\begin{equation}
(2\pi f_{\rm ECR})\tau = \pi/2,
\label{eq.timelength}
\end{equation}

where $\tau$ is the total time length of the square echoed CR pulse. In experiment, one can use Gaussian flat-top CR pulsesGaussian pulse with rising and falling edges, where $\tau_0$ is defined to be the length of the flat-top part of each CR pulse. This $\pi$ rotation includes 20 ns derivative removal via adiabatic gate (DRAG)DRAG pulse, and is buffered by two 10-ns delays. The CR pulses are rounded square with equivalent rise and fall times of 20 ns. Thus we define the two-qubit gate length to be $t_g=2\tau_0+160$ ns, where $\tau_0$ is the average flat-top length of each CR pulse. Due to the finite rise and fall time, we have $\tau>\tau_0$, e.g., for $\tau_0=0$, $\tau$ is non-zero. In the weak driving regime, the linear oscillation frequency can be expressed as

\begin{equation}
f_{\rm ECR} \approx \gamma(f)\Omega,
\end{equation}

with a flux-dependent coefficient $\gamma(f)$, e.g., $\gamma(0.5) \approx 0.1$. The exact flux-dependent $\gamma(f)$ can be found from Eq. (<ref>). By substituting this approximation into Eq. (<ref>), we obtain the following expression for the CR amplitude to make a $\rm ZX_{90}$ gate:

\begin{equation}\label{eq:Omega}
\Omega(f, \tau)=1/[4\gamma(f)\tau].
\end{equation}

§ TWO-QUBIT GATE ERROR

To qualify the gate performance, we compute two-qubit error per gate in an echoed CR pulse sequence for implementing a $\rm ZX_{90}$ gate by considering the density matrix starting in the ground state in the Pauli basis.  CR gate errors can come from several aspects: (1) Limited coherence times compared to operation time. (2)  Classical crosstalk. (3) Parasitic ZZ interaction. Here, ZZ interaction is assumed to be a global error, and for each time step we apply actual gate unitary transformation as well as decoherence terms. The total map following the gate sequences is

\begin{equation}
\rho_{f}=\Lambda_{T1,T2,Q1}\circ\Lambda_{T1,T2,Q2} \circ\Lambda_{\rm ZZ}\circ \Lambda_{\rm XI} \circ \Lambda_{\rm CR-}\circ \Lambda_{\rm XI}\circ \Lambda_{\rm CR+}[\rho_i],
\end{equation}

and each map is defined by <cit.>

\begin{equation}
\begin{split}
\Lambda_{\rm ZZ}[\rho]&=U_{\rm ZZ} \cdot \rho \cdot U_{\rm ZZ}^{\dagger},\\
\Lambda_{\rm XI}[\rho]&=\rm XI \cdot \rho \cdot \rm XI ,\\
\Lambda_{\rm CR\pm}[\rho]&=U_{\rm CR\pm} \cdot \rho \cdot U_{\rm CR\pm}^{\dagger},\\
\Lambda_{T_1,T_2}[\rho]&=\frac{1-e^{-t/T_2}}{2} {\rm{Z}} \cdot \rho \cdot {\rm{Z}} +\frac{1+e^{-t/T_2}}{2} \rho \\
&+\frac{1-e^{-t/T_1}}{2}\left|0\right>\left<1\right| \cdot \rho \cdot \left|1\right>\left<0\right|-\frac{1-e^{-t/T_1}}{2}\left|1\right>\left<1\right| \cdot \rho \cdot \left|1\right>\left<1\right|,
\end{split}
\end{equation}

where the unitary operators $U_{\rm ZZ}$ and $U_{\rm CR\pm}$ are defined as $U_{\rm ZZ}=e^{-i 2 \pi \alpha_{\rm ZZ}t_g \rm ZZ/4}$ and $U_{\rm CR\pm}=e^{-i 2 \pi \tau H_{\rm CR}^{\rm{ct}}(\pm\Omega)}$ with $\alpha_{\rm ZZ}$ defined in Eq. (<ref>) and $H_{\rm CR}^{\rm ct}$ defined in Eq. (<ref>).

To calculate the CR gate fidelityfidelity, we can use a different but efficient visual representation of process maps, the Pauli transfer matrixPauli transfer matrix $\mathcal{R}$ <cit.>, which maps the input density matrix components to output density matrix components in the basis of Pauli operators. In contrast to process matrix representation <cit.>, $\mathcal{R}$ has
several useful properties. It consists of only real numbers and so contains exactly the same number of parameters as the gate, with no redundancy due to Hermiticity. Besides, since it represents the action of a process of density matrix evolution, it can help establish a number of properties of the underlying process which are otherwise hidden in the standard representation. Given a Hilbert space of $d$ dimensions $\mathcal{H}_d$ and space of linear operators $\mathcal{L}(H_d)$, any map $\Lambda$: $\mathcal{L}(H_d)\rightarrow\mathcal{L}(H_d)$ can be represented by the Choi matrix <cit.>

\begin{equation}
\rho_{\Lambda}=\frac{1}{d}\sum_{i,j}E_{ij}\otimes \Lambda(E_{ij}),
\label{eq.choi}
\end{equation}

with $E_{ij}$ being a matrix with 1 in the $ij^{\rm th}$ entry and 0's elsewhere. Now the density matrix is changed to the space $\mathcal{H}_{d}^{(A)}\otimes\mathcal{H}_{d}^{(B)}$. In the following, we assume $A=B$. In case of the Pauli transfer matrix, the qubit state basis is fixed to Pauli matrices $P_i\in\{\rm I, X,Y,Z\}$, and then $\mathcal{R}$ matrix elements are defined as

\begin{equation}
\mathcal{R}_{ij}=\frac{1}{d}{\rm Tr}[P_i\Lambda(P_j)].
\end{equation}

Combining with Eq. (<ref>) changes the expression to

\begin{equation}
\mathcal{R}_{ij}={\rm Tr}[\rho_{\Lambda}P_j^T\otimes P_i],
\end{equation}

\begin{equation}
\rho_{\Lambda}=\frac{1}{d^2}\sum_{i,j}\mathcal{R}P_j^T\otimes P_i.
\end{equation}

The average gate fidelity thus can be found as <cit.>

\begin{equation}
F=\frac{d\rm Tr[\rho_{\rm ideal}\rho]+1}{d+1}=\frac{{\rm Tr}[\mathcal{R}_{\rm ideal}^T\mathcal{R}]/d+1}{d+1}.
\label{eq.fidelity}
\end{equation}

Since no active cancellation is performed on the final experiment, we need to model the classical crosstalk with amplitude $R(f, \tau)\Omega(f,\tau)$, where $R(f,\tau)$ is a flux and gate-length dependent scaling factor. The experimental data on the CSFQ-transmon device revealed that the classical crosstalk on the target qubit is much more significant with increasing gate length. This implies that $R(f, \tau)$ is an increasing function of the gate time length, which is consistent with the assumption that more power is pumped in the system if the gate length is longer.
For simplicity, we consider that flux and gate-length dependence are separable, and fit $R(f, \tau)\approx\alpha(f)\tau^{2/3}$ which agrees well with the experimental data. The nonlinearity with respect to $\tau$ was introduced otherwise the IY Pauli coefficient is consistently larger than the result from measured CR tomography, for example, in Fig. <ref>. The average two-qubit error per gate for the experimental CR gate was measured via standard randomized benchmarkingrandomized benchmarking (RB) <cit.> by varying the flux $f$ and gate length $t_g$ shown in Fig. <ref>.

Average error per two-qubit gate plotted vs. flux for four representative gate lengths. Device parameters of the hybrid CSFQ-transmon circuit are shown in Tab. <ref> Dashed lines indicate theoretical coherence-limited two-qubit gate errors with no ZZ interactions; full theory simulations shown by solid lines.

The flux-dependence of $\alpha(f)$ was extracted by performing separate CR tomography measurements <cit.>. Our collaborators  performed such an active cancellation experiment for a fixed gate length at three different flux points during previous measurement, and found that away from the sweet spot, the classical crosstalk amplitude $R$ followed a nearly linear decrease. To be consistent with previous measurement results, we assume the fitting function of the classical crosstalk to be $R=(0.07-40|f-0.5|^{1.2})\tau^{2/3}$. For the simulation of two-qubit gate error at the flux sweet spotsweet spot (SS), we calculate $R=(0.0123,0.0220,0.0322,0.0383)$ and $\Omega/2\pi=\{70,30,17,12\}$ MHz for the gate length $t_g=(200, 300, 440, 560)$ ns, respectively.

Figure <ref> shows that with increasing gate length, a characteristic “W”-shaped pattern develops with flux, both from theory and experiment, showing the largest errors at sweet spot and two minima at the two flux points $f = 0.496, 0.504$, where static ZZ interaction becomes zero.
The fastest gate measured in this experiment with smallest error rate $1.6\times10^{-2}$ is realized at $t_g = 200$ ns. This behavior can be described by the interplay between fidelity loss from the ZZ interaction and classical crosstalk on the one hand, and fidelity gain from longer coherence times near the sweet spot on the other hand. Away from the sweet spot, the ZZ interaction and classical crosstalk decrease and the gate fidelity approaches the coherence limit.

One of the most prominent advantages of a CSFQ-transmon pair over a transmon-transmon pair is that the static ZZ interaction can be cancelled by carefully choosing qubit parameters.  To make a comparison between hybrid CSFQ-transmon circuits and transmon-transmon circuits, we calculate the gate error in three samples: current CSFQ-transmon device (1), a state-of-the-art tested transmon-transmon device (2) <cit.>, and an ideal CSFQ-transmon device (3) with flux threading CSFQ at sweet spot. On CSFQ-transmon sample (1), we compare four scenarios: in presence of classical crosstalk and nonzero ZZ, only ZZ interaction, static ZZ freedom, only coherence limit. On transmon-transmon sample (2), we compare experimental results and theoretical prediction with ZZ interaction, as well as coherence limit with experimental coherence timescoherence time and ideal coherence times.
On ideal CSFQ-transmon sample (3), we assume both CSFQ and transmon have very long coherence times, and compare the results of static ZZ freedom with coherence limit. In the ideal circuit, static ZZ freedom at the sweet spot can be realized by only changing Josephson energy $E_J$. In practice, such a device could be made by potentially keeping CSFQ at sweet spot, while making the transmon slightly tunable <cit.>.  Corresponding device parameters are listed in Tab. <ref>.

Device     $T_{1}^{(1)}$    $T_{2}^{(1)}$    $T_{1}^{(2)}$    $T_{2}^{(2)}$   $\tilde{\omega}_1$    $\tilde{\omega}_2$    $\delta_1$   $\delta_2$   $\eta$     $\mu$s   $\mu$s    $\mu$s    $\mu$s    GHz    GHz    MHz    MHz   1/MHz(1)    18   15   40   45   5.051   5.286   $+593$   $-327$   $6.0\times 10^{-5}$ (2)   40   54   43   67   5.114   4.914   $-330$   $-330$     $1.6\times 10^{-5}$(3)    200   200   200   200   5.094   5.286   $+593$   $-327$   $8 \times 10^{-6}$
Coherence time, dressed frequency, anharmonicity and nonlinear ZZ interaction rate in Eq. (<ref>) for the current device (1), a transmon-transmon device (2) <cit.>, and an ideal CSFQ-transmon device (3), respectively. $\eta$ is fitted using the data from Fig. <ref>.

By varying the gate length, we plot the gate error in Fig. <ref>. Here (a) represents the device with nonzero static ZZ interaction, while (b) represents the device with static ZZ freedom. Classical crosstalk is only considered in the device marked with $*$. Figure <ref> shows that the CSFQ-transmon device should achieve coherence times at least as long as this experimental device, while maximizing $T_2$ of the CSFQ thus enabling a gate error as the device (1b) comparable to the simulated transmon-transmon results on device (2). For the projected longer coherence times 200 $\mu$s <cit.>, the gate error on device (3b) subject to elimination of classical crosstalk can reach $1\times10^{-3}$. This level is inaccessible for a transmon-transmon device, even with the projected longer coherence times of device (3). Separate comparison of each device can be found in Appendix <ref>.

While coherence-limited gate errors denoted by dashed lines in Fig. <ref> decrease monotonically with gate length, the total error reaches a minimum at an optimum gate length. This is a universal behavior, even in the absence of static ZZ or classical crosstalk e.g. device (3b). It can be explained by the dynamic ZZ that arises from strong CR drive. Since larger CR amplitude is required for shorter gate length and the dynamic ZZ scales quadratically with CR amplitude, at short gate times a large ZZ interaction can still exist, which thus limits the minimum gate error. In next section, we will focus on the additional ZZ component produced by CR drive, and added on top of static ZZ interaction. The new term is a device-dependent quantity, so we could find a way to change it, thus making the total ZZ interaction suppressed.

Experimental data and theory simulation for two-qubit gate error vs. gate length for our present CSFQ-transmon (a), static ZZ-free CSFQ-transmon (b), and a transmon-transmon device with non-zero ZZ (thick lines). The CSFQ was placed at sweet spot. Square CR pulses were used in theory simulation. Three sets of coherence times used in simulation were color-coded in blue, black, and red, and numbered by $n=\{1,2,3\}$. “Limit (n)" represents coherence-limited gate error. Classical crosstalk is not included except $(\rm 1a)^*$. Blue squares and black diamond are experimental data points from present device and Ref. <cit.>, respectively.

§ DYNAMICAL ZZ FREEDOM

In previous section we study the CR gate error and predict that at larger driving amplitude, the dynamical ZZ induced by CR drive leads to a dramatic decrease in the gate fidelityfidelity. In this section, we take several samples of CSFQ-transmon and transmon-transmon pairs by varying the two-qubit detuning as well as anharmonicity of the two qubits to explore the possibility of zeroing total ZZ interaction at a large domain of CR pulse amplitudes. The general idea is to tune the circuit parameters and driving amplitude to make the dynamical part cancel out the static part, meaning that dynamical ZZ should at least has opposite sign with static part. We find that CR amplitude can control the magnitude of dynamic part and allows for vanishing total ZZ strength.  Moreover the freedom is persistent as long as CR gate is active and then improves the CR gate fidelity.

As discussed, when qubits are driven, an additional component is added on top of the static ZZ interaction described in Eq. (<ref>) with $\eta$ being the dynamical quadratic factor. To achieve dynamical ZZ freedom, the minimum requirement is that dynamical part should have opposite sign with static part. Static ZZ interaction can be easily calculated from either full Hamiltonian or effective Hamiltonianeffective Hamiltonian in the dispersive regime in chapter <ref>. In order to evaluate $\eta$, we first derive it from perturbation theory. Although perturbation theory is not a proper approximation for such a driving Hamiltonian, especially at strong driving amplitude, it enables us to find the relation between the dynamical factor and circuit parameters. By block diagonalizing the Hamiltonian in Eq. (<ref>) using SW transformation, dynamical quadratic factor $\eta$ is given by

\begin{equation}
\eta=\frac{J_{01}^2}{2\Delta^2\delta_2(\Delta+\delta_2)^2(\Delta-r\delta_2)^3(2\Delta-r\delta_2)}\sum_{i=0}^6 A_i(r,\gamma)\Delta^i\delta_2^{(6-i)},
\label{eq.eta}
\end{equation}

with $\gamma$ being the ratio of $J_{10}$ and $J_{01}$ in Eq. (<ref>) and $ r\equiv \delta_1/\delta_2$. Detailed $A_i(r,\gamma)$ can be found in Appendix <ref>.

To see how accurate the perturbative result is, we make a comparison between perturbative approach SW transformationSchrieffer-Wolff (SW) transformation and non-perturbative approach least action (LA) principle in terms of Pauli coefficients of ${\rm ZZ}$ and ${\rm ZX}$. Given that the other unwanted terms IX, IY and ZY in the CR Hamiltonian Eq. (<ref>) are eliminated by applying active cancellation pulse on the target qubit and calibrating the global phase of CR pulse to $\pi$, the Hamiltonian is only left with $\rm ZX$ and $\rm ZZ$ terms. Here we take an example of CSFQ-transmon circuit and plot the corresponding Pauli coefficients in Fig. <ref>. One can see that for weak driving amplitude perturbative and non-perturbative methods give consistent results, however as expected they deviate from each other within strong driving domain. This is because in Eq. (<ref>) higher order corrections denoted by $O(\Omega^3)$ starts to contribute in strong driving limit. In the next we keep a record of the both sets of coupling strengths and compare them in the results of $\eta$.

ZZ and ZX coupling strengths in a CSFQ-transmon pair versus CR  amplitude $\Omega$, using Schrieffer-Wolff transformation (dashed) and Least Action transformation (solid).  $\Delta=$0.1 GHz and the other parameters similar to Fig. <ref>(a).

$\eta$ in Eq. (<ref>) as a function of qubit frequency detuning, (a)  CSFQ-transmon devices similar to  Fig. <ref>(a), and  (b)  transmon-transmon devices similar to Fig. <ref>(a) with $g_{12}=2.5$ MHz. Cross points are numerical results from LA transformation and solid line from SW transformation.

In Fig. <ref> we plot the dynamical quadratic factor $\eta$ in two setups using perturbative approach up to the second order in solid lines as well as LA principle in cross points. Figure <ref>(a) shows that in a CSFQ-transmon pair $\eta$ is always positive under current circuit parameters, which makes CR drive to add up positive dynamic ZZ component on top of the static part. This may result in suppression of total ZZ strength if the static part is negative.
While in Fig. <ref>(b) a transmon-transmon pair carries both positive and negative $\eta$ separated by a divergence limit at certain detuning. Perturbation theory shows that in Eq. (<ref>) there exists poles at detuning values $\Delta=-\delta_2, 0, \delta_1/2, \delta_1$, which originates from unexpected higher-level resonance. However LA transformation finds that the divergence is nonphysical and that ZZ strength remains finite, similar to Fig. <ref>.

§.§ Dynamical ZZ cancellation

The opposite sign of dynamical part and static part makes it possible to realize ZZ freedom when the two-qubit are driven. To further explore the possibility, we study five  CSFQ-transmon samples labeled from 1 to 5, and five transmon-transmon samples labeled from 6 to 10, the relative relations of the energy diagrams among the samples are plotted  in Fig. <ref>.

(a) Energy diagrams for  five CSFQ-transmon circuits 1-5. (b) Energy diagrams for five transmon-transmon circuits 6-10.

Let us first talk about CSFQ-transmon devices. In Fig. <ref>(a) the energy levels of $|010\rangle$, $|100\rangle$, and $|001\rangle$ show their differences in the harmonic resonator and the two qubit frequencies, noncomputational levels $|002\rangle$ and $|200\rangle$ are in two sides of $|101\rangle$. Applying CR pulse produces desired ZX entanglement between the two qubits accompanied with unwanted ZZ term. To get precise prediction we determine dynamical ZZ interaction only from non-perturbative LA transformation. ZX strengths in CSFQ-transmon devices have been plotted In Fig. <ref>(a). For each device the strength of ZX coupling increases with the CR amplitude, but monotonously decreases with increasing two-qubit detuning, which can be seen by comparing devices 2 to 5. Furthermore, devices 2 has a larger bus resonator frequency, and shows that smaller qubit-resonator detuning can lead to a fast ZX rate in contrast to device 1. Total ZZ strengths in CSFQ-transmon devices are plotted in Fig. <ref>(b). In samples 1-3 the static ZZ, i.e. at $\Omega=0$, are negative and in 4 and  5 positive. The dynamical part added on top of static part is always positive under current circuit parameters, which is consistent with what has been shown in Fig. <ref>(a). Adding the positive dynamic ZZ component can mitigate the negative static ZZ in samples 1-3, making the total ZZ to be zero at certain driving amplitude $\Omega$.

The coupling strength under CR gate using non-perturbative least action transformation for: (a) ZX interaction in devices 1-5 and (b) ZZ interaction in 1-5. Common  parameters in 1-5 are $\delta_{1/2}=0.6,-0.33$ GHz, $g_{1c/2c}=80$ MHz, and $g_{12}=0$. In 1: $\Delta=70$ MHz and $\Delta_2=1.1$ GHz.  In 2-5: $\Delta=70, 105, 150, 180$ MHz  and $\Delta_2=1.2$ GHz.

Similar calculation is also performed in five transmon-transmon samples labeled from 6 to 10. The corresponding energy levels are depicted in Fig. <ref>(b). In contrast to CSFQ-transmon devices, the noncomputational states $|002\rangle$ and $|200\rangle$ in transmon-transmon circuits are both below $|101\rangle$. Figure <ref>(a) shows that ZX rate is disordered in terms of qubit-qubit detuning, this is because the existence of the divergence-like condition is satisfied at $\Delta=\delta/2=-165$ MHz where multilevel resonance can take place. Samples 6 and 7 are in the two sides of this particular detuning, and device 7 is closer to the point, then the ZX rate is further suppressed, this can be explained by working out the higher order contributions proportional to $\Omega^3$ <cit.>. Since in these examples the coupler frequency is far detuned from qubits, the repulsions between $|101\rangle$ and noncomputational levels in transmon-transmon devices have the same sign and sum together. Dynamical ZZ freedomdynamical ZZ freedom takes place in transmon-transmon circuits as it can be seen in devices 8-10, and the cancellation driving amplitude decreases as the bus resonator frequency is farther away from the qubits by comparing devices 9 with 10. Note that device 6 is the IBM experimental circuit used in Ref. <cit.> and we can see it does not show total ZZ freedom since static part has the same sign with dynamical part. One interesting device is 7 where the total ZZ can approach to zero but never cross it, behaving as a minimum at certain driving amplitude $\Omega$, this is because higher order corrections e.g. $O(\Omega^3)$ at larger amplitude becomes predominant and cannot be ignored.

The coupling strength under CR gate using nonperturbative least action transformation for:  (a) ZX interaction in 6-10 and (b) ZZ interaction in 6-10.  Common parameters in 6-10 are $\delta_{1/2}=-0.33$ GHz, $g_{1c/2c}=98, 83$ MHz, and $g_{12}=2.5$ MHz. In 6-9: $\Delta=-200, -150, -100, -50$ MHz and $\Delta_2=1.4$ GHz. In 10: $\Delta=-70$ MHz and $\Delta_2=2$ GHz.

To further explore the condition of dynamical ZZ cancellation, we can use the dynamical factor $\eta$ in Eq. (<ref>) with sufficient accuracy for weak driving. By solving $\alpha_{\rm ZZ}=\zeta+\eta\Omega^2=0$, the condition for dynamical ZZ freedomdynamical ZZ freedom in the first order of $\Delta/\delta_2$ can be solved at the particular CR amplitude in the limit of $\Delta/\delta_2\ll 1$:
Ω^* =|Δ|√( 2  (r+γ^2)/r+γ(2+γ)) √( 1 - CΔ/δ_2 ),

\begin{equation}
C\equiv \frac{ 1/2+2\gamma+ \gamma^2 + r^2  +  r \gamma (2+\gamma) + \gamma^2 (1+2 \gamma^2)/2r}{ (r +  \gamma^2) [r  + \gamma (2+\gamma) ]}.
\end{equation}

Table <ref> compares the ten samples including both CSFQ-transmon and transmon-transmon pairs, and corresponding cancellation CR amplitude $\Omega^*$ at which dynamical part cancels out the static ZZ interaction. The amplitude $\Omega^*$ is determined using three different methods: In the row labeled by LA we use non-perturbative least action method to determine total ZZ and find where it is zero; In $O(n)$ row we use the SW-evaluated  static ZZ coupling  $\zeta$ of Eq. (<ref>) and the SW-evaluated $\eta$ in Eq. (<ref>) and substitute them in Eq. (<ref>) to find the solution at which amplitude ZZ becomes zero; Below it we present the results from Eq. (<ref>) and in the last row we evaluate the ratio of $\Delta/\delta_2$ in each device. One can see the results are better consistent in the limit of    $\Delta/\delta_2\ll 1$.

2-11 3-11 4-11 5-11 6-11 7-11 8-11 9-11 10-11 11-11
(1)     (2)     (3)     (4)     (5)     (6)     (7)     (8)     (9)     (10)LA     42      30     24     No      No     No     No     115      61     82 $O(n)$     41     31     24     No     No     No     71     83     46     62Eq(<ref>)     41     34     40     20     No     110     104     81     46     61$\Delta/\delta_{2}$     0.11     0.11     0.17     0.25     0.3     0.61     0.45     0.3     0.15     0.21

$\Omega^*$ from different methods in devices 1-10 in unit of MHz. `No' indicates devices with no dynamic ZZ freedom.

Cancellation amplitude of dynamical ZZ freedom versus qubit detuning, (a) for CSFQ-transmon with parameters similar to circuit 2-5 devices, 1 with qubit anharmonicity $\delta_{1/2}=0.6, -0.33$ GHz and 2 with $\delta_{1/2}=0.41, -0.39$ GHz,  (b)  for transmon-transmon with parameters similar to 6-9 devices, 3 with qubit anharmonicity $\delta_{1/2}=-0.33,-0.33$ GHz and 4 with $\delta_{1/2}=-0.41,-0.33$ GHz. Lines are perturbative results, dots and triangles are from nonperturbative least action. Shaded area are the validity domain of perturbation theory.

Let us further study the needed cancellation CR amplitudes at which dynamic freedom is achieved. In what comes next we work with the parameters from CSFQ-transmons devices 2-5  and transmon-transmons devices 6-9. We plot the numerical result  of dynamic ZZ freedom condition within a large range of qubit-qubit detuning $\Delta$ in Fig. <ref>(a) and <ref>(b).  Dots show non-perturbative results using LA transformation and lines are SW perturbative results.  Shaded area is the validity domain within dispersive regime in terms of $\Omega/\Delta$.
Although the ZZ cancellation on samples takes place below 150 MHz, it does not mean that it is limited by the two-qubit detuning. To illustrate this, we add three more triangular points using LA transformation in each setup by changing CSFQ and transmon anharmonicity accompanied with perturbative fitting lines. In CSFQ-transmon pairs,  cancellation CR amplitude is small so perturbative and non-perturbative results agree well with each other.  However, in transmon-transmon pairs one can see perturbation theory is a crude approximation, this is mainly because the static ZZ strength in transmon-transmon pairs is usually large and then cancelling it requires strong driving amplitudes, which falls outside of the dispersive regime. One of the noticeable characteristics of dynamic ZZ freedom in CSFQ-transmon pairs as seen in Fig. <ref>(a)  is that by increasing detuning frequency first cancellation amplitude increases, and then the amplitude squeezes. In CSFQ-transmon devices set $\rm I$,
cancellation CR amplitude is small, but becomes larger by changing qubit anharmonicity, as shown in devices set $\rm II$. In transmon-transmon pairs of Fig. <ref>(b) the amplitude monotonically changes in a wide domain and then starts to drop both in device sets $\rm III$ and $\rm IV$. These behaviors are consistent with what we found above for the way how $\zeta$ and $\eta$ scale with detuning $\Delta$.

§.§ CR gate error

To quantify the performance of echoed-CR gate with respect to the total ZZ interaction, we numerically simulate the CR gateCR gate for several CSFQ-transmon and transmon-transmon devices in Fig. <ref> and Fig. <ref>, respectively. Here we consider the CR pulse is round square with negligible rise and fall times, and keep $\pi$ pulse 40 ns long as in the experiment. When performing a CR gate, the flat-top of the single CR tone $\tau$ satisfies $\tau=1/8\alpha_{\rm ZX}$ for small CR driving amplitude to achieve $\rm ZX_{90}$ rotation, therefore the total gate duration is $t_g=(2\tau+80)$ ns. Assuming that active cancellation pulse eliminate all unwanted terms except ZZ, then the Hamiltonian is only left with desirable ZX and the error source ZZ. We compute the two-qubit error per gate by evaluating how the unitary evolution of the echoed CR gate evolves an initial state as shown in Eq. (<ref>),  and project it into the Pauli basis using Pauli transfer matrixPauli transfer matrix.

Figure <ref>(a) shows the CR gate error caused by ZZ interaction as a function of gate length in CSFQ-transmon pair with different qubit-qubit detuning. In these plots the decoherence effect on the gate is negligible as we assume that qubits can have desirably long coherence times $T_1$ and $T_2$. In CSFQ-transmon devices 2 and 3 the error drops to almost zero at certain gate times, this is exactly where total ZZ can be dynamically set to zero. Figure <ref>(a) and <ref>(b) show that in device 2 the dynamic freedom takes place at $\Omega\sim30$ MHz, where corresponding ZX rate is   $\alpha_{\rm ZX}\sim 2.7$ MHz.  Such a ZX rate requires the flap-top length to be $\tau\sim 46$ ns for each CR pulse to perform $\pi/2$ ZX gate.  The total echoed-CR pulse length then is around 172 ns, which is consistent with the plot. For device 3 zero ZZ interaction takes place at $\Omega\sim21$ MHz, where $\alpha_{\rm ZX}$
is smaller by a factor of 1/1.7 compared to device 2, this causes the prolongation of the gate length of the single CR pulse and makes the total gate time to be 235 ns. While the gate error in devices 4 and 5 monotonously decreases as the gate becomes longer, which corresponds to weak driving amplitude and then smaller total ZZ as the dynamical contribution has the same sign with static part. The gate error in absence of decoherence is in the scale of $10^{-3}$.

ZZ error of the echo CR gate as a function of gate length and two-qubit detuning in (a) CSFQ-transmon devices (b) transmon-transmon devices.

Figure  <ref>(b) presents the gate error in transmon-transmon devices with expected similar behavior as the CSFQ-transmon pair. The time length of perfect ZX gate in devices 8 and 9 are shorter compared to CSFQ-transmon pairs. The reason for such improvement is that static ZZ is larger in these transmon-transmon devices, then the dynamic cancellation requires stronger amplitude where ZX rate is larger. For device 8 the dynamical ZZ freedom takes place at $\Omega\sim120$ MHz, where $\alpha_{\rm ZX}\sim 3.5$ MHz.  Such a ZX rate requires the flat-top length to be $\tau\sim 36$ ns for each CR pulse to perform $\pi/2$ ZX gate, making the total time to be 152 ns. While cancellation amplitude is smaller $\sim 60$ MHz in device 9, but the ZX rate increases a little faster to 3.7 MHz since the two-qubit detuning is far away from fake divergence point $-\delta/2$, resulting in a shorter gate at 148 ns. However, one can see that transmon-transmon devices 7, 8, and 9 show some cutoff in their minimum gate length for ${\rm ZX}_{\pi/2}$ gate. The answer can be found from the saturation of ZX in Fig. <ref>(a), where $\alpha_{\rm ZX}$ rate starts to saturate after some amplitudes and cannot increase anymore. This saturation limits the flat-top length $\tau$ such that it cannot become shorter than a minimum $\tau_{\rm min} = 1/8\alpha_{\rm ZX}^{\rm max}$. More preciously, this cutoff can be calculated as $t_g^{\rm min}=(1/4\alpha_{\rm ZX}^{\rm max}+80)$ ns. For instance device 7 in Fig. <ref>(b) reaches to a maximum at $\sim 2.5$ MHz and this introduces a gate time cutoff below $\sim 180$ ns as shown in Fig. <ref>(b). Moreover the device 7 shows a finite minimum at certain gate time where dynamic ZZ is suppressed to a minimum of $\sim$20 kHz, which results from the higher order contribution for stronger drive. Although th error cannot be eliminated, it can be reduced to $10^{-5}$ without decoherence error.  The worst scenario can be seen in device 6 where the gate error increases with gate time as total ZZ cannot be eliminated.

Following the experiment, we study what the CR gate is and how to constitute such a gate. We evaluate the gate performance with respect to ZZ interaction and classical crosstalk. Our theory also predicts that the CR gate with 99.9% fidelity is achievable in the absence of static ZZ interaction with longer coherence times. The CR pulse produces new ZZ component adds on top of the static part, thus leading to a new strategy to cancel the total ZZ interaction, namely dynamical ZZ freedom. We show that this freedom is applicable in both transmon-transmon circuits and CSFQ-transmon circuits. In the next chapter, we will continue the discussion of parasitic ZZ interaction and propose a new gate at which both static and dynamical ZZ freedom is realized throughout the operation.

CHAPTER: SOME NOVEL GATES

In chapter <ref> we presented that static ZZ freedom is achievable experimentally on a superconducting circuit consisting of a pair of opposite sign anharmonicity at certain parameters. These parameters in the case of our experimental device <cit.> could be met by tuning external flux of CSFQ slightly away from the sweet spot. In that device CSFQ offers another degree of freedom, which is proved to be efficient for mitigating parasitic unwanted ZZ interaction. One can also achieve requirements for static ZZ freedom by changing coupling strength, which requires tuning coupler parameters. However, this tunability always challenges coherence characteristics of the coupler, as what has been seen in the CSFQ-transmon experiment that tuning flux away from sweet spot leads to a dramatic decrease in the decoherence time and then restricts the CR gate fidelity. Therefore suppressing flux noise while keeping tunability is the key factor to a coupler.

Alternatively,  there are several types of tunable couplers which have previously been both designed in theory and tested in experiment <cit.>, can also provide a new degree of freedom. One of the promising candidates is the asymmetric transmon <cit.>, which plays the same role as a tunable device, meanwhile the coherence timecoherence time can slightly decrease to keep the balance of the two properties. Such an asymmetric transmon can either serve as a control/target qubit or a coupler instead of a harmonic oscillator. This type of qubit has been implemented in several devices and enables the improvement of entangling gates performance, by which desired entanglement can be enhanced or unwanted terms can be suppressed <cit.>.

In this chapter we first introduce the properties of asymmetric transmons including frequency tunability and flux noise sensitivity. By including the flux-controlled tunable coupler with a continuous tunability in superconducting circuits, we take advantage of the cancellation for unwanted both static and dynamical ZZ interaction as described in chapter <ref> and chapter <ref>,  then propose a new gate. Later we show how to take advantage of wanted ZZ and make it stronger to constitute a novel CZ gateCZ gate.

§ PROPERTIES OF AN ASYMMETRIC TRANSMON

Let us revisit the discussion in chapter <ref>, the asymmetric transmonasymmetric transmon consists of two Josephson junctionsJosephson junction with external flux threading the loop, plasma frequency is then changed to

\begin{equation}
\omega(f)\approx\sqrt{8E_CE_J(f)}=\sqrt{8E_CE_{J\Sigma}\sqrt{\cos^2(\pi f)+d^2\sin^2(\pi f)^2}},
\label{eq.plas}
\end{equation}

where $E_{J\Sigma}=E_{J1}+E_{J2}$ and $d=(a-1)/(a+1)$ with $a$ being the ratio of Josephson energy $E_{J1}$ and $E_{J2}$. Equation (<ref>) is a periodic function in terms of flux, meaning that plasma frequency oscillates with $f
$. Here we consider external flux is limited to half a period such that $f\in[0,0.5]$ as it has covered the whole tunable window with two boundary frequencies calculated by

\begin{eqnarray}
\omega(f=0.5)&=&\sqrt{8dE_CE_{J\Sigma}},\\
\omega(f=0)&=&\sqrt{8E_CE_{J\Sigma}},
\end{eqnarray}

and $\omega(f=0)>\omega(f=0.5)$ since $0<d<1$. Such two boundaries of the frequency window indicate that larger ratio $a$ leads to a narrow tunable domain. In the next we take an example to illustrate how the frequency is tuned.

Given that the asymmetric transmon has $E_C=0.337$ GHz and $E_{J\Sigma}=17.5$ GHz, external flux $f$ is tuned between 0 and 0.5, we first plot the bare frequency $f_{01}$ as a function of the ratio $a=1,5,9,13$ in Fig. <ref>(a). One can see that the tunable frequency window becomes narrower with increasing ratio $a$, from more than 1.2 GHz at $a=1$ to around 500 MHz at $a=13$.

Asymmetric transmon properties (a) frequency $f_{01}$ versus flux (b) dephasing rate versus flux for different $a=E_{J1}/E_{J2}$, the ratio of two Josephson energies.

However, the new degree of freedom also brings some troubles such as flux noise.

Dephasing rate of a qubit is proportional to the frequency gradient as a function of flux $D_{\Phi}=|\partial f_{01}/\partial\Phi|$, details can be found in Appendix <ref>. This means that tuning the flux away from the sweet spotsweet spot (SS)  leads to decrease in coherence time. In other words, large tunability causes more loss. To assess the effect of flux noise on dephasing, we assume the flux noise power spectrum is identical for all different ratios, which allows background dephasing to be the same for each ratio. Another assumption is that the asymmetric transmon has the same modified expression of dephasing rate as the CSFQ with $\Gamma_{\Phi}=(0.00288{\rm m\Phi_0}) D_{\Phi}+0.039\mu s^{-1}$. By calculating $D_{\Phi}$ using data in Fig. <ref>(a) and substituting the derivative into $\Gamma_{\Phi}$, we obtain the dephasing rate and plot it for the four ratios in Fig. <ref>(b). It can be seen that at two ends dephasing rate is the lowest for all ratios, and the two points are called sweet spots of the asymmetric transmon. The dephasing rate can be approximated as parabolic function of flux, and the sensitivity to $1/f$ flux noise appears to be suppressed with increasing ratio $a$, i.e. $\Gamma_{\varphi}$ is comparably flat across the entire tuning range with a maximum value less than 0.1 $\mu s^{-1}$ when the ratio is 13.

§ THE PARASITIC FREE (PF) GATE

In chapter <ref> we introduce the dynamical ZZ freedom in driven circuits with a fixed-frequency bus resonator, which requires non-zero static ZZ interaction when qubits are idle. However, qubits are not always active or driven in a large quantum processor, therefore the crosstalk from surrounding qubits does affect the working qubits.

One way to deal with this problem is replacing the harmonic bus cavity with a tunable coupler such as the asymmetric transmon. One can first tune the coupler frequency to turn off static parasitic interaction in idle status, and turn it on accordingly when qubits are driven to dynamically cancel out ZZ interaction. In this way we make a gate where ZZ freedom is achieved throughout the gate length, namely parasitic free (PF) gateparasitic free (PF) gate <cit.>. In the following, we discuss the case that two fixed-frequency transmons are coupled via the tunable coupler.  This circuit can produce perfect $\rm{ZX}_{90}$ entanglement when qubits are driven by the CR-type pulses, and is free of static ZZ interaction when qubits are idle.

The circuit scheme is shown in Fig. <ref>. In the pairwise interacting quantum circuit, two fixed-frequency transmons are directly coupled via a capacitor $C_{12}$, and indirectly couple to each other mediated by the asymmetric transmon through $C_{1c}$ and $C_{2c}$, respectively.

Circuit schematic of two fixed-frequency transmon qubits directly coupled to a tunable coupler via $C_{1c}$ and $C_{2c}$, and capacitively couple to each other via $C_{12}$.

Usually $C_{12}$ is much smaller than $C_{1c}$ and $C_{2c}$, such that qubit-coupler interaction is much stronger than the qubit-qubit direct couplingdirect coupling, i.e. $g_{1c}, g_{2c}\gg g_{12}$. The system Hamiltonian is written in the form of multilevel systems as

\begin{equation}\label{eq.hcoupler}
\begin{split}
H&=\sum_{i=1,2,c}\sum_n\omega_i(n_i)|n_i+1\rangle\langle n_i+1|+\sum_{i<j}\sum_n \sqrt{(n_i+1)(n_j+1)}\\
&\times g_{ij}(|n_i\rangle\langle n_{i+1}|-|n_{i+1}\rangle\langle n_i|)\otimes(|n_j\rangle\langle n_{j+1}|-|n_{j+1}\rangle\langle n_j|),
\end{split}
\end{equation}

where $\omega_i(n_i)=E_i(n_i+1)-E_i(n_i)$ with $E_i(n_i)$ being the bare energy for subsystem $i~(i=1,2,c)$. There are two ways to describe this Hamiltonian: one is the sum of difference photon number blocks with interaction terms in the off-block part, the other is the sum of the same excitation number blocks with no interaction in between if counter-rotating terms are ignored <cit.>. Since the coupler is assumed to be far detuned from fixed-frequency transmons, here we choose the first approach and write the Hamiltonian in matrix form as

\begin{equation}\label{eq.hmatrix}
H=\left( \begin{array}{cccccc}
0_p & G_{01} & G_{02} &&&\\
G_{01} & 1_p & G_{12} &&&\\
G_{02} & G_{12} & 2_p& && \\
& & & &&.\end{array}\right). \\
\end{equation}

where $i_p$ is the block containing $i$ photons and $G_{ij}$ includes all interaction between block $i$ and block $j$.

To simplify the Hamiltonian, we can apply SW transformationSchrieffer-Wolff (SW) transformation to decouple $0_p$ block from higher photon excitation numbers, the system thus reduces to the two-qubit basis. As the coupler is an asymmetric transmon and the levels are anharmonic, so at least $2_p$ should be included in the full Hamiltonian before decoupling. After block diagonalizing Eq. (<ref>), the computational subspace can be fully diagonalized and then has the same form as Eq. (<ref>). In the dispersive regimedispersive regime that $g_{ij}/\Delta_{ij}\ll 1$ with $\Delta_{ij}=\omega_i-\omega_j$, we can calculate the eigenvalues of each level using perturbation theory up to fourth order correction, and then derive static ZZ interaction as

\begin{eqnarray}
\zeta&=&\zeta^{(2)}+\zeta^{(3)}+\zeta^{(4)}\label{eq.zetap},\label{eq.zetatot}\\
\zeta^{(2)}&=&2g_{12}^2\left(\frac{1}{\Delta_{12}-\delta_2}-\frac{1}{\Delta_{12}+\delta_1}\right)\label{eq.zeta2}, \\
\zeta^{(3)}&=&2g_{1c}g_{2c}g_{12}\left[ \frac{2}{(\Delta_{12}-\delta_2)\Delta_{1c}}-\frac{2}{(\Delta_{12}+\delta_1)\Delta_{2c}}\right.\nonumber\\
\end{eqnarray}

\begin{eqnarray}
\zeta^{(4)}&=&g_{1c}^2g_{2c}^2\left[2\left(\frac{1}{\Delta_{1c}}+\frac{1}{\Delta_{2c}}\right)^2\frac{1}{\Delta_{1c}+\Delta_{2c}-\delta_c}  \right.\nonumber\\
&&-\left.\frac{2}{\Delta_{2c}^2(\Delta_{12}+\delta_1)} +\frac{2}{\Delta_{1c}^2(\Delta_{12}-\delta_2)}\right.\nonumber\\
&&-\left. \left(\frac{1}{\Delta_{2c}}+\frac{1}{\Delta_{12}} \right)\frac{1}{\Delta_{1c}^2}- \left(\frac{1}{\Delta_{1c}}-\frac{1}{\Delta_{12}} \right)\frac{1}{\Delta_{2c}^2}\right].\label{eq.zeta4}
\end{eqnarray}

It is worth noting that higher excitation number in the coupler i.e. the second excited level with respect to the anharmonicity $\delta_c$ has already been included in the fourth order correction.

Circuit parameters are assumed as follows: the frequency and anharmonicity of qubit 1 and 2 are fixed at $\omega_1=5.12$ GHz, $\delta_1=-0.322$ GHz, and $\omega_2=5.0$ GHz, $\delta_2=-0.322$ GHz, separately, the tunable coupler frequency domain is $\omega_c=5.8-6.8$ GHz, qubit-coupler coupling strength $g_{1c}=g_{2c}=120$ MHz and direct couplingdirect coupling strength $g_{12}=8$ MHz by considering a small direct capacitance. By substituting all parameters into Eqs (<ref>)$-$(<ref>), static ZZ interaction for three coupler anharmonicity is evaluated and plotted in Fig. <ref>. We calculate static ZZ using two approaches: perturbation theory in Eq. (<ref>) with solid line and full Hamiltonian in Eq. (<ref>) with dashed line.

One can see that the two approaches are consistent when coupler frequency is large while anharmonicity is small, but deviate from each other at lower coupler frequency with larger anharmonicity. This disagreement is due to the inaccuracy of perturbation theory on the edge of the dispersive limit, which is most evident at lower coupler frequency with anharmonicity $\delta_c=-350$ MHz. For instance, at $\omega_c=5.8$ GHz, $|020\rangle$ with bare energy 11.25 GHz, interacts with $|110\rangle$ with bare energy 10.92 GHz and $|011\rangle$ with bare energy 10.8 GHz, via the coupling strength $\sqrt{2}g_1=\sqrt{2}g_2\approx170$ MHz, leading to $g/\Delta\sim0.35$, which falls outside of the consistency domain <cit.>.

OFF status: Figure <ref> shows that static ZZ freedomstatic ZZ freedom is achievable in all devices. As the anharmonicity is more negative, the required coupler frequency for zeroing ZZ interaction is lower. Such coupler frequency for ZZ freedom when qubits are in idle status corresponds to the “off” point of the flux sequence threading the asymmetric transmon.

Static ZZ interaction as a function of coupler frequency for three different coupler anharmonicity $\delta_c$. Solid line shows the results obtained from perturbation theory up to four order, dashed line are the results from full Hamiltonian model with off points marked by $\ast$.

ON status: Now let us move to driven regime. Based on the discussion in CR gates we know that fast ZX rate takes place at stronger driving amplitude, at which dynamical ZZ interaction is also larger. In such a device static ZZ should be designed large so that required cancellation driving amplitude is also strong. In the next we will focus on the device with anharmonicity $\delta_c=-150$ MHz as the static ZZ is the largest compared to the other two curves. The off frequency is about 6.345 GHz read from Fig. <ref> when the qubits are in idle status. The system is driven by an echoed CR pulse, with which ZX gate is produced with angle $\theta=\alpha_{\rm ZX} t$ and other terms except ZZ can be eliminated via active cancellation pulse as described in section <ref>. External CR drive adds dynamical ZZ on top of the static component, and makes it possible to achieve dynamical ZZ freedomdynamical ZZ freedom if static and dynamical ZZ have the opposite sign, as shown in Fig. <ref>.

(a) Total ZZ interaction, the sum of static part and CR drive induced dynamical part.  (b) ZX rate versus CR driving amplitude $\Omega$ for four coupler frequencies with anharmonicity $\delta_c=-150$ MHz. Dynamical ZZ freedom takes place around $\Omega=75$ MHz.

Figure <ref> shows corresponding ZX and ZZ strength for four different coupler frequencies using non-perturbative least action transformation. Total ZZ reduces to the static part at $\Omega=0$ which is consistent with Fig. <ref>. Dynamical ZZ cancellation in all four devices can take place around driving amplitude $\Omega=80$ MHz, at which ZX component becomes larger with decreasing coupler frequency. For instance, if the “on” frequency is 5.8 GHz, the dynamic freedom takes place at $\Omega\sim75$ MHz, where corresponding ZX rate is   $\alpha_{\rm ZX}\sim 7.5$ MHz.  Such a ZX rate requires the flat-top length to be $\tau\sim 17$ ns for each CR pulse to perform $\pi/2$ ZX gate. By considering the each of the echo CR pulse is followed by a 40 ns $\pi$ rotation, the total pulse length is around 114 ns with respect to a square-shape CR pulse.

Another feature of the PF gate is that flux noise can be suppressed throughout the operation. As mentioned before, the asymmetric transmon suffers from $1/f$ noise when tuning the flux away from sweet spot. In order to reduce the error caused by flux noise, the circuit design and qubit control are needed to be optimized. As described in Fig. <ref>(b) the flux noise is suppressed most around the sweet spots $f=0$ and $f=0.5$. If we can find a device where the two on and off points are around sweet spots with  lower dephasing rate $\Gamma_\varphi$, the gate performance will undoubtedly be improved. Here we show one example of such an ideal device. The coupler can be designed as follows: $E_C=0.142$ GHz, $E_{J1}/E_{J2}=10$ and $E_{J\Sigma}=37.6$ GHz. Figure <ref> plots the coupler frequency as a function of external flux. Dashed lines mark on and off frequency with about 50 MHz gap to the frequencies at sweet spots, which provides room for circuit parameters variations.

Coupler frequency  versus external flux. On and off frequencies are designed to be close to sweet spot with a gap about 50 MHz.

In such a device where the coupler frequency is tunable, we can realize the universal ZZ freedom. This means that in idle status tuning coupler frequency to the off point can decouple the ZZ interaction, with which the single-qubit operation accuracy can be improved; In driven status, first by tuning the coupler frequency to the on point, a non-zero static parasitic interaction is obtained, then applying external pulses especially driving control qubit with target  frequency adds exactly the same amount of ZZ but with an opposite sign. The total effect will make the dynamical ZZ cancellation possible. Compared to the devices described in chapter <ref>, the new circuit with the asymmetric transmon can produce a larger ZX rate, which can effectively reduce the gate length.

§ THE TUNABLE CZ GATE

In previous discussion we always aim to suppress or eliminate this parasitic ZZ interaction. In fact, without any control the free evolution of a system in presence of ZZ coupling can be used to constitute a controlled-Z (CZ) gateCZ gate <cit.>. In a large quantum processor, we want a pair of interacting qubits to be free of ZZ interaction in idle status but preserve strong ZZ when performing a CZ gate. Generally speaking, decoherence time of a superconducting qubit is a couple of microseconds. To realize high-performance gates the operation time should be shorter, i.e. several Hundreds of nanosecond, which limits the ZZ interaction to be in the magnitude of megahertz. Then it comes to the question how can we choose the frequency domain to fast tune the ZZ interaction from zero to several megahertz? A few experiments have been performed by replacing the harmonic resonator with the asymmetric transmon in transmon-transmon pairs <cit.>, and successfully realized shorter CZ gates, but none of them can tune ZZ interaction to the exact zero.

Here we introduce a device on which an asymmetric transmon coupled to a CSFQ via a harmonic bus resonator. As discussed, CSFQ-transmon pair is a good device where static ZZ freedom can be achieved without driving meanwhile entangling two qubits, although CSFQ suffers from shorter decoherence time as flux is away from sweet spotsweet spot (SS). One way to deal with flux noise is fixing CSFQ at the sweet spot, and utilizing the asymmetric transmon instead to tune ZZ interaction. Thus we can totally turn off ZZ interaction in idle status, and turn it on to realize the CZ gate.

The circuit is designed as follows: an asymmetric transmon couples to a CSFQ mediated by a bus resonator, each qubit is measured by a readout cavityreadout cavity. The frequency is arranged that bus resonator is in between two qubits. Detailed circuit scheme is shown in Fig. <ref>. Compared to the experimented hybrid CSFQ-transmon circuit in chapter <ref> and <ref>, Josephson junctionJosephson junction in transmon is replaced by a DC SQUID,  two more capacitors are included using ANSYS Q3D software, which efficiently performs the 3D and 2D quasi-static electromagnetic filed simulations required for the extraction of RLCG parameters from an interconnect structure.

Circuit scheme of CSFQ coupled to an asymmetric transmon via a bus resonator.

Before we go deeper into the analysis of the circuit and parameters, one question should be answered, that is how we can strengthen ZZ interaction and remain the zero point in the same device. When the bus resonator is far detuned between two qubits, maximally ZZ is less than 1 MHz as shown in Fig. <ref>. Perturbed derivation of ZZ interaction in Eq. (<ref>) indicates that increasing $J$ coupling can effectively enlarge it, and the most applicable method is to reduce the bus resonator frequency. Therefore the qubit-bus detuning should be changed, i.e. less than 1 GHz when the gate is on. Circuit Hamiltonian has the same form as Eq. (<ref>), with corresponding circuit parameters are listed in Tab. <ref>.

Figure <ref> shows frequency tuning window is about 700 MHz, anharmonicity changes slowly from $-385$ MHz to $-394$ MHz. Solid blue curve is the numerical derivation and dashed red is the result from fitting function expressed as

\begin{eqnarray}
\omega_1&=&6.51262 - 6.7 f^2 + 3.5 f^3 + 8.1 f^4,\label{eq.omega1}\\
\delta_1&=&-0.385807 - 0.0635 f^2 + 0.124 f^4.
\end{eqnarray}

4|c|Capacitance (fF)     2c|Josephson energy (GHz)$C_{12}$   $3.4$     $C_{78}$     $4.44$     $E_{JT1}$     $15.75$$C_{20}$   $52.8$     $C_{70}$     $68$     $E_{JT2}$      $1.75$$C_{30}$   $43.9$     $C_{50}$      $57.5$      $a$     $9$$C_{ shT}$   $21$      $C_{shCSFQ}$   $27.5$    $E_{J1}=E_{J2}=E_J$     $111.70$$C_{T1}$   $4$      $C_1=C_{2}=C$   $4$    $E_{J3}=\alpha E_J$    $48.03$$C_{T2}=C_{T1}$   $4/9$     $C_{3}=\alpha C$      $1.72$     $\alpha$    $0.43$$C_{34}$    $26.52 $     $C_{45}$     $32.6$   2c|Inductance (nH)$C_{24}$    $2.17$     $C_{47}$     $2.24$   $L_{rT}$   1.14$C_{rT}$    $454.73$    $C_{rCSFQ}$     $461.32$     $L_{rCSFQ}$     $1.15$$C_{R}$    $600.59$              $L_{R}$     $1.50$

Circuit parameters design values that were extracted using ANSYS Q3D Extractor simulation of the qubit layout.

(a) Asymmetric transmon frequency $f_{01}$ as a function of flux $f$(b) Asymmetric transmon anharmonicity as a function of flux $f$. Solid line represents the result from numerical simulation, while dotted line represents corresponding fitting function.

Let us consider $C_{24}\ll C_{34}$ and $C_{47}\ll C_{C45}$, thus contribution from the two additional capacitors will not affect the expression in Eq. (<ref>), and all $g$ couplings can be obtained by simply replacing $a-h$ with 1-8 correspondingly. By substituting circuit parameters into the expression in chapter <ref>, CSFQ bare frequencies, anharmonicity, bus frequency and coupling strengths are calculated in Tab. <ref>.

The variation of coupling strength due to tunability of the coupler is tiny and then ignored. Note that CSFQ-bus detuning is only 400 MHz while corresponding coupling strength is 167 MHz. Although the system is still within the dispersive regime, the accuracy of perturbation theory is not enough. In the following we will denote the asymmetric transmon as Q1, CSFQ as Q2 and deal with the system using full Hamiltonian model.

CSFQ freq.      4.9 GHz
CSFQ anharm.      $577.6$ MHz

Bus  freq.     5.3 GHz
Bus-CSFQ coupling     167 MHz
Bus-Trans. coupling      161 MHz

CSFQ readout     6.9 GHz
CSFQ readout g     42 MHz
trans. readout     7.0 GHz
trans. readout g     40.6  MHz
direct coupling     10.1 MHz

Frequency scales on device with CSFQ at the sweet spot.

To easily illustrate how the CZ gate is operated, we use its energy-eigenstates $|Q1,R,Q2\rangle$ to describe the system. Figure <ref> shows system eigenvalues as a function of Q1 frequency, and only states in the two-excitation manifold are plotted. Although the anti-crossing is beyond tunable domain, it is also worth noticing that two-photon process between $|101\rangle$ and $|020\rangle$ is possible especially when the gap is small at lower transmon frequency.
For an ideal CZ gate, the computational state $|101\rangle$ should adiabatically evolve from the off point to the on point at which ZZ is strengthened, then back to the off point with no leakageleakage to non-computational states. This can be realized by tuning transmon frequency, during this process non-zero ZZ leads to a CZ gate if the total phase accumulation of Q1 is $\pi$. Here the adiabatic processadiabatic process means unwanted transitions should be avoided. Particularly, the minimum gap between |$101\rangle$ and other states is a key factor in determining adiabaticity.

System eigenvalues as a function of the asymmetric transmon frequency. Only states in the two-excitation manifold are shown.

In our device, the minimum gap occurs between $|101\rangle$ and $|020\rangle$ states, and can be evaluated through near-degenerate perturbation theory when the two states are near resonance <cit.>. There is a two-fold degeneracy before interaction Hamiltonian is switched on, then we can separate the eigenstates into two sets: $|m\rangle$ denotes the unperturbed eigenstates of the degenerate energy $E_d$ with $m\in d$, and $|l\rangle$ is the set of unperturbed eigenstates with different energies. Note that the two sets do not coincide with each other. Define projectors $Q_d=\sum_{m\in d}|m\rangle\langle m|$ and $P_d=1-Q_d$, then second order perturbation theory in the degenerate subspace gives rise to <cit.>

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

where $H_0$ is the unperturbed Hamiltonian, $V$ is the interaction Hamiltonian and $E_k$ is the eigenvalues. By substituting all parameters into the equation above, we find the minimum gap between two states is $\sim$ 20 MHz, corresponding to $2g_{101\leftrightarrow020}$, similar to what we found in Eq. (<ref>). When operating at transmon frequency away from the degenerate point, the condition $g_{101\leftrightarrow020}t>1$ should be satisfied, meaning that the CZ gate should be slow enough to avoid such a transition, i.e. $t>100$ ns.

Similarly the full Hamiltonian can be diagonalized and rewritten as

\begin{equation}
H=\tilde{\omega}_1|100\rangle\langle 100|+\tilde{\omega}_2|001\rangle\langle 001|+(\tilde{\omega}_1+\tilde{\omega}_2+\alpha_{\rm ZZ})|101\rangle\langle 101|,
\end{equation}

with $\alpha_{\rm ZZ}$ being plotted as a function of transmon frequency in Fig. <ref>.

ZZ interaction as a function of asymmetric transmon frequency, on and off frequencies are 5.981 GHz and 6.257 GHz, respectively. Solid line represents the result from numerical simulation, while dotted line represents corresponding fitting function.

One can see that under current parameters ZZ coupling can be completely turned off at $\omega_1=6.257$  GHz. Corresponding fitting function marked by dashed line is

\begin{equation}\label{eq.zfit}
\alpha_{\rm ZZ}(\omega_1)=\left[\frac{1}{0.73(\omega_1-5.7)}+1.75(\omega_1-5.75)^2-2.92\right] \rm MHz.
\end{equation}

To speed up CZ gate while maintaining no leakage to higher levels, we choose a relatively large ZZ coupling strength to be 2 MHz as the working point at $\omega_1=5.981$ GHz. Corresponding flux can be evaluated from fitting functions Eq. (<ref>) as
$f=0.214$ OFF and $f=0.338$ ON, separately.

To further suppress the leakage to noncomputational levels, an optimized control pulse is needed to be implemented such as flat-top Gaussian pulse <cit.> and Slepian-based optimal control <cit.>. In each case the pulse is not square, then accumulated conditional phase is expressed as

\begin{equation}\label{eq.zint}
\phi_{\rm ZZ}=\int\alpha_{\rm ZZ}(t)dt,
\end{equation}

and a CZ gate is realized when $\phi_{\rm ZZ}=\pi$. In the following we will take an example of tanh-shaped pulse to estimate the gate length. The pulse is expressed as

\begin{equation}
\label{eq.pulse}
\begin{split}
f(f_0,f_{\rm end},x,s,t)&=[f_0+\tanh(xst)(f_{\rm end}-f_0)][\sign(1/s-t)+1]/2\\
&+\{f_0+\tanh[xs(2/s-t)](f_{\rm end}-f_0)\}[\sign(t-1/s)+1]/2,
\end{split}
\end{equation}

where $s=1/2t_g$ with $t_g$ being the gate length, $f_0=0.214$, $f_{\rm end}=0.338$ and $x$ determines the flat-top length. In arbitrary unit this pulse shape is plotted in Fig. <ref>.

Control pulse with different flat top length versus time (arbitrary unit). $f$ is the flux threading the loop, and $x$ determines the flat top length as shown in Eq. (<ref>).

By combining all fitting functions and substituting all parameters, one can obtain the gate length. For instance, if the pulse take the shape of $x=5$, the pulse is only a function of gate length $t_g$. We first substitute Eq. (<ref>) into Eq. (<ref>), and then substitute Eq. (<ref>) into Eq. (<ref>) to get the gate-length dependent ZZ interaction, finally we can integrate ZZ interaction from $t=0$ to $t=t_g$ as shown in Eq. (<ref>). By solving the integration $\phi_{\rm ZZ}=\pi$, a CZ gate is achieved at the gate length $t_g\sim419$ ns.

Compared to other schemes to perform CZ gates, this system can completely turn off parasitic ZZ interaction although the gate length is longer. Different from other protocols in which the asymmetric transmon serves as a coupler <cit.>, the main leakageleakage source in this device is the transition between $|101\rangle$ and $|020\rangle$ since transmon frequency is far detuned from CSFQ and bus cavity. This transition needs to  be realized by two-photon processtwo-photon process, therefore is comparably weak. The CZ gate length can be further reduced by pushing transmon frequency closer to the bus cavity, meanwhile avoiding the unwanted transition by optimizing the control pulse.

We  have studied two novel gates based on the asymmetric transmon. The first one, PF gate, is realized in a circuit where two fixed-frequency transmons couple to each other via the asymmetric transmon. On such a gate we take advantage of the cancellation for unwanted ZZ interaction throughout the operation, namely static ZZ freedom in idle status and dynamical ZZ freedom in driven qubits. The second one, the tunable CZ gate, is performed in a circuit where the asymmetric transmon serves as a qubit, and couples to a CSFQ via a harmonic bus resonator. In this device wanted ZZ can be completely switched off when the gate is inactive, but wanted ZZ should be further strengthened to make the gate length shorter when it is active.

CHAPTER: THREE-QUBIT INTERACTION

In previous chapters we have analyzed the characteristics of wanted & unwanted two-qubit interaction in superconducting qubits. However, as quantum processors with many qubits are developed, each qubit is inevitably exposed to interacting neighbors.  Interaction with nearby  idle  qubits will undoubtedly cause unwanted entanglement and accumulate errors across the system in the presence or absence of a two-qubit gate. Such interactions do not obviously affect the resting qubits, but can give rise to gate parameters change and lower gate performance. For instance, errors arising from coupling between control/target qubit of a CR gate and their surrounding idle qubits lead to reduction in the gate fidelity <cit.>.

For a larger quantum processor, i.e. Sycamore <cit.> which uses 53 qubits and 86 couplers as shown in Fig. <ref>,  qubit at one vertex marked by red is connected with four neighboring qubits marked by green via adjustable couplers, thus four pairs of two-body interactions will change the qubit parameters and can accumulate errors to related quantum gates. Furthermore, each neighboring qubit is surrounded by another three idle qubits marked by yellow, thus there are 12 non-pairwise additive three-body interaction with respect to the qubit at red vertex. Although three-qubit interaction is usually weaker than two-body interaction <cit.>, but in total the contribution to one qubit is nontrivial and can even destroy quantum gates.

Layout of Sycamore processor <cit.>. Given a qubit marked by red, there are four qubits marked by green next to it, and each of which is surrounded by another three qubits marked by orange.

This chapter we study the characteristics of such three-body interaction in two complicated circuit schemes: triangle and planar geometries of three qubits <cit.>. In triangle case there are three couplers among the three qubits and in planar one the number of coupler is two. These circuits can be seen as the continuation of previous two-qubit model by adding a third qubit. To study how the neighboring qubits affect two-qubit gates, we briefly describe a three-qubit model with respect to two-body interaction, and then propose a protocol for static three-qubit Hamiltonian tomography to find out the dependency of two- and three-body coupling strengths. Finally we present several examples with simulation results to illustrate the impacts and make a comparison with the two-qubit model.

§ TRIANGLE GEOMETRYTRIANGLE GEOMETRY OF  THREE-QUBIT COUPLING

First, let us consider a triangle circuit which consists of three superconducting qubits and three couplers as shown in Fig. <ref>. Here circles represent qubits and rectangles represent couplers. Each pairwise qubits are connected to each other via a coupler in the coupling strength $g_{qc_i}$ between qubit $q$ and coupler $c_i$ with $q,i=1,2,3$, respectively. Meanwhile, they are also capacitively coupled to each other via direct coupling $g_{qq'}$ between qubit $q$ and $q'$. For simplicity, all couplers are assumed to be harmonic oscillators.

Triangle geometry of three qubit interaction. Each pair of qubits is indirectly coupled to each other via a coupler, and directly via a capacitor.

Following the convention used in chapter <ref>, bare qubit $q$ frequency is $\omega_q(n_q)={E}_q(n_q+1)-{E}_q(n_q)$ with bare energy $E_q(n_q)$ at Fock state $|n_q\rangle$ by setting $\hbar\equiv1$, and coupler $c_i$ frequency is $\omega_{c_i}$. We can ignore the weak interaction between different resonators, and then write the circuit Hamiltonian $H$ in the basis of multilevels following the strategy used in the two-qubit model as

\begin{equation}
\begin{split}
H&=\sum_{i=1}^3\omega_{c_{i}}c_{i}^{\dagger}c_{i}+\sum_{q=1}^3\sum_{n}\omega_q(n_q)\left|n_q+1\right\rangle \left\langle n_q+1\right|\\
\left|n_q+1\right\rangle \left\langle n_q\right|-\left|n_q\right\rangle \left\langle n_q+1\right|\right)\\
&+\sum_{q\neq q'}\sum_{n m}\sqrt{n_q+1}\sqrt{m_{q'}+1}g_{qq'}\\
&\times\left(|n_q\rangle\langle n_q+1|-|n_q+1\rangle\langle n_q|\right)\otimes\left(|m_{q'}\rangle\langle m_{q'}+1|-|m_{q'}\rangle\langle m_{q'}+1|\right).
\end{split}
\label{eq.threeH}
\end{equation}

The first line in Eq. (<ref>) is the unperturbed Hamiltonian of qubits and resonators, second line is qubit-coupler interacting Hamiltonian, and the rest is direct capacitive coupling part. In the case that coupler frequencies are far detuned from qubits, higher or lower, we can decouple all couplers from the circuit using SW transformation, then the qubit Hamiltonian $H_Q$ in the dispersive regimedispersive regime is reduced to

\begin{equation}
\label{eq.threeQf}
\begin{split}
H_{Q}&=\sum_{q}\sum_{n}\sum_{i}\bar{\omega}_{q}(n_q)\left|n_q+1\right\rangle\left\langle n_q+1\right|-\sum_{q\neq q'}\sum_{nm}\sum_{i}\sqrt{n_q+1}\sqrt{m_{q'}+1}\\
&\times J_{c_{i}qq'}^{n_qm_{q'}}\left(|n_q\rangle\langle n_q+1|-|n_q+1\rangle\langle n_q|\right)\otimes\left(|m_{q'}\rangle\langle m_{q'}+1|-|m_{q'}\rangle_{q'}\langle m_{q'}+1|\right),
\end{split}
\end{equation}

where dressed frequency and effective virtual photon exchange rate are

\begin{align}
\bar{\omega}_q(n_q)=&\omega_q(n_q)-\sum_i \frac{n_qg_{qc_{i}}^{2}}{\omega_{c_i}-\omega_{q}({n_q})},\label{eq.omegaq}\\
\end{align}

with $\Delta_{qc_i}=\omega_{c_i}-\omega_{q}(n_q)$ and $\Sigma_{qc_i}=\omega_{c_i}+\omega_{q}(n_q)$.  One can see that effective $J$ coupling depends on the transition states $n$ and $m$ of pairwise interacting qubits $q$ and $q'$. To illustrate how strong the two- and three-qubit interactions are, we calculate them using the following six approaches.

RWA-PT: The first approach is perturbation theory (PT) with RWA by ignoring fast-rotating terms. If the three qubits are the same species  with same sign anharmonicity, i.e. all transmons, the coupling strength $J_{c_iqq'}^{n_qm_{q'}}$ can be approximately state-independent for a pair of interaction as shown in Fig. <ref>(c) and Fig. <ref>(d) since the anharmonicity of a transmon is small compared to the frequency. The Hamiltonian then can be rewritten in terms of creation and annihilation operators as

\begin{equation}
H_{Q}=\sum_{q=1}^3\left( \bar{\omega}_{q}a_{q}^{\dagger}a_{q}+\frac{\delta_{q}}{2}a_{q}^{\dagger}a_{q}^{\dagger}a_{q}a_{q}\right) +\sum_{q<q'}J_{qq'}(a_{q}^{\dagger}a_{q'}+a_{q'}^{\dagger}a_{q}),
\label{eq.threeQ}
\end{equation}

where $\bar{\omega}_q={\bar\omega}_q(0)$, $\bar{\delta}_q={\bar\omega}_q(1)-{\bar\omega}_q(0)\approx\delta_q$ and $J_{qq'}\approx J_{r_iqq'}^{01}$.  Note that here we drop all counter-rotating terms. The Hamiltonian in Eq. (<ref>) describes an interacting three-body problem, but we can treat it as three pairs of two-body interaction, and tackle them using similar methods as introduced in previous chapters.

The most straightforward way is the perturbation theory which gives the dressed energy of each level, then all Pauli coefficients can be derived.  The Hamiltonian in the computational subspace can thus be rewritten in terms of Pauli matrices as

\begin{equation}
H_{\rm eff}=\rm \alpha_{ZII}\frac{ZII}{2}+ \alpha_{IZI}\frac{IZI}{2}+ \alpha_{IIZ}\frac{IIZ}{2}+ \alpha_{ZZI}\frac{ZZI}{4}+\alpha_{ZIZ}\frac{ZIZ}{4}+\alpha_{IZZ}\frac{IZZ}{4}+ \alpha_{ZZZ}\frac{ZZZ}{8},
\end{equation}

and Pauli coefficients from perturbation theory up to third order are

\begin{eqnarray}
{\rm \alpha_{ZII}}&=&\frac{1}{4}\left(E_{000}+E_{011}+E_{001}+E_{010}-E_{100}-E_{101}-E_{110}-E_{111}\right)\nonumber\\
&=&-\omega_1-\frac{J_{12}^2}{\Delta_{12}}-\frac{J_{13}^2}{\Delta_{13}}-\frac{2J_{12}J_{13}J_{23}}{\Delta_{12}\Delta_{13}}-\frac{\alpha_{\rm ZZI}}{2}-\frac{\alpha_{\rm ZIZ}}{2}-\frac{\alpha_{\rm ZZZ}}{4},\\
{\rm \alpha_{IZI}}&=&\frac{1}{4}\left(E_{000}+E_{001}+E_{100}+E_{101}-E_{010}-E_{011}-E_{110}-E_{111}\right)\nonumber\\  &=&-\omega_2-\frac{J_{12}^2}{\Delta_{21}}-\frac{J_{23}^2}{\Delta_{23}}-\frac{2J_{12}J_{13}J_{23}}{\Delta_{21}\Delta_{23}}-\frac{\alpha_{\rm ZZI}}{2}-\frac{\alpha_{\rm IZZ}}{2}-\frac{\alpha_{\rm ZZZ}}{4},\\
{\rm \alpha_{IIZ}}&=&\frac{1}{4}\left(E_{000}+E_{100}+E_{010}+E_{110}-E_{001}-E_{011}-E_{101}-E_{111}\right)\nonumber\\
&=&-\omega_3-\frac{J_{13}^2}{\Delta_{31}}-\frac{2J_{23}^2}{\Delta_{32}}-\frac{2J_{12}J_{13}J_{23}}{\Delta_{31}\Delta_{32}}-\frac{\alpha_{\rm ZIZ}}{2}-\frac{\alpha_{\rm IZZ}}{2}-\frac{\alpha_{\rm ZZZ}}{4},\\
{\rm \alpha_{ZZI}}&=&\frac{1}{2}\left(E_{000}+E_{110}-E_{010}-E_{100}+E_{001}+E_{111}-E_{101}-E_{011}\right)\nonumber\\  &=&\frac{4J_{12}^2(\delta_1+\delta_2)}{(\Delta_{12}+\delta_1)(\Delta_{12}-\delta_2)}-\frac{4J_{12}J_{13}J_{23}(A_{123}+B_{123}+C_{123})}{(\Delta_{12}+\delta_1)(\Delta_{12}-\delta_2)},\label{eq.zzi}\\
{\rm \alpha_{ZIZ}}&=&\frac{1}{2}\left(E_{000}+E_{101}-E_{100}-E_{001}+E_{010}+E_{111}-E_{011}-E_{110}\right)\nonumber\\  &=&\frac{4J_{13}^2(\delta_1+\delta_3)}{(\Delta_{13}+\delta_1)(\Delta_{13}-\delta_3)}-\frac{4J_{12}J_{13}J_{23}(A_{132}+B_{132}+C_{132})}{(\Delta_{13}+\delta_1)(\Delta_{13}-\delta_3)},\label{eq.ziz}\\
{\rm \alpha_{IZZ}}&=&\frac{1}{2}\left(E_{000}+E_{011}-E_{001}-E_{010}+E_{100}+E_{111}-E_{101}-E_{110}\right)\nonumber\\  &=&\frac{4J_{23}^2(\delta_2+\delta_3)}{(\Delta_{23}+\delta_2)(\Delta_{23}-\delta_3)}-\frac{4J_{12}J_{13}J_{23}(A_{231}+B_{231}+C_{231})}{(\Delta_{23}+\delta_2)(\Delta_{23}-\delta_3)},\label{eq.izz}\\
{\rm \alpha_{ZZZ}}&=&E_{000}+E_{011}-E_{001}-E_{010}+E_{101}+E_{110}-E_{100}-E_{111}\nonumber\\  &=&\frac{8J_{12}J_{13}J_{23}(B_{123}+C_{123})}{(\Delta_{12}+\delta_1)(\Delta_{12}-\delta_2)},\label{eq.zzz}
\end{eqnarray}

where $E_{ijk}$ is the eigenvalues of state $|ijk\rangle$, parameters $A$, $B$ and $C$ are defined as

\begin{eqnarray}
\end{eqnarray}

with qubit-qubit detuning $\Delta_{ij}=\bar\omega_i-\bar\omega_j$ $(i,j=1,2,3)$. In contrast with two-qubit architecture, one can see that another term proportional to all three couplings is subtracted from  two-qubit ZZ interaction in Eq. (<ref>). Now the single-qubit flipping frequency is the combination of dressed qubit frequency with corresponding two- and three-qubit interactions. The new three-body interactionthree-body interaction term is created in the form of ZZZ, corresponding coefficient is obviously smaller than two-body interaction as it is proportional to $J^3/\Delta^2$. In other words, the nonzero value  first shows up in the third order correction.

It is worth mentioning that near-degenerate case, in which noncomputational levels could be in the vicinity of computational levels, is excluded in the calculation. But if conditions of detrimental multi-qubit frequency collisions are satisfied, e.g. $\omega_q(n)=\omega_{q'}(m)$, we can still deal with the problem using similar method in Eq. (<ref>).

RWA-SSW: The second approach is simplified SW (SSW) transformationSchrieffer-Wolff (SW) transformation by considering the Hamiltonian in Eq. (<ref>), together with RWArotating wave approximation (RWA) by ignoring fast-rotating terms. Since the Hamiltonian in matrix form is not in the ascending order of energies, so we need to reorder it as

\begin{equation}
H=\left( \begin{array}{ccc|ccc}
E_{000} && &&& \\
& \ddots&  && {\makebox(0,0){\it{\huge J}}}&\\
&& E_{111}&  &&\\ \cline{1-6}
& & & E_{002}&& \\
& {\makebox(0,0){\it{\huge J}}}& & &\ddots& \\
& & & &&E_{222}\\
\end{array}\right)
\end{equation}

The upper block contains all eight computational levels which only allow transition from 0 to 1, the other higher levels are in the lower block with $J$ representing all couplings between the two blocks. To decouple higher levels, a second SW transformation is applied, therefore the Hamiltonian is reduced to a $8\times8$ matrix. By fully diagonalizing this matrix, all Pauli coefficients can be obtained.

RWA-SW: The third approach is SW transformation by considering the Hamiltonian in Eq.(<ref>), together with RWA by ignoring fast-rotating terms. Compared to RWA-SSW, here effective coupling discrepancy is included, the other diagonalization processes are the same.

NRWA-PT: The fourth approach is perturbation theory with NONRWA by including all counter-rotating terms in the calculation. Detailed formula of fast oscillation contribution can be found in Appendix <ref>.

NRWA-SSW: The fifth approach is simplified SW by assuming effective $J$ couplings are only qubit-dependent, and all counter-rotating terms are included in the calculation. Numerical simulation is the same as RWA-SSW.

NRWA-SW: The last approach is SW by including the state-dependent $J$ discrepancy, and all counter-rotating terms in the calculation. Numerical simulation is the same as RWA-SW. Note that NRWA-SW gives the most accurate Pauli coefficients among the six results since it includes both coupling discrepancy and counter-rotating terms.

Now let us take two examples  in a circuit with fixed-frequency transmons, and name them Device 1 and Device 2, to show what the difference is among the six approaches. Frequency, anharmonicity and coupling strength of each subsystem in Device 1 are listed in Tab. <ref>.

5|lDevice 1
4r|unit: GHz $\omega_1$   $\omega_2$     $\omega_3$   $\delta_1$   $\delta_2$   $\delta_3$    $\omega_{c_1}$      $\omega_{c_2}$   $\omega_{c_3}$  4.9    5.0     5.1    $-0.33$   $-0.33$    $-0.33$    6.0     6.3    6.6  $g_{1c_1}$   $g_{1c_3}$    $g_{2c_1}$   $g_{2c_2}$   $g_{3c_2}$   $g_{3c_3}$    $g_{12}$     $g_{13}$   $g_{23}$  0.08   0.08    0.08   0.08   0.08   0.08    0    0   0

Device 1 parameters.

In Device 1, all coupler frequencies are way larger than qubits. With these numbers we evaluate Pauli coefficients both analytically and numerically using the six approaches.
Figure <ref> shows that three-body ZZZ interaction is around 100 kHz while two-body ZZ interaction is at least double of that.  By comparing the RWA with NRWA results, we find that counter rotating terms have the same sign with co-rotating terms and make the total two- and three-body interaction stronger. State-independent $J$ for PT and SSW leads to a lager two-body interaction than the numerical SW by a few tens of kilohertz. In contrast to ZZI component, IZZ interaction is weaker although the two subsystems have same 100 MHz qubit-qubit detuning. This is because coupler C2 in between Q2 and Q3 has a higher frequency than C1 in between Q1 and Q2, which results in  $|J_{12}|>|J_{23}|$. This phenomenon is similar to what we found in Fig. <ref> for the two-qubit models.

Another scenario we explore is that one coupler frequency is below the qubit while keeping the other parameters of Device 1 unchanged. Frequency, anharmonicity and coupling strength of each subsystem are listed in Tab. <ref>.

5|lDevice 2
4r|unit: GHz $\omega_1$   $\omega_2$     $\omega_3$   $\delta_1$   $\delta_2$   $\delta_3$    $\omega_{c_1}$      $\omega_{c_2}$   $\omega_{c_3}$  4.9    5.0     5.1    $-0.33$   $-0.33$    $-0.33$    6.0     6.3    3.4  $g_{1c_1}$   $g_{1c_3}$    $g_{2c_1}$   $g_{2c_2}$   $g_{3c_2}$   $g_{3c_3}$    $g_{12}$     $g_{13}$   $g_{23}$  0.08   0.08    0.08   0.08   0.08   0.08    0    0   0

Device 2 parameters.

Here  the third coupler frequency is changed from $6.6$ GHz to $3.4$ GHz, and corresponding Pauli coefficients are shown in Fig. <ref>. One can see that now ZZZ coefficient is negative since  the sign of $J_{13}$ in Eq. (<ref>) changes to positive under the condition that $\omega_{c_3}<\omega_1,\omega_3$. For RWA-PT and RWA-SSW, the decrease in coupler C3 frequency leads to increase in Q1 and Q3 dressed frequencies due to the sign change in Eq. (<ref>). From the perturbation theory we find that third order correction has the same sign with second order under current condition that only $J_{13}$ is negative. ZZI and IZZ components then get strengthened in contrast to Fig. <ref>.  As for ZIZ, increasing dressed frequency results in reduction of second order correction term, which is now added by a positive third order correction, the total effect is that we see a decrease in ZIZ interaction.

Pauli coefficients of triangle geometry three qubit interaction in Device 1, with $\omega_{c_i}>\omega_j$ $(i, j=1, 2, 3)$ using six approaches. Perturbation theory (PT) and simplified SW (SSW) results are obtained by assuming effective qubit-qubit coupling $J$ is state-independent, while the most accurate result is for SW takes into account $J$ discrepancy. The most accurate result is from NRWA-SW, marked by black box.

Pauli coefficients of triangle geometry three qubit interaction in  Device 2 using six approaches in the condition that only C3 frequency is lower than qubits. Perturbation theory (PT) and simplified SW (SSW) results are obtained by assuming effective qubit-qubit coupling $J$ is state-independent, while SW takes into account $J$ discrepancy. The most accurate result is from NRWA-SW, marked by black box.

Now let us revisit the Sycamore layout in Fig. <ref>. Given that three-qubit interaction has the same amplitude as what we show in Fig <ref> and Fig. <ref>, in total there are 12 non-pairwise three-body interactions accumulating phase errors on the state of the central qubit. Such total three-body interaction is even comparable to the total two-body ZZ interaction if they are linearly sum together, e.g. 1 MHz, and becomes more obvious in a complex quantum process, then should be cancelled out.

§ PLANAR GEOMETRYPLANAR GEOMETRY OF THREE-QUBIT  CHAIN

A simpler version of three-qubit coupling scheme is a qubit chain, as shown in Fig. <ref>. This circuit is equivalent to stretch the triangle geometry in Fig. <ref> to a line by removing one coupler.

Geometry of three qubit chain Qubits Q1 and Q3 are coupled to Q2 via couplers C1 and C2, separately. Each pair of qubits are also capacitively coupled to each other.

Here we also explore two examples, and name them Device 3 and Device 4. To make a comparison with triangle geometry, we use the same device parameter in Tab <ref> for Device 3, and assume $\omega_{c_3}$, $g_{1c_3}$ and $g_{3c_3}$ to be zero. Corresponding Pauli coefficients are plotted in Fig. <ref>.

Pauli coefficients of planar geometry three qubit interaction in Device 3. Perturbation theory (PT) and simplified SW (SSW) results are obtained by assuming effective qubit-qubit coupling $J$ is state-independent, while SW takes into account $J$ discrepancy. The most accurate result is from NRWA-SW, marked by black box.

Compared to the triangle geometry, now the effective coupling $J_{13}$ between Q1 and Q3 is zero since direct capacitive coupling $g_{13}=0$. Thus two-body  interaction obtained from perturbation theory reduces to the formula of static ZZ interaction that we have derived in Eq. (<ref>). Three-body ZZZ interaction in Eq. (<ref>) also turns out to be zero. PT gives exact zero ZIZ component, but RWA-SSW and RWA-SW show $\sim$ 1 kHz contribution. This is because we expand the SW transformation to the second order, in which cross terms like $J_{12}^2J_{23}^2/\Delta_{12}\Delta_{23}^2$ are included and lead to two orders weaker ZIZ interaction. In contrast to Fig. <ref>, coefficients of IZZ and ZZI become larger since that the third order correction now is approximated to be zero.

Note that in the discussion above we assume direct coupling $g_{13}$ to be zero. But in reality the two qubits can also capacitively interact with each other, leading to a nonzero direct couplingdirect coupling. In this case we explore the Device 4 in which direct coupling $g_{13}=2$ MHz, other parameters are the same as Device 3.  Accordingly, Pauli coefficients are plotted in Fig. <ref>.

Pauli coefficients of planar geometry three qubit interaction in Device 4 by assuming direct coupling $g_{13}=2$ MHz using six approaches. Perturbation theory (PT) and simplified SW (SSW) results are obtained by assuming effective qubit-qubit coupling $J$ is state-independent, while SW takes into account $J$ discrepancy. The most accurate result is from NRWA-SW, marked by black box.

One can see that both two-body interaction ZIZ and three-body interaction ZZZ are nonzero.
This is because nonzero direct coupling $g_{13}$ leads to the total effective coupling $J_{13}=2$ MHz as shown in Eq. (<ref>). It is clear that direct coupling plays an important role in the two- and three-body interaction although it is small. ZIZ component increases from zero to 60 kHz while ZZZ is about $-50$ kHz. The sign of ZZZ interaction is changed to negative since now only $g_{13}$ is positive.

By comparing the triangle geometry and planar geometry of three-qubit interaction, we show that the sign of three-body ZZZ interaction can be changed to the opposite by either changing coupler frequency in triangle geometry or increasing direct coupling in planar geometry. This provides two potential ways to eliminate ZZZ interaction. In other words, it is possible to find out a boundary between the positive region and negative region by changing device parameters, on which ZZZ interaction is always zero, like what we can see in Fig. <ref>.

In a chain with more than three qubits, there are only 2 non-pairwise three-body interactions changing the state of a particular qubit. As shown Fig <ref>,  in the presence of nonzero direct coupling between the two ends of a three-qubit chain, the total three-body interaction is much weaker than two-body interaction, which can be safely ignored when performing quantum gates in this processor.

In brief, we explore three-qubit interaction in two setups: triangle geometry and a qubit chain. We derive the Pauli coefficients using both analytical and numerical approaches, and present several examples to illustrate how the two- and three-qubit interactions are affected by the circuit geometry and parameters. We show that the sign of ZZZ interaction can be changed to the opposite by either changing coupler frequency in triangle geometry or increasing direct coupling in the three-qubit chain. This provides two potential ways to eliminate ZZZ interaction.

CHAPTER: CONCLUSION

Superconducting qubits have become one of the most prominent candidates for the realization of scalable quantum computing. However, the performance of quantum logic gates still needs to be improved. One limiting factor for the two-qubit gate errors is parasitic interaction between qubits. In this book we mainly focus on ZZ interaction between pairwise coupled qubits, and develop several protocols to either eliminate or utilize it to perform high-fidelity quantum gates.

In chapter <ref>, we show the static ZZ interaction is originated from transitions between computational and noncomputational levels. We characterize such interaction in two circuits of qubit pairs: a hybrid CSFQ-transmon pair and a pair of interacting transmons. The hybrid pair with opposite anharmonicity between the qubits allows for the elimination of static ZZ interaction meanwhile keeps two qubits entangled. The same-specie pair can only achieve static ZZ freedom by almost decoupling two qubits. We also derive the condition of static ZZ freedom  from perturbation theory, and use this model to simulate the first hybrid CSFQ-transmon experiment performed by our collaborators, in which zero static ZZ interaction was observed. Our theoretical results are consistent with the experimental data.

In chapter <ref>, we follow the experiment and study the performance of a cross-resonance (CR) gate in the hybrid system. The  CR gate is realized by driving the control qubit CSFQ with the frequency of target qubit transmon. Our analysis shows that zeroing static ZZ interaction can effectively improve the CR gate fidelity, although the CR pulse produces additional dynamical component on top of it. We predict that two-qubit gate error rate of the order of 0.001 is feasible if the coherence times can be prolonged to 200 $\mu$s and static ZZ freedom is realized. Inspired by the experiment, we notice that static and dynamical ZZ interaction can cancel each other. Therefore we  derive the condition of dynamical ZZ freedom for CSFQ-transmon pair as well as transmon-transmon pair, and show that this type of freedom can lead to a large  increase in the CR gate fidelity.

Chapter <ref> describes our proposal for two novel two-qubit gates by including a tunable coupler: the parasitic free (PF) gate and the tunable controlled-Z (CZ) gate. We first study the parasitic free (PF) gate in a circuit of two fixed-frequency transmons coupled via a frequency-tunable coupler. In the circuit both static ZZ and dynamical ZZ freedom can be realized by tuning the flux threading the frequency-tunable coupler. By fine designing the pulse we achieved ZZ freedom at two sweet spots with extremely reduced flux noise. The second gate we propose is a tunable CZ gate in a circuit of a CSFQ coupled to a frequency-tunable transmon threading by external flux via a bus resonator. In this circuit ZZ interaction can be totally turned off at one flux point, and dramatically increased to several MHz at another flux point.

In last chapter, we study three-qubit circuits in triangle  and chain geometry. In each setup we study and evaluate the two-body ZZ and three-body ZZZ interaction between qubits. We show that the sign of ZZZ interaction can be changed to the opposite by either changing coupler frequency in triangle circuit or increasing direct coupling in the chain. This provides two potential ways to eliminate such ZZZ interaction.

CHAPTER: CSFQ HAMILTONIAN QUANTIZATION

In contrast to a transmon, higher order terms (>4) in the expansion of the CSFQ potential also contributes to the eigenvalues, which will change the zero point fluctuation to an unknown number. To be more precise, the Hamiltonian can be quantized in terms of field operators, e.g. $n=i(\hat{a}-\hat{a}^{\dagger})/2\xi$ and  $\phi=\xi(\hat{a}+\hat{a}^{\dagger})$ with $\xi$ being the expansion parameter which will minimize the total energy, the normal ordered Hamiltonian then can be written as
H    =    -E_c/ξ^2 (a^†-a)^2+ ∑_u=0^∞ξ^2u+2 ∑_v=0^u U_2u+2(φ_0)/2^u-v(u-v)!
×∑_w=-(v+1)^v+1  (a^†)^v+1+w(a)^v+1-w/(v+1+w)! (v+1-w)!

+    ∑_u=0^∞ξ^2u+1 ∑_v=0^u U_2u+1(φ_0)/2^u-v(u-v)!
×∑_w=-(v+1)^v (a^†)^v+1+w(a)^v-w/(v+1+w)!(v-w)!

\begin{eqnarray}
U(\varphi_0)&\equiv& -2E_{J}\cos\varphi_0/2-\alpha E_{J}\cos\left(2\pi f-\varphi_0\right)
\end{eqnarray}

where $\varphi_0$ is the phase of minimum $U(\varphi_0)$, i.e. $\varphi_0=-2 \pi \alpha (\delta f)/(1/2-\alpha)$, which vanishes at sweet spot.
By solving Schrödinger equation, the eigenenergies $E_n$ can be obtained using perturbation theory. Unperturbed eigenvalues $E_n^{(0)}$ and first three order corrections are given by

\begin{equation}
\begin{split}
E_{n}^{(2)}&=\sum_{k\neq n}^{n+L}\frac{V_{nk}^{2}}{E_{n}^{(0)}-E_{k}^{(0)}}\\
E_{n}^{(3)}&=\sum_{k\neq n}^{n+L}\sum_{m\neq n}^{n+L}\frac{V_{nm}V_{mk}V_{kn}}{\left(E_{n}^{(0)}-E_{k}^{(0)}\right)\left(E_{n}^{(0)}-E_{m}^{(0)}\right)}
\label{E}
\end{split}
\end{equation}

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

Sum up unperturbed energy and all three orders corrections, one can obtain the analytical formula of CSFQ eigenvalues, then frequency $\omega_{01}=(E_1-E_0)/\hbar$ and anharmonicity $\delta=(E_2-2E_1+E_0)/\hbar$ can be evaluated. Here is an example to illustrate how to find these parameters. Consider a CSFQ with $E_C=0.292$ GHz, $E_J=108.9$ GHz, $\alpha=0.43$ and $f=0.5$. Firstly, expand the potential to 20th order (L=10) at the sweet spot and plot the frequency $f_{01}$ as a function of $\xi$.
Determine $\xi$ by finding the minimum as shown in Fig. <ref> (approximately $\xi\approx\phiz/\sqrt{2}$), then substitute $\xi$ back to the Eq. (<ref>), the corresponding frequency and anharmonicity spectra are shown in Fig. <ref>.

Frequency as a function of $\xi$ at the sweet spot. Simulation parameters $E_C=0.292$ GHz, $E_J=108.9$ GHz, $\alpha=0.43$ and $f=0.5$.

Frequency (a) and anharmonicity (b) as a function of external flux $f$ using perturbation theory.

CHAPTER: HAMILTONIAN OF EXPERIMENTAL CIRCUIT

It is worth noting that we put the two redundant phase degrees of freedom in the first place, which will be removed next. In Fig. <ref> we plot the potential energies associated with the readout resonator coupled to the transmon (a), the bus resonator (b), the readout resonator coupled to the CSFQ (c), the fixed frequency transmon (d), the CSFQ at the sweet spot (e), and away from the sweet spot (f).

Potential profiles. (a) Readout resonator for transmon. (b) Bus resonator. (c) Readout resonator for CSFQ. (d) Transmon. (e) CSFQ along the $\varphi_m$ direction at $f=0.5$. (f) CSFQ along the $\varphi_m$ direction at $f=0$.

Since $\varphi_b$ and $\varphi_e$ are not included in the potential, and  $\varphi_p$ can also be neglected as discussed in the section of CFSQ, we can use Cholesky decomposition <cit.> to safely remove these three phases, and thus reduce the matrix size of the circuit Hamiltonian from $8\times 8$ to the following $5 \times 5$:

\begin{equation}
\end{equation}

where $\overrightarrow{n}=(n_a, n_T, n_d, n_m, n_h)$ is the canonical term of $\overrightarrow{\varphi}$, and

\begin{equation}
\mathbf{\mathbf{C'}}=\left(\begin{array}{ccccc}
C_{a}' & -\frac{C_{ab}C_{dT}}{C_{T0}}& -\frac{C_{ab}C_{cd}}{C_{T0}}& 0 & 0\\
-\frac{C_{ab}C_{dT}}{C_{T0}} & C_{T}' & \frac{C_{a0}C_{cd}}{C_{T0}} & 0 & 0\\
-\frac{C_{ab}C_{cd}}{C_{T0}} & \frac{C_{a0}C_{cd}}{C_{T0}} & C_{r}' & -\frac{2C_{de}C_{h0}}{C_{m0}} & -\frac{C_{de}C_{gh}}{C_{m0}}\\
0 & 0 & -\frac{2C_{de}C_{h0}}{C_{T0}} & C_{m}' & \frac{2C_{dm}C_{gh}}C_{m0}\\
0 & 0 & -\frac{C_{de}C_{gh}}{C_{m0}} &\frac{ 2C_{dm}C_{gh}}{C_{m0}} & C_{h}'
\end{array}\right).
\end{equation}

where the new notations are defined from existing ones in Fig. <ref> as

\begin{equation}
\begin{split}
C_{T0}	&=C_{ab}+C_{b0}+C_{c0}+C_{cd},C_{h0}	=C_{g0}+C_{gh}\\
C_{m0}	&=C_{de}+C_{e0}+C_{g0}+C_{gh}, C_{dm}=C_{de}+C_{e0}\\
C_{dT}	&=C_{cd}+C_{c0},C_{a0}=C_{ab}+C_{b0}\\
C_{T}'	&=C_{dT}+C_{shT}+C_{T}-{C_{dT}^{2}}/{C_{T0}}\\
C_{m}'	&=2C_J+4(C_{3}+C_{shCSFQ})-4C_{h0}^{2}/C_{m0}+4C_{h0}\\
C_{r}'	&=-C_{cd}^{2}/C_{T0}+C_{cd}+C_{de}+C_{r}-{C_{de}^{2}}/{C_{m0}}\\
C_{a}'	&=-C_{ab}^{2}/C_{T0}+C_{ab}+C_{rT}\\
C_{h}'    &=-C_{gh}^{2}/{C_{m0}}+C_{gh}+C_{rCSFQ}.
\end{split}
\end{equation}

The relationships between the various coupling strengths $g_{ij}$ and the relevant capacitances are given by the following expressions:

\begin{equation}
\begin{split}
g_{hm}	&\propto -\frac{2C_{gh}C_{dm}}{\left(C_{gh}+C_{rCSFQ}\right)\left(C_{gs}C_{m0}-4C_{h0}^{2}\right)} \\
g_{aT}&\propto\frac{C_{ab}C_{dT}}{\left(C_{ab}+C_{rT}\right)\left(C_{gT}C_{T0}-C_{a0}^{2}\right)} \\
g_{rT}&\propto -\frac{C_{cd}C_{a0}}{C_{cder}\left(C_{dT}^{2}-C_{gT}C_{T0}\right)}\\
\end{split}
\end{equation}

where $C_{gs}=2C_{0}+4C_{g0}+4C_{gh}+4(C_{3}+C_{shCSFQ})$,  $C_{gT}=C_{cd}+C_{c0}+C_{shT}+C_{T}$, and $C_{cder}=C_{cd}+C_{de}+C_{R}$.

CHAPTER: CSFQ COHERENCE VERSUS FLUX

(a) at 9 410

(b) at 9 410

(a) Pure dephasing rate $\Gamma_\varphi$ of the CSFQ vs. qubit frequency gradient $D_\Phi$. Red solid line is a linear fit to the linear portion of data. Black dashed line is the modified pure dephasing rate to account for the reduction in gate errors measured with randomized benchmarking when dephasing is dominated by $1/f$ noise, as discussed in text. (b) Hahn-echo $T_2$ vs. flux and the effective $T_2$ calculated using our modified dephasing model.

For understanding the map of coherence, we must know the coherence times $T_1$ and $T_2$ of each qubit. Transmon frequency is fixed, therefore its $T_1$ and $T_2$ can be extracted  with standard Ramsey sequence. To simulate the gate error for different flux biases, we must model the appropriate flux-dependent dephasing of the CSFQ. Experimental $T_2$ from Ramsey measurement with a single echo refocusing pulse is shown in Fig. <ref>. However, such a dephasing rate overestimates the gate errors away from the sweet spot. It means that the effective $T_2$ must be longer than what is measured with the standard protocol. This behavior may be due to the depolarizing effect from twirling the $1/f$ noise, which is typically the dominant contribution to dephasing in flux tunable CSFQ for bias points away from a sweet spot <cit.>. If we consider Gaussian noise, the random phase accumulated at time $t$ is $\Delta\varphi=D_{\Phi}\int_{0}^{t}dt'\delta\Phi(t')$ with respect to flux $D_{\Phi}=\partial f_{01}/\partial \Phi$, and one can calculate the decay law of the free induction (Ramsey signal) as

\begin{equation}
f_{z,R}(t)=\left\langle \exp(i\Delta\varphi)\right\rangle =\exp\left(-\frac{1}{2}D_{\Phi}^{2}\int_{0}^{t}\int_{0}^{t}dt'dt''\left\langle \delta\Phi(t')\delta\Phi(t'')\right\rangle \right).
%\exp(-1/2\left\langle \Delta\varphi^{2}\right\rangle )
\end{equation}

Note that only in the equilibrium, $\left\langle \delta\Phi(t')\delta\Phi(t'')\right\rangle $ reduces to power spectrum $ S_{\Phi}(\left|t'-t''\right|)$  which depends on the time difference, otherwise, time point $t'$ and $t''$ should be considered. In the equilibrium the free induction decay is simplified as

\begin{equation}
\end{equation}

Fourier transformation of the power spectrum in the frequency domain is written as

\begin{equation}
\end{equation}

By substituting this Fourier transformation, the induction decay changes to

\begin{equation}
=\exp\left[-\frac{t^{2}}{2}D_{\Phi}^{2}\int_{-\infty}^{+\infty}\frac{d\omega}{2\pi}S_{\Phi}\left(\omega\right){\rm {sinc}}^{2}\frac{\omega t}{2}\right].
\end{equation}

here $S_{\Phi}\left(\omega\right)=2\pi A_{\Phi}/|\omega|$ and $\omega_{ir}<|\omega|<\omega_{c}$, the infra-red cutoff $\omega_{ir}$ is usually determined by the measurement protocol and can be set and controlled in experiments <cit.>. For $1/f$ noise, at times $t\ll1/\omega_{ir}$, the free induction (Ramsey) decay is dominated by the frequencies $\omega<1/t$, so the formula above reduces to

\begin{equation}
f_{z,R}(t)	%=\exp\left[-t^{2}D_{\Phi}^{2}\int_{\omega_{ir}}^{\omega_{c}}\frac{d\omega}{2\pi}\frac{2\pi A_{\Phi}}{\omega}\right]
\end{equation}

So pure dephasing rate is

\begin{equation} \Gamma_{\varphi}=2\pi D_{\Phi}\sqrt{A_{\Phi}\left|\ln\omega_{ir}t\right|}.
\end{equation}

The modeled pure dephasing rate can be extracted $\Gamma_\varphi=(0.00288\,{\rm m}\Phi_0)D_\Phi+0.039\ \mu\rm s^{-1}$ for $D_\Phi \ge 0$ shown as the black dashed line in Fig. <ref>(a). Using this modified $\Gamma_\varphi$, we calculate the effective $T_2$ as a function of flux as shown in Fig. <ref>(b). This approach to accounting for gate error measurements in the presence of $1/f$ noise results in calculated coherence-limited gate error vs. flux curves that agree reasonably well with the experimental data.

CHAPTER: CR GATE ERROR

§ ERROR OF EXPERIMENTAL CIRCUIT

To explore the impact of classical crosstalk and ZZ interaction on the gate fidelity, we plot the gate error of echoed CR pulses vs. the flux threading CSFQ at $t_g=560$ ns in Fig. <ref>.

Error per gate for $t_g=560$ ns with three different cases of error sources. Thick dashed line corresponds to coherence-limited gate error, which sets the lower bound for error per gate. Thin dashed line shows the gate error when ZZ contribution is added in the simulation. Solid line shows the case where both ZZ and classical crosstalk are included.

Two-qubit gate error for three sets of coherence times.  $\left\{T_1^{(1)}, T_2^{(1)}, T_1^{(2)}, T_2^{(2)}\right\}$, where the superscripts indicate qubit label, are (18, 15, 40, 45) for the experimental chip, (40, 54, 43, 67) for the transmon-transmon device in Ref. <cit.>, and (200, 200, 200, 200) for an ideal CSFQ-transmon device (all times in $\mu$s). (a) Present CSFQ-transmon device. (b) Transmon-transmon device. (c) Ideal CSFQ-transmon device. Note that all three figures share the same legend as in (a). CSFQ is assumed to be at the sweet spot in the simulation.

The flux-dependent gate error is calculated from the theory model for three cases: only coherence-limited error marked with thick dashed line, ZZ but no classical crosstalk included marked with thin dashed line, and both ZZ and classical crosstalk included marked with solid line. One can see that the gate error due to unwanted terms is the most at the flux sweet spot, despite at which coherence times are the longest and  the ZZ interaction is maximal as shown in Fig. <ref>. Away from the sweet spot, all unwanted terms become suppressed, and therefore the gate fidelity approaches its coherence limit.

To explore how classical crosstalk, static ZZ interaction and coherence times impact the CR gate both in the CSFQ-transmon circuits with transmon-transmon circuits, we plot the CR gate errors in three samples: experimental CSFQ-transmon device (1), transmon-transmon device (2) and ideal CSFQ-transmon device (3). And plot it in Fig. <ref>. One can see that eliminating both classical crosstalk and static ZZ interaction while prolonging the coherence times can suppression the CR gate error to be around $0.1\%$.

§ DYNAMICAL QUADRATIC FACTOR

Dynamical quadratic factor is obtained in Eq. (<ref>) using perturbation theory, and corresponding coefficients are given by

\begin{eqnarray}
\end{eqnarray}

CHAPTER: NONRWA PERTURBATION THEORY

Pauli coefficients are calculated from two parts: rapid counter rotating terms (CRW) and slow co-rotating terms. RWA neglects the first term while NRWA includes the contribution from fast oscillating terms. Therefore the NRWA gives rise to

\begin{equation}
\alpha_{\rm UVW}^{\rm tot}=\alpha_{\rm UVW}+\alpha_{\rm UVW}^{\rm coun}
\end{equation}

where U, V and W $\in$$ \{\rm I, X, Y, Z\}$. RWA results have been derived in chapter <ref>, so here we only discuss contribution from counter-rotating terms. Similarly, these terms can also be defined from perturbation theory as

\begin{equation}
\begin{aligned}
4\alpha_{\rm ZII}^{\rm coun}&=E_{000}^{\rm coun}+E_{011}^{\rm coun}+E_{001}^{\rm coun}+E_{010}^{\rm coun}-E_{100}^{\rm coun}-E_{101}^{\rm coun}-E_{110}^{\rm coun}-E_{111}^{\rm coun}\\
4\alpha_{\rm IZI}^{\rm coun}&=E_{000}^{\rm coun}+E_{001}^{\rm coun}+E_{100}^{\rm coun}+E_{101}^{\rm coun}-E_{010}^{\rm coun}-E_{011}^{\rm coun}-E_{110}^{\rm coun}-E_{111}^{\rm coun}\\
4\alpha_{\rm IIZ}^{\rm coun}&=E_{000}^{\rm coun}+E_{100}^{\rm coun}+E_{010}^{\rm coun}+E_{110}^{\rm coun}-E_{001}^{\rm coun}-E_{011}^{\rm coun}-E_{101}^{\rm coun}-E_{111}^{\rm coun}\\
2\alpha_{\rm ZZI}^{\rm coun}&=E_{000}^{\rm coun}+E_{110}^{\rm coun}-E_{010}^{\rm coun}-E_{100}^{\rm coun}+E_{001}^{\rm coun}+E_{111}^{\rm coun}-E_{101}^{\rm coun}-E_{011}^{\rm coun}\\
2\alpha_{\rm ZIZ}^{\rm coun}&=E_{000}^{\rm coun}+E_{101}^{\rm coun}-E_{100}^{\rm coun}-E_{001}^{\rm coun}+E_{010}^{\rm coun}+E_{111}^{\rm coun}-E_{011}^{\rm coun}-E_{110}^{\rm coun}\\
2\alpha_{\rm IZZ}^{\rm coun}&=E_{000}^{\rm coun}+E_{011}^{\rm coun}-E_{001}^{\rm coun}-E_{010}^{\rm coun}+E_{100}^{\rm coun}+E_{111}^{\rm coun}-E_{101}^{\rm coun}-E_{110}^{\rm coun}\\
\alpha_{\rm ZZZ}^{\rm coun}&=E_{000}^{\rm coun}+E_{011}^{\rm coun}-E_{001}^{\rm coun}-E_{010}^{\rm coun}+E_{101}^{\rm coun}+E_{110}^{\rm coun}-E_{100}^{\rm coun}-E_{111}^{\rm coun}
\end{aligned}
\end{equation}

with $E_{ijk}^{\rm coun}$ being energy shift in state $|ijk\rangle$ $(i,j,k=1,2,3)$ due to CRW terms. For brevity's sake, we define such notations $\Delta_{ij}=\omega_i-\omega_j$ and $\Sigma_{ij}=\omega_{i}+\omega_j$ for qubit frequency detuning and summation, separately. If we consider more than three order perturbation theory, this correction is not purely fast oscillating terms but is a combination of both of the two terms. For instance, the third correction in $|101\rangle$ state contains such a term $\langle101|V|202\rangle\langle202|V|112\rangle\langle112|V|101\rangle$, where the first and third transitions are fast oscillating terms while the second is the slow rotating term. Energy shift from CRW in the computational levels are given by

\begin{align}
E_{000}^{\rm coun}&=-\frac{J_{12}^{2}}{\Sigma_{12}}-\frac{J_{13}^{2}}{\Sigma_{13}}-\frac{J_{23}^{2}}{\Sigma_{23}}+\frac{2J_{12}J_{13}J_{23}}{\Sigma_{12}\Sigma_{13}\Sigma_{23}}(\Sigma_{12}+\Sigma_{13}+\Sigma_{23})\\
E_{001}^{\rm coun}&=-\frac{J_{12}^{2}}{\Sigma_{12}}-\frac{2J_{13}^{2}}{\Sigma_{13}+\delta_3}-\frac{2J_{23}^{2}}{\Sigma_{23}+\delta_3}+2J_{12}J_{13}J_{23}\left[\frac{2}{\Sigma_{12}(\Sigma_{13}+\delta_3)}\right.\nonumber\\
E_{010}^{\rm coun}&=-\frac{2J_{12}^{2}}{\Sigma_{12}+\delta_2}-\frac{J_{13}^{2}}{\Sigma_{13}}-\frac{2J_{23}^{2}}{\Sigma_{23}+\delta_2}+2J_{12}J_{13}J_{23}\left[\frac{2}{\Sigma_{13}(\Sigma_{12}+\delta_2)}\right.\nonumber\\
E_{100}^{\rm coun}&=-\frac{2J_{12}^{2}}{\Sigma_{12}+\delta_1}-\frac{2J_{13}^{2}}{\Sigma_{13}+\delta_1}-\frac{J_{23}^{2}}{\Sigma_{23}}+2J_{12}J_{13}J_{23}\left[\frac{2}{\Sigma_{23}(\Sigma_{13}+\delta_1)}\right.\nonumber\\
E_{011}^{\rm coun}&=-\frac{2J_{12}^{2}}{\Sigma_{12}+\delta_2}-\frac{2J_{13}^{2}}{\Sigma_{13}+\delta_3}+\frac{J_{23}^{2}}{\Sigma_{23}}-\frac{4J_{23}^{2}}{\Sigma_{23}+\delta_2+\delta_3}+2J_{12}J_{13}J_{23}\left[\frac{1}{\Sigma_{23}\Delta_{21}}\right.\nonumber\\
&\left.-\frac{2}{\Sigma_{12}+\delta_2}\left(\frac{1}{\Delta_{31}}+\frac{1}{\Delta_{32}-\delta_2}\right) -\frac{2}{\Sigma_{13}+\delta_3}\left(\frac{1}{\Delta_{21}}+\frac{1}{\Delta_{23}-\delta_3}\right)\right.\nonumber\\
E_{101}^{\rm coun}&=-\frac{2J_{12}^{2}}{\Sigma_{12}+\delta_1}+\frac{J_{13}^{2}}{\Sigma_{13}}-\frac{4J_{13}^{2}}{\Sigma_{13}+\delta_1+\delta_3}-\frac{2J_{23}^{2}}{\Sigma_{23}+\delta_3}+2J_{12}J_{13}J_{23}\left[\frac{1}{\Sigma_{13}\Delta_{12}}\right.\nonumber\\
&\left.-\frac{2}{\Sigma_{12}+\delta_1}\left(\frac{1}{\Delta_{32}}+\frac{1}{\Delta_{31}-\delta_1}\right) -\frac{2}{\Sigma_{23}+\delta_3}\left(\frac{1}{\Delta_{12}}+\frac{1}{\Delta_{13}-\delta_3}\right)\right.\nonumber\\
E_{110}^{\rm coun}&=\frac{J_{12}^{2}}{\Sigma_{12}}-\frac{4J_{12}^{2}}{\Sigma_{12}+\delta_1+\delta_2}+\frac{2J_{13}^{2}}{\Sigma_{13}+\delta_1}-\frac{2J_{23}^{2}}{\Sigma_{23}+\delta_2}+2J_{12}J_{13}J_{23}\left[\frac{1}{\Sigma_{12}\Delta_{13}}\right.\nonumber\\
&\left.-\frac{2}{\Sigma_{13}+\delta_1}\left(\frac{1}{\Delta_{23}}+\frac{1}{\Delta_{21}-\delta_1}\right) -\frac{2}{\Sigma_{23}+\delta_2}\left(\frac{1}{\Delta_{13}}+\frac{1}{\Delta_{12}-\delta_2}\right)\right.\nonumber\\
\end{align}

\begin{align}
E_{111}^{\rm coun}&=\frac{J_{12}^{2}}{\Sigma_{12}}-\frac{4J_{12}^{2}}{\Sigma_{12}+\delta_1+\delta_2}+\frac{J_{13}^{2}}{\Sigma_{13}}-\frac{4J_{13}^{2}}{\Sigma_{13}+\delta_1+\delta_3}+\frac{J_{23}^{2}}{\Sigma_{23}}-\frac{4J_{23}^{2}}{\Sigma_{23}+\delta_2+\delta_3}\nonumber\\
&\left.+\frac{4}{\Sigma_{12}+\delta_1+\delta_2}\left(\frac{1}{\Delta_{13}+\delta_1}+\frac{1}{\Delta_{23}+\delta_2} \right) +\frac{4}{\Sigma_{13}+\delta_1+\delta_3}\left(\frac{1}{\Delta_{12}+\delta_1}+\frac{1}{\Delta_{32}+\delta_3} \right)\right.\nonumber\\
&\left.+\frac{4}{\Sigma_{23}+\delta_2+\delta_3}\left(\frac{1}{\Delta_{31}+\delta_3}+\frac{1}{\Delta_{21}+\delta_2} \right)\right.\nonumber\\
\end{align}

In fact, these cross terms such as $\Sigma_{ij}\Delta_{jk}$ are dominant compared to pure CRW transitions like $\Sigma_{ij}\Sigma_{jk}$.

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

abstractIn this paper a microscopic quantum mechanical model of computers as represented by Turing machines is constructed. It is shown that for each numberN and Turing machineQ there exists a HamiltonianHNQ and a class of appropriate initial states such that if c is such an initial state, thenψQN(t)=exp(−1HNQt)ψQN(0) correctly describes at timest3,t6,⋯,t3N model states that correspond to the completion of the first, second, ⋯, Nth computation step ofQ. The model parameters can be adjusted so that for an arbitrary time intervalΔaroundt3,t6,⋯,t3N, the “machine”part ofψQN(t) is stationary.
journaltitleJournal of Statistical Physics
titleThe computer as a physical system: A microscopic quantum mechanical Hamiltonian model of computers as represented by Turing machines

journaltitleInternational Journal of Theoretical Physics
titleSimulating physics with computers

journaltitleProceedings of the Royal Society of London. A. Mathematical and Physical Sciences
titleQuantum theory, the Church-Turing principle and the universal quantum computer

booktitleProceedings 35th Annual Symposium on Foundations of Computer Science
titleAlgorithms for quantum computation: discrete logarithms and factoring

booktitleProceedings of the twenty-eighth annual ACM symposium on Theory of computing
titleA fast quantum mechanical algorithm for database search

abstractA critical question for quantum computing in the near future is whether quantum devices without error correction can perform a well-defined computational task beyond the capabilities of supercomputers. Such a demonstration of what is referred to as quantum supremacy requires a reliable evaluation of the resources required to solve tasks with classical approaches. Here, we propose the task of sampling from the output distribution of random quantum circuits as a demonstration of quantum supremacy. We extend previous results in computational complexity to argue that this sampling task must take exponential time in a classical computer. We introduce cross-entropy benchmarking to obtain the experimental fidelity of complex multiqubit dynamics. This can be estimated and extrapolated to give a success metric for a quantum supremacy demonstration. We study the computational cost of relevant classical algorithms and conclude that quantum supremacy can be achieved with circuits in a two-dimensional lattice of 7 ×7 qubits and around 40 clock cycles. This requires an error rate of around 0.5% for two-qubit gates (0.05% for one-qubit gates), and it would demonstrate the basic building blocks for a fault-tolerant quantum computer.
journaltitleNature Physics
titleCharacterizing quantum supremacy in near-term devices

Wiley Online Library

journaltitleFortschritte der Physik: Progress of Physics
titleThe physical implementation of quantum computation

American Physical Society

journaltitlePhysical Review Letters
titleHigh-Fidelity Preparation, Gates, Memory, and Readout of a Trapped-Ion Quantum Bit

American Physical Society

journaltitlePhysical Review Letters
titleHigh-Fidelity Quantum Logic Gates Using Trapped-Ion Hyperfine Qubits

AIP Publishing LLC

journaltitleApplied Physics Reviews
titleTrapped-ion quantum computing: Progress and challenges

abstractEntanglement lies at the heart of quantum mechanics, and in recent years has been identified as an essential resource for quantum information processing and computation1,2,3,4. The experimentally challenging production of highly entangled multi-particle states is therefore important for investigating both fundamental physics and practical applications. Here we report the creation of highly entangled states of neutral atoms trapped in the periodic potential of an optical lattice. Controlled collisions between individual neighbouring atoms are used to realize an array of quantum gates, with massively parallel operation. We observe a coherent entangling–disentangling evolution in the many-body system, depending on the phase shift acquired during the collision between neighbouring atoms. Such dynamics are indicative of highly entangled many-body states; moreover, these are formed in a single operational step, independent of the size of the system5,6.
titleControlled collisions for multi-particle entanglement of optically trapped atoms

abstractAt nanokelvin temperatures, ultracold quantum gases can be stored in optical lattices, which are arrays of microscopic trapping potentials formed by laser light. Such large arrays of atoms provide opportunities for investigating quantum coherence and generating large-scale entanglement, ultimately leading to quantum information processing in these artificial crystal structures. These arrays can also function as versatile model systems for the study of strongly interacting many-body systems on a lattice.
titleQuantum coherence and entanglement with ultracold atoms in optical lattices

abstractThe long-term promises of quantum simulators are far-reaching. The field, however, also needs clearly defined short-term goals.
journaltitleNature Physics
titleGoals and opportunities in quantum simulation

American Physical Society

journaltitlePhysical Review Letters
titleDefect-Free Assembly of 2D Clusters of More Than 100 Single-Atom Quantum Systems

American Physical Society

journaltitlePhysical Review A
titleQuantum computation with quantum dots

journaltitlePhysical Review Letters
titleManipulation of a single charge in a double quantum dot

journaltitleReviews of Modern Physics
titleSpins in few-electron quantum dots

John Wiley & Sons

titleSpins in optically active quantum dots: concepts and methods

Nature Publishing Group

journaltitleNature Nanotechnology
titleStoring quantum information for 30 seconds in a nanoelectronic device

IOP Publishing

journaltitlePhysica Scripta
titleQuantum coherence with a single Cooper pair

Nature Publishing Group

titleCoherent control of macroscopic quantum states in a single-Cooper-pair box

American Physical Society

journaltitlePhysical Review B
titleSuperconducting persistent-current qubit

American Association for the Advancement of Science

abstractA qubit was designed that can be fabricated with conventional electron beam lithography and is suited for integration into a large quantum computer. The qubit consists of a micrometer-sized loop with three or four Josephson junctions; the two qubit states have persistent currents of opposite direction. Quantum superpositions of these states are obtained by pulsed microwave modulation of the enclosed magnetic flux by currents in control lines. A superconducting flux transporter allows for controlled transfer between qubits of the flux that is generated by the persistent currents, leading to entanglement of qubit information.
titleJosephson Persistent-Current Qubit

American Physical Society

journaltitlePhysical Review Letters
titleRabi Oscillations in a Large Josephson-Junction Qubit

American Association for the Advancement of Science

abstractWe report the generation and observation of coherent temporal oscillations between the macroscopic quantum states of a Josephson tunnel junction by applying microwaves with frequencies close to the level separation. Coherent temporal oscillations of excited state populations were observed by monitoring the junction’s tunneling probability as a function of time. From the data, the lower limit of phase decoherence time was estimated to be about 5 microseconds.
titleCoherent Temporal Oscillations of Macroscopic Quantum States in a Josephson Junction

American Physical Society

journaltitleReviews of Modern Physics
titleNon-Abelian anyons and topological quantum computation

American Association for the Advancement of Science

titleTopological quantum computation—from basic concepts to first experiments

journaltitlePhysical Review B
titleFlux-controlled quantum computation with Majorana fermions

journaltitlePhysical Review B
titleDetecting Majorana modes in one-dimensional wires by charge sensing

American Association for the Advancement of Science

titleSilica-on-silicon waveguide quantum circuits

American Association for the Advancement of Science

titleQuantum walks of correlated photons

American Association for the Advancement of Science

titleUniversal linear optics

booktitle2016 International Conference on Optical MEMS and Nanophotonics (OMN)
titleSilicon quantum photonics

Optical Society of America

journaltitleJOSA B
titleLaser cooling of trapped ions

journaltitlePhysical Review A
titleTechniques for microwave near-field quantum control of trapped ions

IOP Publishing

journaltitleQuantum Electronics
titleLaser system for Doppler cooling of ytterbium ion in an optical frequency standard

American Association for the Advancement of Science

titleMicrowave-induced cooling of a superconducting qubit

Nature Publishing Group

journaltitleNature Physics
titleSisyphus cooling and amplification by a superconducting qubit

journaltitlePhysical Review Letters
titleThermal and residual excited-state population in a 3D transmon qubit

abstractQuantum technology promises revolutionizing applications in information processing, communications, sensing and modelling. However, efficient on-demand cooling of the functional quantum degrees of freedom remains challenging in many solid-state implementations, such as superconducting circuits. Here we demonstrate direct cooling of a superconducting resonator mode using voltage-controllable electron tunnelling in a nanoscale refrigerator. This result is revealed by a decreased electron temperature at a resonator-coupled probe resistor, even for an elevated electron temperature at the refrigerator. Our conclusions are verified by control experiments and by a good quantitative agreement between theory and experimental observations at various operation voltages and bath temperatures. In the future, we aim to remove spurious dissipation introduced by our refrigerator and to decrease the operational temperature. Such an ideal quantum-circuit refrigerator has potential applications in the initialization of quantum electric devices. In the superconducting quantum computer, for example, fast and accurate reset of the quantum memory is needed.
journaltitleNature Communications
titleQuantum-circuit refrigerator

journaltitleApplied Physics Letters
titleFast control of dissipation in a superconducting resonator

American Physical Society

journaltitlePhysical Review B
titleTunable refrigerator for nonlinear quantum electric circuits

American Physical Society

journaltitlePhysical Review B
titleMultilevel quantum description of decoherence in superconducting qubits

American Physical Society

journaltitlePhysical Review Letters
titleQuasiparticle Relaxation of Superconducting Qubits in the Presence of Flux

IOP Publishing

journaltitleSuperconductor Science and Technology
titleThe effect of environmental coupling on tunneling of quasiparticles in Josephson junctions

American Physical Society

journaltitlePhysical Review B
titleDynamics of parametric fluctuations induced by quasiparticle tunneling in superconducting flux qubits

IOP Publishing

journaltitleSuperconductor Science and Technology
titleRate of tunneling nonequilibrium quasiparticles in superconducting qubits

AIP Publishing LLC

journaltitleApplied Physics Reviews
titleA quantum engineer's guide to superconducting qubits

American Physical Society

journaltitlePhysical Review B
titleNoise and microresonance of critical current in Josephson junction induced by Kondo trap states

American Physical Society

journaltitlePhysical Review Letters
titleDecoherence in Josephson Phase Qubits from Junction Resonators

IOP Publishing

abstractTwo-level ionic systems, where quantum information is encoded in long-lived states (quantum bits, qubits), are discussed extensively for quantum information processing. We present a collection of measurements which characterize the stability of a qubit based on the S1/2 – D5/2 transition of single 40Ca+ ions in a linear Paul trap. We find coherence times of ≃ 1 ms, discuss the main technical limitations and outline possible improvements.
journaltitleJournal of Physics B: Atomic, Molecular and Optical Physics
titleThe coherence of qubits based on single Ca ions

Nature Publishing Group

journaltitleNature Photonics
titleSingle-qubit quantum memory exceeding ten-minute coherence time

Nature Publishing Group

journaltitleNature Communications
titleSingle ion qubit with estimated coherence time exceeding one hour

AIP Publishing LLC

journaltitleApplied Physics Letters
titleCharacterization and reduction of capacitive loss induced by sub-micron Josephson junction fabrication in superconducting qubits

titleManufacturing low dissipation superconducting quantum processors

abstractThe superconducting transmon qubit is a leading platform for quantum computing and quantum science. Building large, useful quantum systems based on transmon qubits will require significant improvements in qubit relaxation and coherence times, which are orders of magnitude shorter than limits imposed by bulk properties of the constituent materials. This indicates that relaxation likely originates from uncontrolled surfaces, interfaces, and contaminants. Previous efforts to improve qubit lifetimes have focused primarily on designs that minimize contributions from surfaces. However, significant improvements in the lifetime of two-dimensional transmon qubits have remained elusive for several years. Here, we fabricate two-dimensional transmon qubits that have both lifetimes and coherence times with dynamical decoupling exceeding 0.3 milliseconds by replacing niobium with tantalum in the device. We have observed increased lifetimes for seventeen devices, indicating that these material improvements are robust, paving the way for higher gate fidelities in multi-qubit processors.
journaltitleNature Communications
titleNew material platform for superconducting transmon qubits with coherence times exceeding 0.3 milliseconds

abstractHere we report a breakthrough in the fabrication of a long lifetime transmon qubit. We use tantalum films as the base superconductor. By using a dry etching process, we obtained transmon qubits with a best T1 lifetime of 503 μs. As a comparison, we also fabricated transmon qubits with other popular materials, including niobium and aluminum, under the same design and fabrication processes. After characterizing their coherence properties, we found that qubits prepared with tantalum films have the best performance. Since the dry etching process is stable and highly anisotropic, it is much more suitable for fabricating complex scalable quantum circuits, when compared to wet etching. As a result, the current breakthrough indicates that the dry etching process of tantalum film is a promising approach to fabricate medium- or large-scale superconducting quantum circuits with a much longer lifetime, meeting the requirements for building practical quantum computers.
journaltitlenpj Quantum Information
titleTowards practical quantum computers: transmon qubit with a lifetime approaching 0.5 milliseconds

* article entry 'wang2021transmon' (mythesis.bib): Invalid format '2022/01/13' of date field 'date' - ignoring

Nature Publishing Group

titleSpin–orbit qubit in a semiconductor nanowire

Nature Publishing Group

journaltitleNature Physics
titleUltrafast coherent control and suppressed nuclear feedback of a single quantum dot hole qubit

IOP Publishing

journaltitleEurophysics Letters
titleManipulation and decoherence of acceptor states in silicon

abstractElectron-spin qubits have long coherence times suitable for quantum technologies. Spin–orbit coupling promises to greatly improve spin qubit scalability and functionality, allowing qubit coupling via photons, phonons or mutual capacitances, and enabling the realization of engineered hybrid and topological quantum systems. However, despite much recent interest, results to date have yielded short coherence times (from 0.1 to 1 μs). Here we demonstrate ultra-long coherence times of 10 ms for holes where spin–orbit coupling yields quantized total angular momentum. We focus on holes bound to boron acceptors in bulk silicon 28, whose wavefunction symmetry can be controlled through crystal strain, allowing direct control over the longitudinal electric dipole that causes decoherence. The results rival the best electron-spin qubits and are 104 to 105 longer than previous spin–orbit qubits. These results open a pathway to develop new artificial quantum systems and to improve the functionality and scalability of spin-based quantum technologies.
journaltitleNature Materials
titleEngineering long spin coherence times of spin–orbit qubits in silicon

Russian Academy of Sciences, Steklov Mathematical Institute of Russian …

journaltitleUspekhi Matematicheskikh Nauk
titleQuantum computations: algorithms and error correction

American Physical Society

journaltitlePhysical Review A
titleSingle-qubit-gate error below ${\mathbf{10}}^{\ensuremath{-}\mathbf{4}}$ in a trapped ion

journaltitlePhysical Review Letters
titleMeasuring and suppressing quantum state leakage in a superconducting qubit

Nature Publishing Group

journaltitlenpj Quantum Information
titleAchieving high-fidelity single-qubit gates in a strongly driven charge qubit with 1/f charge noise

American Physical Society

journaltitlePhysical Review Letters
titleQuantum Computations with Cold Trapped Ions

American Association for the Advancement of Science

abstractTo build a universal quantum computer—the kind that can handle any computational task you throw at it—an essential early step is to demonstrate the so-called CNOT gate, which acts on two qubits. Zajac et al. built an efficient CNOT gate by using electron spin qubits in silicon quantum dots, an implementation that is especially appealing because of its compatibility with existing semiconductor-based electronics (see the Perspective by Schreiber and Bluhm). To showcase the potential, the authors used the gate to create an entangled quantum state called the Bell state.Science, this issue p. 439; see also p. 393Single-qubit rotations and two-qubit CNOT operations are crucial ingredients for universal quantum computing. Although high-fidelity single-qubit operations have been achieved using the electron spin degree of freedom, realizing a robust CNOT gate has been challenging because of rapid nuclear spin dephasing and charge noise. We demonstrate an efficient resonantly driven CNOT gate for electron spins in silicon. Our platform achieves single-qubit rotations with fidelities greater than 99%, as verified by randomized benchmarking. Gate control of the exchange coupling allows a quantum CNOT gate to be implemented with resonant driving in  200 nanoseconds. We used the CNOT gate to generate a Bell state with 78% fidelity (corrected for errors in state preparation and measurement). Our quantum dot device architecture enables multi-qubit algorithms in silicon.
titleResonantly driven CNOT gate for electron spins

abstractElectron spins in semiconductors are promising qubits because their long coherence times enable nearly 109 coherent quantum gate operations. However, developing a scalable high-fidelity two-qubit gate remains challenging. Here, we demonstrate an entangling gate between two double-quantum-dot spin qubits in GaAs by using a magnetic field gradient between the two dots in each qubit to suppress decoherence due to charge noise. When the magnetic gradient dominates the voltage-controlled exchange interaction between electrons, qubit coherence times increase by an order of magnitude. Using randomized benchmarking, we measure single-qubit gate fidelities of  99%, and through self-consistent quantum measurement, state, and process tomography, we measure an entangling gate fidelity of 90%. In the future, operating double quantum dot spin qubits with large gradients in nuclear-spin-free materials, such as Si, should enable a two-qubit gate fidelity surpassing the threshold for fault-tolerant quantum information processing.
journaltitlenpj Quantum Information
titleHigh-fidelity entangling gate for double-quantum-dot spin qubits

American Physical Society

journaltitlePhysical Review Letters
titleHigh-Fidelity Two-Qubit Gates Using a Microelectromechanical-System-Based Beam Steering System for Individual Qubit Addressing

American Physical Society

journaltitlePhysical Review Letters
titleExperimental Implementation of Universal Nonadiabatic Geometric Quantum Gates in a Superconducting Circuit

journaltitlePhysical Review A
titleQuantum energy teleportation with trapped ions

journaltitlePhysical Review A
titleImplementation schemes for unsharp measurements with trapped ions

Nature Publishing Group

journaltitleNature Nanotechnology
titleSelf-consistent measurement and state tomography of an exchange-only spin qubit

journaltitlePhysical Review A
titleProtocol for fermionic positive-operator-valued measures

IOP Publishing

journaltitleLaser Physics Letters
titleRobust quantum state transfer between two superconducting qubits via partial measurement

journaltitlePhysical Review Letters
titleWeak Measurement of a Superconducting Qubit Reconciles Incompatible Operators

American Physical Society

journaltitlePhysical Review B
titleFully microwave-tunable universal gates in superconducting qubits with linear couplings and fixed transition frequencies

journaltitlePhysical Review Letters
titleSimple all-microwave entangling gate for fixed-frequency superconducting qubits

abstractWe experimentally constructed an all-microwave scheme for the controlled-NOT (cNOT) gate between two superconducting transmon qubits in a three dimensional cavity. Our cNOT gate is based on the microwave-activated phase (MAP) gate, which requires an additional procedure to compensate the accumulated phases during the operation of the MAP gate. We applied Z-axis phase gates using microwave hyperbolic secant pulse on both qubits with adequate rotation angles systematically calibrated by separate measurements. We evaluated the gate performance of the constructed cNOT gate by performing two-qubit quantum process tomography (QPT). Finally, we present the experimental implementation of the Deutsch-Jozsa algorithm using the cNOT gate.
journaltitleScientific Reports
titleConstruction of controlled-NOT gate based on microwave-activated phase (MAP) gate in two transmon system

abstractQuantum supremacy is demonstrated using a programmable superconducting processor known as Sycamore, taking approximately 200 seconds to sample one instance of a quantum circuit a million times, which would take a state-of-the-art supercomputer around ten thousand years to compute.
titleQuantum supremacy using a programmable superconducting processor

American Physical Society

journaltitlePhysical Review Letters
titleMacroscopic quantum tunneling in magnetic proteins

American Physical Society

journaltitlePhysical Review A
titleTheory of fault-tolerant quantum computation

abstractA two-dimensional quantum system with anyonic excitations can be considered as a quantum computer. Unitary transformations can be performed by moving the excitations around each other. Measurements can be performed by joining excitations in pairs and observing the result of fusion. Such computation is fault-tolerant by its physical nature.
journaltitleAnnals of Physics
titleFault-tolerant quantum computation by anyons

American Physical Society

journaltitlePhysical Review Letters
titleFault-Tolerant Quantum Computation with High Threshold in Two Dimensions

American Physical Society

journaltitlePhysical Review Letters
titleCharacterization of Addressability by Simultaneous Randomized Benchmarking

American Physical Society

journaltitlePhysical Review Applied
titleParametrically Activated Entangling Gates Using Transmon Qubits

American Association for the Advancement of Science