vec{l}=2\pi n=(\varphi_a-\varphi_b)+(\varphi_b-\varphi_c)+(\varphi_c-\varphi_d)+(\varphi_d-\varphi_a).
\end{equation}

From the definition of the gauge invariant phase, the phase difference is given by

\begin{eqnarray}
\varphi_a-\varphi_b&=&\varphi_1+\frac{2\pi}{\Phi_0}\int_a^b \vec{A}\cdot d\vec{l},\\
\varphi_c-\varphi_d&=&-\varphi_2+\frac{2\pi}{\Phi_0}\int_c^d \vec{A}\cdot d\vec{l},
\end{eqnarray}

with $\vec{A}$ being the magnetic
vector potential. In superconductors the supercurrent equation gives

\begin{eqnarray}
\varphi_b-\varphi_c&=&\Lambda\int_b^c\vec{J}\cdot d\vec{l}+\frac{2\pi}{\Phi_0}\int_b^c \vec{A}\cdot d\vec{l},\\
\varphi_d-\varphi_a&=&\Lambda\int_d^a\vec{J}\cdot d\vec{l}+\frac{2\pi}{\Phi_0}\int_d^a \vec{A}\cdot d\vec{l}.
\end{eqnarray}

where $\vec{J}$ is the supercurrent density and $\Lambda$ depends on the number of Cooper pairs. Often the contour can be deep within the superconductor where $J = 0$, in this case flux quantization condition is

\begin{equation}
\varphi_1-\varphi_2=2\pi n+\frac{2\pi}{\Phi_0}\oint\vec{A}\cdot d\vec{l}=2\pi n+2\pi\frac{\Phi_{\rm ext}}{\Phi_0}.
\end{equation}

This equation reduces to $\varphi_1-\varphi_2=2\pi f$ at $n=0$ with $f=\Phi_{\rm{ext}}/\Phi_0$.

The potential of the asymmetric transmon can be rewritten by defining $\varphi_p=(\varphi_1+\varphi_2)/2$ and $a=E_{J1}/E_{J2}$ as $U=-E_J(f)\cos\varphi$, here

\begin{eqnarray}\label{eq.asEJ}
\varphi&=&\varphi_p+\arctan[d\tan(\pi f)],\\
E_J(f)&=&E_{J\Sigma}\sqrt{\cos^2(\pi f)+d^2\sin^2(\pi f)},
\end{eqnarray}

with $E_{J\Sigma}=E_{J1}+E_{J2}$ and $d=(a-1)/(a+1)$, indicating that the effective Josephson energy can be tuned by threading a magnetic flux through the loop. Therefore the Hamiltonian of the asymmetric transmon is similar to Eq. (<ref>).

Usually the asymmetric transmon have large enough $E_J/E_C$ such that $n_g$ can be safely dropped as shown in Fig. <ref>.
In this case the variance of variable $\varphi$ is small, so it is safe to expand the potential of a transmon up to fourth order in $\varphi$ as

\begin{equation}
H\approx 4E_Cn^2-E_J\left(1-\frac{1}{2}\varphi^2+\frac{1}{24}\varphi^4\right).
\end{equation}

Similar to LC resonator, a transmon canonical operators are defined as

\begin{equation}\label{eq.qac}
\begin{split}
{\varphi}&=\phiz (a +a^{\dagger}),\\
n&= i \nz (a -a^{\dagger}),
\end{split}
\end{equation}

with the zero-point fluctuations $\phiz$ and $\nz$, satisfying the minimum uncertainty $\phiz  \nz=1/2$. The fluctuations can be determined by the characteristic impedance $Z_t=\sqrt{E_C/E_J}$,  i.e. $\phiz = \sqrt{\hbar Z_t/2 }$. A transmon’s small anharmonicity allows to approximate the periodic potential energy in the vicinity of minimum with a 4th degree polynomial, making a Duffing oscillator ($\hbar\equiv1$)

\begin{equation}\label{eq.Hse}
H=\omega_0 a^{\dagger}a+\frac{\delta}{12}(a^{\dagger}+a)^4,
\end{equation}

with $\omega_0=\sqrt{8E_CE_J}$ being the plasma frequency and $\delta=-E_C$ being anharmonicity. Higher order corrections are  known in Ref. <cit.>. For $\omega\gg|\delta|$ the rotating wave approximation (RWA)rotating wave approximation (RWA) is valid, only excitation number conservation terms are kept since the other terms rapidly average out and then can be neglected. Then transmon Hamiltonian reduces to

\begin{equation}
H_{\rm tr}=\omega a^{\dagger}a+\frac{\delta}{2}a^{\dagger}a(a^{\dagger}a-1),
\label{eq.Htrf}
\end{equation}

with $\omega=\omega_0+\delta$.

§ FLUX QUBITS

Another type of superconducting qubit is the persistent current qubit, namely flux qubit <cit.>. The circuit scheme is depicted in Fig. <ref>.

Circuit scheme of three-junction flux qubit.

Two in-series identical JJ's with $E_{J1}=E_{J2}=E_J$ and $C_1=C_2=C_J$ are in a closed loop with a third JJ that has critical current $\alpha  I_0$, capacitance  $C_3=\alpha C_J$  and Josephson energy $E_{J3}=\alpha E_J$ with $\alpha<1$.  In this qubit two supercurrents circulate with opposite directions. This type of qubit is insensitive to background charges and the states can be manipulated with magnetic fields. Based on the analysis of JJ, the potential of a flux qubit can be written as

\begin{equation} U=-E_{J}(\cos\varphi_1-\cos\varphi_2+\alpha\cos\varphi_3),
\end{equation}

with $\varphi_i$ being the phase across junction $i$ $(i=1, 2, 3)$. Similarly, integrating $\nabla \varphi$ around the loop gives the flux quantization condition $\varphi_1-\varphi_2+\varphi_3=2\pi f$.

To investigate how the flux influences the Josephson energy, we can take the flux derivative and get

\begin{equation}
\frac{\partial U}{\partial f}=-2\pi E_J \sin(\varphi_1-\varphi_2-2\pi f).
\end{equation}

One can see flux-noise-insensitive configurations take place at $f=n/2$ with $n$ being an integer.  These points are referred to as “sweet spots” (SS)sweet spot (SS). At the first nontrivial SS $f=1/2$, we plot the potential for different $\alpha$ in Fig. <ref>. Qubit states are located inside the single and double well potential.

Contour plot of potential for $f=0.5$ at (a) $\alpha=0.4$ and (b) $\alpha=0.8$. The nested nearly circular shapes mark the maxima in the potential, the near square and  hourglass-shaped contours enclose one and two minima, respectively.

To find the minimum points of the potential, one can calculate  stable solutions of the following system of equations:

\begin{equation}
\begin{split}
\frac{\partial U}{\partial \varphi_1}&=\sin\varphi_1+\alpha\sin(2\pi f+\varphi_1-\varphi_2)=0,\\
\frac{\partial U}{\partial \varphi_2}&=\sin\varphi_2-\alpha\sin(2\pi f+\varphi_1-\varphi_2)=0.
\end{split}
\end{equation}

The solutions $(\varphi_1^*,\varphi_2^*)$ is given by combining the two equations

\begin{eqnarray}
\sin\varphi_1^*&=&-\sin\varphi_2^*=\sin\varphi^*,\\
\sin\varphi^*&=&-\alpha\sin(2\pi f+2\varphi^*).
\end{eqnarray}

At sweet spot when $\alpha<1/2$ there is only one minimum of the potential at $(0,0)$, while $\alpha\geq1/2$ there are two pairs of solutions $(\varphi^*,-\varphi^*)$ and $(-\varphi^*,\varphi^*)$, this is consistent with the trend of potential as shown in Fig. <ref>. For $\alpha>1/2$, the two minima in the eight-shaped unit cell of the double well can tunnel across the barrier between them as it is lower than that from the minimum in one unit cell to the neighboring unit cell. This makes energy levels anti-cross at sweet spot, which is not suitable to be considered as a qubit. More details can be found in Ref. <cit.>.

§.§ CSFQ

In the last decade, a variation of flux qubit capacitively shunted flux qubit (CSFQ) has been developed by shunting a capacitance to the smaller JJ, and beneficial for the improvement in the coherence timecoherence time to the microsecond magnitude <cit.>. The circuit scheme is shown in Fig. <ref>.

Circuit scheme of CSFQ with shunting capacitance $C_s$.

Similar to the asymmetric transmon, we can reduce the degrees of freedom by defining $ \varphi_m=(\varphi_1-\varphi_2)/2$ and $ \varphi_p=(\varphi_1+\varphi_2)/2$, then Lagrangian of a CSFQ can be written as

\begin{equation}
\mathcal{L}=\frac{1}{2}\left(\frac{\Phi_0}{2\pi} \right)^2 C_J\left[\left( 2+4\beta\right)\dot\varphi_m^2+ 2\dot\varphi_p^2\right]+2E_J\cos\varphi_m\cos\varphi_p+\alpha E_J\cos(2\pi f-2\varphi_m),
\end{equation}

where $\beta=\alpha+C_s/C_J$ with $C_s\gg C_J$.
Hamiltonian can be derived following Legendre transformation

\begin{equation}
\begin{split}
H&=\sum_{i=m,p}\frac{\partial \mathcal{L}}{\partial \dot\varphi_i}\dot\varphi_i-\mathcal{L}\\&=4E_{Cm}n_m^2+4E_{Cp}n_p^2-2E_J\cos\varphi_m\cos\varphi_p-\alpha E_J\cos(2\pi f-2\varphi_m),
\end{split}
\end{equation}

where $E_{Cm}=e^2/2C_J(1+2\beta)$ and $E_{Cp}=e^2/2C_J$. The larger shunting capacitance permits the $\varphi_p$ direction to behave as a fast free oscillation and can be ignored, also confirmed using the Born-Oppenheimer analysis <cit.>. For simplicity we drop the subscript $m$ from variables, and the Hamiltonian reduces to
\begin{equation}
H=4E_{C}n^2-2E_J\cos\varphi-\alpha E_J\cos(2\pi f-2\varphi),
\end{equation}

with potential reduced to $U=-2E_J\cos\varphi-\alpha E_J\cos(2\pi f-2\varphi)$. Similar to the three-junction flux qubit, corresponding minimum phase satisfies the relation $\sin\varphi_{\rm min}=-\alpha\sin2\varphi_{\rm min}$. At $\alpha>1/2$ the potential has two minima while at $\alpha\leq1/2$ it conveys one minimum. Due to the only one degree of freedom, the potential at SS is one-dimensional and plotted in Fig <ref> for different $\alpha$.

Potential of CSFQ as a function of phase at $\alpha=0.4$ and $\alpha=0.8$

In the following we only study the case that $\alpha<1/2$ in which the potential can be used as a qubit because each period conveys one minimum, not more. In this case energy barrier from the minimum in one unit cell to the neighboring unit cell is large enough to guarantee a discrete energy levels. If the external flux $f=1/2+\delta f$ is away from SS with $\delta f \ll 1/2-\alpha$, the minimum point, denoted by $\varphi_0$, can be found by setting the derivative of potential to zero. Since $\varphi_0$ and $\delta f$ are very small, we assume $\sin\varphi_0\approx\varphi_0$ and $\sin(2\pi\delta f-2\varphi_0)\approx\sin\left(2\pi\delta f\right)-2\cos\left(2\pi\delta f\right)\varphi_0$, these assumptions yield

\begin{equation}
\varphi_0=-\frac{\alpha\sin\left(2\pi\delta f\right)}{1-2\alpha\cos\left(2\pi\delta f\right)}\approx -2\pi \delta f\frac{\alpha}{1-2\alpha}.\\
\end{equation}

Similarly, Hamiltonian of CSFQ can also be rewritten as Eq. (<ref>) in the vicinity of minimum. This potential has the simple form of a Duffing oscillator at SS, i.e. $\delta f=0$,  with the periodic frequency and anharmonicity

\begin{eqnarray}
&&\omega= \sqrt{8 E_J E_C (1/2-\alpha)}+\delta,\\
&&\delta=E_C (8\alpha-1)/(2-4\alpha),
\end{eqnarray}

In contrast to a transmon,  a CSFQ with  $1/8<\alpha<1/2$ has positive anharmonicity. When $f$ is away from SS the less symmetric potential can be approximated to

\begin{equation}
U(f)=\left(1 -2 \alpha \right) E_{J}    \Delta \varphi ^2  + \left( 8\alpha -1 \right)  E_{J} \frac{\Delta \varphi ^4}{4!}   +  2 \pi (\delta f) \alpha E_{J}\left( 2\Delta \varphi  -\frac{ (2\Delta\varphi )^3}{3!}\right)+\cdots
\end{equation}

with $\Delta \varphi=\varphi-\varphi_0$. However higher order terms cannot be simply ignored due to nonzero flux shift. More accurate calculation can be found in Appendix <ref>.

§ THE SCHRIEFFER-WOLFF TRANSFORMATION

An open question in dealing with quantum many-body systems is how to decouple lower-energy dynamics from higher-energy degrees of freedom. The Schrieffer-Wolff (SW) transformationSchrieffer-Wolff (SW) transformation has been proved to effective and consistent
to address this problem <cit.>. In mathematical language, SW transformation is unitary operationunitary operator which can remove generalized off-diagonal terms and hence serves as a way of diagonalization to determine effective qubit Hamiltonian. Before stepping into the section of circuit QED, we briefly introduce SW transformation which can block or fully diagonalize a Hamiltonian in the dispersive regime.

In general, qubit-resonator interacting Hamiltonian can be decomposed into

\begin{equation}
H=H_0+H'=H_0+\epsilon V,
\end{equation}

with $H_0$ being the unperturbed component and $H'$ being the perturbation part. Unperturbed $H_0$ can be divided into two subspace A and B with assumption that A is separated from B of the spectrum by a gap $\Delta$. Let us consider sufficiently small $\epsilon$ so that weak perturbations are not strong enough to cover the gap, in other words the following limitation should be satisfied

\begin{equation}
\epsilon\leq\frac{\Delta}{2||V||}.
\end{equation}

By translating the Hamiltonian into operator language, $H_0$ is diagonal while $H'=H_1+H_2$ is off-diagonal with $H_1$ being block diagonal and $H_2$ being block off-diagonal. The unitary operator $U=e^{-S}$ with anti-Hermitian $S$  is defined to diagonal or block-diagonal the Hamiltonian, therefore the choice of $S$ is critical to this transformation.
Effective Hamiltonianeffective Hamiltonian is given by

\begin{equation}
H_{\rm eff}=e^{-S}H e^{S}=\sum_{j=0}^{\infty}\frac{1}{j!}[H,S]^{(j)},
\end{equation}

here Campbell-Baker-Hausdorff formula is applied and $[H,S]^{(j)}=[[H,S]^{(j-1)},S]$.  The commutation relation between block diagonal and block off-diagonal operators gives rise to a block off-diagonal matrix and the commutator of two block off-diagonal results in a block diagonal matrix, therefore the following relation is obtained

\begin{equation}
H_{\rm eff}^{\rm off}=\sum_{m=0}^{\infty}\frac{1}{(2m+1)!}[H_0+H_1,S]^{(2m+1)}+\sum_{m=0}^{\infty}\frac{1}{2m!}[H_2,S]^{(2m)}{\overset{!}{=}}0.
\end{equation}

One can expand $S$ in terms of $\epsilon$ as

\begin{equation}
S=\sum_{j=0}^{\infty}\epsilon^j S_j.
\end{equation}

Since $H_1$ and  $H_2$ are of the first order in $\epsilon$, then Hermitian operator $S$ satisfies

\begin{eqnarray}
\end{eqnarray}

Effective Hamiltonian is expressed as

\begin{equation}
H_{\rm eff}=\sum_{j=0}^{\infty}\epsilon^j (f_j(H_0,\{S_n\}_{n=1}^j )+f_{j-1}(H',\{S_n\}_{n=1}^{j-1})),
\end{equation}

where $f_j(A,\{S_n\}_{n=1}^j)=\sum_n^j i^n/n!{\mathcal F}(A,\{S_n\}_{n=1}^j,j,n)$ with ${\mathcal F}$
being the function that takes all n-fold commutation relations between $A$ and $S$. For instance, we list the several orders of expansion as

\begin{equation}
\begin{split}
\end{split}
\end{equation}

Up to second order the effective Hamiltonian reduces to

\begin{equation}
\begin{split}
H_{\rm eff}&=H_0+[H_0,S_1]+[H_0,S_2]+\frac{1}{2}[[H_0,S_1],S_1]+H'+[H',S_1]\\
\end{split}
\end{equation}

Next we take two examples to present how to utilize the SW transformation in the form of matrix calculation <cit.>. The first one is to diagonalize a Hamiltonian. Since $H_0$ is unperturbed and diagonal, one can separate the effective Hamiltonian into $H_{\rm}=H_0+H^{(m)}$ with higher order corrections

\begin{equation}
\end{equation}

where $H_x^{(m)}$ includes all other commutations. The diagonalized effective Hamiltonian requires non-diagonal elements must be zero, namely

\begin{equation}
\begin{split}
\left<p\right|H^{(m)}\left|q\right>=0
\end{split}
\end{equation}

with $p\neq q$ and $H_0\left|p\right>=E_p^{(0)}\left|p\right>$. $S^{(m)}$ then is solved as

\begin{equation}
\left<p\right|S^{(m)}\left|q\right>=-\frac{\left<p\right|H_x^{(m)}\left|q\right>}{E_p^{(0)}-E_q^{(0)}}.
\end{equation}

Each order of $S$ can be calculated in this way and then higher order corrections are obtained.

The second example is to block diagonalize a Hamiltonian, similar approach is applied. Considering $H^{(0)}=H_0$ is block diagonal, higher order terms can be expressed as <cit.>

\begin{equation}
H^{(m)}=H_1^{(m)}\oplus\cdots \oplus H_k^{(m)}\cdots,
\end{equation}

with $\oplus$ being the direct matrix sum. Since $S$ is block off-diagonal, similarly $S_{j,k}^{(m)}$ is defined as

\begin{equation}
\left<p\right|S_{j,k}^{(m)}\left|q\right>=-\frac{\left<p\right|H_{x_{j,k}}^{(m)}\left|q\right>}{\left<p\right|H_j^{(0)}\left|p\right>-\left<q\right|H_k^{(0)}\left|q\right>}.
\end{equation}

An alternative approach is black box quantization in which the circuit is more complex and might even be a ‘black box’ whose properties are unknown. One can first consider the black box as a linearized problem with respect to the coupling between qubit and resonators, and then account for the weak anharmonicity of the Josephson effect. This method can determine the qubit parameters numerically much easier and more accurately compared to perturbation theory <cit.>.

§ CIRCUIT QED

Now let us discuss the circuit QED. Analog to cavity QED which describes the interaction between light and matter at the quantum level, Circuit QED can describe the interaction between superconducting circuits behaving as artificial atoms and resonators. This section will build up the general model of qubit-resonator interaction based on circuit quantization, and revisit some ideas borrowed from cavity QED.

§.§ Single transmon coupled to a resonator

First we consider  the interaction between a harmonic oscillator and one transmon as shown in Fig. <ref>.

One transmon couple to one resonator.

Compared to Fig. <ref>, the voltage source is replaced by a resonator. Lagrangian of the circuit is written as

\begin{equation}
\mathcal{L}=\left(\frac{\Phi_0}{2\pi} \right)^2\frac{1}{2}\left[C\dot\varphi_r^2+(C_s+C_J)\dot\varphi^2+ C_g(\dot\varphi_r-\dot\varphi)^2\right]-E_J\cos\varphi-\frac{\varphi_r^2}{2L},
\end{equation}

with $\varphi$ and $\varphi_r$ being the phase difference across the junction and the harmonic resonator, respectively. In the case that $C\gg C_g$, the corresponding conjugate terms $n$ and $n_r$ can be approximated as

\begin{equation}
\left(\begin{array}{c}
\dot\varphi \\ \dot\varphi_r
\end{array}\right)=\left(\begin{array}{cc}
\frac{1}{C}&\frac{C_g}{CC_{\Sigma}}\\\frac{C_g}{CC_{\Sigma}}&\frac{1}{C_{\Sigma}}\\
\end{array}\right)\left(\begin{array}{c}
\end{array}\right).
\end{equation}

with $C_{\Sigma}=C_s+C_J+C_g$. Legendre transformation gives the Hamiltonian

\begin{equation}\label{eq.Hcqed1}
\end{equation}

where $E_C=e^2/2C_{\Sigma}$ and $E_r=e^2/2C$. By expanding Josephson energy to the 4th order and utilizing the creation and annihilation operators defined in Eq. (<ref>) and Eq. (<ref>), the Hamiltonian can be quantized as

\begin{equation}\label{eq.Hcqedq}
H=\omega_r r^{\dagger}r+\omega_q a^{\dagger}a+\frac{\delta_q}{2}a^{\dagger}a(a^{\dagger}a-1)-g(a-a^{\dagger})(r-r^{\dagger}),
\end{equation}

where $\omega_r=1/\sqrt{LC}$ is the resonator frequency, $\omega_q\approx\sqrt{8E_CE_J}+\delta$ is the qubit frequency with anharmonicity $\delta\approx-E_C$ and coupling strength is

\begin{equation} g=\frac{1}{2}\frac{C_g}{\sqrt{CC_\Sigma}}\sqrt{\omega_q\omega_r}
\label{eq.g}
\end{equation}

Fast oscillation terms, also called counter-rotating terms can be ignored based on the rotating-wave approximation (RWA), then the Hamiltonian simplifies to

\begin{equation}
H=\omega_r r^{\dagger}r+\omega_q a^{\dagger}a+\frac{\delta_q}{2}a^{\dagger}a(a^{\dagger}a-1)+g(ar^{\dagger}+a^{\dagger}r).
\label{eq.gJC}
\end{equation}

By restricting the transmon to the first two levels and replacing creation and annihilation operators with Pauli matrix such that  $a^\dagger\rightarrow\sigma^+=\left|e\right>\left<g\right|$ and $a\rightarrow\sigma^-=\left|g\right>\left<e\right|$, the famous Jaynes-Cummings HamiltonianJaynes-Cummings Hamiltonian is obtained

\begin{equation}
H=\omega_r r^{\dagger}r+\frac{\omega_q} {2}\sigma^z+g(\sigma^-r^{\dagger}+\sigma^+r),
\label{eq.JC}
\end{equation}

with the convention $\sigma^z=\left|e\right>\left<e\right|-\left|g\right>\left<g\right|$.

Previously, we perturbatively expand the cosine potential to fourth order to quantize the qubit Hamiltonian. However, to proceed accurate calculation, multilevel form of transmon Hamiltonian is needed,  Eq. (<ref>) is thus rewritten as

\begin{equation}
H=\omega_r r^{\dagger}r+\sum_j\omega_j \left|j\right>\left<j\right|+\sum_j \sqrt{j+1}g_j\left(r^{\dagger}\left|j\right>\left<j+1\right|+r\left|j+1\right>\left<j\right|\right),
\label{eq.Hfull}
\end{equation}

where $\left|j\right>$ is the bare eigenvector of the transmon and $g_j$ is the coupling strength, and $\omega_j$ is the corresponding bare energy by assuming $\hbar\equiv1$. With respect to the dispersive regime  $|g_j/(\omega_r-\omega_j)|\ll1$, the Hamiltonian can be full-diagonalized using SW transformation. Given that the first order in $S$ has the form of

\begin{equation}
S_1=\alpha r^{\dagger} \sum_j\sqrt{j+1}\left|j\right>\left<j+1\right|-\alpha^* r\sum_j\sqrt{j+1}\left|j+1\right>\left<j\right|.
\end{equation}

Substituting this ansatz into Eq. (<ref>) yields

\begin{equation}
S_1=- \sum_j \frac{\sqrt{j+1}g_j}{\omega_r-\omega_j}\left(r^{\dagger}\left|j\right>\left<j+1\right|- r\left|j+1\right>\left<j\right|\right).
\end{equation}

Calculating the first order correction and ignoring the two-photon processes leaves

\begin{equation}
\frac{1}{2}[H_2,S_1]=\sum_j[r^{\dagger}r(g_{j+1}^2\mu_{j+1}-g_j^2\mu_j)-g_j^2\mu_j]\left|j\right>\left<j\right|,
\end{equation}

where $\mu_j=j/(\omega_r-\omega_j+\omega_{j-1})$. Summing it together with the unperturbed part $H_0$ gives the effective Hamiltonian

\begin{equation}
H_{\rm eff}=\left( \omega_r+\sum_j\chi_j\left|j\right>\left<j\right|\right) r^{\dagger}r+\sum_j\tilde{\omega}_j\left|j\right>\left<j\right|,
\end{equation}

with $\chi_j=g_{j+1}^2\mu_{j+1}-g_j^2\mu_j$ and $\tilde{\omega}_j=\omega_j-g_j^2\mu_j$. The shift in resonator frequency is called AC Stark shiftAC-Stark shift, as the nonlinear shift is qubit-state dependent, it is also known as cross-Kerrcross-Kerr effect; While the shift in qubit frequency is called Lamb shift, as it arises from nonlinearity of JJ's, it is also known as self-Kerrself-Kerr effect <cit.>.

Generally speaking, the coupling strength $g_j$ cannot be readout and measured directly, but it is worth noting that dressed resonator frequency in the dispersive regime relies on the state of transmon. Therefore  if the resonator is measured with a swapping frequency, there should be two peaks in the spectrum, with which the coupling strength can be estimated. More accurate derivation can be found in Ref. <cit.>.

§.§ Two transmons coupled via a resonator

A more complicated example is that two transmons are coupled to each other via a bus resonator as shown in Fig. <ref>.

Two transmon qubits couple to a LC resonator.

Similarly, by using standard circuit quantization, circuit Lagrangian can be written as

\begin{equation}
\begin{split}
\mathcal{L}&=\frac{1}{2}(C_{J1}+C_{s1})\dot{\phi}_1^2+\frac{1}{2}(C_{J2}+C_{s2})\dot{\phi}_2^2+\frac{1}{2}C_r\dot{\phi}_r^2+\frac{1}{2}C_{1r}(\dot{\phi_1}-\dot{\phi_r})^2\\
&+\frac{1}{2}C_{2r}(\dot{\phi_r}-\dot{\phi_2})^2+E_{J1}\cos\left( {\frac{2\pi}{\Phi_0}\phi_1}\right)+E_{J2}\cos\left( {\frac{2\pi}{\Phi_0}\phi_2}\right)-\frac{\phi_r^2}{2L_r},
\end{split}
\label{eq.cqed}
\end{equation}

where $\phi_i =(\Phi_0/2\pi)\varphi_i$ with $i=1,2,r$ is the flux across each island node. The kinetic part can also be expressed in matrix form as

\begin{equation}
\label{eq.tcqed}
\end{equation}

with $\vec{\varphi}=(\varphi_1,\varphi_r,\varphi_2)$ being the phase at each node and capacitance matrix given by

\begin{equation}
\mathbf{C}=\begin{pmatrix}C_{J1}+C_{s1}+C_{1r} & -C_{1r} & 0\\
-C_{1r} & C_r+C_{1r}+C_{2r}& -C_{2r} \\
0& -C_{2r} & C_{J2}+C_{s2}+C_{2r}
\end{pmatrix}.
\end{equation}

One can rewrite the Hamiltonian in terms of the conjugate term of phase $\varphi$ as

\begin{equation}\label{eq.Hmatrix}
\end{equation}

with Cooper pair number operator $\overrightarrow{n}=(n_1,n_r,n_2)$. By defining $C_1=C_{J1}+C_{s1}$, $C_2=C_{J2}+C_{s2}$, and assuming $C_{1r},C_{2r}\ll C_{1},C_2,C_r$, the inverse of the matrix can be simplified to

\begin{equation}
\mathbf{C}^{-1}\approx\begin{pmatrix}\frac{1}{C_1} & \frac{C_{1r}}{C_1C_r} &\frac{C_{1r} C_{2r}}{C_1C_2C_r} \\[6pt]
\frac{C_{1r}}{C_1C_r}& \frac{1}{C_r}&\frac{C_{2r}}{C_2C_r}\\[6pt]
\frac{C_{1r} C_{2r}}{C_1C_2C_r}&\frac{C_{2r}}{C_2C_r}&\frac{1} {C_2}
\end{pmatrix}.
\end{equation}

Expanding the Hamiltonian in Eq. (<ref>) to the general form gives

\begin{equation}
\begin{split}
\end{split}
\end{equation}

where $E_{Ci}=e^2/2C_i$ with $i=   1,2,r$. Second quantization of the Hamiltonian by introducing creation and annihilation operators yields

\begin{equation}
\begin{split}
H&=\omega_1a_1^{\dagger}a_1+\frac{\delta_1}{2}a_1^{\dagger}a_1(a_1^{\dagger}a_1-1)+\omega_2a_2^{\dagger}a_2+\frac{\delta_2}{2}a_2^{\dagger}a_2(a_2^{\dagger}a_2-1)+\omega_r r^{\dagger}r\\
\end{split}
\label{eq.fullH1}
\end{equation}

where the $a_q$ ($a_q^{\dagger}$) and $r$ ($r^{\dagger}$) are annihilation (creation) operators of qubits and the resonator, respectively. Frequency, anharmonicity and coupling strength are given by

\begin{eqnarray}
\omega_q&=&8\sqrt{E_{Cq} E_{Jq}}+\delta_q,\\
\delta_q&=&-E_{Cq},\\
\omega_r&=&4\sqrt{E_{Cr} E_{Lr}},\\
g_{q r}&=&\frac{1}{2}\frac{C_{q r}}{\sqrt{C_r C_q}}\sqrt{\omega_q\omega_r}\label{eq.gir},\\
\label{eq.g12}
\end{eqnarray}

To decouple the resonator from the system, the SW transformation should be applied. Unitary operator can be solved using Eq. (<ref>) as

\begin{equation}
U=\exp\left\{\sum_{q=1,2}\left[\frac{g_{q r}}{\Delta_q}(a_q^{\dagger}r-a_qr^{\dagger})+\frac{g_{q r}}{\Sigma_q}(a_q^{\dagger}r^{\dagger}-a_qr)\right]\right\},
\end{equation}

with $\Delta_q=\omega_r-\omega_q$ and $\sum_q=\omega_r+\omega_q$. Effective Hamiltonian then can be written as

\begin{equation}
H_{\rm eff}=\tilde{\omega}_1a_1^{\dagger}a_1+\frac{\tilde{\delta}_1}{2}a_1^{\dagger}a_1(a_1^{\dagger}a_1-1)+\tilde{\omega}_2a_2^{\dagger}a_2+\frac{\tilde{\delta}_2}{2}a_2^{\dagger}a_2(a_2^{\dagger}a_2-1)+J(a_1^{\dagger}a_2+a_2^{\dagger}a_1),
\label{eq.Heff1}
\end{equation}

where parameters in the dressed frame are

\begin{eqnarray}
\tilde{\omega}_q&=&\omega_q-g_{q r}^2\left( \frac{1}{\Delta_q}+\frac{1}{\sum_q}\right), \\
\tilde{\delta}_q&\approx&\delta_q,\\
J&=&g_{12}-\frac{g_{1r}g_{2r}}{2}\left(\frac{1}{\Delta_1}+\frac{1}{\Delta_2}+\frac{1}{\sum_1}+\frac{1}{\sum_2} \right).
\end{eqnarray}

Note that the above expression for $J$ is only valid for single-mode oscillators <cit.>. In the following chapters, we turn to applications of the circuit QED on two- and three-qubit interactions.

CHAPTER: STATIC ZZ FREEDOM

An arbitrary unitary operationunitary operator can be decomposed into a set of single-qubit and two-qubit gates. In today's quantum processors made up of superconducting qubits, high-performance two-qubit gates are challenging to realize compared to fast single qubit rotations <cit.>. State-of-the-art two-qubit gate error rates remain $0.4\%-1\%$,  which is still above the threshold for error correction <cit.>, despite tremendous improvements in coherence and optimal control <cit.>. One limiting factor is the intrinsic parasitic interaction between a pair of interacting qubits. Due to the multilevel system of the physical qubits,  computational and noncomputational energy levels can interact with each other.  This coupling shifts the levels up or down, resulting in the always-on ZZ interaction. Such an interaction can accumulate quantum phase in states with Z being the Pauli operator $\sigma_{\rm z}$. On one hand, the ZZ interaction is unwanted for most gates and degrades the performance of desired entanglement <cit.>. Therefore eliminating it is crucially important to achieve a high-contrast on/off operation modes. On the other hand, we can utilize  ZZ interaction and strengthen it to implement high-speed controlled-phase (CPHASE) gates such as controlled-Z (CZ) gates, which will be introduced in chapter <ref>.

In this chapter we propose several strategies for freeing two idle qubits from the ZZ interaction, more distinctly, static ZZ without driving. We demonstrate that in a circuit consisting of the same-sign-anharmonicity superconducting qubits, e.g. transmon qubits, the static ZZ interaction vanishes only by switching off qubit-qubit coupling strength. However applying two-qubit gates on such a noninteracting pair does not let them entangle, therefore static ZZ freedom in transmon-transmon pair is not applicable.  We also present a more interesting case, in which we let qubits eliminate their static ZZ in the presence of qubit-qubit coupling, therefore can produce parasitic free entanglement. For this aim, We combine qubits with different-sign anharmonicity,  and show that ZZ freedom can take place under certain circumstance.

This chapter is based on some parts of our two publications Ref. <cit.> published in Physical Review Letters and Ref. <cit.> published in Physical Review Applied. Other parts will be discussed in next chapter. First we introduce circuit models used to study the qubit-qubit coupling problem, both perturbatively and nonperturbatively.  Using circuit QED theory we then evaluate all interactions between qubits, including the parasitic interaction. We use delicated theory that static ZZ is considered to be 2 or 3 orders of magnitude weaker than normal coupling between qubits. We search for conditions
that warrant the elimination of static ZZ interaction between idle qubits. We separate our analysis into transmon-transmon pair with the same sign anharmonicity, and CSFQ-transmon pair with opposite sign anharmonicity. Lastly, we use our theory model to simulate an experimental hybrid CSFQ-transmon circuit  performed by our collaborators at IBM and Syracuse University.

§ GENERAL MODEL

Two qubits can interact with each other in two ways: (i) direct capacitive couplingdirect coupling, (ii) indirect couplingindirect coupling via a coupler. Figure <ref> shows a circuit with both couplings combined together. Qubits 1 and 2 are coupled via a coupler C with coupling strength $g_{1c}$ and $g_{2c}$, and also capacitively coupled to each other with coupling strength $g_{12}$. The two qubits can be the same species e.g. transmon-transmon pair or different species e.g. CSFQ/transmon pair.

Two qubits coupled via a coupler C with indirect coupling  $g_{1c}$ and $g_{2c}$, and also capacitively coupled to each other with direct coupling $g_{12}$. Each qubit is measured by a readout resonator.

To study the circuit Hamiltonian shown in Fig. <ref> , we can either keep the coupler throughout our analysis
or decouple it and then work in the two-qubit basis.  This in fact refers to two approaches of studying the circuit QED: full Hamiltonian model and effective Hamiltonianeffective Hamiltonian model. The full Hamiltonian model evaluates higher excitations contribution in the coupler, which is more eivdent if the coupler is tunable. While the  effective Hamiltonian is usually applied if the coupler is harmonic and is far detuned from qubits. To make our analysis more precise, we consider both co-rotating terms and counter-rotating terms in our calculation.

§.§ Full Hamiltonian Model

By considering both qubits and the coupler as  multilevel systems, the circuit Hamiltonian can be quantized as

\begin{align}
H =&\sum_{i, n_i} \omega_i(n_i) |n_i+1\rangle \langle n_i+1| - \sum_{j (\neq i)}{g}_{ij}(a_i-a^{\dagger}_i)(a_j-a^{\dagger}_j),
\label{eq.ham}
\end{align}

where $i=1,2$ is the qubit labels and $i=c$  the coupler label, $a_i^{\dagger} =\sum_{n_i} \sqrt{n_i+1} |n_i+1\rangle \langle n_i |$ and $a_i = \sum_{n_i} \sqrt{n_i} |n_i\rangle \langle n_i +1 |$ are the creation and annihilation operators with $|n\rangle$ being the Fock states of each element in the circuit, separately. The eigenstates of the noninteracting Hamiltonian compose of the bare basis. Here we denote bare energy of each level by $E_1({n_{1}})$ and $E_2({n_2})$ for qubits  and $E_c({n_c})$ for the coupler.  $\omega_i(n_i)$ is the energy gap between the two energy levels $E_i({n_i+1})$ and $E_i({n_i})$ of subsystem $i$. A qubit bare frequency is the difference between the lowest two levels i.e. $\omega_{1/2}(0)$ or simply $\omega_{1/2}$. The full Hamiltonian of the circuit contains all possible interactions even the fast oscillating terms, also called co-rotating terms between levels in the bare basis of all circuit elements.  In the presence of the non-zero coupling strength, bare energy levels will be shifted up or down, and this causes modification of qubit frequencies and interaction strengths, sometime appearance of new unwanted interactions. The new values for such quantities are labeled as “dressed” values to make them distinct from bare ones.

Once these dressed states are found, and their energies are known, the dynamics of the system is simple: the eigenstate is a superposition of these bare states with the amplitudes being constant. To understand how the coupling terms change the bare basis, let us apply the full Hamiltonian in Eq. (<ref>)  on the bare states $\left|n_1,n_c,n_2\right>$ for Q1, coupler and Q2, respectively. It is worth noticing that  in Eq. (<ref>) only one excitation is allowed in each qubit, but in some cases multi-photon processes can also take place. Generally speaking, one can consider the first order of $g_{ij}$ with $i\neq j$ and $i,j=1,2,c$, and expand Eq. (<ref>) in terms of single particle exchange terms:

\begin{equation}
\begin{split}
\hat{H}{|n_1,n_{c},n_2\rangle} &=  [\omega_1(n_1)+\omega_{c}(n_{c}) +\omega_2(n_2) ]  {|n_1,n_{c},n_2\rangle}  \\
&+ g_{12}  \left[   \sqrt{(n_1+1) n_2} |n_1+1,n_{c},n_2-1\rangle +  \sqrt{n_1 (n_2+1)}  |n_1-1,n_{c},n_2+1\rangle \right.   \\
&\left.+\sqrt{n_1 n_2} |n_1-1,n_{c},n_2-1\rangle
+ \sqrt{(n_1+1) (n_2+1)} |n_1+1,n_{c},n_2+1\rangle \right]\nonumber
\end{split}
\end{equation}

\begin{equation}
\begin{split}
& + g_{1c}   \left[    \sqrt{(n_1+1) n_{c}} |n_1+1,n_{c}-1,n_2\rangle +  \sqrt{n_1 (n_{c}+1)} \ |n_1-1,n_{c}+1,n_2\rangle \right.   \\
&\left.+\sqrt{n_1 n_{c}} |n_1-1,n_{c}-1,n_2\rangle
+ \sqrt{(n_1+1) (n_{c}+1)} |n_1+1,n_{c}+1,n_2\rangle \right]\\
&   + g_{2c}   \left[  \sqrt{(n_2+1) n_{c}} |n_1,n_{c}-1,n_2+1\rangle     +  \sqrt{n_2 (n_{c}+1)} |n_1,n_{c}+1,n_2-1\rangle \right. \\
& \left.+ \sqrt{n_2 n_{c}} |n_1,n_{c}-1,n_2-1\rangle + \sqrt{(n_2+1) (n_{c}+1)} |n_1,n_{c}+1,n_2+1\rangle\right].\label{eq.fullH}
\end{split}
\end{equation}

The equation above shows different ways of energy transfer between two circuit elements. We can show energy level-crossing diagram as a function of qubit-qubit detuning, see Fig. <ref>.

Energy diagram of the circuit in Fig. <ref>. Red heads indicate avoided crossing by co-rotating terms with one or two particles exchange, while green marks with white heads indicate counter-rotating interactions.

Let us assume a fixed frequency Q2 is detuned from Q1 by $\Delta$, i.e. $\omega_1=\omega_2-\Delta$. Now we consider more than one particle exchange cases. Given that an eigenstate of unperturbed Hamiltonian is $|n_1,n_{c},n_2\rangle$, interacting Hamiltonian provides transition to $ |n_1\pm k,n_{c}\mp k,n_2\rangle$ via the coupling strength $(g_{1c})^k$, to $|n_1,n_{c}\mp k,n_2\pm k\rangle$ via the coupling strength  $(g_{2c})^k$ , to $ |n_1\pm k,n_{c},n_2\mp k\rangle$ via the coupling strength $(g_{12})^k$ , to $|n_1\pm 1,n_{c}\mp2,n_2\pm1\rangle$ via the strength  $g_{1c}g_{2c}$, also to $|n_1\pm1,n_{c}\pm1,n_2\rangle$ via counter rotating terms with the strength $g_{1c}$, to $|n_1,n_{c}\pm1,n_2\pm1\rangle$ via counter rotating terms with the strength $g_{2c}$, to $|n_1\pm1,n_{c},n_2\pm1\rangle$ via counter rotating terms with the strength $g_{12}$,  and so on.   Figure <ref>  marks these transitions with red and green heads either at intersections or on dashed-coupled lines, and label them by maximum two photon couplings. Green marks with white heads indicate counter rotating interactions such as $a_ia_j$, and red heads represent the co-rotating terms like $a_i^{\dagger}a_j$. One example is that the transition between  $|200\rangle$ and  $|101\rangle$ takes place either by direct coupling $g_{12}$ or via the intermediate state $| 110\rangle$ with coupling $g_{1c}$ and $g_{2c}$. In absence of direct coupling the avoided crossing between the two levels has been found to be $\sqrt{2} g_{1c}g_{2c}/\Delta_{2} (\Delta_{1}+\delta)$ in Ref <cit.>. The seemingly non-interacting levels may stay so, or may interact via $n$ photons for $n>2$ or under external gates although the interaction may be way smaller than the first order coupling.

The full Hamiltonian model is complicated and usually performed numerically. By writing the Hamiltonian in the bare basis, namely the matrix form, and fully diagonalizing it, one can find all dressed energies. In this way, the parasitic ZZ interaction is defined as

\begin{equation}
\zeta={\tilde E}_{101}+{\tilde E}_{000}-{\tilde E}_{100}-{\tilde E}_{001},
\label{eq.fzz}
\end{equation}

with ${\tilde E}_{n_1n_cn_2}$ being the dressed energy and subscripts indicating the state of $\left|n_1,n_c,n_2\right>$. This is the original definition of static ZZ interaction without any approximation.

§.§ Effective Hamiltonian Model

The coupler frequency $\omega_{c}$ is usually assumed to be far detuned from qubits in order to warrant no backaction from itself. Therefore one can treat the Hamiltonian perturbatively in the dispersive regimedispersive regime if  $g_{ij}\ll\Delta_{ij}$ with $\Delta_{ij} \equiv \omega_i-\omega_j$ being frequency detuning <cit.>. Within this regime, the coupler can become decoupled from the system via Schrieffer-Wolff (SW) transformation. Then the circuit Hamiltonian $H$ is block diagonalized into the qubit subspaces associated with different photon number of the coupler, the same as Eq. (<ref>).  In the multilevel two-qubit basis, the qubit-qubit Hamiltonian reduces to
H   =    ∑_q∑_n_q ω̅_q(n_q) |n_q+1⟩⟨n_q+1| + ∑_n_1,n_2 √((n_1+1)(n_2+1))
×   J_n_1,n_2   (| n_1, n_2+1⟩⟨n_1+1,n_2| +h.c. ),

where dressed frequency $\bar{\omega}_q({n_q})$ is the difference between $\bar{E}_q({n_q+1})$ and $\bar{E}_q({n_q})$ in dressed basis associated to interacting Hamiltonian, and expressed as

\begin{equation}
\bar{\omega}_q({n_q}) = \omega_q({n_q}) -g_{qc}^{2}(n_q+1)/\Delta_{q}({n_q}),
\end{equation}

with $g_{qc}$ being the coupling strength between qubit $q$ and the coupler $c$, $\Delta_q(n_q)=\omega_c-\omega_q(n_q)$ being the qubit-coupler frequency detuning. We can generalize the definition of anharmonicity to

\begin{equation}
\bar{\delta}_q=\bar{E_q}(2)+\bar{E_q}(1)-\bar{E_q}(0)=\delta_q[1-2g_{qc}^2/\Delta_{q}(\Delta_{q}-\delta_q)],
\end{equation}

and the qubit-qubit coupling strength within two photons limit is
J_n_1n_2≡g_12 -g_1cg_2c/2 ∑_q=1,2[ 1/Δ_q(n_q)+1/Σ_q(n_q)],

with $\Sigma_q(n_q) \equiv  \omega_{c} + \omega_q(n_q)$. For simplicity, $n_q=1$ will be dropped from all notations, e.g.  $\bar{\omega}_q(n_q=1)=\bar{\omega}_q$, $\Delta_q(n_q=1)=\Delta_q$ and $\Sigma_q(n_q=1)=\Sigma_q$.

Second round of SW diagonalization can separate the computational subspace from higher levels. Within the dispersive regime qubit-qubit Hamiltonian in the the computational form turns out to have the following operator structure:

H_   =ω̃_1|10><10|+ω̃_2|01><01|+(ω̃_1+ω̃_2+ζ)|11><11|

=-ω̃_1+ζ/2/2 ZI-ω̃_2+ζ/2/2 IZ+ ζ/4ZZ,

where static ZZ interaction in Eq. (<ref>) reduces to
ζ= Ẽ_11-Ẽ_10-Ẽ_01+Ẽ_00.

Let us emphasize that Eq. (<ref>) is a variant definition of static ZZ interaction in two-qubit basis compared to Eq. (<ref>). SW transformation can evaluate the following $\zeta$ and doubly-dressed frequencies within the dispersive regime
ζ=2J_10^2/(Δ̅-δ̅_1)-2J_01^2/(Δ̅+ δ̅_2),

ω̃_1=ω̅_1-J_00^2/Δ̅,    and   ω̃_2=ω̅_2+J_00^2/Δ̅,

with $\bar\Delta\equiv \bar{\omega}_2-\bar{\omega}_1$. One should notice that perturbation theory predicts there are a number of divergences in Eq. (<ref>) and Eq. (<ref>) at $\bar{\Delta}=-\bar{\delta}_2,0,\bar{\delta}_1,\cdots$. However the original definition Eq. (<ref>) is free of such divergence due to finiteness of energy levels and the divergences are mathematical artefacts of perturbation theory. Therefore ZZ interaction from such theory in the vicinity of these divergences is inaccurate.

To illustrate the absence of these fake divergences, let us numerically evaluate the first five nontrivial eigenvalues of free and interacting Hamiltonian by diagonalizing Eq. (<ref>). Figure <ref> shows the energy dispersion ($E_{\rm dress}-E_{\rm bare}$) versus qubit-qubit detuning in a CSFQ-transmon and a transmon-transmon circuit. The detuning is tuned by changing Q1 frequency while keeping Q2 and coupler fixed. At the divergence point $\Delta=0$ two qubits have the same bare frequency, however the computational dressed energy levels are changed by small shift, which corresponds to a finite ZZ interaction.

Dispersion between free and interacting energy levels obtained from Hamiltonian (<ref>) vs.  qubit-qubit detuning in (a) a CSFQ-transmon device and (b) a transmon-transmon device.  Circuit parameters are the same as those used in Fig. <ref> and Fig. <ref>.

Usually the frequency shift due to the interaction is much smaller than the  energy difference of the adjacent levels, so there will be no change in the relative position between computational and noncomputational subspaces. Intuitively, in a CSFQ-transmon pair with coupler frequency far detuned from both qubits, as well as within the limit of $|\Delta|<|\delta_{1/2}|$, $E_{11}$ falls in between noncomputational levels $E_{02}$ and $E_{20}$. $E_{11}$ can interact with the two levels at  the effective coupling strength $J_{01}$ and $J_{10}$, respectively.  The lower frequency decreases and the higher increases, this is called level repulsion. In this device, it is possible to make the two repulsions cancel each other at certain frequency detuning. However, in a transmon-transmon pair, the two noncomputational levels $E_{02}$ and $E_{20}$ are in one side of $E_{11}$, which makes the two repulsions to sum and not cancelled.

§ ZEROING STATIC ZZ INTERACTION

Perturbation theory shows that parasitic ZZ interaction relies on the effective $J$ coupling defined in Eq. (<ref>). In particular, the frequency shifts depend on the effective coupling strength $J_{00}$, which causes $|01\rangle \leftrightarrow \langle10|$ transition with excitation from $|0\rangle$ to $|1\rangle$ in one qubit and absorption in the other qubit from $|1\rangle$ to $|0\rangle$. As one can see in Eq. (<ref>), the two dressed levels are further separated by the additional gap $2J_{00}^2/\bar{\Delta}$ with $\bar{\Delta}$ being the dressed qubit-qubit detuning. $J_{00}^2/\bar{\Delta}$ is added on the dressed frequency of one qubit, and the same amount is subtracted from the other qubit.
Therefore based on the original definition Eq. (<ref>), $J_{00}$  has no contribution to the static ZZ coupling strength. However in higher levels the scenario is different; $J_{01}$ couples $|02\rangle \leftrightarrow \langle11|$, and $J_{10}$ does $|02\rangle \leftrightarrow |11\rangle$ transition.  The two interaction can cancel one another and free ZZ interaction of $E_{11}$, only  if $J_{01}$ and $J_{10}$ become zero simultaneously or the two noncomputational levels are on both sides of $E_{11}$.

§.§ Perturbative ZZ Freedom Condition

Now let us explore the evolution of the computational basis $|00\rangle$, $|01\rangle$, $|10\rangle$ and $|11\rangle$ by applying time-evolution operator in the form of $U=e^{-iH_{\rm eff}t}$. In contrast to the free Hamiltonian, static ZZ interaction can accumulate additional phase error relying on the states. Evolving the idle quantum states $|00\rangle$ and  $|11\rangle$ after time $t$ results in the phase error $\exp(+i\zeta t/4)$, while the states $|01\rangle$ and  $|10\rangle$ return a opposite-sign phase $\exp(- i\zeta t/4)$. This error will be transmitted across a circuit and finally lower the two-qubit gate fidelity.

Based on the expression of ZZ in Eq. (<ref>), we can search for the possibility of eliminating the phase error for interacting qubits. One easy solution is to set $J_{01}\approx J_{10}\approx0$. A more interesting solution, as discussed before, is to enable the elimination by finding ways to cancel level repulsions between computational and noncomputational levels while keeping two qubits coupled. To find out such a nontrivial solution, we solve the equation $\zeta=0$ and obtain the following condition:
Δ̅=δ̅_1+δ̅_2 γ^2/1- γ^2,

where $\gamma \equiv J_{10}/J_{01}$ with $J_{01}$ and $J_{10}$ defined in Eq. (<ref>). Counter-rotating terms and direct coupling are one order smaller than the co-rotating terms and indirect coupling, separately, therefore can be ignored. This simplifies $\gamma$  to the following expression
γ≈1-δ_1/ (2Δ_2 +Δ)/1-δ_2/(2 Δ_2 +Δ) 1-δ_2/Δ_2/1-δ_1/(Δ_2+Δ),

with $\Delta_2=\omega_c-\omega_2$ being the frequency difference between qubit 2 and coupler. The condition of the static ZZ freedomstatic ZZ freedom in Eq. (<ref>) makes it possible to explore such a  possibility for certain circuit parameters.

§.§ Examples of Static ZZ Freedom

Now we study the elimination of ZZ interaction in two types of circuits: (a) a CSFQ-transmon pair with the opposite-sign anharmonicity coupled via a coupler; (b) a transmon-transmon pair with same-sign anharmonicity coupled via a coupler.

§.§.§ (a) CSFQ-transmon pair

This is an example of opposite-sign anharmonicity pair. Combining a positive and a negative anharmonic qubits makes it possible to realize static ZZ freedom. This freedom can take place in the presence of finite $J$ interaction between qubits that let them entangle once they are driven by gates. Here we search for circuit characteristics to achieve static ZZ freedom.

Let us consider Q2 in Fig. <ref> is a transmon with  fixed frequency $\omega_2$  coupled to Q1, a CSFQ with $\Delta$ detuned frequency, i.e. $\omega_1=\omega_2-\Delta$.  We write the Hamiltonian Eq. (<ref>) in matrix form with maximum excitation number being four, i.e. all levels from $E_{00}$ up to $E_{04}$ and $E_{40}$. We block diagonalize the matrix into two blocks: computational subspace and noncomputational subspace. Using Eq. (<ref>) we can evaluate the repulsion between computational and noncomputational levels, namely the static ZZ interaction. Figure <ref> shows ZZ  over a wide range of the frequency detuning $\Delta$ in unit of gigahertz on $x$ axis, and CSFQ anharmonicity $\delta_1$ on $y$ axis. The ZZ interaction can be negative or positive with a nontrivial borderline marked by solid black line between the two regions on which it vanishes. The marked circle indicates similar  parameters of the CSFQ-transmon circuit experimented which will be introduced in section <ref>.

Static ZZ interaction in a CSFQ-transmon device. Q1 has the anharmonicity $\delta_1$ and is detuned from a transmon Q2 by $\Delta$. Static ZZ interaction dependence on CSFQ anharmonicity and qubit qubit detuning. The two qubits are uncoupled directly, but they are indirectly coupled via a bus resonator with frequency $\omega_{c}=6.492$ GHz and coupling strengths $g_{1c}=g_{2c}=g=80$ MHz with Q2 frequency and anharmonicity $\omega_2=5.292$ GHz, $\delta_2=-0.33$ GHz. The effective ZZ interaction is positive (negative) in red (blue) areas and it vanishes on the solid lines.

Figure <ref>(a) compares two different approaches to determine the static ZZ for CSFQ-transmon pair as a function of different  frequency detuning: dashed line calculating ZZ  from effective Hamiltonianeffective Hamiltonian in Eq. (<ref>), solid line obtained from numerical analysis by fully diagonalizing the Hamiltonian in Eq. (<ref>).

(a) Static ZZ interaction with $\delta_1=0.6$ GHz using the effective (dashed magenta) and full model (solid brown) individually. (b) Static ZZ freedom criteria via the effective (solid), analytical zeroth-order (dotted), first order (dashed) and full Hamiltonian model (cross).

Comparing the two methods reveals that: both effective and full Hamiltonian models show the consistent  presence of ZZ freedom, and perturbation theory becomes more accurate as detuning $\Delta$ increases.  Figure <ref>(b) shows the parameters criteria at which the static ZZ is zero. The parameter landscape explored here are the normalized frequency detuning $\Delta$ by transmon-coupler detuning $\Delta_{2}$ and the magnitude of anharmonicity ratio by keeping the others unchanged. Crossed points show exact results from full Hamiltonian model, and the solid line is obtained from Eq. (<ref>). The two dashed lines are analytical solutions of Eq. (<ref>) in the zeroth and first order in $|\delta_2/\Delta_{2}|$.  For obtaining these analytical solutions we rewrite ZZ interaction in terms of  $\delta_1=  k \delta$,  $\delta_2=-\delta$ with  $k, \delta>0$. We consider no direct coupling between qubits and the universal qubit-coupler coupling strength $g_1=g_2=g$, and $\Delta=b\Delta_{2}$. By substituting these parameters in Eq.  (<ref>), $\gamma$ is simplified to
γ= (1+a)(1+b)(ak - b -2)/(2+b+a) (ak -b -1),
with $a\equiv \delta/\Delta_{2}$. In the absence of $a$, static ZZ cancellation condition in Eq. (<ref>) then reduces to
i.e. zeroth order solution.  This result gives similar trend as the other two numerical methods, but the precision is quite low. Adding the first order of $a$ increases the solution accuracy. One can see for increasing absolute ratio of CSFQ and transmon anharmonicity the analytical approximation is more trustable.

§.§.§ (b) transmon-transmon pair

This is an example of same-sign anharmonicity pair. Typically there are many choices of pairs of qubits coupled to each other, but here we only study the most popular one, transmon. In such a pair both anharmonicities are negative. Let us start with the trivial solution that $J_{01}\approx J_{10}\approx0$, and revisit the definition of $J$ coupling in Eq. (<ref>). One can find that zeroing it requires that direct coupling $g_{12}$ cancels out the indirect couplings. This freedom has been realized recently in several experiments <cit.>.

However non-interacting qubits cannot be operated to achieve entanglement at such an operating point and therefore are not useful for quantum computation. To perform gates on such a transmon-transmon pair with high fidelity, one can suppress ZZ interaction by tuning circuit parameters. For this purpose we consider a transmon-transmon pair with almost the same anharmonicity and different frequencies.

We simulate this device and extract static ZZ and $J$ coupling by diagonalizing the full Hamiltonian.

Static ZZ interaction in a transmon-transmon device with $\omega_2=4.914$ GHz, $\omega_c=6.31$ GHz, $\delta_1=\delta_2=-0.33$ GHz and direct coupling $g_{12}=0, 2.5, 5$ MHz as a function of (a) qubit-qubit detuning with $g_1=98$ MHz, $g_2= 83$ MHz, (b) transmon-resonator coupling strength $g_{2c}$ with $g_{1c}=98$ MHz and $\Delta=-0.1$ GHz, separately. Corresponding $J_{01}$ and $J_{10}$ are plotted in (c) and (d).

Figure <ref>(a) shows static ZZ interaction is near-symmetric and decreases by lowering the magnitude of detuning frequency $\Delta$ without direct coupling. On top of this we also study for the other two nonzero direct couplings $g_{12}$. At $g_{12}=5$ MHz the ZZ interaction can be dramatically suppressed to a few tens of kilohertz, and at specific detuning static ZZ freedom is realized with almost zero $J$ coupling as shown in Fig. <ref>(c). Apart from the static ZZ freedomstatic ZZ freedom, parasitic interaction increases slowly while $J$ coupling can go higher and become non-zero.

Interestingly the weaker $g_{12}=2.5$ MHz case shows that suppression of  ZZ interaction can be achieved within a rather large domain of detuning frequency  $\Delta$, making it possible to design a transmon-transmon circuit with suppressed ZZ interaction within the dispersive regime $\Delta \gg J$.  Figure <ref>(b) shows how ZZ coupling changes by varying transmon-resonator coupling $g_{2c}$.  Although the zero ZZ points belong to $J$-interaction-free domain, however in their vicinity one can see a large class of transmons that not only interact but also their unwanted ZZ interaction is suppressed. To strengthen $J$ around this region, one can vary the coupler frequency.

A few months after we post this work on arXiv <cit.>, the idea of suppressing static ZZ interaction in transmon-transmon devices was tested and verified in a IBM paper <cit.>. In that paper, they strengthened the direct coupling by adding a direct coupler to make static ZZ freedom possible in the all frequency-fixed transmons circuit. In the vicinity of static ZZ freedom shaded in Fig. <ref>. They improved $J_{\rm eff}$ and performed an experiment at the frequency marked by the star. They achieved an effective $J$ of 3.5 MHz with 26 kHz ZZ interaction, and demonstrated a CNOT gateCNOT gate fidelity of 99.77(2)%.

(a) ZZ vs the mean qubit frequency at different qubit-qubit detunings. The experimental data is highlighted by the star. The dashed line is the ZZ for a pair of qubits with $\Delta = 60$ MHz coupled via  a direct coupler. (b) The effective $J$ at different qubit-qubit detunings, the experiment value is the star. The dotted line is the effective $J$ for a $\Delta = 60$ MHz direct coupler device. The shaded region represents the frequency region where the multi-path coupler shows an improvement in $J_{\rm eff}$ <cit.>.

Searching for other possibilities of ZZ freedom in the presence of $J$ coupling gives that qubit-coupler coupling  $g_{qc}$ is equal or larger than the qubit-resonator frequency detuning $\Delta_{q}$, and this goes beyond the dispersive regime and therefore  perturbation theory fails. Exact numerical result from full Hamiltonian presents consistent non-divergent solution and does not show any possibility for ZZ freedom in the transmon-transmon pair within the domain of quantum computation. Another approach is to replace frequency-fixed harmonic resonator with a tunable couplertunable coupler i.e. an asymmetric transmon, making it possible
to vary static ZZ interaction close to zero <cit.>. Furthermore, a ZZ-free iSWAP gate is implemented experimentally with the fidelity close to the coherence limit via optimal control <cit.>.

§ EXPERIMENTAL SETUP

In this section, we present the detailed theoretical model of the first experimented hybrid CSFQ-transmon device. The device was fabricated at IBM, and consists of one fixed-frequency transmon, one bus resonator, one CSFQ, and readout resonators for each qubit. The measurement was taken by our collaborators at Syracuse University. More details can be found in our publication Ref. <cit.>.

§.§ Impact of Measurement

Let us briefly discuss the impact of measurementmeasurement of qubits. Readout measurement usually takes place by means of weakly coupling a qubit to a resonator with the interaction $H_{qR}=g_{qR}(\hat{a}_q+\hat{a}_q^\dagger) (\hat{a}_R+\hat{a}^\dagger_R)$.  In a circuit where qubits are both coupled to the coupler and readout resonator, the process of eliminating the readout resonatorreadout cavity can take place before or after eliminating the coupler. Since both of the couplings are within the dispersive regime, SW are insensitive to the decoupling orders. However in reality the two orders sometime show inconsistent results <cit.>. To study the difference of the two decoupling orders, we derive corresponding dressed frequency and anharmonicity.  Decoupling first readout and then bus resonator in the dispersive regime gives rise to

\begin{equation}
\tilde\omega_{1/2}=\breve\omega_1\pm\frac{\left[g_{12}-\frac{g_{1r}g_{2r}}{2}\left(\frac{1}{\omega_r-\bar\omega_1}+\frac{1}{\omega_r-\bar\omega_2}\right)\right]^2}{\breve\omega_1-\breve\omega_2},
\end{equation}

\begin{equation}
\begin{split}
\breve\omega_q&\equiv\bar\omega_q-\frac{g_{qr}^2}{\omega_r-\bar\omega_q},\\
\bar\omega_q&\equiv\omega_q-\frac{g_{qR}^2}{\omega_R-\omega_q}.
\end{split}
\end{equation}

Similarly, first decoupling bus and then readout gives

\begin{equation}
\tilde\omega'_{1/2}=\check\omega_1\pm\frac{\left[g_{12}-\frac{g_{1r}g_{2r}}{2}\left(\frac{1}{\omega_r-\omega_1}+\frac{1}{\omega_r-\omega_2}\right)\right]^2}{\check\omega_1-\check\omega_2},\\
\end{equation}

\begin{equation}
\check\omega_q\equiv\omega_q-\frac{g_{qr}^2}{\omega_r-\omega_q}-\frac{g_{qR}^2}{\omega_R-\omega_q-\frac{g_{qr}^2}{\omega_r-\omega_q}}.
\end{equation}

By assuming zero direct coupling $g_{12}=0$ and universal indirect coupling $g_{qR}=g_{qr}=g$, we derive the inconsistency in dressed qubit frequency $\textup{\dj} \omega\equiv\tilde\omega_1-\tilde\omega_1'=\tilde\omega_2-\tilde\omega_2'$ and  anharmonicity $\textup{\dj} \delta$ as
ω̃_2/g^6=- ω̃_1/g^6= (Δ+2Δ_2)^2(Δ^2+ΔΔ_2+Δ_2^2)/2ΔΔ_2^4 (Δ+Δ_2)^4,

δ̃_1/2/2 g^6=-  (Δ_2-δ_1/2)^3+(δ_1/2+Δ_2)Δ_2^2/(Δ_2^3 Δ∓δ_1/2)(Δ_2-δ_1/2)^4 ±2/ΔΔ_2^4,

with universal qubit-resonator detuning $\Delta_2=\omega_{r/R}-\omega_2$, two qubit detuning $\Delta=\omega_2-\omega_1$, and $\delta_{1},\delta_2,\Delta\ll\Delta_{2}$. In the dispersive regime, the difference is in the order of $g^6/\Delta_2^4\Delta$ and can be ignored. In the following discussion, we decouple readout resonator first then bus resonator to reduce the Hamiltonian.

§.§ Modeling the Experimental Circuit

First we build a full-circuit Hamiltonian from a lumped-element circuit model for our CSFQ-transmon device in Fig. <ref>, and the corresponding design parameters are listed in Tab. <ref>. The circuit contains three subsystems: transmon and its readout cavity, bus cavity, CSFQ and its readout cavity. By following standard circuit quantization, Lagrangian of the total system can be obtained. Note that we label all seven nodes from $a$ to $h$, and kinetic energy should include all charging energy stored in the capacitors.

Readout at 40 80
Cavity at 40 65
Transmon at 140 190
Bus cavity at 240 190
CSFQ at 360 190
Readout at 485 80
Cavity at 485 65

Circuit model for CSFQ-transmon device. Each qubit has its own readout resonator and they are coupled via a bus resonator.

Similarly, we define $\varphi_T\equiv\varphi_b-\varphi_c$, $\varphi_m\equiv(\varphi_e-\varphi_g)/2$ and  $\varphi_p\equiv(\varphi_e+\varphi_g)/2-\varphi_f$. The superscript and subscript $T$, $m/p$ represent transmon and CSFQ, respectively. Particularly, $m$ means the phase difference between two large junctions in CSFQ while $p$ is the phase sum. The Lagrangian is then  simplified into the following form:

\begin{equation}
\begin{split}
\mathcal{L}&=\frac{1}{2}\left(\frac{\Phi_{0}}{2\pi}\right)^{2}\left[C_{rT}\dot{\varphi_{a}}^2+C_{ab}(\dot{\varphi_{a}}-\dot{\varphi_{b}})^{2}+C_{b0}\dot{\varphi_{b}}^{2}+C_{cd}(\dot{\varphi_{b}}-\dot{\varphi_{T}}-\dot{\varphi_{d}})^{2}\right.\\
&+\alpha E_{J}\cos\left(2\pi f-2\varphi_{m}\right)-\left(\frac{\Phi_{0}}{2\pi}\right)^{2}\left(\frac{\varphi_{a}^{2}}{2L_{rT}}+\frac{\varphi_{d}^{2}}{2L_{R}}+\frac{\varphi_{h}^{2}}{2L_{rCSFQ}}\right),
\end{split}
\end{equation}

where $f=\Phi/\Phi_0$ is the normalized magnetic flux, $C_1=C_2\equiv C_J$, and $ E_{J1}=E_{J2}\equiv E_J$. The Hamiltonian is calculated from the Legendre transformation of the Lagrangian

\begin{eqnarray}
H&=&\sum_{i}\dot{\varphi_{i}}\frac{\partial \mathcal{L}}{\partial\dot{\varphi_{i}}}-\mathcal{L}=T+U,\\
U&=&E_{La}\varphi_{a}^{2}+E_{Ld}\varphi_{d}^{2}+E_{Lh}\varphi_{h}^{2}+E_{JT}\cos\varphi_{T} \\ \nonumber
&-&2E_{J}\cos{\varphi_{p}}\cos{\varphi_{m}}-\alpha E_{J}\cos\left(2\pi f-2\varphi_{m}\right),
\end{eqnarray}

where the phase vector in the circuit is defined as $\vec{{\varphi}}^{T}=\left({\varphi_{b}},{\varphi_{e}},{\varphi_{a}},{\varphi_{T}},{\varphi_{d}},{\varphi_{m}},{\varphi_{p}},{\varphi_{h}}\right)$, and the energies stored in the readout resonators for the transmon, the CSFQ, and the bus resonator are $E_{La}=\Phi_0^2/8\pi^2 L_{rT} $, $E_{LCSFQ}=\Phi_0^2/8\pi^2 L_{rCSFQ}$, and $E_{LR}=\Phi_0^2/8\pi^2 L_{R} $, respectively.

4|c|Capacitance (fF)     2c|Josephson energy (GHz)$C_{rT}$    $452^*$    $C_{rCSFQ}$      $439^*$     $E_{J1}=E_{J2}$     $109^*$$C_{ab}$   $6.6$     $C_{gh}$     $6.6$     $E_{J3}=\alpha E_{J1}$     $46.8^*$$C_{b0}$   $66$     $C_{g0}$      $66$      $E_{JT}$     $13.7^*$$C_{shT}$   $26$      $C_{shCSFQ}$   $26$    $\alpha$     $0.43^*$$C_{T}$   $0.5-1^*$      $C_1=C_2$   $1.2-2.3^*$     2c|Inductance (nH)$C_{c0}$   $63$     $C_{e0}$      $63$     $L_{R}$     $1.3^*$$C_{cd}$    $16 $     $C_{de}$     $16$    $L_{rT}$     $1.2^*$$C_R$    $469^*$     $C_3=\alpha C_1$     $0.5-1^*$    $L_{rCSFQ}$     $1.2^*$

Circuit parameters corresponding to each circuit element in Fig. [fig:csfq-transmon]<ref>. The parameters with $*$ symbol are calculated based on experimental data, while the parameters without $*$ are design values that are extracted using ANSYS Q3D Extractor simulation of the qubit layout.

We can decouple the first two redundant phase degrees of freedom since they are not included in the potential, and ignore the phase sum of CSFQ as  discussed in section <ref>. More details can be found in Appendix <ref>. Now let us rewrite the Hamiltonian in terms of canonical conjugate $n$ and $\varphi$ as

\begin{equation}
\begin{split}
H&=4E_{Ca}n_{a}^{2}+E_{La}\varphi_{a}^{2}+4E_{Cd}n_{d}^{2}+E_{Ld}\varphi_{d}^{2}+4E_{Ch}n_{h}^{2}+E_{Lh}\varphi_{h}^{2}\\&+4E_{CT}n_{T}^{2}-E_{JT}\cos\varphi_{T}+4E_{Cm}n_{m}^{2}-2E_{J}\cos\varphi_{m}-\alpha E_{J}\cos\left(2\pi f-2\varphi_{m}\right)\\&+\lambda_{dm}n_{d}n_{m}+\lambda_{hm}n_{h}n_{m}+\lambda_{aT}n_{a}n_{T}+\lambda_{dT}n_{d}n_{T}+\lambda_{mT}n_{m}n_{T},
\end{split}
\label{eq.ge}
\end{equation}

with charging and Josephson energy listed in Tab. <ref>.

Description     Symbol     Value
Transmon Josephson energy     $E_{JT}$     13.7 GHz
Transmon charging energy      $E_{CT}$     0.286 GHz
CSFQ Josephson energy     $E_{Jm}$     109  GHz
CSFQ charging energy     $E_{Cm}$     0.292  GHz
CSFQ critical current ratio     $\alpha$     0.43

Junction parameters and charging energy of the CSFQ and transmon. CSFQ Josephson energy is for the larger junctions. The two transmon parameters were calculated using the measured dressed qubit frequency. Meanwhile, the three CSFQ parameters were obtained by fitting spectroscopy data of the dressed qubit frequencies, $\tilde{\omega}_1(0)/2\pi$ and $\tilde{\omega}_1(1)/2\pi$ vs. flux with 1D potential approximation <cit.>.

This Hamiltonian can be decomposed into two parts: free part including readout resonators, qubits and bus cavity; interacting parts containing all the interaction. As discussed before, accurate frequency and anharmonicity of transmon and CSFQ can be derived from Ref. <cit.> and Appendix <ref>, separately. By introducing several pairs of creation and annihilation operators, the circuit Hamiltonian can be quantized as

\begin{equation}
\begin{split}
H_{\rm circuit}&=\omega_{a}a^{\dagger}a-g_{aT}\left(a-a^{\dagger})(T-T^{\dagger}\right)+\omega_{h}h^{\dagger}h-g_{hm}\left(h-h^{\dagger})(m-m^{\dagger}\right)\\
&+\omega_{r}r^{\dagger}r+\sum_j\omega_T^j\left|j\right>\left<j\right|+\sum_k\omega_m^k\left| k \right>\left< k\right|-g_{rm}\left(r-r^{\dagger})(m-m^{\dagger}\right)\\
\end{split}
\label{eq.Heff}
\end{equation}

where $a$ and $h$ represent the readout resonators, $r$ is the bus resonator, $\omega_T^j$ is the transmon bare energy at level $j$, and $\omega_m^k$  is the transmon bare energy at level $k$. Corresponding annihilation operators are defined as $m=\sum_k\sqrt{k+1}\left|k\right>\left<k+1\right|$  for CSFQ and  $T=\sum_j\sqrt{j+1}\left|j\right>\left<j+1\right|$ for transmon.

We use the notation $\omega_1^b(j)$ to denote the bare transition frequency between the energy levels, $j+1$ and $j$ in the transmon; similarly, $\omega_2^b(k)$, the bare transition frequency between the energy levels, $k+1$ and $k$ in the CSFQ.  Figure <ref> shows the theoretical flux dependence of the bare CSFQ frequency and anharmonicity.

Bare CSFQ frequency and anharmonicity versus flux. (a) The first three bare transition frequency $\omega_1^b(n)$ between levels $n$ and $n+1$. (b) The first three bare CSFQ anharmonicity $\delta_1^b(n)=\omega_1^b(n+1)-\omega_1^b(n)$.

In the dispersive regime we first eliminate the readout resonators and then the bus, the Hamiltonian in Eq. (<ref>) reduces to qubit-qubit Hamiltonian. For simplicity, we replace the labels $j$, $k$ with $n_1$ for transmon and $n_2$ for CSFQ. The Hamiltonian in the form of multilevel systems is then written as

\begin{equation}\label{eq:twoQ_Hamil}
\begin{split}
H_{\rm circuit}&=H_r+H_q\\
&\times J_{n_1,n_2}\left(\left|n_1+1,n_2\right>\left<n_1,n_2+1\right|+\left|n_1,n_2+1\right>\left<n_1+1,n_2\right|\right),
\end{split}
\end{equation}

where the dressed bus frequency is $\tilde{\omega}_r=\omega_r+\Sigma_q \chi_{n_q}^q\left|n_q\right\rangle \left\langle n_q\right|$ with $\chi$ being the dispersive AC-Stark shiftAC-Stark shift of the resonator frequency. $\omega_q(n_q)$ is the energy difference between levels $n_q+1$ and $n_q$ for qubit $q$. $J_{n_1, n_2}$ is the virtual photon exchange rate calculated from $J_{n_1,n_2}=J^{\text{dir}}+J_{n_1,n_2}^{\text{indir}}$ with the direct coupling being $J^{\text{dir}}=g_{mT}$, and the indirect coupling $J_{n_1,n_2}^{\text{indir}}$ being <cit.>

\begin{equation}
\begin{split}
J_{n_1,n_2}^{\text{indir}}&=-\frac{g_{rm}g_{rT}}{2}\left(\frac{1}{\Delta_1(n_1)}+\frac{1}{\Delta_2(n_2)}+\frac{1}{\Sigma_1(n_1)}+\frac{1}{\Sigma_2(n_2)}\right) \\
\Delta_1(n_1)&=\omega_r-\omega_1(n_1), \\
\Delta_2(n_2)&=\omega_r-\omega_2(n_2),\\
\Sigma_1(n_1)&=\omega_r+\omega_1(n_1), \\
\Sigma_2(n_2)&=\omega_r+\omega_2(n_2),
\end{split}
\end{equation}

where counter-rotating terms have already been included in the indirect coupling.

In the dispersive regime that $|\Delta|\gg J$, the Hamiltonian in Eq. (<ref>) can be fully diagonalized via SW transformationSchrieffer-Wolff (SW) transformation using a unitary operatorunitary operator $U$ as:

\begin{equation}
\tilde{H_q}=U^{\dagger}H_q U=\sum_{q=1,2}\sum_{n_q}\tilde{{\omega}}_q(n_q)\left|n_q\right>\left<n_q\right|.
\label{eq.diagH}
\end{equation}

where $U$ can be derived from Eq. (<ref>). The dressed qubit frequencies, anharmonicity, bare bus frequency, coupling strength, and two-photon exchange rate are presented in Tab. <ref>. Note that we define $\tilde{\omega}_q\equiv\tilde{\omega}_q(0)$.

Description     Symbol     Frequency

CSFQ bare freq.    $\omega_1^b(0)/2\pi$     5.0616 GHz
CSFQ dressed  freq.    $\tilde{\omega}_1(0)/2\pi$     5.0511 GHz
CSFQ anharmonicity     $\delta_1/2\pi$     +592.7 MHz
CSFQ bare readout  freq.     $\omega_{h}/2\pi$     6.9065  GHz
CSFQ dressed readout  freq.     $\tilde{\omega}_{h}/2\pi$     6.9074 GHz
CSFQ-readout coupling     $g_{hm}/2\pi$     34 MHz
CSFQ-readout freq. shift     $\chi_{hm}/2\pi$     550 kHz
Trans. bare  freq.     $\omega_2^b(0)/2\pi$     5.2920 GHz
Trans. dressed freq.     $\tilde{\omega}_2(0)/2\pi$     5.2855 GHz
Trans. anharmonicity     $\delta_2/2\pi$     -326.6 MHz
Trans. bare readout freq.     $\omega_a/2\pi$     6.8050 GHz
Trans. dressed readout freq.    $\tilde{\omega}_{a}/2\pi$     6.8059 GHz
Trans.-readout coupling     $g_{aT}/2\pi$     36.2 MHz
Trans.-readout freq. shift     $\chi_{aT}/2\pi$     200 kHz
Bus bare  freq.    $\omega_r/2\pi$     6.3062 GHz
Bus dressed  freq.     $\tilde{\omega}_r/2\pi$     6.3226 GHz
Bus-Trans. freq. shift     $\chi_{rT}/2\pi$     -2.2 MHz
Bus-CSFQ freq. shift     $\chi_{rm}/2\pi$     5.9 MHz
Bus-CSFQ coupling     $g_{rm}/2\pi$     111.7 MHz
Bus-Trans. coupling     $g_{rT}/2\pi$     76.4 MHz
virtual exchange coupling     $J_{00}/2\pi$     6.3  MHz & fit
direct coupling     $g_{mT}/2\pi$     -2.7  MHz

Frequency scales on device with CSFQ at the sweet spot. Dressed qubit frequencies are measured from qubit spectroscopy, readout resonator frequencies and qubit-readout coupling strengths are from resonator measurement. Bare frequencies are calculated using the same method in Ref. <cit.>. Bus dressed frequencies and qubit-bus coupling strengths are from bus spectroscopy <cit.>. Virtual exchange rate is extracted from CSFQ spectroscopy as well as theory fit, and direct coupling is estimated using Eq. (<ref>).

To illustrate two-qubit levels in the computational subspace, we plot the frequency diagram in Fig. <ref>(a). The measured primary qubit frequency and anharmonicity  are 5.051 GHz and $+593$ MHz for the CSFQ at SSsweet spot (SS), 5.286 GHz and $-327$ MHz for the transmon, and thus the qubit-qubit detuning is 235 MHz. The flux dependence of the CSFQ spectrum in Fig. <ref>(b) allows us to explore a range of qubit-qubit detuning in the following experiments. By tuning CSFQ frequency to be exactly the same as transmon, we can extract $J_{00}$ coupling within the computational subspace. This is easy to be proved by truncating the Hamiltonian in Eq. (<ref>) to a $4\times4$ matrix

\begin{equation}
0 & 0 & 0 &0\\
0 & \omega_1 & J_{00} & 0\\
0 & J_{00} & \omega_2 & 0\\
0 & 0 & 0 & \omega_1+\omega_2
\end{array}\right).
\end{equation}

Please note that this truncation does not influence the eigenvalues of $|01\rangle$ and $|10\rangle$ since all interaction has been considered. In the dressed basis eigenvalues can easily be found as

\begin{equation}\label{eq.Jde}
\tilde{\omega}_{01/10}=\frac{1}{2}\left( \omega_1+\omega_2\pm\sqrt{(\omega_1-\omega_2)^2+4J_{00}^2}\right).
\end{equation}

When $\omega_1=\omega_2$ the difference between the two levels is $2J_{00}$. We fit the anti-crossing between CSFQ and transmon in <ref>(b) (inset) to obtain the zeroth-order exchange coupling strength $J_{00}= 6.3$ MHz.

(a) at 15 465
(b) at 335 465

(a) Frequency diagram of transmon and CSFQ at flux sweet spot in the hybrid circuit. (b) CSFQ qubit frequency spectrum vs. external magnetic flux. Orange dots indicate flux points where static ZZ becomes zero. (Inset) Anti-crossing of transmon and CSFQ with fit (red solid line).

In a CSFQ-transmon circuit the second term of static ZZ interaction in Eq. (<ref>) can be negative, due to the large and positive anharmonicity of the CSFQ. Note that for a hybrid CSFQ-transmon device, $J_{01}$ is quite different from $J_{10}$, in contrast to a transmon-transmon pair where they are almost the same. This allows the hybrid CSFQ-transmon pair to be free of ZZ interaction. This interaction is measured using a Joint Amplification of ZZ (JAZZ) experiment <cit.>, which is a modified Bilinear Rotational Decoupling (BIRD) sequence <cit.>. The standard BIRD sequence used in nuclear magnetic resonance (NMR) is a RamseyRamsey interferometry experiment on one qubit, with echo pulses on both the measured qubit and the coupled qubit. In the JAZZ experiment, this sequence is performed twice, for each initial state of the control qubit <cit.>. Static ZZ $\zeta$ is equal to the frequency difference found between the two experiments. The JAZZ experiment is shown in the Fig. <ref>(a) and measured two frequencies are shown in Fig. <ref>(b).

(a) Joint Amplification of ZZ (JAZZ) <cit.> pulse sequences: Ramsey on one qubit with echo pulses on both qubits. (b) Measured frequencies with (without) $\pi$ rotation on control qubit CSFQ at the sweet spot.

Static ZZ measured as a function of flux. Error bars are comparable with or smaller than the size of the data symbols. Red solid line represents a theory calculation using Eq. (<ref>). Static ZZ becomes zero at two flux points (0.496, 0.504), and the corresponding device parameters are shown in the plot.

Based on the measured and estimated parameters, static ZZ interaction has been extracted and fitted in Fig. <ref>. The maximum ZZ is around 140 kHz at the SS, away from this point it decreases and eventually crosses zero symmetrically near $\Phi/\Phi_0$ = 0.496 and 0.504. Corresponding circuit parameters are listed in the plot. It is worth noting that $J_{10}$ is much larger than $J_{01}$ in the hybrid circuit. Our theory agrees well with the experiment except the vicinity of zero-crossing points, where experimental data exhibit a kink. We speculate the kink could be caused by the breakdown of perturbation theory used in Eq. (<ref>). CSFQ frequency goes higher when the flux is away from the SS, in other words, qubit-qubit detuning decreases while $J_{10}$ increases, thus pushing the ratio $J/\Delta$ beyond the dispersive regime. A framework for treating such situation is discussed in Ref. <cit.>.

So far, we have studied the origin of parasitic static ZZ interaction and the possibility of freeing qubits from it. We show that the static ZZ freedom can take place in two-types of circuits: 1) In transmon-transmon pair zero ZZ is found with no entanglement by canceling indirect couplingindirect coupling via increasing direct couplingdirect coupling; 2) In CSFQ-transmon pair, eliminating ZZ interaction is realized by combining opposite sign anharmonicity meanwhile keeping two qubits entangled. We also use our theory to model the experimental hybrid  CSFQ-transmon circuit and demonstrate the presence of static ZZ freedom. In the next chapter, we will study how the ZZ interaction impacts quantum gates when qubits are driven.

CHAPTER: CROSS-RESONANCE GATE AND DYNAMICAL ZZ FREEDOM

Static ZZ interaction is one of the major problems to prevent fast and  high-performance two-qubit entanglement such as MAP gate <cit.>, iSWAP gate <cit.> and cross-resonance gate <cit.> in superconducting circuits. In last chapter we discussed the cancellation of static ZZ in several devices. This chapter we focus on the cross-resonance gate and show how the ZZ freedom can impact the gate fidelity. The main content is from the other parts of our two publications Ref. <cit.> published in Physical Review Letters and  Ref. <cit.> published in Physical Review Applied.

§ CROSS RESONANCE GATE

A cross-resonance (CR) gateCR gate is enabled by driving the control qubit at the frequency of the target qubit, and this CR drive causes the target qubit state to rotate in a direction depending on the control qubit state, and thus corresponds to a ZX gate <cit.>. This is a general method to implement CNOT gateCNOT gate by $\rm CNOT=ZX_{\pi/2}IX_{-\pi/2}ZI_{-\pi/2}$ with additional single-qubit rotations. The CR driving Hamiltonian needs to be transferred to the same regime as the the qubit Hamiltonian in Eq. (<ref>). In the dressed frame it is written as

\begin{equation}
\tilde{H_d}=U^{\dagger}H_{d}U=\Omega\cos(\omega_d t) \sum_{n_1} U^{\dagger}(\left|n_1\right\rangle \left\langle n_1+1\right|+\left|n_1+1\right\rangle \left\langle n_1\right|)U,
\end{equation}

where $U$ is a unitary operator that diagonalizes the circuit Hamiltonian to Eq. (<ref>), $\Omega$ is the driving amplitude at driving  frequency $\omega_d$. Now let us move into the rotating frame at the frequency  $\omega_d$ on both qubits and ignore the counter rotating terms, total Hamiltonian then becomes

\begin{equation}
H_r=W^{\dagger}(\tilde{H_q}+\tilde{H_{ d}}) W-i W^{\dagger}\dot{W},
\label{eq.Hr}
\end{equation}

where $W=\sum_n\exp(-i\omega_{ d} t \hat{n})\left|n\right\rangle \left\langle n\right|$ with $\omega_d={\tilde
\omega}_2$. External pulses sometime are strong enough such that $0.1<\Omega/\Delta<1$ or even $\Omega/\Delta>1$ with $\Delta$ being the qubit-qubit detuning.  Since effective Hamiltonian approach is valid only in the dispersive regime, as shown in Ref. <cit.>, and cannot be performed on qubits in the limit of strong drives.  Therefore we should treat block diagonalization of driving Hamiltonian with a nonperturbative method, other than SW.

§.§ Principle of Least Action

One approach to explore strong driving impacts is the so called “least action method” described in Ref. <cit.>.  This method aims to find a unitary operator $T$ that is closest to the identity operation and can perform a block diagonalization. The least action unitary operator $T$ that satisfies $H_{BD}=T^{\dagger} H_r T$ is given by

\begin{equation}
T=S S_{BD}^{\dagger}S_P^{-\frac{1}{2}},
\label{eq.leastaction}
\end{equation}

where $S$ is the nonsingular eigenvector matrix of $H_r$, $S_{BD}$ is the block-diagonal matrix of $S$, and $S_P=S_{BD}S_{BD}^{\dagger}$.  However, only part of the full matrix is required when block diagonalizing a Hamiltonian, and $T$ can be simplified as follows:  given a general $d\times d$ Hermitian Hamiltonian, it can be transformed into a block diagonal matrix $\mathscr{H}$ using Eq. (<ref>)  with  two blocks $\mathscr{H}_{nn}$ and $\mathscr{H}_{mm}$, where $n$ and $m$ are the dimensions of the blocks and satisfy $d=n+m$ and $n\leq m$. The corresponding matrix of all eigenvectors $S$ can also be divided into 4 blocks, namely
S =

S_nn    S_nm

S_mn    S_mm
As described in Ref. <cit.>, the unitary transformation $T$ can be simplified as $T=U(U^{\dagger}U)^{-1/2}$ where

1    X

-X^†    1
,  U^†U=
1 +X X^†   0

0    1+X^†X
with  $X=-(S_{mn}S_{nn}^{-1})^{\dagger}=S_{nm}S_{mm}^{-1}$. Therefore, only part of the eigenvector matrix $S$ is needed, for example, the first $n$ eigenvectors.

§.§ Effective CR Hamiltonian

To study the impact of the driving pulse on the ZX gate, we block-diagonalize the total Hamiltonian in the rotating frame into two individual qubit blocks within the computational subspace and one block for all higher excited levels  to decouple them. In the case that driving amplitude is large enough to go beyond the dispersive regime, the block diagonalization should be applied under the principle of least action: First two blocks $4\times4$ and $11\times11$, and then repeat this approach in the $4\times4$ matrix to divide it into two 2$\times$2 blocks, each associated to a qubit. Finally, the effective CR Hamiltonian in the computational subspace can be written as

\begin{equation}
H_{\rm CR}=\alpha_{\rm ZI}\frac{\rm ZI}{2}+\alpha_{\rm IX}\frac{\rm IX}{2}+\alpha_{\rm ZX}\frac{\rm ZX}{2}+\alpha_{\rm ZZ}\frac{\rm ZZ}{4}.
\label{eq.Hcr1}
\end{equation}

For a CR gate, only ZX term in the Hamiltonian is desired, the others such as ZZ, IX, and ZI are unwanted interactions. ZI and IX can be cancelled out by echoed CR sequences <cit.>, leaving ZX term accompanied with accumulated phase from ZZ interaction, resulting in the rotation of the target qubit with errors on the Bloch sphere.
On top of the static ZZ interaction $\zeta$, which solely comes from the contribution of higher excitation in the qubit-qubit interaction,  the CR drive with the amplitude $\Omega$ introduces an additional ZZ interaction that depends quadratically on $\Omega$. The two together produce the total ZZ interaction as follows

\begin{equation}
\alpha_{\rm ZZ}=\zeta + \eta\Omega^2+O(\Omega^3),
\label{eq.totZZ}
\end{equation}

where $\eta\Omega^2$ is what we refer to as the dynamical ZZ interaction.

§.§ Classical Crosstalk and Active Cancellation

When applying microwave on one qubit, some additional classical changes happen to another qubit, which is undesirable and should be cancelled, this is called classical crosstalk. In a CR gate, classical crosstalk is assumed to produce new unwanted interactions due to targeted drives on non-local elements. The new unwanted terms have been demonstrated to be existing in experiments <cit.>. In the presence of classical crosstalkclassical crosstalk, the normal driving Hamiltonian is modified by adding another term as

\begin{equation}\label{eq:cc_driving_Hamil}
\begin{split}
H_{d}^{\rm ct}&=\Omega\cos(\omega_{ d} t+\phi_0)\sum_{n_1}(\left|n_1\right\rangle \left\langle n_1+1\right|+\left|n_1+1\right\rangle \left\langle n_1\right|)\\
&+R\Omega\cos(\omega_{ d} t+\phi_0+\phi_R)
\sum_{n_2}(\left| n_2 \right\rangle \left\langle n_2+1\right|+\left|n_2+1\right\rangle \left\langle n_2\right|),
\end{split}
\end{equation}

where $R$ is a scaling factor for classical crosstalk amplitude, and depends on both two-qubit gate length and flux. $\phi_0$ is the new phase of the CR drive to the control qubit and $\phi_R$ is the phase to the target qubit. When the Hamiltonian with classical crosstalk is taken to the dressed frame and block diagonalized, one can find the new terms IY and ZY in the effective CR Hamiltonian:

\begin{equation}
H_{\rm CR}^{\rm{ct}}=\beta_{\rm ZI}\frac{\rm ZI}{2}+\beta_{\rm ZX}\frac{\rm ZX}{2}+\beta_{\rm ZY}\frac{\rm ZY}{2}+\beta_{\rm IX}\frac{\rm IX}{2}+\beta_{\rm IY}\frac{\rm IY}{2}+\beta_{\rm ZZ} \frac{\rm ZZ}{4}.
\label{eq.ctp}
\end{equation}

Our collaborators also applied CR drive on the CSFQ to perform the CR gate and investigate the characteristic behavior of the CR effect. Here we present the theory analysis of the same chip under CR drive: CSFQ is the control qubit while transmon is the target qubit. On the experimental chip,  an external flux threads CSFQ and sometime is tuned away from the SS. By ignoring higher order perturbative results in $\Omega$, e.g. $O(\Omega^4)$, Pauli coefficients can be approximated as

\begin{equation}\label{eq:Pauli_coeff}
\begin{split}
\beta_{\rm ZX}&\approx(B_f\Omega+C_f\Omega^3)\cos\phi_0,\\
\beta_{\rm ZY}&\approx(B_f\Omega+C_f\Omega^3)\sin\phi_0,\\
\beta_{\rm IX}&\approx(D_f\Omega+E_f\Omega^3)\cos\phi_0+RK_f\Omega\cos(\phi_0+\phi_R)\\
%&= M_f(\Omega)\cos\phi_1,\\
\beta_{\rm IY}&\approx(D_f\Omega+E_f\Omega^3)\sin\phi_0+RK_f\Omega\sin(\phi_0+\phi_R)\\
%&= M_f(\Omega)\sin\phi_1,\\
\beta_{\rm ZI}&\approx\alpha_{\rm ZI},\\
\beta_{\rm ZZ}&\approx\alpha_{\rm ZZ},
\end{split}
\end{equation}

where $B_f$, $C_f$, $D_f$, $E_f$  are flux-dependent quantities that can be evaluated numerically. The phase shift can be extracted using quantum state tomography. One intermediate result during the experiment is shown in Fig. <ref>.

Pauli coefficients as a function of CR variable extracted from quantum state tomography (dotted) and fitted using effective Hamiltonian (solid) for the hybrid circuit.

All Pauli coefficients at a certain driving amplitude are extracted by varying the CR phase $\phi_x$ added on top of $\phi_0$. Note that the phase difference between IX and IY, ZX and ZY remains $\pi/2$ which is consistent with Eq. (<ref>).  $\phi_0$ can be find at which the ZX component is maximized and ZY is zero. When IY component is zero, the phase $\phi_R$ is also extracted by setting $\beta_{\rm IY}=0$. These terms except desired ZX will accumulate errors, therefore must be eliminated in the process. $\phi_0$ can be calibrated to $0$ or $\pi$ for achieving the maximum. IX and IY can be removed by either applying an active cancellation pulse on the target qubit <cit.> or optimized rotary echoes <cit.>. Here we introduce the theory behind the active cancellation.

Active cancellation is accomplished by introducing a second microwave drive tone on the target qubit. The new pulse has the similar form as classical crosstalk, i.e. $A\cos(\omega_d t+\phi)\sum_{n_2}(\left| n_2 \right\rangle \left\langle n_2+1\right|+\left|n_2+1\right\rangle \left\langle n_2\right|)$, and also adds new components on top of Pauli coefficients. By assuming $\phi_0$ is calibrated to $\pi$, then ZY term vanishes and ZX is maximum negative. The additional pulse on the target qubit is required to cancel the existing IX and IY terms, namely the following conditions should be satisfied

\begin{equation}
\begin{split}
(D_f\Omega+E_f\Omega^3)+RK_f\Omega\cos\phi_R&=A K_f\Omega\cos\phi,\\
\end{split}
\end{equation}

The next step is to set proper amplitude of the cancellation pulse to remove both IX and IY simultaneously. In experiment, this is realized by sweeping the cancellation pulse amplitude. If IX and IY are not zero at the same cancellation amplitude  then the cancellation phase is incorrect and has to be calibrated again <cit.>.

From Fig. <ref> the phase of CR drive to the control qubit can be calibrated as $\phi_0=\pi$ and $\phi_R$ is measured as $\phi_R=\pi+0.4$. Due to the limit of experimental apparatus, active cancellation pulse was not performed in the final experiment, therefore IX and IY will be kept in the Hamiltonian. Using quantum state tomography experimental Pauli coefficients at the sweet spotsweet spot (SS) are plotted as a function of driving amplitude in Fig. <ref>, together with our theory prediction. One can see that the unwanted ZY vanishes in the device as expected. It is also clear to see that components IZ and ZZ are small and insensitive to weak CR amplitude. The measured Pauli coefficients ZX and ZZ are consistent with the results predicted by the effective Hamiltonian (<ref>).

Pauli coefficients vs. CR amplitude at the sweet spot: experimental tomographic measurements (points) and theoretical curves (solid lines). The parameters in Eq. (<ref>) used for simulation are $R=0.0125$, $\phi_0=\pi$ and $\phi_R=\pi+0.4$.

In brief, the active cancellation process includes two parts: phase calibration can effectively eliminate ZY term, while active cancellation pulse applied on the target qubit can totally remove both IX and IY terms, leaving the effective CR Hamiltonian only with Pauli components ZX and ZZ.

§.§ Echoed CR Rate

Although active cancellation can remove ZY, IX and IY terms, ZI remains large in the Hamiltonian (<ref>). The standard way to eliminate such a term is to use echoed CR pulse <cit.>, which involves two Gaussian flat-top CR pulsesGaussian pulse  with $\pi$ phase difference, and a $\pi$-pulse on the control qubit after each CR pulse as shown inset of Fig. <ref>. If the ZZ interaction can also be eliminated, then an ideal CR gate will effectively behave like a two-qubit gate corresponding to $\rm ZX_{\theta}$ <cit.>:

\begin{equation}
{\rm ZX}_{\theta}=\exp{\left[-i \theta \left(\rm ZX/2\right)\right]}=\left(\begin{array}{cccc}
\cos(\theta/2)&-i\sin(\theta/2)&0&0\\
\end{array}\right).
\end{equation}

By choosing $\theta=\pi/2$ together with single qubit rotations, the CNOT gate can be realized.  However in reality the ZZ interaction is always on, the frequency of the echoed CR oscillation, i.e. $f_{\rm ECR}$, thus can be determined from the following relation:

\begin{equation}
\begin{split}
f_{\rm ECR}&=\sqrt{\left(\beta_{\rm ZX}+\beta_{\rm IX}\right)^2+\left(\beta_{\rm ZY}+\beta_{\rm IY}\right)^2+\left(\beta_{\rm ZZ}/2\right)^2}\\
&+\sqrt{\left(\beta_{\rm ZX}-\beta_{\rm IX}\right)^2+\left(\beta_{\rm ZY}-\beta_{\rm IY}\right)^2+\left(\beta_{\rm ZZ}/2\right)^2}.
\end{split}
\label{eq.fCR}
\end{equation}

If all unwanted terms are eliminated, Eq. (<ref>) reduces to 2$\rm \beta_{ZX}$. Figure <ref> shows the experiential oscillation frequency $f_{\rm ECR}$ of the transmon as a function of the CR amplitude at different flux bias $\Phi/\Phi_0$'s.

Echoed CR rate vs. CR amplitude at four representative flux points. Device parameters are shown in Tab. <ref>. The corresponding qubit-qubit detunings are (234, 217, 166) in MHz. Solid lines correspond to theoretical model.  (Inset) Color density plot of the oscillation of target qubit driven with various CR amplitudes at sweet spot with CSFQ (control) in ground state. Colorbar represents first excited state probability of target qubit. Echoed CR pulse sequence illustrated above inset plots.

At weak driving amplitude, $f_{\rm ECR}$ increases almost linearly, which corresponds to the first order perturbation theory. And by increasing $\Omega$, $f_{\rm ECR}$ deviates from linearity due to the off-resonant drive on CSFQ <cit.>. As CR amplitude becomes stronger,  increase of $f_{\rm ECR}$ slows down, eventually changes the sign of slope to the opposite. This is because the energy levels $E_{11}$ and $E_{02}$ get closer to each other and at certain driving amplitude the anti-crossing takes place as shown in Fig. <ref>. The resulting theoretical curves for $f_{\rm ECR}$ vs. CR amplitude under the principle of least action are consistent with the experimental points.

(a) at 5 380
(b) at 215 380
(c) at 5 175
(d) at 215 175

Dressed energy levels $E_{11}$ and $E_{02}$ vs. CR amplitude for flux points: (a) $f=0.5$, (b) $f=0.5026$, (c) $f=0.5036$, and (d) $f=0.505$. Device parameters are shown in Tab. <ref>.

The $\rm ZX_{90}$ gate produces a Rabi oscillation occurred on the target, where the axis of oscillation will depend on the state of the control. The time length of CR pulse and the oscillation frequency of the target qubit satisfy