Department of Applied Physics CQT2023-15

Performing Non-Local Phase Estimation with a Rydberg-Superconducting Qubit
Hybrid

by

J.C. Boschero

MSC THESIS

Assessment committee

Member 1 (chair): | Dr.ir., E.J.D, Vredenbregt
---|---
Member 2: | Dr.ir. J. van Dijk
Member 3: | Prof.dr. A. Todri-Sanial
Advisory member: | MSc. N.M.P Neumann

Graduation

Program: | Applied Physics
---|---
Capacity group: | Coherent Quantum Technology
Supervisor: | Dr. Ir., E.J.D, Vredenbregt
Date of defense: | October 5, 2023
Student ID: | 1244356
Study load (ECTS): | 45
Track: | Nano, Quantum and Photonics
External supervisor(s): | MSc. N.M.P. Neumann
| MMath. W.E. van der Schoot

The research of this thesis has been carried out in collaboration with _TNO_.

This thesis is public and Open Access.

This thesis has been realized in accordance with the regulations as stated in
the TU/e Code of Scientific Conduct.

Disclaimer: the Department of Applied Physics of the Eindhoven University of
Technology accepts no responsibility for the contents of MSc theses or
practical training reports.

## Abstract

Distributed quantum computation is the key to high volume quantum computation
in the NISQ era. This investigation explores the key aspects necessary for the
construction of a quantum network by numerically simulating the execution of
the distributed phase estimation algorithm in a proposed novel
superconducting-resonator-atom hybrid system. The phase estimation algorithm
is used to estimate the phase or eigenvalue of a given unitary operator and is
a sub-process of many other known quantum algorithms such as Shor’s algorithm
and the quantum counting algorithm . An entangling gate between two qubits is
utilised in the distributed phase estimation algorithm, called an $E2$ gate
which provides the possibility to transfer quantum information from one
quantum computer to another, which was numerically shown to have a
construction time of 17ns at a fidelity of 93%. This investigation
analytically derives the Hamiltonian dynamics as well as the noise sources of
each system (flux qubit, resonator and Rydberg atom) and utilizes quantum
optimal control (QOC), namely the gradient ascent pulse engineering (GRAPE)
algorithm, to minimize fidelity error in the corresponding systems gate
construction. The GRAPE algorithm showed very accurate engineering of Rydberg
atom single and multi-qubit gates with fidelities higher than 90% while the
flux qubit suffered greatly from noise with multi-qubit gate fidelities lower
than 90%. The manipulation of the C-shunt factor was shown to decrease the
noise of the flux qubit which in turn increased the probability of accurately
estimating the phase using 4 counting qubits. The study showed a trade off
between the number of time steps used in the descent/ascent and the number of
GRAPE iterations ran on the optimisation for low C-shunt factors
$0<\zeta<1000$. For $\zeta=1000$, the GRAPE algorithm showed effectiveness for
a large number of time steps and large amount of GRAPE iterations by reaching
estimation accuracies <90% .

###### Contents

  1. 1 Introduction and Purpose of the Investigation
  2. 2 Fundamentals of Quantum Information and Quantum Circuit Theory
    1. 2.1 Qubits and Quantum Circuits
    2. 2.2 The Phase Estimation Algorithm
    3. 2.3 Distributed Algorithms and the Non-Local Phase Estimation Algorithm
    4. 2.4 QuTiP Implementation of the Non-Local Phase Estimation Algorithm
  3. 3 The Hybrid Quantum System
    1. 3.1 The Rydberg Qubit
      1. 3.1.1 Rydberg Interactions
      2. 3.1.2 Rydberg Blockade and Transitions for Information Processing
      3. 3.1.3 Rydberg Atom System
    2. 3.2 The Flux Qubit
      1. 3.2.1 Introduction to Flux Qubits
      2. 3.2.2 Flux Qubit System
    3. 3.3 Hybrid Quantum Computer Architecture
    4. 3.4 Forming the Non-Local GHZ State
  4. 4 Modeling Noise and Decoherence in Quantum Systems
    1. 4.1 Rydberg Atom Noise
    2. 4.2 Flux Qubit Noise
    3. 4.3 Hybrid System Noise
  5. 5 Quantum Algorithm Simulation and Analysis with Gradient Ascent Pulse Engineering
    1. 5.1 Introduction to GRAPE with BFGS
    2. 5.2 Quantum Process Tomography and Implementation of GRAPE Optimised Gates
    3. 5.3 GRAPE Optimised Distributed Phase Estimation
  6. 6 Final Insights and Concluding Remarks
    1. 6.1 Conclusion
    2. 6.2 Discussion on Limitations and Further Research
  7. A Quantum Logic Gates
    1. A.1 Single Qubit Gates
    2. A.2 Multiple Qubit Gates
  8. B Proofs and Concepts in Quantum Computation
    1. B.1 Quantum Fourier Transform Product Representation
    2. B.2 Introduction to Open Quantum Systems and the Lindbladian Derivation
    3. B.3 Measuring Expected Values with the $\chi$ Matrix for a Hadamard Gate
  9. C Code & Functions
    1. C.1 Distributed Phase Estimation
    2. C.2 Flux-Resonator-Atom Hybrid Numerical Simulation
    3. C.3 Creation of GRAPE optimised gates
    4. C.4 Values for Parameters
    5. C.5 Diagram of Simulation Processes
  10. D Quantum Principles in Rydberg Systems
    1. D.1 Wavefunctions for Alkali Atoms
    2. D.2 Non-Degenerate Perturbation Theory
    3. D.3 Stark Effect
    4. D.4 Anti-Crossing
    5. D.5 Förster Energy Defect
    6. D.6 Rydberg Light Interaction Hamiltonian
  11. E Flux Qubit Calculations
    1. E.1 Collective Wavefunction in Superconducting Qubits
    2. E.2 Quantum LC Oscillator
    3. E.3 Potential Energy Stability
    4. E.4 Flux-Resonator Hamiltonian
    5. E.5 BFGS Algorithm
  12. F Hybrid System

###### List of Figures

  1. 2.1 Two-level system represented on a Bloch sphere. The two levels are seen on the $z$ axis where $-z$ is the excited state or computational state $\ket{1}$ and $z$ is the ground state or computational state $\ket{0}$. Given a qubit in a superposition state $\ket{\psi}$, it can be represented as a point on the sphere when plotted.
  2. 2.2 Representation of a general two qubit circuit with initial states $\ket{b}$ and $\ket{a}$. The dot parallel to the square signifies that the corresponding qubit is the control. $G_{M}$ on the other hand corresponds to an operation involving many gates and spans the qubits that are involved in the operation. The gates at the end of the sequence for each qubit denote the measurement operation for the specific qubit.
  3. 2.3 The (inverse) quantum Fourier transform circuit portraying the phase gate positions and final qubit states from eq. (2.11) or (2.12) with the phase gates described by $R_{n}$ ($R_{n}^{-1}$).
  4. 2.4 The quantum phase algorithm circuit diagram. The black dots represent the control for the controlled U gates in the $\ket{\psi}$ register [112].
  5. 2.5 (a) Non-local circuit diagram of the entangling gate (E2 gate) for two qubits equaling its local counterpart [137]. (b) Non-local circuit diagram of the entangling gate for three qubits (E3 gate) equaling its local counterpart [137].
  6. (a) Entangling gate for two qubits ($E_{2}$ gate).
  7. (b) Entangling gate for three qubits ($E_{3}$ gate).
  8. 2.6 Implementation of a non-local controlled $U$-gate between $\ket{\psi}$ and $\ket{\phi}$ where the dotted box shows the different quantum computers and the dotted lines denote classical communication [87, 138, 137].
  9. 2.7 A section of a distributed quantum Fourier transform where $n$-local operations on $n$ qubits are performed using one ebit. The double lines between the gate denote the classical communication channel. The control is $\ket{j_{n+k}}$ acting on the targets $\ket{j_{1}}...\ket{j_{n}}$. The remaining $k$ qubits in the second quantum computer are omitted [87].
  10. 2.8 Illustration of the workflow of the pulse-level noisy quantum circuit simulation. First, a quantum circuit is defined in QuTiP (or imported from QASM). Then, the circuit is compiled into control pulses for each control Hamiltonian (blue represents single qubit gates while red represents two-qubit gates). Noise is then generated depending on the chosen processor and finally, the dynamics of the system is solved. This image was taken from [76].
  11. 2.9 Approximate deconstruction of the CPHASE gate using CNOT gates and Z rotation gates. Inspired by [43].
  12. 2.10 Phase estimation circuit where there are a total of seven qubits across both systems in which four are counting qubits. The computer containing the eigenstate $\ket{\psi}$ contains a single counting qubit while the rest are on the other system. Each system is denoted by gray dotted boxes.
  13. 2.11 The distributed circuit responsible for applying the controlled phases from the qubit containing the $\ket{\phi}$ state to the counting qubits. The $E_{2}$ circuit is seen below.
  14. 2.12 The distributed quantum Fourier transform circuit used on the counting qubits where the $R_{i}$ gates $i\subset\mathcal{I}$ refer to the $R^{iQFT_{n}}$ seen in Eq. (2.14)
  15. 2.13 Final computational state probabilities of the distributed quantum phase algorithm with different qubit configurations without noise in the computational basis for different starting phases. The x-axis represents the measured phase in the computational basis, i.e $01=1/2^{2}$. a.) Results of a configuration in which Bob has a qubit that stores the phase, a counting qubit and a communication qubit (3 qubits) whereas Alice has a communication qubit and a counting qubit (2 qubits). b.) Same set up as a. but the starting phase has a $1/2$ factor added. c.) Results of a configuration in which Bob has a qubit that stores the phase, a counting qubit and a communication qubit (3 qubits) whereas Alice has a communication qubit and two counting qubits (3 qubits).
  16. 3.1 (a) Energy levels with corresponding transition channels contributing to the dipole-dipole interaction. (b) Rydberg blockade for two atoms at distance R. [133]
  17. 3.2  Rydberg-Rydberg interactions used to construct quantum gates. (a) Each atom possesses a pair dipole coupled states which have transition dipole moments $\mu_{ac}$, $\mu_{bd}$. (b) For $\Delta_{F}\approx 0$ the Rydberg-Rydberg interactions have a dipolar exchange $\ket{ab}\bra{cd}$ with interaction strength $1/R^{3}$. (c) When $|\Delta_{F}|>|\mu_{ac}\mu_{bd}|/R^{3}$ the interactions are Van de Waals form $\ket{ab}\bra{cd}$ with interaction strength $1/R^{6}$ [85].
  18. 3.3  CZ gate implementation where three pulses are consecutively executed on the qubit. Each panel depicts the different scenarios depending on the initial state. The step number within each panel are shown in blue circles. The bold red lines indicate that the transition was successful while the dashed red lines indicate that the transition was unsuccessful [116].
  19. 3.4 (a) Schematic detailing the Rydberg energy states paired to the respective computational states of the proposed Rydberg computer atom array with the energy states of the communication qubit depicted in the red dotted square. The atoms in the atom array contain two computational levels in the hyperfine $\ket{5S_{1/2},F=1,2}$ states, the Rydberg auxiliary state in the $\ket{70S_{1/2}}$ state and an extra auxiliary $6P_{3/2}$ to facilitate the transfer to the $\ket{70S_{1/2}}$ state. The communication qubit in the red dotted box shows the two energy levels used in the work done by Yu et al. whereby two energy states, $\ket{22D_{5/2}}$ and an arbitrary superposition state $\ket{\Phi_{n=20}}=\ket{n=20,l\geq 3,j,m=5/2}$, are used to denote the computational $\ket{1}$ and $\ket{0}$ states for the $E_{2}$ gate respectively [141]. (b) Schematic of the system whereby an Rubidium-87 atom is held between two spherical capacitors (with capacitance $C$) connected to an LC resonator with inductance $L$. This atomic states are manipulated by an electrostatic field $E$ in the $z$-direction using parallel plates with imposed voltage difference $U$. The atom forms part of Rydberg quantum computer shown as an atom array (red dots in the purple block).
  20. 3.5 The potential energy $U=\frac{1}{2}LI_{q}^{2}-E_{J}\mathrm{cos}(\frac{2\pi\Phi}{\Phi_{0}})$ plotted against $\Phi/\Phi_{0}$ [44].
  21. 3.6 (a) Flux qubit with detailing the three Josephson junctions each denoted by a cross. Each Josephson junction possesses a capacitance $C_{i}$ that is related to the gauge invariant phase difference of each junction. (b) The ground state $\ket{0}$ and the excited energy state $\ket{1}$ of the qubit versus the frustration near $f_{\varepsilon}$ = 0.5. As $\alpha$ becomes smaller, $\Delta$ increases while the persistant current $I_{p}$ increases [94].
  22. 3.7 (a) Top: Phase diagram of $\varphi_{1}$ and $\varphi_{2}$ with magnetic frustration $f=0.50$ and $\alpha=0.5$. The circular shapes are the maxima of the potential while the sponge like contours enclose two minima. Bottom: At a magnetic frustration $f=0.52$, the sponge like contour lines becomes tilted wherein the right side potential is deeper. (b) Top: $U/E_{J}$ plotted against $\varphi^{*}$ where $\pm\varphi^{*}=\varphi_{1}=-\varphi_{2}$ for $f=0.50$. Bottom: $f=0.52$.
  23. 3.8 (a) Representation of Flux qubits (each qubit represented by $Q_{i}$) on a coplanar waveguide (in blue). (b) A contour plot of the potential energy $U/E_{J}$ at $f_{\varepsilon}=0.53$ for $\alpha=0.8$. The measured potential well is seen highlighted with the purple line. (c) A more specified outline of the C-shunt 3JJ flux qubit (circuit on the left) mutually coupled to a resonator, or in this case, the co-planar wave-guide (circuit on the right). (d) The potential function plotted against $\phi^{*}=\phi_{1}=-\phi/2$ at $\alpha=0.8$ and $f_{\varepsilon}=0.53$ for the potential well seen in (c). The asymmetric bias gives rise to the $\ket{L}$ and $\ket{R}$ states.
  24. 3.9 Hybrid system composed of a chain of three-JJ flux qubits, an LC resonator and an atom. The flux qubits have a tunable coupling mitigated by Josephson coupling while the Rydberg atom is Rydberg coupled to a chain of Rydberg atoms. The flux qubit, biased by an external flux $\Phi_{ex}$ is inductively coupled to the resonator with a mutual inductance $M$. The atom is placed in the middle point between two spheres with an oscillating intraresonator electric field with amplitude $\varepsilon$ in the $z$-axis. An additional electrostatic field $E$ runs across the Rydberg qubit in the $z$-axis generated by the parallel-plate capacitors. This model is inspired by [139].
  25. 3.10 (a) Stark map of Rb around $22^{2}D_{5/2}$. (b) The atom qubit is formed by the ground state $\ket{g}$ and the excited state $\ket{e}$ at $E^{(1)}=520.0$ V/cm on two adiabatic energy lines starting with the $22^{2}D_{5/2}$ and manifold state $\ket{\psi_{1}}$ composed of a set of $\ket{n=20,l\geq 3,j=l\pm\frac{1}{2},m=\frac{5}{2}}$ at $E=0$. The auxiliary state $\ket{u}$ is on another manifold state $\ket{\psi_{2}}$ yet on the same set as $\ket{n=20,l\geq 3,j=l\pm\frac{1}{2},m=\frac{5}{2}}$. There are two avoided crossings, $A_{1}$ and $A_{2}$ with energy separations $\Omega$ and $\Omega^{^{\prime}}$ respectively. (c) Energy spectrum ($\omega_{f-LC}$ of the flux qubit-resonator interface vs. the parameter $\gamma_{q}$. The interaction between the flux qubit anf the resonator is negligible at $\gamma_{q}^{(2)}=-5\times 10^{-3}$. At $\gamma_{q}=0$, an anticrossing exists with spacing $\Delta=2\pi\times 5$ GHz. Another avoided crossing with spacing $\delta=2\pi\times 0.1$ GHz induced by the resonant flux qubit-resonator coupling which happens at $\gamma_{q}^{(1)}=-3.06\times 10^{-3}$. (d) Anticrossing seen with spacing $\delta$. This image was taken from [141], authored by Yu et al..
  26. 3.11 The probability (denoted as a color) of the density matrix of the specific systems $\rho=\ket{\psi}\bra{\psi}$ at different steps of the GHZ implementation. (a) Density state of the atom system after increasing $E$ from $E^{(1)}$ to $E^{(5)}$ where the system transitions to the $\frac{1}{\sqrt{2}}(\ket{R,0,e}+\ket{R,0,g})$ with 0.99 fidelity. (b) Density state of the atom-resonator system after reducing $E$ from $E^{(5)}$ to $E^{(2)}$ nonadiabatically. After this step, the system is at $\frac{1}{\sqrt{2}}(\ket{R,1,u}+\ket{R,0,g})$ with 0.95 fidelity. (c) Density state of atom system after $E$ is increased to $E^{(3)}$ quickly wherein the final system state is $\frac{1}{\sqrt{2}}(\ket{L,0,e}+\ket{R,0,g})$ with 0.97 fidelity. (d) The final GHZ density state of the system with 0.93 fidelity.
  27. 4.1 Potential noise sources that can arrise in superconducting qubits. Inspired by [98].
  28. 5.1 Representation of an arbitrary control amplitude $u_{k}(t)$ over time $t$ divided in $N$ discrete steps of duration $\Delta t$. During each step $l$, the control amplitude is equal to $u_{k}(l)$. The vertical arrows represent gradients $\partial\Phi_{0}/\partial u_{k}(l)$ which gives an indication of how to modify $u_{k}(l)$ in the next iteration to improve $\Phi_{0}$.
  29. 5.2 The $\chi_{re}$ matrix evaluating the construction of the Hadamard gate for both the flux and Rydberg systems (each depicted on a row) against GRAPE iterations and time step. Each constructed gate contains the calculated fidelity and the axes represent $i$ as the identity $I$ and $x,y,z$ as the Pauli $\sigma_{x},\sigma_{y},\sigma_{z}$, respectively.
  30. 5.3 (a) Visualization depicting the expected contributions of $\sigma_{x},\sigma_{y}$ and $\sigma_{z}$ (labelled as Expectation value $X,Y.Z$) in the evolution of the flux qubit system over time optimised with GRAPE (seen in purple) and without GRAPE (seen in blue). The red star depicts the ideal Hadamard gate.(b) The plot depicting the distance from the ideal Hadamard gate over time in $\mu$s seen in (a), in arbitrary units, for both the GRAPE optimised and none-GRAPE optimised case.
  31. 5.4 Quantum process tomography evaluating the construction of the CNOT gate for both the flux and Rydberg systems (each depicted on a row) against GRAPE iterations and time step. Each constructed gate contains the calculated fidelity and the axes represent $i$ as the identity $I$ and $x,y,z$ as the Pauli $\sigma_{x},\sigma_{y},\sigma_{z}$, respectively. The notation $ii$ refers to a tensor product $I\otimes I$.
  32. 5.5 Graph depicting the fidelity of a Hadamard gate for the flux qubit over time for two different GRAPE optimisation configurations. The orange line shows the evolution of the gate with 500 GRAPE iterations while the blue line shows the evolution of the gate with 100 GRAPE iterations. The 100 GRAPE iteration evolution finished with a fidelity of 0.97 while the blue line finished with a fidelity of 0.92.
  33. 5.6 Probability of measuring the $\varphi=3/16$ or $0011$ computational state after 10 executions of the phase estimation algorithm on the noisy hybrid circuit for ranging GRAPE iterations and time steps for a shunt factor of $\zeta=$ 10\. Depicted on the left is a graph showing the result in 2D while on the right is the same result on a 3D plane.
  34. 5.7 Identical concept to figure 5.6 for a shunt factor of $\zeta=$ 100\. Depicted on the left is a graph showing the result in 2D while on the right is the same result on a 3D plane.
  35. 5.8  Identical concept to figure 5.6 for a shunt factor of $\zeta=$ 1000\. Depicted on the left is a graph showing the result in 2D while on the right is the same result on a 3D plane.
  36. 6.1 Quantum process tomography evaluating the construction of the CNOT gate for the flux qubit with $\alpha=0.49$ GRAPE iterations and changes in time step number. Each constructed gate contains the calculated fidelity and the axes represent $i$ as the identity $I$ and $x,y,z$ as the Pauli $\sigma_{x},\sigma_{y},\sigma_{z}$, respectively. The notation $ii$ refers to a tensor product $I\otimes I$.
  37. 6.2 Probability of measuring the $\varphi=3/16$ or $0011$ computational state after 10 executions of the phase estimation algorithm on the noisy hybrid circuit with the flux qubit modifed per total GRAPE iteration number per time step number for a shunt factor of $\zeta=$ 1000 and $\alpha=0.49$. Depicted on the left is a graph showing the result in 2D while on the right is the same result on a 3D plane.
  38. B.1 Diagram demonstrating how tracing out the environment from the unitary evolution of the total state leads to $\mathcal{E}(t,0)$ [105, 18, 48].
  39. C.1 Detailed schematic of the steps involved when simulating the distributed phase estimation with GRAPE optimised gates.
  40. D.1 The probability of the radial part of the wavefunction for the state $\ket{22D_{5/2},j=5/2,s=1/2}$ as a function of the atomic distance.
  41. F.1 The interaction strength as a function of the pair-state relative energy between two atoms in state $\ket{70S_{1/2}}$. The Rydberg blockade effect can be seen before the 4.1$\mu m$ distance where the phenomena causes the mixing of states resulting in an exponential relative energy. This image was produced using [115].
  42. F.2 Fitted model function of the interaction strength as a function of the pair-state relative energy between two atoms in state $\ket{70S_{1/2}}$ in the vicinity of the characteristic radius. The black line shows the calculated characteristic radius $R_{vdW}$ equal to $3.5\mu$m. This image was produced using [115].

###### List of Tables

  1. 3.1 Initial and final states of the CZ pulse sequence in the computational basis.
  2. C.1 List of physical parameters used for the master equations.
  3. F.1 List of physical parameters mentioned in the three-qubit hybrid structure [141].

##  Chapter 1 Introduction and Purpose of the Investigation

The number of transistors per area on a microchip has been steadily increasing
every year as predicted by Moore, in 1965 [84] stating that every two years,
this value doubles. Currently, the market is looking to produce transistors
with a gate pitch of 48nm and a gate length of 24nm [31, 109]. Although
roadmaps exist to reduce transistor size in the coming years [31], continuing
this trend requires lithographic wavelengths far beyond extreme ultraviolet
that are more expensive and physically harder to reach. Furthermore,
transistors require insulating layers to be above a certain size
(approximately 1nm) to avoid electrons tunneling through them. Hence,
depending solely on classical computing for a sustained exponential boost in
computational capacity over the long run is not dependable. Therefore, an
alternative approach to achieving significant computational capacity is
required.

In 1981, Richard Feynman proposed an alternative to simulating physical
systems using classical computation. This alternative, became known as quantum
computation [101]. With his proposal, the amount of quantum computer elements
capable of simulating a large physical system are proportional to the space-
time volume of the physical system. Classically this requires exponential
resources [101]. Quantum computers use "qubits" describing this quantum
computational element [110].

Feynman’s vision of a quantum computer relied purely on its ability to
simulate physical systems instead of performing computational tasks. Benioff
possessed a different vision of a quantum computer, whereby he introduced the
idea of a quantum computer as a quantum turing machine [8]. This was later
cemented through the emergence of quantum solutions to oracle-based problems
through the likes of Deutsch’s algorithm [35] in 1985, the Bernstein-Vazirani
algorithm [9] in 1993 and Simon’s algorithm in 1994 [117]. Peter Shor famously
built on top of these results with his algorithm coined Shor’s algorithm [113]
in 1994, where he demonstrated that with a sufficiently large amount of stable
qubits, one could break RSA encryption. As of now, however, quantum hardware
cannot run these algorithms at a sufficiently high quality.

Currently, quantum computing is in the noisy intermediate-scale (NISQ) era
[100]. The NISQ era is characterized by quantum processors containing up to
1000 noisy qubits [21]. These noisy devices can only outperform classical
devices on specific tasks. Furthermore, no single architecture exists to build
these devices. Different quantum computers have been studied as well as
constructed, all with their own set of advantages and disadvantages: trapped
ions [91] and atoms [11, 107], spin systems [119] and superconducting circuits
[64, 93]. The quantum architectures currently under development include,
magnonic qubits [70] and skyrmion qubits [102].

IBM have recently announced the construction of a 433 qubit quantum computer
[25] but with limited circuit depth (the amount of operations a qubit
undergoes). Scaling this further poses physical and economical difficulties
similar to the difficulties classical computing faced, in which the amount of
logic gates capable of fitting on a chip was limited. This was solved by
introducing multiprocessor computing in which processes were distributed
throughout many processors instead of having a single processor complete one
complex task. This same idea is possible in quantum computing called
distributed quantum computing. Distributed quantum computing involves the
collaboration of quantum computers possessing a restricted number of qubits to
collectively solve computational tasks that surpass the computational
resources achievable within a single quantum device [23].

Distributed quantum computing opens the doors to the possibility of a quantum
computer network, in which quantum computers can collaborate on executing a
task over large distances. Such network has been developing in European
research since 2005 and so far progress has been made in the area [68]. In
June 2018, BT announced that it had built a secure quantum network capable of
transferring quantum information between Cambridge, UK and BT’s laboratory in
Ipswich [132]. This network was established using quantum computers with the
same architecture. However, given the evolution of various quantum
architectures, an important question arises: Can two quantum devices with
distinct physical architectures collaborate to solve a distributed
computational task? In addressing this question, this investigation divides it
into three smaller sub-questions.

Can two quantum devices with distinct physical architectures collaborate to
solve a distributed computational task? 1. How can one distribute a quantum
task or algorithm? 2. How can a viable connection be established between two
disparate quantum architectures? 3. Is it possible for such distributed system
to complete a distributed task accurately?

The first sub-question will be addressed by distributing the phase estimation
algorithm. The quantum phase estimation algorithm was first proposed in 1995
and is used to estimate the phase corresponding to an eigenvalue of a given
unitary operator [63]. The phase estimation algorithm was chosen because it is
a prominent subprocess used in Shor’s algorithm [113], the quantum algorithm
for solving linear systems of equations [51], and the quantum counting
algorithm [16].

In order to investigate the feasibility of connecting two computers with
differing quantum architectures (second sub-question), it is important to
understand the coupling (or entanglement) mechanics of one device and then
translate it to another [134]. Coupling resonances on different architectures
is not a "one-size-fits-all" problem. For this reason, the investigation will
focus on two specific quantum architectures, namely, Rydberg atoms and super
conducting qubits, which shall be called a Rydberg-superconducting qubit
hybrid system.

Rydberg atom computers rely on the interactions of alkali atoms with the
valence electron in a highly excited state to execute quantum algorithms.
Rydberg atom computers are constructed on an optical table utilizing an array
of optical lenses and lasers. Rydberg atom quantum systems as a cloud service
have emerged in the last 2 years, with the two notable companies being PASQAL
[73] and PennyLane [127]. Furthermore, one of TU Eindhoven’s goals for the
KAT-1 project, a subsidiary of the Quantum Delta NL ecosystem, is to create a
Rydberg atom computer [39, 90].

Superconducting (SC) qubit computers were chosen to be the other quantum
device because they are arguably the most popular type of qubit. They are
constructed using electric circuits and are kept at very low temperatures (sub
Kelvin). Because the qubits are built on an electrical chip, the fabrication
techniques are analogous to that of classical circuits, consequently making
the quantum hardware cheaper and more intuitive when compared to other quantum
architectures. The circuit is placed at the bottom of a large structure mainly
responsible for its cooling which resembles that of a "golden chandelier".
IBM’s quantum computers, readily accessed through the cloud by researchers and
interested parties, use superconducting qubits [45]. The SC qubit’s popularity
is even attributed beyond the realm of the scientific community wherein the
image of a quantum computer in pop culture is denoted by this "golden
chandelier" [4].

The accuracy of general quantum algorithms depends on the number of operations
the qubits undergo. Thus, accurately running a distributed algorithm
possessing noisy qubits (third sub-question), requires external optimisation
techniques. This is done through Quantum Optimal Control (QOC), in which a
laser pulse is tuned to construct a specific quantum state using a noisy state
[5]. Algorithms such as CRAB (Chopped RAndom Basis) [38], Krotov [47] and
GRAPE (GRadient Ascent Pulse Engineering) [62] exist in order to optimize
dynamical systems. In this study, GRAPE has been scoped and utilized due to
its robustness, efficiency and resource management techniques it provides when
coupled to the limited-memory BFGS (L-BFGS) algorithm when optimizing pulses
for limited qubit systems.

Chapter 2 will answer sub-question (1) by first introducing quantum
information theory in Section 2.1 and then describing the steps needed to
distribute the phase estimation algorithm, seen in Sections 2.2 and 2.3.
Furthermore, its implementation on QuTiP, a Python framework used to model
quantum systems [58], will be explained in Section 2.4.

Chapter 3 will develop the system required to answer sub-question (2). This
chapter begins with an introduction of both the Rydberg atom quantum hardware
(Section 3.1.1) and the introduction of the flux qubit, a specific
superconducting qubit that is used (Section 3.2). The dynamics of both systems
will be explored and modelled. Once both models have been introduced, Section
3.3 will denote the specifications of the device required to couple both
systems. The chapter finalizes with numerical simulations of the sequence
required to establish the communication protocoal between both systems
(Section 3.4).

Chapter 4 details the noise sources that potentially arise from the systems,
discussed in chapter 3, which will be used in Chapter 5 to answer sub-question
(3). The system will split the noise effects into three parts, the Rydberg
atom system, seen in Section 4.1, the flux qubit system, seen in Section 4.2,
and the coupled resonator system 4.3, seen in Section 4.3.

Chapter 5 addresses the optimisation method and its implementation in order to
answer the final sub-question. The optimization method, GRAPE, is presented in
Section 5.1, and subsequently employed for optimizing quantum gates, as
described in Section 5.2. Utilizing the optimised gates, Section 5.3 applies
the optimised gates in the whole distributed algorithm. The accuracy of the
Rydberg-superconducting qubit hybrid system that executes the phase estimation
algorithm with GRAPE-optimised gates is subsequently assessed.

Chapter 6 will include a thorough examination of the model’s limitations,
along with suggested directions for future research in Section 6.2.
Furthermore, Section 6.1 will reiterate the results required to answer sub-
questions (1)-(3), which will help answer the main question determining if two
quantum devices with distinct physical architectures can jointly solve a
distributed computational task.

##  Chapter 2 Fundamentals of Quantum Information and Quantum Circuit Theory

This chapter details the fundamentals of quantum information as well as the
local and non-local phase estimation algorithm explored in this research. The
goal of this chapter is to introduce the non-local phase estimation algorithm
used in the research as well as its numerical implementation. Section 2.1
details the mathematical and physical foundation of what a qubit is and how it
propagates through a quantum circuit as well as an explanation on what quantum
gates are and how they are represented in a circuit. Section 2.2 details the
concepts and intricacies within the phase estimation algorithm such as the
quantum Fourier transform as well as the phase estimation algorithm itself.
Using the theory discussed on section 2.2, section 2.3 details the
construction of non-local circuits while also describing the non-local phase
estimation that will be used throughout the entirety of the research. Finally,
section 2.4 highlights the essential components required to implement the
phase estimation algorithm using QuTiP.

###  2.1 Qubits and Quantum Circuits

The underlying concept of a qubit is inspired by a classical bit in the sense
that they both represent a piece of information, most notably 1 or 0. While a
classical bit can only attain the values 0 or 1, a qubit can be in a
superposition of both values. Although a qubit will always result in either a
0 or a 1 when measured, the probability of measuring either a 0 or a 1 can be
modified. Using Dirac notation (often called bra-ket notation), one can
represent a qubit as a wave function $\ket{\psi}$,

$\ket{\psi}=\alpha\ket{0}+\beta\ket{1}=\begin{pmatrix}\alpha\\\
\beta\end{pmatrix}$ (2.1)

where $\\{\alpha,\beta\\}\subset\mathbb{C}$ and $|\alpha|^{2}+|\beta|^{2}=1$.
Thus, a qubit can take on any value that is a normalized linear combination of
the computational basis states $\\{\ket{0},\ket{1}\\}$. A basis state can
represent the different energy levels or states a quantum system can reside
in. The normalization of this states allows for a more fruitful representation
of the quantum state [89],

$\ket{\psi}=e^{i\gamma}\bigg{(}\mathrm{cos}\frac{\theta}{2}\ket{0}+e^{i\varphi}\mathrm{sin}\frac{\theta}{2}\ket{1}\bigg{)},$
(2.2)

where $\gamma$ is the global phase while both $\theta$ and $\varphi$ represent
the polar and azimuthal angles, respectively, in spherical coordinates.
Visualizing and measuring qubit states in a superposition, for example the one
seen in Eq. (2.2), requires an understanding of the mathematical vector space
they are represented by.

Qubits mathematically inhabit a Hilbert space. A Hilbert space is a vector
space $H$ that contains an inner product such that the norm defined by
$|f|=\sqrt{\langle f,f\rangle}$ turns $H$ into a complete metric space [118,
130]. Being a complete metric space means that the Hilbert space has a
complete inner product. Hence, any point in a Hilbert space can be represented
as an infinite sequence of coordinates or as a vector with finite components.
In quantum mechanics, the observables such as position, momentum and spin are
self-adjoint [96]. Self adjointness is often exhibited in the representation
of quantum information, or the Dirac convention whereby the inner product on a
Hilbert space can simply be represented by matrix representation. In order to
grasp the self-adjoint principle, it is important to first assume that,
$\ket{w}=\sum_{i}w_{i}\ket{i}$ and $\ket{v}=\sum_{j}v_{j}\ket{j}$ are the
representations of vectors $\ket{w}$ and $\ket{v}$ with respect to some
orthonormal basis $\ket{i}$, where $\braket{i}{j}=\delta_{ij}$, the inner
product is said to equal,

$\braket{v}{w}=\bigg{(}\sum_{i}v_{i}\ket{i},\sum_{j}w_{j}\ket{j}\bigg{)}=\sum_{ij}v_{i}^{*}w_{j}\delta_{ij}=\sum_{i}v_{i}^{*}w_{i}=\begin{bmatrix}v^{*}_{1}&\cdots&v^{*}_{n}\end{bmatrix}\begin{bmatrix}w_{1}\\\
\vdots\\\ w_{n}\end{bmatrix}.$ (2.3)

Assuming $a$ is a linear operator residing in a Hilbert space, $a$ is said to
be self-adjoint if there exists a linear operator $a^{{\dagger}}$ on the
Hilbert space such that for the vectors $\ket{v},\ket{w}$,

$(\ket{v},a\ket{w})=(a^{\dagger}\ket{v},\ket{w}).$ (2.4)

The linear operator $a^{\dagger}$ is known as the adjoint or Hermitian
conjugate of the operator $a$ [89]. To simplify, a linear operator or matrix
is called self-adjoint if it is equal to its own adjoint. When analyzing
qubits inhibiting a quantum state space (or Hilbert space) the operator
$a^{\dagger}$ is commonly referred to as the raising or creation operator
while $a$ is the lowering or annihilation operator.

Measuring of some state $\ket{\psi}$ requires a measurement operator $M_{m}$.
The probabilities of some outcome $m$ occurring are
$\bra{\psi}M^{\dagger}_{m}M_{m}\ket{\psi}$. Incorporating a global phase to
the quantum state, $e^{i\gamma}\ket{\psi}$, and measuring once more yields,
$\bra{\psi}e^{-i\gamma}M^{{\dagger}}_{m}M_{m}e^{i\gamma}\ket{\psi}=\bra{\psi}M^{\dagger}_{m}M_{m}\ket{\psi}$.
Therefore, from an observational point of view, the global phase can be
ignored in Eq. (2.2), which leaves the state,

$\ket{\psi}=\mathrm{cos}\frac{\theta}{2}\ket{0}+e^{i\varphi}\mathrm{sin}\frac{\theta}{2}\ket{1}.$
(2.5)

The expression acquired in Eq. (2.5) is essential for the creation of qubit
gates. Modifying $\theta$ and $\varphi$ such that the phase change alters the
superposition of the quantum state, is the basis of quantum computation. A
Bloch sphere is a geometric representation of the pure state space of a two-
level system. Thus, the modification of phase can be represented using a Bloch
sphere. This can be seen depicted in figure 2.1.

The Bloch sphere is also used to represent single qubit gates. The most
general single qubit gate is the U-gate, which takes the form,

$U(\theta,\varphi,\gamma)=\begin{bmatrix}\mathrm{cos}(\frac{\theta}{2})&-e^{i\gamma}\mathrm{sin}(\frac{\theta}{2})\\\
-e^{i\varphi}\mathrm{sin}(\frac{\theta}{2})&e^{i(\varphi+\gamma)}\mathrm{cos}(\frac{\theta}{2})\end{bmatrix}.$
(2.6)

An example is the Hadamard gate, a gate responsible for placing a single qubit
in a superposition state between $\ket{0}$ and $\ket{1}$, which is denoted as,

$H\equiv U(\frac{\pi}{2},0,\pi)=\frac{1}{\sqrt{2}}\begin{bmatrix}1&1\\\
1&-1\end{bmatrix}.$ (2.7)

Figure 2.1: Two-level system represented on a Bloch sphere. The two levels are
seen on the $z$ axis where $-z$ is the excited state or computational state
$\ket{1}$ and $z$ is the ground state or computational state $\ket{0}$. Given
a qubit in a superposition state $\ket{\psi}$, it can be represented as a
point on the sphere when plotted.

A general circuit can be seen represented in figure 2.2 where each qubit
travels along an arbitrary synchronous line that ends in a read-out or
measurement gate. The gates are presented by squares, either on a single line
to denote single qubit gates or spread over many lines denoting a multi-qubit
gate. In classical computation, logic gates are physical devices integrated
into circuits, where electrons are inputted and outputted. Within the realm of
quantum computing, logic gates are applied to qubits by employing controlled
pulses at specific frequencies, as opposed to the physical gates used in
classical computing. Although in practice logic gates in quantum computing are
very different to those in classical computing, the circuit representations
are very similar.

Much like classical computing, quantum computing requires the use of multi-
qubit gates. This requires extending the single qubit basis to a multi-qubit
basis. Begin by assuming two separated qubits with two states,

$\ket{a}=\begin{bmatrix}a_{0}\\\
a_{1}\end{bmatrix},\quad\ket{b}=\begin{bmatrix}b_{0}\\\ b_{1}\end{bmatrix},$
(2.8)

the collective state can mathematically be described by applying a tensor
multiplication between both states [17],

$\ket{ba}=\ket{b}\otimes\ket{a}=\begin{bmatrix}b_{0}a_{0}\\\ b_{0}a_{1}\\\
b_{1}a_{0}\\\ b_{1}a_{1}\end{bmatrix}.$ (2.9)

In figure 2.2, the multi-qubit gate, $G_{c}$, is shown to be a controlled gate
where the qubit in state $\ket{b}$ is the target and the qubit in state
$\ket{a}$ is the control. A controlled gate works by only applying a gate to
the target qubit if the control qubit is in the excited state or $\ket{1}$
state. This type of gate in particular will be used extensively throughout
this research. A list of common single and multi-qubit logic gates can be seen
in Appendix A.

Figure 2.2: Representation of a general two qubit circuit with initial states
$\ket{b}$ and $\ket{a}$. The dot parallel to the square signifies that the
corresponding qubit is the control. $G_{M}$ on the other hand corresponds to
an operation involving many gates and spans the qubits that are involved in
the operation. The gates at the end of the sequence for each qubit denote the
measurement operation for the specific qubit.

###  2.2 The Phase Estimation Algorithm

Combining a variety of quantum gates correctly with the combination of a
measurement ultimately resulting in an answer (often described in the
computational basis). Changing the notion of the measurements, one can extract
more complex information from the system. The phase estimation algorithm is an
example of such phenomenon. Suppose a unitary operator $U$ has an eigenvector
$\ket{u}$ with eigenvalue $e^{2\pi i\varphi}$, where the observable, $\varphi$
is not known [89]. The goal of the phase estimation algorithm is to estimate
the value of $\varphi$. It finds applications in encryption breaking [113] and
many other algorithms.

Understanding the phase estimation algorithm requires the understanding of the
quantum Fourier transform (QFT), as it constitutes a significant component of
the phase estimation algorithm. Parsing multiple qubits through a Fourier
transform essentially entangles all qubits at once. The QFT is extremely
powerful because it exponentially speeds up factoring algorithms [7, 30]. The
classical discrete Fourier transform takes complex components
$\\{f(0),f(1),...,f(N-1)\\}$ and transforms them into
$\\{\tilde{f}(0),\tilde{f}(1),...,\tilde{f}(N-1)\\}$ using [7],

$\tilde{f}(j)=\frac{1}{\sqrt{N}}\sum^{N-1}_{k=0}e^{2\pi i\frac{jk}{N}}f(k).$
(2.10)

The quantum Fourier transform works in the same fashion. It is defined on a
quantum register of $n$ qubits (with a total of $N=2^{n}$ states) as the
unitary operator $F$ whose action on the states of the computational basis is
given by [7],

$F(\ket{j})=\frac{1}{\sqrt{2^{n}}}\sum^{2^{n}-1}_{k=0}e^{2\pi
i\frac{jk}{2^{n}}}\ket{k}$ (2.11)

where an arbitrary state $\ket{\psi}=\sum_{j}f(j)\ket{j}$ transforms into
$\ket{\tilde{\psi}}=F\ket{\psi}=\sum^{2^{n}-1}_{k=0}\tilde{f}(k)\ket{k}$ [7,
30]. The inverse quantum transform is simply given as,

$F^{-1}(\ket{j})=\frac{1}{\sqrt{2^{n}}}\sum^{2^{n}-1}_{k=0}e^{-2\pi
i\frac{jk}{2^{n}}}\ket{k}.$ (2.12)

Because the implementation of both the quantum Fourier transform and the
quantum inverse Fourier transform differs only by a negative phase, the
circuit structures for these operations are nearly identical. The goal now is
to convert the algebraic representation of the Fourier transform to circuit
form as seen in section 2.1. To do so, it is helpful to write state $\ket{j}$
in binary form, $j=j_{1}j_{2}\cdots j_{n}$, or more formally,
$j=j_{1}2^{n-1}+j_{2}2^{n-2}+\cdots+j_{n}2^{0}$. Adopting the notation
$0.j_{l}j_{l+1}\cdots j_{m}$ to represent the binary fraction
$j_{l}/2+j_{l+1}/4+\cdots+j_{m}/2^{m-l+1}$ allows the quantum Fourier
transform to be given by the product representation [89],

$\ket{j_{1},\cdots,j_{n}}\rightarrow\frac{\big{(}\ket{0}+e^{2\pi
i0.j_{n}}\ket{1}\big{)}\big{(}\ket{0}+e^{2\pi
i0.j_{n-1}j_{n})}\ket{1}\big{)}\cdots\big{(}\ket{0}+e^{2\pi
i0.j_{1}j_{2}\cdots j_{n}}\ket{1}\big{)}}{2^{n/2}}.$ (2.13)

This representation allows the construction of the efficient quantum circuit
and a proof as to why the Fourier transform is unitary. The proof as to how
the product representation in Eq. (2.13) follows from the formal definition in
Eq. (2.12) can be seen in Appendix B.1. Figure 2.3 shows the circuit of the
quantum Fourier transform where the $R_{n}$ gates represent a controlled Z
rotation gate seen in eq. (2.14) which follows from the unitary
transformations seen in (2.13). The inverse Fourier transform simply follows
by replacing $R_{n}$ by $R^{-1}_{n}$,

$R_{n}=\begin{pmatrix}1&0&0&0\\\ 0&e^{-2\pi i/2^{n}}&0&0\\\ 0&0&1&0\\\
0&0&0&e^{2\pi i/2^{n}}\end{pmatrix},R^{-1}_{n}=\begin{pmatrix}1&0&0&0\\\
0&e^{2\pi i/2^{n}}&0&0\\\ 0&0&1&0\\\ 0&0&0&e^{-2\pi i/2^{n}}\end{pmatrix}.$
(2.14)

Figure 2.3: The (inverse) quantum Fourier transform circuit portraying the
phase gate positions and final qubit states from eq. (2.11) or (2.12) with the
phase gates described by $R_{n}$ ($R_{n}^{-1}$).

The inverse Fourier transform can be seen in the context of the phase
estimation algorithm in figure 2.4 as $QFT^{-1}_{n}$ where it is
compartmentalized as a subroutine between $n$ qubits.

Figure 2.4 shows the circuit for the phase estimation algorithm. The easiest
method of interpreting the diagram is to distinguish two registers, the target
register with $n$ qubits denoted as a series of $\ket{0}$ states and the
control register with $m$ qubits necessary to store state $\ket{\psi}$. The
eigenvector $\ket{\psi}$ is the one targeted for measurement, with the
objective of determining its corresponding eigenvalue.

Figure 2.4: The quantum phase algorithm circuit diagram. The black dots
represent the control for the controlled U gates in the $\ket{\psi}$ register
[112].

The $C-U^{2^{j}}$ gate is a specific type of controlled U gate (seen in
Appendix A). The action of the gate $C-U^{2^{j}}$ on a given
$\frac{1}{\sqrt{2}}(\ket{0}+\ket{1})\ket{\psi}$ state is given by,

$C-U^{2^{j}}\frac{1}{\sqrt{2}}(\ket{0}+\ket{1})\ket{\psi}=\frac{1}{\sqrt{2}}(\ket{0}\ket{\psi}+\ket{1}U^{2^{j}}\ket{\psi})=\frac{1}{\sqrt{2}}(\ket{0}+\ket{1}e^{i2^{j}\varphi})\ket{\psi},$
(2.15)

and can be described in matrix form as,

$C-U^{2^{j}}=\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&1&0\\\ 0&e^{2\pi
i\varphi 2^{j}}\\\ \end{pmatrix},$ (2.16)

which follows that of a traditional controlled U-gate where global phase is
ignored ($\gamma=0$) and only the $z$ component is taken into account
($\theta,\lambda=0$). The controlled U-gate described in Eq. (2.16) is used to
map the phase of the $\ket{\psi}$ state onto the qubits that will then go
through the inverse Fourier transform. The output for the quantum phase
circuit before the application of the inverse Fourier transform is given by
[7, 87, 89],

$\frac{1}{2^{t/2}}(\ket{0}+e^{i2\pi 2^{t-1}\varphi}\ket{1})(\ket{0}+e^{2\pi
2^{t-2}\varphi}\ket{1})\cdots(\ket{0}+e^{2\pi
i2^{0}\varphi}\ket{1})\ket{\psi}=\frac{1}{2^{t/2}}\sum^{2^{t}-1}_{k=0}e^{2\pi
i\varphi k}\ket{k}\ket{\psi},$ (2.17)

where $t$ is the amount of counting qubits. To observe how the phase is
estimated, it is first important to express $\varphi$ as
$\varphi=0.\varphi_{1}\cdots\varphi_{t}$ which leaves Eq. (2.17) as [89],

$\frac{1}{2^{t/2}}(\ket{0}+e^{2\pi i0.\varphi_{t}}\ket{1})(\ket{0}+e^{2\pi
i0.\varphi_{t-1}\varphi_{t}}\ket{1})\cdots(\ket{0}+e^{2\pi
i0.\varphi_{1}\varphi_{2}\cdots\varphi_{t}}\ket{1}).$ (2.18)

One can see that applying the inverse Fourier transform seen in Eq. (2.13)
outputs the product state $\ket{\varphi_{1}\cdots\varphi_{t}}$, which can then
be measured in the computational basis to give the exact $\varphi$.

The description of the special case where $\varphi=0.a_{1}\cdots a_{m}$ can be
obtained by applying the inverse QFT, which produces $\ket{a_{1},\cdots
a_{m}}$ exactly (and hence $\varphi$). This is because the QFT maps between
the state space and the phase space. However $\varphi$ is not in general a
fraction of a power of two. For such case, it is important to apply and derive
the accuracy of the inverse quantum Fourier transform by giving the best
$n$-bit estimate of $\varphi$ with high probability. By conveniently writing
$\varphi=2\pi\Big{(}\frac{a}{2^{n}}+\delta\Big{)}$ where
$a=a_{n-1}a_{n-2}...a_{0}$ and $\delta$ is the phase error defined by
$|2^{n}\delta|\leq 1/2$, it is possible to see that $2\pi a/2^{n}$ is the best
$n$ bit approximation of $\varphi$ [7, 28].Thus, in the general case,
$\delta\neq 0$, the best $n$-bit estimate of $\phi$ is given by $a$. Utilizing
the approximations, applying an inverse QFT as seen in Eq. (2.17), the
expression for the best $n$ \- bit estimate (with the omitted $\ket{\psi}$
state) is,

$\frac{1}{2^{n}}\sum^{2^{n}-1}_{k=0}\sum^{2^{n}-1}_{l=0}e^{2\pi
i(a-k)l/2^{n}}e^{2\pi i\delta l}\ket{k}.$ (2.19)

The next step is to perform a measurement on the $\ket{k}$ state. This is
obtained from a standard measurement of the first register containing
probability $p_{a}=|c_{a}|^{2}$ where $c_{a}$ comes from the decomposition
$\sum_{k=0}^{2^{n}-1}c_{a}\ket{k}$ [7, 28, 63],

$c_{a}=\sum^{2^{n}-1}_{l=0}e^{2\pi i(a-k)l/2^{n}}e^{2\pi i\delta l},$ (2.20)

which can finally be evaluated to,

$|c_{a}|^{2}=\frac{1}{2^{n}}\bigg{[}\frac{1-e^{2\pi i\delta 2^{n}}}{1-e^{2\pi
i\delta}}\bigg{]}.$ (2.21)

Employing Euler’s form, $|1-e^{2\pi i\delta 2^{n}}|=2|\mathrm{sin}(\pi\delta
2^{n})|$ and using that $2|\mathrm{sin}(\pi\delta 2^{n})|\geq 4|\delta|2^{n}$
as well as $2|\mathrm{sin}(\pi\delta)|\leq 2\pi|\delta|$ (since for any
$z\in[0,1/2]$, $2z\leq\mathrm{sin}(\pi z)\leq\pi z$). These two inequalities
can be inserted into eq. (2.21) to get,

$|c_{a}|^{2}\geq\frac{4}{\pi^{2}}\approx 0.405.$ (2.22)

This result shows that the best $n$-bit estimate of $\theta$ can be achieved
up to this precision [28, 63, 87]. An intuitive way of understanding this
value is by first assuming two qubits (thus a precision of 1/4) are used to
try to estimate a phase equal to 1/8. Although the qubits lack precision to
output the proper result in the computational basis, the results 00 and 01
(0/4 and 1/4 in the computational basis) will have at least 40.5% probability
of being outputted since 1/8 lies between both values. By adding a number of
extra qubits in the order of $O(\mathrm{log}(1/\epsilon))$ and truncating the
extra qubits, the probability can increase to $1-\epsilon$ [28]. A more
general interpretation of this phenomenon is sought by thinking in terms of
classical bits. Increasing the amount of bits used to represent a number means
increasing the precision the number possesses. Thus, increasing the register
$n$, increases the precision of the phase estimation.

###  2.3 Distributed Algorithms and the Non-Local Phase Estimation Algorithm

(a) Entangling gate for two qubits ($E_{2}$ gate).

(b) Entangling gate for three qubits ($E_{3}$ gate).

Figure 2.5: (a) Non-local circuit diagram of the entangling gate (E2 gate) for
two qubits equaling its local counterpart [137]. (b) Non-local circuit diagram
of the entangling gate for three qubits (E3 gate) equaling its local
counterpart [137].

Having revisited both quantum circuit theory and the phase estimation
algorithm, it is now possible to expand these concepts into a non-local or
distributed quantum computing basis. As opposed to local implementations of
quantum algorithms where the interaction between qubits is implied, a non-
local representation requires the teleportation or communication of states
between channels. These channels are represented by qubits called channel
qubits which can be sent back and forth over a network [137].

Eisert et al. have shown that setting up these channel qubits relies on the
implementation of a distributed CNOT gate [40]. On top of that, Eisert et al.
shows that the distributed CNOT gate, much like the local CNOT gate seen in
Appendix A, is a native gate that can create other operations such as
controlled gates [40]. Implementing a distributed CNOT gate requires a shared
entangled qubit pair (ebit) between two quantum computers and two classical
bits (cbits). The generation of the ebit can be seen in figure 2.5(a), where
an ebit is constructed simply using a Hadamard gate and a CNOT gate. To
properly establish an ebit pair, however, one must use an $E_{2}$-gate which
explicitly distributes the ebit pair between two devices. This is done by
having the remote computer send a channel qubit to the local machine. The
local machine then entangles the channel with one of the local channel qubits.
Finally, one qubit of the pair is sent back to the remote machine resulting in
a state $\frac{1}{2}(\ket{00}+\ket{11})$ called a GHZ state [137]. However,
for simplicity, this investigation will refer to the entangling gate in figure
2.5(a) as the $E_{2}$ gate. Figure 2.5(b) shows the same concept but with
three qubits which in turn form a GHZ state.

The subsequent stage in non-local computation involves sharing the information
of a qubit from one party with another. Yimsiriwattana and Lomonaco have
identified a set of primitive operations that allow one party to share the
information with another [138, 137, 103]. Suppose a user, specifically Alice,
holds a superposition state and wishes for another party to execute a complex
computation on it. In order to share this state, Alice first applies a CNOT
gate between her state and the ebit in her system and then measures the
corresponding ebit. If the measurement reveals a $\ket{1}$ state, Alice
applies a NOT gate (also known as X-gate seen in Appendix A) and classically
communicates this information to the other party, specifically Bob. Bob then
applies a NOT gate to his ebit only if Alice has shown a $\ket{1}$. This
procedure is shown as the first five gates in the non-local U-gate
implementation seen in figure 2.6 where Alice’s and Bob’s quantum computers
are shown in dotted black boxes and the dotted lines denote the use of
classical communication. In this specific case, Alice is shares the state
$\ket{\phi}$ with Bob.

Bob is now free to use the quantum information on his ebit to perform
computations locally using the Alice’s shared qubit state. In this case, it is
assumed that Bob performs a single controlled U-gate (defined in equation
2.16) between the ebit and the qubit with state $\ket{\psi}$ in his system,
which can be seen in figure 2.6 on the $\ket{\psi}$ state. Bob now wishes to
share the resulting state with Alice. To do so, Bob first applies a hadamard
gate across his ebit and then measures it. If a $\ket{1}$ state is measured, a
NOT gate is applied by Bob on his ebit and the result is classically
communicated to Alice. Assuming the output is $\ket{1}$, Alice applies a Z
gate (seen in Appendix A)) on the qubit she shared in order to obtain the
outcome of Bob’s computation.

Figure 2.6: Implementation of a non-local controlled $U$-gate between
$\ket{\psi}$ and $\ket{\phi}$ where the dotted box shows the different quantum
computers and the dotted lines denote classical communication [87, 138, 137].

This investigation will use the protocol depicted in figure 2.6 in order to
distribute information between two systems. Section 2.2 has emphasized that
the precision of the phase estimation algorithm improves as you employ a
greater number of counting qubits. Therefore, this study will delve into the
scenario in which one quantum computer retains the state requiring estimation
of the phase, accompanied by a sole counting qubit. Simultaneously, the other
quantum computer will be responsible for managing the counting qubits, thereby
enhancing the precision of the estimation process. Thus, the two main
distributed processes used in the research will be the distributed QFT and the
distributed controlled U-gate.

The distributed QFT is similar to that of the controlled U-gate. One method of
performing the distributed QFT is by replacing each controlled rotation gate
with a non-local rotation gate. However, it is also possible to use a single
shared entangled state to perform multiple non-local gates, by grouping all
operations on one computer that are all controlled by a single qubit from the
other computer [87]. This fundamentally leads to the distributed version of
the quantum Fourier transform depicted in figure 2.7 [87].

Figure 2.7: A section of a distributed quantum Fourier transform where
$n$-local operations on $n$ qubits are performed using one ebit. The double
lines between the gate denote the classical communication channel. The control
is $\ket{j_{n+k}}$ acting on the targets $\ket{j_{1}}...\ket{j_{n}}$. The
remaining $k$ qubits in the second quantum computer are omitted [87].

###  2.4 QuTiP Implementation of the Non-Local Phase Estimation Algorithm

The simulation and execution of the quantum algorithms will be constructed on
QuTiP [58]. QuTiP, built on Python, offers many quantum information processing
packages. Furthermore, it uses quantum dynamic solvers and control
optimization features that will be utilized throughout this research. More
specifically, QuTiP can run quantum circuits on simulated quantum processors.
The workflow is described in figure 2.8.

As seen from figure 2.8, the main benefit QuTiP offers when simulating quantum
circuits is the ability to simulate the relevant noise depending on the
physical architecture. For simplicity, the processor used will be the spin
chain model. The spin chain model models the general physics involved in any
given system since it describes both a general single qubit interaction and a
multi-qubit interaction. The spin chain model acts in accordance with the one
dimensional Ising model. The 1D Ising model describes the energy configuration
of a lattice containing a finite amount of particles (in this case qubits)
allocated in sites that can each be in a spin configuration
$\sigma_{k}\in\\{-1,+1\\}$ representing the spin.

Figure 2.8: Illustration of the workflow of the pulse-level noisy quantum
circuit simulation. First, a quantum circuit is defined in QuTiP (or imported
from QASM). Then, the circuit is compiled into control pulses for each control
Hamiltonian (blue represents single qubit gates while red represents two-qubit
gates). Noise is then generated depending on the chosen processor and finally,
the dynamics of the system is solved. This image was taken from [76].

For any two adjacent sites $i,j$ in the lattice, there is an interaction
energy with strength $J_{ij}$. Additionally, an extra factor is added to
describe the external field interaction $h_{j}$, which is often manipulated to
incorporate quantum gates. The full Ising model can be seen in Eq. (2.23)
described as a Hamiltonian, a function denoting the energy dynamics of a
system [76].

$H(\sigma)=-\sum_{ij}J_{ij}\sigma_{i}\sigma_{j}-\mu\sum_{j}h_{j}\sigma_{j}.$
(2.23)

Knowing the dynamics of the system in use is important because it allows one
to determine the native qubit gates. Although this model is very general, it
is possible to derive the fact that a controlled U gate is not possible to
reproduce using the set of dynamics stated [54]. Creating a controlled U gate
is possible by extending the model and introducing more parameters, this can
be seen in [52]. In this research, the controlled U-gate will be simplified to
controlled phase gate due to the simplicity of the phase in the eigenvalue.
The controlled phase gate is similar to the definition used in Eq. (2.16),
where $j=0$ and the $2\pi$ factor in the exponent is omitted, thus leaving,

$\mathrm{CPHASE}(\varphi)=\begin{bmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&1&0\\\
0&0&0&e^{i\varphi}\end{bmatrix}.$ (2.24)

The CPHASE gate is still not possible to construct in this model, thus, an
approximation will be used to create this gate using native gates. The
approximation can be constructed using the circuit seen in figure 2.9. The
function for the creation of this gate can be seen in Appendix C.1 as
cphase_gate.

Figure 2.9: Approximate deconstruction of the CPHASE gate using CNOT gates and
Z rotation gates. Inspired by [43].

With the comprehensive description of all gates and processes now provided, it
is crucial to gain a clear understanding of the final circuit employed. The
research will concentrate on two quantum computers. One of these computers
will consist of three qubits, including the qubit containing the phase to be
estimated, a single counting qubit, and a communication qubit. The other
computer will comprise three counting qubits and a single communication qubit,
forming the entangled bit. The total circuit can be seen in figures 2.10,
2.11, and 2.12. The QuTiP functions can be seen in Appendix C.1. Figure 2.10
begins by performing hadamard gates on all the counting qubits, which can be
seen in function form as perform_hadamards.

The following step is to perform the distributed phase gates, or DCP block on
the circuit, from the three qubit computer to the four qubit computer. This is
seen in figure 2.11 where three non-local controlled phases and a single local
controlled phase are performed. This process is described by
pulse_phase_sequence, which takes a sub-process responsible for performing the
specific non-local gate, perform_non_local_phase. Furthermore, the $E_{2}$
gate is initialized using initialize_E_gate.

In the non-local functions, it is necessary to measure a single qubit. To do
so, one must specify a list of measurement vectors called positive operator-
valued measure (POVM). As mentioned in Section 2.1, given a quantum system
$\ket{\psi}$, the probability of an outcome $m$ is given by
$p(m)=\bra{\psi}M^{\dagger}M\ket{\psi}$. Defining $E_{m}=M^{\dagger}M$, it
follows that $\sum_{m}E_{m}=I$ where $I$ is the identity matrix and
$p(m)=\bra{\psi}E_{m}\ket{\psi}$ which means that the set of operators $E_{m}$
are sufficient to determine the probabilities of the different measurement
outcomes [89]. The set of $\\{E_{m}\\}$ are the POVM which are required to be
explicitly defined when making single qubit measurements. This is seen
incorporated as perform_measurement which takes a qubit number as input.

The inverse QFT circuit is then performed, which is expanded upon in figure
2.12. As can be observed, only a single non-local operation is required since
one of the four total counting qubits are on a separate circuit. This process
is described by the recursive function iqft_rotations, which, once again,
takes a sub-process responsible for performing the specific non-local gates,
perform_non_local_iqft.

Figure 2.10: Phase estimation circuit where there are a total of seven qubits
across both systems in which four are counting qubits. The computer containing
the eigenstate $\ket{\psi}$ contains a single counting qubit while the rest
are on the other system. Each system is denoted by gray dotted boxes.  Figure
2.11: The distributed circuit responsible for applying the controlled phases
from the qubit containing the $\ket{\phi}$ state to the counting qubits. The
$E_{2}$ circuit is seen below. Figure 2.12: The distributed quantum Fourier
transform circuit used on the counting qubits where the $R_{i}$ gates
$i\subset\mathcal{I}$ refer to the $R^{iQFT_{n}}$ seen in Eq. (2.14)

This circuit will be used throughout the entire investigation. However, the
results shown in figure 2.13 shows the QuTiP simulation runs with varying the
amount of counting qubits in the computer without the phase register. This is
done for simplicity to see the difference in varying initial starting phases.
The chosen phase factor $\varphi$ changes each run and the resulting graph
denotes the probability of each vector state (measured in the computational
basis). Recalling section 2.2, the final estimate is represented using
$\phi/2^{N}$ where $\phi$ is the measured state in the computational basis and
$N$ is the total amount of counting qubits. Thus, figures 2.13a and 2.13c
demonstrate that without noise, the distributed algorithm manages to calculate
the phase perfectly for $N=2$ and $N=3$ respectively when the phase is divided
by the integer $2^{N}$. However, even when $\delta=0.5$ (seen in Eq. (2.20)),
the best estimate of $\varphi$ given by $\phi$ has to have a probability
$\mathrm{Pr}(\phi)\geq 4/\pi^{2}\approx 40.5\%$. This result can be seen in
figure 2.13b. where the estimated computational basis states $\phi$ with the
highest probabilities are $\phi=\varphi\pm 0.5$. It is thus shown that the
resulting probabilities are in accordance with the inequality observed in Eq.
(2.22).

To conclude, the design and implementation of a distributed phase estimation
circuit has been realized. Chapters 3 and 4 details the Hamiltonians required
execute the noisy version of this algorithm. The execution and implementation
of the distributed noisy phase estimation algorithm with four counting qubits
using optimization techniques will be seen in Chapter 5.

Figure 2.13: Final computational state probabilities of the distributed
quantum phase algorithm with different qubit configurations without noise in
the computational basis for different starting phases. The x-axis represents
the measured phase in the computational basis, i.e $01=1/2^{2}$. a.) Results
of a configuration in which Bob has a qubit that stores the phase, a counting
qubit and a communication qubit (3 qubits) whereas Alice has a communication
qubit and a counting qubit (2 qubits). b.) Same set up as a. but the starting
phase has a $1/2$ factor added. c.) Results of a configuration in which Bob
has a qubit that stores the phase, a counting qubit and a communication qubit
(3 qubits) whereas Alice has a communication qubit and two counting qubits (3
qubits).

##  Chapter 3 The Hybrid Quantum System

The future of distributed quantum computation requires the coupling and
manipulating of qubits in both the flux qubit and Rydberg atom quantum
architectures [49]. The establishment of coupling or entanglement between two
qubits possessing distinct physical properties, here referred to as hybrid
qubits, becomes essential in enabling the interconnection of quantum computers
with varying architectures. Furthermore, a hybrid quantum structure consisting
of a variation of different physical realizations has the ability to inherit
the advantages of each component [13, 126, 134]. Hence, similar to classical
computers, which feature multiple chips for executing distributed processes,
quantum computers can likewise operate in a comparable manner with the
advantage of employing diverse qubits. The goal of this chapter is to
introduce the physical characteristics of both quantum computers, which will
then be implemented into the circuit seen in Section 2.4. Section 3.1
introduces the Rydberg qubit, which is then utilized to construct the model of
the Rydberg computer. Similarly, Section 3.2 introduces the flux qubit, which
is then utilized to construct the model of the flux qubit computer. Section
3.3 details the coupling of both architectures, which establishes the physical
implementation of the $E_{2}$ gate. This system is then subsequently subjected
to numerical evaluation in Section 3.4, with the objective of gauging the
effectiveness of the gate.

###  3.1 The Rydberg Qubit

Rydberg atoms are atoms that are excited into a high energy level.
Specifically, Rydberg atoms possess a single valence electron in the highest
energy level. This phenomenon leads to interactions that can be exploited to
carry and propagate quantum information between atoms. Hence, Rydberg atom
quantum computers use Alkali atoms as mediators of quantum information [106,
107]. It is important to note that Sections 2.1, 3.2.1 and 3.3 detail three
distinct sets of Hamiltonians that will be used in the simulation of the phase
estimation.

Rydberg atoms were first discovered in 1885 by J.J Balmer, where he verified
that the energy levels $n$ are analogous to that of Hydrogen for $n<16$ [14,
20, 122]. The early 1970s saw the advent of tunable lasers and with it further
studies on Rydberg atom ensembles, which lead to the exploration of Rydberg
atoms in the context of external electric fields. The scaling laws governing
Rydberg atoms were revealed to be bizarre in nature as the dipole moment was
found to scale as $n^{2}$. Thus, for high $n$ Rydberg atoms at $n=30$ energy
level, the dipole moment was found to equate to $10^{3}$Å as seen derived in
Appendix D.1 [97, 6, 66, 3]. The Rydberg atom emerged as a compelling
candidate for qubit applications, since the dipole moments at this scale were
found to propagate and interact with other atoms at large distances ($\mu$m),
as well as the fact that Rydberg atoms could strongly couple to external
fields [20].

The basis of quantum information processes with trapped Rydberg atoms is to
encode quantum information in the internal states of single atoms with
interactions mediated via the highly excited Rydberg states [56, 79]. Using
laser trapping techniques and laser cooling, these atoms can be well isolated
and arranged in different geometries that can potentially exaggerate coupling
between atoms [85]. In this section, a detailed explanation will be provided
on how Rydberg interactions enable atoms to function as qubits, along with a
description of the architecture of the Rydberg part in the hybrid quantum
computer.

####  3.1.1 Rydberg Interactions

The main interaction at play when preparing or using Rydberg atoms for quantum
computing is the dipole-dipole interaction, which is observed when atoms
interact with each other. When an atom interacts with an external electric
field, it undergoes a phenomenon known as the Stark effect, resulting in the
splitting of its spectral lines. The interaction makes use of the
characteristic scaling, which is denoted as $n^{*}=n-\delta_{nlj}$ where
$\delta_{nlj}$ is the quantum defect dependent on the principal quantum number
$n$, the orbital angular momentum quantum number $l$ and the total angular
momentum quantum number $j$. For the rest of the investigation, the effective
principal quantum number $n^{*}$ and principal quantum number $n$ are
interchangeable when describing Rydberg atoms. The mean radius, $r$, of the
Rydberg atom scales as $\langle r\rangle\propto n^{2}a_{0}$ ($a_{0}$ being the
Bohr radius).

The single valence electron allows the Rydberg atom to be treated as a dipole
which couples strongly to external electric fields. A dipole $\mu$ in an
external electric field $E$ has an interaction potential,

$V_{dip}=\boldsymbol{\mu}\cdot\boldsymbol{E}=-\mu_{z}E,$ (3.1)

where the field is chosen to be in the $z$ direction [36]. The dipole moment
of an atom is given by $\boldsymbol{\mu}=-e\langle\boldsymbol{r}\rangle$ where
$\boldsymbol{r}$ is the radial distance of the electron from the atom and $e$
is the electron charge. For pure angular momentum states, the mean radial
distance $\langle r\rangle$ is always zero. Since the external electric field
perturbates the well defined angular momentum states,
$\langle\boldsymbol{r}\rangle\neq 0$. Hence the atomic states are coupled by a
transition dipole moment given by
$\mu_{1,2}=-e\bra{\psi_{1}}\boldsymbol{r}\ket{\psi_{2}}$ which is a second
order coupling consequently leading to the second order Stark effect [36,
120],

$\Delta E=-\frac{1}{2}\alpha E^{2},$ (3.2)

where $\alpha$ is the polarizability. The polarizability as well as the Stark
effect can be seen derived in sections D.2 and D.3 [24, 55, 108]. Allowing
$\boldsymbol{e}_{R}$ to be the unit vector along the relative coordinate
$\boldsymbol{R}$ between two atoms and both $\boldsymbol{r_{1}}$ and
$\boldsymbol{r_{2}}$ to be the relative displacement from the ion to the
Rydberg electron for an atom 1 and atom 2 respectively, the classical dipole-
dipole interaction between two Rydberg atoms becomes,

$V_{dd}=\frac{e^{2}}{4\pi\epsilon_{0}}\frac{\boldsymbol{r}_{1}\cdot\boldsymbol{r}_{2}-3(\boldsymbol{r}_{1}\cdot\boldsymbol{e}_{R})(\boldsymbol{r}_{2}\cdot\boldsymbol{e}_{R})}{R^{3}}.$
(3.3)

Using the fact that Rydberg dipole moments scale $|r|\propto n^{2}$, the
classical interaction strength can be deduced to scale as $V_{dd}\propto
n^{4}$. This however breaks down in the absence of an external electric field.
In this scenario, atoms are unpolarized consequently resulting in a vanishing
dipole moment due to the spatial symmetry of the wavefunction,
$\langle\boldsymbol{r}\rangle=0$ thus resulting in $\langle V_{dd}\rangle=0$
[133]. The spatial symmetry cannot vanish because $\boldsymbol{r}$ has non-
vanishing matrix elements between eigenstates with different parities [133].

A more rigorous quantum description is necessary to describe the interaction
between different Rydberg states in the absence of an external electric field.
In the spherical harmonic basis, the dipole-dipole operator can be expressed
as,

$V_{dd}=\frac{-e^{2}}{4\pi\epsilon_{0}R^{3}}\sqrt{\frac{24\pi}{5}}\sum_{\mu,\nu}C^{1,1,2}_{\mu,\nu,\mu+\nu}Y^{\mu+\nu}_{2}(\theta,\phi)^{*}d_{\mu}^{(1)}d_{\nu}^{(2)},$
(3.4)

where $C^{j_{1},j_{2},J}_{m_{1},m_{2},M}$ is the Clebsch-Gordan coefficient,
$Y^{m}_{l}(\theta,\phi)$ is the spherical harmonic function (seen in Appendix
D.2) and $d_{q}^{(i)}$ represents the spherical component ($q=\pm 1,0$) of the
dipole operator for the $i^{\mathrm{th}}$ atom [133]. Utilizing, Eq. (3.4) one
can derive the energy transfer between two atomic states, known as the Förster
Resonance Energy Transfer, which is explicitly derived in [129]. Knowing this,
it is now possible to consider a single transition channel
$r_{1}+r_{2}\rightarrow r_{1}^{^{\prime}}+r_{2}^{^{\prime}}$ in the absence of
an external electric field. This transition can be seen on Fig. 3.1(a)
depicted as red arrows.

The Förster Resonance Energy Transfer is a technique used in Rydberg atom
computing to transfer quantum information between Rydberg atoms. The energy
transfer is dependent on the potential of both atoms (or the distance between
them). This is simply characterized by an energy difference between the
initial and final state called the energy defect. The energy defect is defined
as $\Delta_{F}$ and can be seen derived in Appendix D.5 [104, 11]. For
simplicity, the dipole-dipole interaction energy will be rewritten to
$V_{dd}=C_{3}(\theta)/R^{3}$, where $C_{3}(\theta)$ is the anisotropic
interaction coefficient [142]. Thus, the Hamiltonian of the model system
dominated by a single channel in the subspace spanned by
$\\{\ket{r_{1}r_{2}},\ket{r_{1}^{^{\prime}}r_{2}^{^{\prime}}}\\}$ can be
written as [125],

$H=\begin{bmatrix}0&C_{3}(\theta)/R^{3}\\\
C_{3}(\theta)/R^{3}&\Delta_{F}\end{bmatrix}$ (3.5)

The eigenvalues for $H$ are given by,

$E_{\pm}=\frac{\Delta_{F}}{2}\pm\frac{1}{2}\sqrt{\Delta_{F}^{2}+4V^{2}},$
(3.6)

from which the induced energy shift is $\Delta
E_{\pm}=E_{\pm}(R)-E_{\pm}(R\rightarrow\infty)$. For $\Delta_{F}=0$, there is
strong coupling between the initial and final state since they are degenerate
leading to the eigenstates
$\ket{\pm}=(\ket{r_{1}r_{2}}\pm\ket{r_{1}^{^{\prime}}r_{2}^{^{\prime}}})/\sqrt{2}$
with eigenenergies proportional to $\pm C_{3}/R^{3}$. This is the so called
Föster resonance. When two atoms are sufficiently close and $\Delta_{F}\ll
V(R)$, the off-diagonal elements of $H$ are still dominant and $\Delta
E_{\pm}\approx\pm C_{3}/R^{3}$.

This concept can be seen in figure 3.2(a), whereby two dipole coupled Rydberg
atoms with states (a,c) and (b,d) have dipole moments $\mu_{ac}$ and
$\mu_{bd}$, respectively. The Förster defect in this scenario is equal to
$\Delta_{F}=E_{c}+E_{d}-(E_{a}+E_{b})$ that offsets the interaction energy
between both pair states, where the subscript $i$ in $E_{i}$ denotes the
energy state. When the electrostatic field is tuned to negate this Förster
defect (make it approximately equal to 0), the Rydberg-Rydberg interactions
scales with $1/R^{3}$ seen in Fig. 3.2(b) [85]. This means that the atomic
states assume a two dipole-coupled pair Rydberg state ($a=d,b=c$) and the pair
state can be treated as a two level system.

On the other hand, when $\Delta_{F}\gg V(R)$, the interaction can be treated
as a perturbation that just slightly shifts the energies of the bare states,
thus giving rise to $\Delta E_{-}=-C_{6}/R^{6}$ for $\ket{r_{1}r_{2}}$ and
$\Delta E_{+}=C_{6}/R^{6}$ for $\ket{r_{1}^{^{\prime}}r_{2}^{^{\prime}}}$ with
$C_{6}=C_{3}^{2}/\Delta_{F}$ [125, 133, 142]. As can be seen, the interaction
in this regime is proportional to $R^{-6}$, conventionally called the van der
Waals interaction. The crossover of the energy shift from $R^{-3}$ to $R^{-6}$
is described by a characteristic radius $R_{vdW}=|C_{6}/\Delta_{F}|^{1/6}$
where $V(R_{vdW})=\Delta_{F}$.

This concept can be seen in 3.2(c), where the interaction is non-resonant if
$|\Delta_{F}|>|\mu_{ac}\mu_{bd}|/R^{3}$. Since the dipole interaction is
treated using second-order perturbation theory, the $\ket{ab}$ state has a
potential proportional to $C_{6}=\frac{\mu_{ac}^{2}\mu_{bd}^{2}}{\Delta_{F}}$
[85]. This concept is utilized in the next subsection to make multi-qubit
gates between Rydberg atoms.

Figure 3.1: (a) Energy levels with corresponding transition channels
contributing to the dipole-dipole interaction. (b) Rydberg blockade for two
atoms at distance R. [133] Figure 3.2:  Rydberg-Rydberg interactions used to
construct quantum gates. (a) Each atom possesses a pair dipole coupled states
which have transition dipole moments $\mu_{ac}$, $\mu_{bd}$. (b) For
$\Delta_{F}\approx 0$ the Rydberg-Rydberg interactions have a dipolar exchange
$\ket{ab}\bra{cd}$ with interaction strength $1/R^{3}$. (c) When
$|\Delta_{F}|>|\mu_{ac}\mu_{bd}|/R^{3}$ the interactions are Van de Waals form
$\ket{ab}\bra{cd}$ with interaction strength $1/R^{6}$ [85].

####  3.1.2 Rydberg Blockade and Transitions for Information Processing

As seen in the previous section, the van der Waals interaction scales as
$V\propto 1/R^{6}$ which means that when the interatomic distance between two
atoms is large enough, this interaction does not occur. However, at close
distances, this interaction governs the system’s Hamiltonian. The goal for
this specific quantum system is to have minimal interaction between atoms when
performing local gates (i.e., little cross-talk) and have large interactions
between atoms when performing multi-qubit gates meaning that this phenomenon
is highly desired. To fully grasp the phenomena, it is important to assume a
two atom system where by each atom possesses two distinct energy levels, a
$\ket{g}$ ground state and an excited state $\ket{r}$.

The simplest case of the Rydberg blockade can be observed in figure 3.1(b)
where the blockade effect is shown for two atoms. It is assumed that the
system starts with the population of both atoms at the ground state, or
written as a single state, $\ket{gg}$. A resonant laser excitation with
frequency $\Omega\sqrt{2}$ (where $\Omega$ is the Rabi frequency) is applied
to the system which drives the state to a symmetric single-atom excited state
$(\ket{gr}+\ket{rg})/\sqrt{2}$. If the atoms are at a distance $R>R_{b}$ then
another resonant pulse will excite the collective state to the double excited
state, $\ket{rr}$. On the other hand, if $R<R_{b}$ holds, the applied laser
pulse will not drive the state to the double excited state since the energy
required is much larger. This means that the double excited state is
essentially blocked.

This follows the Ising configuration of having Rydberg states adhere to a
ground state and a Rydberg excited state [69, 116]. This configuration is
governed by a transition Hamiltonian of the form,

$\mathcal{H}(t)=\sum_{i}\bigg{(}H_{i}^{D}(t)+\sum_{j<i}\frac{C_{6}}{R_{ij}^{6}}n_{i}n_{j}\bigg{)}$
(3.7)

where $n_{i}$ denotes the projector $\ket{r}\bra{r}_{i}$ on the
$i^{\mathrm{th}}$ atom. $H^{D}$ is the driving Hamiltonian between two energy
levels. The driving Hamiltonian is defined as,

$H^{D}(t)=\frac{\hbar}{2}\boldsymbol{\Omega}(t)\cdot\boldsymbol{\sigma},$
(3.8)

where $\boldsymbol{\sigma}=(\sigma^{x},\sigma^{y},\sigma^{z})^{T}$ is the
Pauli vector and
$\Omega(t)=(\Omega(t)\mathrm{cos}(\varphi),-\Omega(t)\mathrm{sin}(\theta),-\delta(t))^{T}$
is the rotation vector in which $\delta(t)$ is the detuning and both $\varphi$
and $\theta$ are the azimuthal and polar angle on the Bloch sphere (figure
2.1), respectively. It is assumed that the laser targeting the transition is
tuned at a resonance frequency $\omega_{ab}=|E_{a}-E_{b}|/\hbar$ with detuning
$\delta(t)=\omega(t)-\omega_{ab}$ where $a$ and $b$ denote the different
energy states. The Pauli vector is defined explicitly as [116],

$\sigma^{x}=\ket{a}\bra{b}+\ket{b}\bra{a},$ (3.9)
$\sigma^{y}=i\ket{a}\bra{b}-i\ket{b}\bra{a},$ (3.10)
$\sigma^{z}=\ket{b}\bra{b}-\ket{a}\bra{a}.$ (3.11)

As can be seen from Eq. (3.7), the Hamiltonian of the system when $R>R_{b}$,
the Hamiltonian can simply be described by the driving Hamiltonian, however,
under the Rydberg blockade effect, the Hamiltonian has an added term. At
$R>R_{b}$, pulsing, evolving the driving Hamiltonian through means of a laser,
a single atom or qubit consequently changes the state of the qubit without
influencing the other qubits around it. This is said to be the digital
approach and is often used to pulse single quantum gates [116]. The digital
approach is often encoded on two hyperfine levels of an atom. These are
referred to as $\ket{g}=\ket{0}$ and $\ket{h}=\ket{1}$. To evolve the driving
Hamiltonian between these states, one needs to define a pulse dictated by the
operator,

$U(\boldsymbol{\Omega},\tau)=T\mathrm{exp}\Big{[}-\frac{i}{2}\int_{0}^{\tau}\boldsymbol{\Omega}(t)\cdot\boldsymbol{\sigma}\mathrm{d}t\Big{]},$
(3.12)

where $T$ is the time-ordering operator and $\tau$ is the pulse time. The
unitary $U$ describes the rotation around the axis $\boldsymbol{\Omega}$. It
is possible to induce rotations around the fixed axis
$\boldsymbol{e}(\varphi)=(\mathrm{cos}\varphi,\mathrm{sin}\varphi,0)$ situated
on the equator of the Bloch sphere, by pulsing a resonant pulse of phase
$\varphi$ ($\delta=0$), there is a rotation angle of [116],

$\theta=\int^{\tau}_{0}\Omega(t)\mathrm{d}t.$ (3.13)

situated on the equator of the Bloch sphere. The unitary operator is thus,

$R_{\boldsymbol{e}(\varphi)}(\theta)=e^{-i\frac{\theta}{2}(\mathrm{cos}(\varphi)\sigma_{x}-\mathrm{sin}(\varphi)\sigma_{y})}=e^{i\frac{\varphi}{2}\sigma_{z}}e^{i\frac{\theta}{2}\sigma_{x}}e^{-i\frac{\varphi}{2}\sigma_{z}}=R_{z}(-\varphi)R_{x}(\theta)R_{z}(\varphi).$
(3.14)

Adding an additional $z$-rotation, which can be achieved virtually through a
shift in the phase reference frame (more detailed overview in [82]), allows
the construction of an arbitrary single-qubit gates,

$U(\theta,\varphi,\gamma)=R_{z}(\gamma+\varphi)R_{\boldsymbol{e}(\varphi)}(\theta)=R_{z}(\gamma)R_{x}(\theta)R_{z}(\varphi).$
(3.15)

as seen in Eq. (2.6). To build a set of universal gates, one requires multi-
qubit gates. Multi-qubit gates use the manipulation of unitaries provided in
equation (3.15) and the Rydberg blockade effect. The native multi-qubit gate
in Rydberg computing is the CZ gate, which is seen in Appendix A. This gate
can then be expanded to create other multi-qubit gates. The CZ Rydberg gate
turns the initial states into the final states seen in table 3.1.

Table 3.1: Initial and final states of the CZ pulse sequence in the computational basis. Initial state | $\ket{00}$ | $\ket{01}$ | $\ket{10}$ | $\ket{11}$
---|---|---|---|---
Final state | $-\ket{00}$ | $-\ket{01}$ | $-\ket{10}$ | $\ket{11}$

To understand how the two qubit CZ gate is pulsed, it is important to take the
computational basis ($\ket{0}$ and $\ket{1}$) that was encoded to the
hyperfine states and add an auxiliary Rydberg state $\ket{r}$. Two qubits
(target and control) are first physically moved to a distance $R_{b}$ from
each other (seen in figure 3.1) where the van der Waals interaction is
observed. Then, a $\pi$ pulse attempts to bring the control qubit in an
initial $\ket{0}$ to the auxiliary Rydberg state. If the qubit is in the
$\ket{1}$ state, it will not be able to transition to the auxiliary state. A
second $2\pi$ pulse is then used to attempt to transition the target qubit to
the auxiliary state and back from the $\ket{0}$ state. This transition will be
successful if the target is initially in the $\ket{0}$ state and if the
control failed to transition to the auxiliary state. This is because only a
single qubit is allowed to be in the auxiliary state as seen in figure 3.1.
Finally, a third pulse is used to attempt to transition the control qubit back
to the $\ket{0}$ state [116]. Applying a global $-1$ factor or a global phase
of $\pi$ to the final states in table 3.1, leads to the CZ gate.

Figure 3.3:  CZ gate implementation where three pulses are consecutively
executed on the qubit. Each panel depicts the different scenarios depending on
the initial state. The step number within each panel are shown in blue
circles. The bold red lines indicate that the transition was successful while
the dashed red lines indicate that the transition was unsuccessful [116].

####  3.1.3 Rydberg Atom System

As described in section 2.3, a distributed quantum algorithm requires the use
of an E2 gate. The E2 gate is required to establish quantum communication
between two quantum computers, in this case, two different quantum
architectures. This subsection will investigate the Rydberg ensemble and how
it pairs to the physical E2 gate. A more detailed description of the physical
E2 implementation will be investigated in section 3.3.

The Rydberg ensemble requires three energy levels, the ground state $\ket{g}$
(encoded to the computational basis state $\ket{0}$), the hyperfine state
$\ket{h}$ (encoded to the computational basis state $\ket{1}$) and the
auxiliary excited state $\ket{r}$. For Rubidium-87, the hyperfine levels are
chosen to be $\ket{5S_{1/2},F=1}=\ket{g}$ and $\ket{5S_{1/2},F=2}=\ket{h}$
[75, 133]. The auxiliary $\ket{r}$ state is encoded to be in the
$\ket{70S_{1/2}}$ state, which will be used to initialize the Rydberg
blockade. The transition from the $\ket{0}$ to the auxiliary Rydberg state
$\ket{r}$ requires frequencies that are optically difficult to achieve (sub
300nm wavelengths). Thus in practice, an auxiliary $\ket{6P_{3/2}}$ state is
used to aid the transfer. The hyperfine levels transition with a frequency of
6.8 GHz while the transition from the $\ket{0}$ state to the $\ket{6P_{3/2}}$
state is optically achievable using a 420nm wavelength laser. Finally, the
transition from the auxiliary state to the Rydberg auxiliary $\ket{r}$ state
is achievable using a 1013nm tuned laser. A schematic detailing the energy
levels in the computational basis can be seen in figure 3.4a.

Figure 3.4: (a) Schematic detailing the Rydberg energy states paired to the
respective computational states of the proposed Rydberg computer atom array
with the energy states of the communication qubit depicted in the red dotted
square. The atoms in the atom array contain two computational levels in the
hyperfine $\ket{5S_{1/2},F=1,2}$ states, the Rydberg auxiliary state in the
$\ket{70S_{1/2}}$ state and an extra auxiliary $6P_{3/2}$ to facilitate the
transfer to the $\ket{70S_{1/2}}$ state. The communication qubit in the red
dotted box shows the two energy levels used in the work done by Yu et al.
whereby two energy states, $\ket{22D_{5/2}}$ and an arbitrary superposition
state $\ket{\Phi_{n=20}}=\ket{n=20,l\geq 3,j,m=5/2}$, are used to denote the
computational $\ket{1}$ and $\ket{0}$ states for the $E_{2}$ gate respectively
[141]. (b) Schematic of the system whereby an Rubidium-87 atom is held between
two spherical capacitors (with capacitance $C$) connected to an LC resonator
with inductance $L$. This atomic states are manipulated by an electrostatic
field $E$ in the $z$-direction using parallel plates with imposed voltage
difference $U$. The atom forms part of Rydberg quantum computer shown as an
atom array (red dots in the purple block).

The apparatus that connects the Rydberg atom system to the flux system follows
the work of Yu et al.. The study denotes a system in which a Rubidium-87 atom
is placed between two spherical capacitors connected to an LC resonator in
which an electric field runs in the $z$-direction This system is further
elaborated upon in Section 3.3. Two states are used in the study, a state
$22D_{5/2}$ populated when the Rydberg qubit is in the computational $\ket{1}$
state and an arbitrary $\ket{n=20,l\geq 3,j,m=5/2}$ superposition state is
populated when the Rydberg qubit is in the $\ket{0}$ computational basis. The
transition frequency between the two states is approximately 3.2GHz. This
research extends Yu et al. by connecting the Rubidium-87 atom in the system to
an additional array of Rubidium-87 atoms.

Transitioning between the atom array to the atom between the capacitor is
necessary to bridge both systems. This means that the $\ket{22D_{5/2}}$ state
needs to transition to the computational $\ket{1}$ state in the hyperfine
$\ket{5S_{1/2},F=2}$. Similarly, the $\ket{n=20,l\geq 3,j,m=5/2}$ (denoted as
$\Phi_{n=20}$) state needs to transition to the computational
$\ket{0}=\ket{5S_{1/2},F=1}$ ground state. The optical wavelength to achieve
these transitions is approximately 300nm, which equates to a THz frequency.
Although, this is theoretically possible, further work should aim to
experimentally analyze the transition frequencies between the hyperfine levels
of the $\ket{5S_{1/2}}$ state and the $\ket{22D_{5/2},\Phi_{n=20}}$ levels of
the communication qubit as well as the effectiveness of the transition.

For the system to work optimally, it is important that the atoms in the array
are far enough apart from each other when processing single qubit operations.
This will limit the amount of unwanted interactions from surfacing that may
influence the decoherence in the energy states. This means avoiding the
dipolar exchange seen in Fig. 3.2b by avoiding a resonant Förster defect
$\Delta_{F}\approx 0$. The potential energy of the interaction was seen to be
$V_{dd}=C_{3}/R^{3}$. Thus, it is important to maximize the interatomic
distance $R$ when performing single qubit operations.

However, for multi-qubit gates, it is important that the atoms are close
enough to experience the exchange interaction. In order to implement the
universal CZ gate depicted in Fig. 3.3, the Rydberg blockade must be in
effect. To do so, the Förster defect needs to be non-resonant,
$\Delta_{F}>|\mu^{2}|/R^{3}|$ where $\mu$ is the transition dipole moment of
the transition $\ket{5S_{1}/2}\rightarrow\ket{70S_{1}/2}$. The energy shift
can thus be calculated using $\Delta E_{\pm}=\pm C_{6}/R^{6}$.

Using ARC [115], the characteristic radius, $R_{vdW}$, was calculated to be
$3.5\mu$m which can be seen in Fig. F.1 and Fig. F.2. The dipole-dipole
coefficient was calculated to be $C_{3}=32.45$ GHz $\mu m^{3}$ and the van der
Waals coefficient was calculated to be $C_{6}=801.98$ GHz $\mu m^{6}$ for the
pair state. Therefore, when executing single qubit gates, it is important to
observe an interatomic distance $R\gtrsim 9\mu$m to avoid substantial dipole-
dipole interactions. On the other hand, when processing multi-qubit gates, the
interatomic distance is chosen to be $R=3.5\mu$m which observes an energy
shift $\Delta E_{\pm}=801.98$ GHz capable of achieving the Rydberg blockade
effect.

###  3.2 The Flux Qubit

####  3.2.1 Introduction to Flux Qubits

The Josephson persistent-current qubit or the flux qubit is a type of
superconducting qubit which consist of an inductance loop shorted by a
Josephson junction [94]. Josephson junctions are made up of two
superconductors sandwiching a thin non-superconducting layer such that
electrons can tunnel through the barrier [88]. This produces what is known as
the Josephson effect which is a current that flows continuously across the
Josephson junction without supplied current. At temperatures close to zero
kelvin, this circuit experiences superconductivity, and in turn, forces the
electrons in the circuit to the lowest energy state. In order to not violate
Pauli’s principle, electrons of opposite spins form a single particle called a
Cooper pair, which exhibits the same properties of a boson. The flux qubit’s
energy states are dependent on the magnetic flux measured in the inductance
loop, which is dependent on the amount of Cooper pairs crossing the Josephson
junction. The wavefunction of the Cooper pair system as well as the magnetic
energy states are described in Appendix E.1.

The energetics of this system can be described by the following Hamiltonian,

$H=\frac{Q^{2}_{J}}{2C_{J}}-E_{J}\mathrm{cos}\Big{(}\frac{2\pi\Phi}{\Phi_{0}}\Big{)}+\frac{1}{2}LI_{q}^{2},$
(3.16)

where $Q_{J}$ is the electric charge on the capacitor, $C_{J}$ is the self
capacitance, $I_{q}$ is the current circulating in the loop, $L$ is the
inductance of the inductor, $\Phi$ is the applied magnetic field, $E_{J}$ is
the energy from the Josephson junction and $\Phi_{0}$ is the magnetic flux
quantum. The Hamiltonian follows that of the quantum LC Hamiltonian seen in
Appendix E.2. Following Eq. (3.16), it is possible to see minimas form at
$\Phi=\pm\Phi_{0}/2$. The two points are called the ’degeneracy points’, in
which the self-inductance of the Josephson junction is approximately equal to
the inductance of the inductive loop $L_{J}\approx L$ [12]. These two values
are associated with equal and opposite flux states formed by a persistent
current $I_{q}$ in the loop. Because the magnetic field is seen to rotate
clockwise or anti-clockwise around the loop the states are labelled left
circle or $\ket{L}$ at $+\Phi_{0}/2$ and right circle or $\ket{R}$ at
$-\Phi_{0}/2$ [12, 94].

The two minimas can be described by analyzing the dependence of $\Phi$ on the
potential energy of the system $U$. This relationship is seen in figure 3.5
where the strength of the interaction between both states is dependent on the
height of the barrier separating them. Both $\ket{R}$ and $\ket{L}$ are
classically isolated from each other when the thermal energy is lower than the
barrier height, however, due to quantum tunneling, there can be interaction
between both states. Unfortunately, this is very difficult to do with a single
junction since the energy splitting (or barrier height) scales exponentially
with $L/L_{J}$. However, to form a double well potential from a single
Josephson junction, $L\approx L_{J}$. The trade-off between the two make it
difficult to form a decent qubit.

Figure 3.5: The potential energy
$U=\frac{1}{2}LI_{q}^{2}-E_{J}\mathrm{cos}(\frac{2\pi\Phi}{\Phi_{0}})$ plotted
against $\Phi/\Phi_{0}$ [44].

The flux qubit was improved by replacing the single Josephson junction with
three Josephson junctions. This is called the 3JJ persistent current qubit
[94]. The purpose of the two junctions is to replace the loop inductance $L$
with the Josephson inductance $L_{J}$ of the two extra junctions seen in
figure 3.6(a). The ratio $L/L_{J}$ is now,

$\alpha=\frac{2I_{3}}{I_{1}+I_{2}},$ (3.17)

where $I_{i}$ is the critical currents of each junction, which follow the
current-phase relation $I_{i}=I_{0}\mathrm{sin}\varphi_{i}$, where
$\varphi_{i}$ are the gauge-invariant phase of each junction $i$ [93]. In this
architecture, the ratio of junction sizes is important as opposed to the
inductance [12].

With the new 3JJ formulation, it is important to derive the new system
dynamics. Figure 3.6(a) shows an additional variable $f_{\varepsilon}$, called
the magnetic frustration, which is simply the magnetic flux measured in units
of the flux quantum $\Phi_{0}$. Two additional parameters of relevance are,
$E_{J}=\Phi_{0}I_{c}/2\pi$ and the charging energy $E_{c}=(2e)^{2}/2C$ [94].
The energy ratio $E_{J}/E_{c}$, determines whether the eigenstates are defined
by states of definite charge or flux (or a combination of the two). This ratio
also determines whether qubits are more susceptible to charge or flux noise.

Figure 3.6: (a) Flux qubit with detailing the three Josephson junctions each
denoted by a cross. Each Josephson junction possesses a capacitance $C_{i}$
that is related to the gauge invariant phase difference of each junction. (b)
The ground state $\ket{0}$ and the excited energy state $\ket{1}$ of the qubit
versus the frustration near $f_{\varepsilon}$ = 0.5. As $\alpha$ becomes
smaller, $\Delta$ increases while the persistant current $I_{p}$ increases
[94].

Following from [93], the fluxoid quantization around the loop containing the
junction is strictly $\varphi_{1}-\varphi_{2}+\varphi_{3}=-2\pi
f_{\varepsilon}$. Using the derived parameters, the total Josephson energy $U$
is $U=\sum_{i}E_{Ji}(1-\mathrm{cos}\varphi_{i})$ where each junction has
energy $E_{Ji}(1-\mathrm{cos}\varphi_{i})$. Combining this with the flux
quantization term, the total Josephson energy is,

$\frac{U}{E_{J}}=2+\alpha-\mathrm{cos}\varphi_{1}-\mathrm{cos}\varphi_{2}-\alpha\mathrm{cos}(2\pi
f_{\varepsilon}+\varphi_{1}-\varphi_{2})$ (3.18)

Thus the Josephson energy is dependent on the two phases $\varphi_{1}$ and
$\varphi_{2}$. These can be modified for a range of magnetic frustration
$f_{\varepsilon}$. There exists two stable solutions around
$f_{\varepsilon}=1/2$, defined by a region $[1/2-f_{c},1/2+f_{c}]$, where
$f_{c}$ is derived in Appendix E.3. The solutions have circulating currents of
opposite direction and are degenerate at $f_{\varepsilon}=1/2$.

It is now crucial to observe the relationship between $f_{\varepsilon}$ and
$\alpha$ in order to make suitable qubit eigenstates. At $\alpha\leq 1/2$, $U$
has a single minimum when $\varphi_{1}=\varphi_{2}=0$ however, above the
critical $\alpha=1/2$ value, the minimum bifurcates into two degenerate
minima, $\pm\varphi^{*}=\varphi_{1}=-\varphi_{2}$ [93, 94]. Displayed on Fig.
3.7(a) the contour of the Josephson energy can be seen as a function of the
phase variables for $\alpha=0.5$. The top graph shows the pattern when
$f_{\varepsilon}=1/2$ (depicted in the figure as $f$) whereby the sponge like
pattern shows the stable energy potential minima while the circular patterns
show the maxima. When the magnetic frustration diverges from
$f_{\varepsilon}=0.5$ point by a $\delta$, the minimum between both phases
becomes biased. The bifurcated minima is seen in the top figure in Fig. 3.7(b)
for different values of $\alpha$ (for $\alpha\geq 1/2$) while the bottom
figure shows the bifurcation effect of a small perturbation ($\delta=0.2$) in
the magnetic frustration. The minimas no longer become degenerate as denoted
by the bottom figure in Fig. 3.7(a), the potential energy deepens on the right
side of the sponge.

Tunneling through the barrier that separates the two potential minima provides
a coupling $\Delta/2$ between the two persistent current states. The tunnel
coupling consequently creates an anti-crossing $\Delta$ between the classical
energy levels as seen in Fig. 3.6(b) due to the eigenstates of the Hamiltonian
at $f_{\varepsilon}=1/2$ becoming symmetric and anti-symmetric superpositions
of the persistent current states [94]. To conclude, it has been observed that
tuning $\alpha$ is important for varying the barrier height and
$f_{\varepsilon}$ is used to control the symmetry between both states when
designing a flux qubit.

Having built upon the fundamental understanding of how $\alpha$ and the
magnetic frustration changes the energy dynamics of the qubit, it is now
important to derive the two-level Hamiltonian in order to model the system
dynamics. In the two-level approximation the qubit Hamiltonian is described in
terms of the magnetic energy $E_{LR}$ and the tunnel coupling $\Delta/2$ that
creates an anti-crossing [94]. In the persistent-current basis
($\ket{L},\ket{R}$) the qubit Hamiltonian is,

$H=-\frac{1}{2}(\varepsilon\sigma_{z}+\Delta\sigma_{x}),$ (3.19)

where $\varepsilon=2E_{R}=2I_{p}(f_{\varepsilon}-1/2)\Phi_{0}$, $I_{p}$ is the
persistent current, and both $\sigma_{x}$ and $\sigma_{z}$ are the Pauli
matrices [94]. The corresponding eigenenergies of the diagonalized Hamiltonian
are $E_{0},E_{1}=\mp\frac{1}{2}\sqrt{\varepsilon^{2}+\Delta^{2}}$ where the
energy level splitting in the computational basis, $\\{0,1\\}$, is
$E_{01}=E_{1}-E_{0}=\sqrt{\varepsilon^{2}+\Delta^{2}}$.

Figure 3.7: (a) Top: Phase diagram of $\varphi_{1}$ and $\varphi_{2}$ with
magnetic frustration $f=0.50$ and $\alpha=0.5$. The circular shapes are the
maxima of the potential while the sponge like contours enclose two minima.
Bottom: At a magnetic frustration $f=0.52$, the sponge like contour lines
becomes tilted wherein the right side potential is deeper. (b) Top: $U/E_{J}$
plotted against $\varphi^{*}$ where $\pm\varphi^{*}=\varphi_{1}=-\varphi_{2}$
for $f=0.50$. Bottom: $f=0.52$.

Dynamically modifying the two level flux qubit system relies on using external
magnetic fields. Revising the Bloch sphere (Fig. 2.1), the external field is
aligned along the $z$-axis. The angle $\theta$ for a flux qubit is determined
by $\theta=\mathrm{arctan}(\Delta/\varepsilon)$. The natural precession of the
qubit around the $z$-axis depends on the Larmor frequency $\omega_{L}=E/\hbar$
where $E$ is the qubit energy [94]. In order to induce transitions, an
oscillating field matching the Larmor frequency has to be applied in the
$xy$-plane. To do so, microwaves are usually applied in the lab by means of a
microwave frequency magnetic flux in the $z$-direction. This is of the form,
$\delta\Phi_{LR}=|\Phi_{LR}|\mathrm{sin}(\omega_{LR}t+\varphi_{LR})$, where
$t$ is the time and $\omega_{LR}$ is the applied microwave frequency between
the two flux states [94]. This modulates the energy bias of the qubit with
$\delta\varepsilon_{LR}=2I_{p}\delta\Phi_{LR}$. The Hamiltonian describing the
two-level flux qubit is thus,

$H^{(01)}_{LR,\varepsilon}=-\frac{\delta\varepsilon_{LR}}{2}\frac{\Delta}{\sqrt{\varepsilon^{2}+\Delta^{2}}}\begin{pmatrix}\varepsilon/\Delta&1\\\
1&-\varepsilon/\Delta\end{pmatrix},$ (3.20)

where the off-diagonal terms determine the strength of driving between the two
qubit states [94].

This section has introduced the 3JJ qubit by analyzing the impact of $\alpha$
and $f_{\varepsilon}$, describing the difference between the flux states
$\\{L,R\\}$ and the computational states $\\{0,1\\}$. The Hamiltonian seen in
Eq. (3.19) with the two level system described in Eq. (3.20) will be used in
the next section to construct the flux qubit architecture.

####  3.2.2 Flux Qubit System

Figure 3.8: (a) Representation of Flux qubits (each qubit represented by
$Q_{i}$) on a coplanar waveguide (in blue). (b) A contour plot of the
potential energy $U/E_{J}$ at $f_{\varepsilon}=0.53$ for $\alpha=0.8$. The
measured potential well is seen highlighted with the purple line. (c) A more
specified outline of the C-shunt 3JJ flux qubit (circuit on the left) mutually
coupled to a resonator, or in this case, the co-planar wave-guide (circuit on
the right). (d) The potential function plotted against
$\phi^{*}=\phi_{1}=-\phi/2$ at $\alpha=0.8$ and $f_{\varepsilon}=0.53$ for the
potential well seen in (c). The asymmetric bias gives rise to the $\ket{L}$
and $\ket{R}$ states.

Section 3.2 introduced the flux qubit mainly in the context of a single qubit.
Flux qubits coupled together via transmission lines or resonators tend to
possess an additional capacitor making them C-shunt flux qubits [12, 136].
This is due to the reduced noise susceptibility that is obtained when
introducing an extra capacitor. The goal of this section will be to introduce
the C-shunt qubit Hamiltonian, which follows that of Section 4.2 and derive
the quasistatics of the system, which will be useful in deriving the noise
fluctuations in Chapter 4. Figure 3.8(c) shows a circuit overview of the 3JJ
C-shunt flux qubit coupled via a resonator to the CPW. This system of
interacting flux qubits will be coupled together through an inductive coupling
to the co-planar wave-guide (CPW). Figure 3.8(a) shows the flux qubits
geometrically on a CPW.

The two-level-system Hamiltonian for an individual C-shunt flux qubit near
flux-degeneracy and coupled to a CPW resonator is,

$H=\frac{\hbar}{2}[\varepsilon\sigma_{z}+\Delta\sigma_{x}]+\hbar\omega(\hat{a}^{\dagger}\hat{a}+\frac{1}{2})+\hbar
g\sigma_{y}(\hat{a}^{\dagger}+\hat{a}),$ (3.21)

where the first part of the equation follows Eq. 3.19 and
$\hat{a}^{\dagger}(\hat{a})$ is the raising (lowering) operator for photons.
This Hamiltonian can be intuitively broken down into three different terms.
The three terms are respectively, the qubit, resonator and qubit-resonator
Hamiltonians [135, 136]. The coupling strength between the resonator and qubit
are dependent on $M$, $L_{r}$, $C_{r}$ and $L_{q}$ as seen in the full
derivation, treated in Appendix E.4 [67, 50].

The goal now is to derive the quasistatic Hamiltonian of the C-shunted flux
qubit in order to understand the benefits of the C-shunt design and to then
derive the noise in Section 4.2. The C-shunt flux qubit has three features
that differentiate it from the normal 3JJ flux qubit. It has a lower critical
current, $I_{c}$, typically $\alpha<0.5$ and the additional shunting junction
has a capacitance $C_{sh}=\zeta C$, where $C$ is the capacitance of the
smallest junction and $\zeta\gg 1$. As seen from Appendix E.2, the 3JJ
capacitively-shunted flux qubit can be described by a Hamiltonian with a
kinetic and potential part,

$\displaystyle H$ $\displaystyle=T+U,$ (3.22) $\displaystyle T$
$\displaystyle=\frac{1}{2}(\boldsymbol{Q}+\boldsymbol{q})^{\intercal}\mathbf{C}^{-1}(\boldsymbol{Q}+\boldsymbol{q}),$
(3.23) $\displaystyle U$
$\displaystyle=E_{J}\\{2+\alpha-\mathrm{cos}\varphi_{1}-\mathrm{cos}\varphi_{2}-\alpha\mathrm{cos}(2\pi
f_{\varepsilon}+\varphi_{1}-\varphi_{2})\\}.$ (3.24)

The charges $\boldsymbol{Q}$ and $\boldsymbol{q}$ are the charges induced
charges on the islands, where the islands refer to the two nodes $A$ and $B$
in figure 3.8(c) [136]. The matrices can be rewritten to,

$\boldsymbol{Q}=-2e\begin{bmatrix}\frac{\partial}{\partial\varphi_{1}}\\\
\frac{\partial}{\partial\varphi_{2}}\end{bmatrix},\quad\boldsymbol{q}=\begin{bmatrix}q_{A}\\\
q_{B}\end{bmatrix},\quad\boldsymbol{C}=C\begin{bmatrix}\zeta+1+\alpha&-(\zeta+\alpha)\\\
-(\zeta+\alpha)&\zeta+1+\alpha\end{bmatrix}.$ (3.25)

The system is less sensitive to charge fluctuations due to the shunt capacitor
reducing the effective charging energy. Choosing
$\varphi_{+}=(\varphi_{1}+\varphi_{2})/2$ and
$\varphi_{-}=(\varphi_{1}-\varphi_{2})/2$ using the Cooper-pair number
operators $\hat{n}_{\sigma}=-i\partial/\partial\varphi_{i}$ where
$i=\\{-,+\\}$, the reduced Hamiltonian becomes,

$H=\frac{1}{2}E_{C,+}\hat{n}^{2}_{+}+\frac{1}{2}E_{C,-}\hat{n}^{2}_{-}+E_{J}\\{2+\alpha-2\mathrm{cos}\varphi_{-}\mathrm{cos}\varphi_{+}-\alpha\mathrm{cos}(2\pi
f_{\varepsilon}+2\alpha_{-})\\},$ (3.26)

where $E_{C,+}=2e^{2}/C$ and $E_{C,-}=(\zeta+\alpha+1)e^{2}/C$ are the
effective charging energy for the $+$-mode and $-$-mode respectively. Ideally,
the $+$-mode can be omitted since $\zeta\gg 1$. Thus, the simplified
Hamiltonian is [136],

$H_{m}=\frac{1}{2}E_{C,-}\hat{n}^{2}_{-}+E_{J}\\{-2\mathrm{cos}\varphi_{-}-\alpha\mathrm{cos}(2\pi
f_{\varepsilon}+2\alpha_{-})\\}.$ (3.27)

This will be used in Section 4.2 to derive the flux and charge noise.

To realize the full potential of a C-shunt flux qubit, one would need to
choose an $\alpha<0.5$ because it reduces the circuits sensitivity to charge
noise and stray fields [12, 53]. However, in this regime, the qubit resembles
a phase qubit where the potential well becomes singular and thus loses
anharmonicity between two energy states. In order to realize the hybrid
$E_{2}$ gate, described in Section 3.3, each flux qubit requires two potential
wells with states $\ket{L}$ and $\ket{R}$ in each [134]. Additionally, the
potential wells must have differing depths as seen on figure 3.8(b) and
3.8(d). Maintaining the anharmonicity between the two levels allows there to
be a very strong and a very weak coupling to a resonator depending on the
energy level. It is important that the flux qubit system only couples to the
Rydberg system when the resonator is in the $\ket{1}$ state, or when the
system needs to communicate. Hence, there is a trade-off between noise and the
coupling to the hybrid system [134]. Thus, the chosen values for the flux
system are $\alpha=0.8$ and $f_{\varepsilon}=0.53$ as seen in figures 3.8(b)
and 3.8(d).

###  3.3 Hybrid Quantum Computer Architecture

Coupling an atomic qubit to a superconducting qubit as of yet remains a
theoretical area of study. The current model relies on trapping atoms on a
superconducting atom chip [60, 86, 126] or placing the atoms close to a
superconducting chip and using a mediator to couple both the superconducting
qubits and Rydberg atoms [139, 140, 141]. This mediator is often chosen to be
a co-planar wave-guide (CPW) or an LC resonator due to their simplicity as
well as their ability to couple many different frequencies [134].

This study will mainly focus on a simplistic model whereby a chain of flux
qubits are connected to a LC resonator on one side while a chain of Rydberg
atoms are connected to the same LC resonator on the other side inspired by
[139]. The system dynamics for the atom-resonator-flux qubit, albeit similar
to the Hamiltonians described in Sections 3.1.3 and 3.2, will use a much
simpler model to describe the system dynamics. This is due to the fact that
the time scale used to realize the $E_{2}$ gate is much smaller than many of
the modelled effects in previous Hamiltonians [141]. This system has been
extensively studied in [141] and will be used in this research to establish
the $E_{2}$ gate.

A diagram of the model can be seen in Fig. 3.9. The mediator of the model is a
superconducting LC resonator operating at mK temperatures. The characteristic
frequency of such resonator is $\omega_{0}=1/\sqrt{LC}$ which is chosen to be
$2\pi\times 20$ GHz. The capacitor composed of spherical identical spheres has
a capacitance $C$ and an inductance $L$. The remaining numerical values as
well as their units can be seen in Table F.1. The Hamiltonian for the
resonator,

$H_{LC}=\frac{\phi^{2}}{2L}+\frac{q^{2}}{2C}=\hbar\omega_{0}(b^{\dagger}b+1/2),$
(3.28)

where $b^{\dagger}$ and $b$ are the creation and annihilation operators, the
magnetic flux $\phi=\sqrt{\hbar/2C\omega_{0}}(b^{\dagger}+b)$ and
$q=i\sqrt{C\hbar\omega_{0}/2}(b^{\dagger}-b)$ is the charge operator [140,
141]. The Rydberg atom is placed in the midpoint between the two spherical
capacitors which couples to the local electric field in the $z$-direction
resulting in the atom-resonator interaction operator [139, 141],

$V_{a}=-iD\varepsilon(b^{\dagger}-b).$ (3.29)

$D$ is the atomic dipole moment and $\varepsilon$ is the amplitude of the
oscillating electric field inside the resonator. It is assumed that the
inhomogeneity of $\varepsilon$ within the atomic wave-pocket is
$|\frac{1}{\varepsilon}\frac{\partial\varepsilon}{\partial\alpha}r|<10^{-3}$
where $\alpha$ are the different dimensions $\alpha=x,y,z$ and $r$ is the
radius of the Rydberg state, consequently affecting the atom-resonator
coupling [141]. Furthermore, the inhomegeneity of the external electric field,
$E$, caused by the screening effect of the spheres is also negligible.

Figure 3.9: Hybrid system composed of a chain of three-JJ flux qubits, an LC
resonator and an atom. The flux qubits have a tunable coupling mitigated by
Josephson coupling while the Rydberg atom is Rydberg coupled to a chain of
Rydberg atoms. The flux qubit, biased by an external flux $\Phi_{ex}$ is
inductively coupled to the resonator with a mutual inductance $M$. The atom is
placed in the middle point between two spheres with an oscillating
intraresonator electric field with amplitude $\varepsilon$ in the $z$-axis. An
additional electrostatic field $E$ runs across the Rydberg qubit in the
$z$-axis generated by the parallel-plate capacitors. This model is inspired by
[139]. Figure 3.10: (a) Stark map of Rb around $22^{2}D_{5/2}$. (b) The atom
qubit is formed by the ground state $\ket{g}$ and the excited state $\ket{e}$
at $E^{(1)}=520.0$ V/cm on two adiabatic energy lines starting with the
$22^{2}D_{5/2}$ and manifold state $\ket{\psi_{1}}$ composed of a set of
$\ket{n=20,l\geq 3,j=l\pm\frac{1}{2},m=\frac{5}{2}}$ at $E=0$. The auxiliary
state $\ket{u}$ is on another manifold state $\ket{\psi_{2}}$ yet on the same
set as $\ket{n=20,l\geq 3,j=l\pm\frac{1}{2},m=\frac{5}{2}}$. There are two
avoided crossings, $A_{1}$ and $A_{2}$ with energy separations $\Omega$ and
$\Omega^{^{\prime}}$ respectively. (c) Energy spectrum ($\omega_{f-LC}$ of the
flux qubit-resonator interface vs. the parameter $\gamma_{q}$. The interaction
between the flux qubit anf the resonator is negligible at
$\gamma_{q}^{(2)}=-5\times 10^{-3}$. At $\gamma_{q}=0$, an anticrossing exists
with spacing $\Delta=2\pi\times 5$ GHz. Another avoided crossing with spacing
$\delta=2\pi\times 0.1$ GHz induced by the resonant flux qubit-resonator
coupling which happens at $\gamma_{q}^{(1)}=-3.06\times 10^{-3}$. (d)
Anticrossing seen with spacing $\delta$. This image was taken from [141],
authored by Yu et al..

The Rydberg atom coupled to the resonator starts at the excited state
$\ket{e}$ on $22^{2}D_{5/2}(m=5/2)$. The Stark map for $22^{2}D_{5/2}(m=5/2)$
can be seen in Fig. 3.10(a). When the electric field $E$ is around 500 V/cm,
the excited state interacts with states in the set $\ket{n=20,l\geq
3,j=l\pm\frac{1}{2},m=\frac{5}{2}}$. These interacting manifold states can be
seen interacting with the $\ket{e}$ in 3.10(b) for different values of $E$.
The anticrossings with manifolds $\psi_{2}$ and $\psi_{1}$ can be seen for
different values of $E$. The states on the manifolds include an auxiliary
state $\ket{u}$ and a ground state $\ket{g}$ with anticrossings labelled A1
and A2. The energy separations are denoted as $\Omega$ and
$\Omega^{^{\prime}}$. Through adiabatic tuning, one can vary through the
states $\ket{\mu=e,g,u}$. The Hamiltonian of the atom is given by,

$H_{a}=\sum_{\mu=e,g,u}\hbar\omega_{\mu}\ket{\mu}\bra{\mu}+\frac{\hbar\Omega}{2}(\ket{e}\bra{g}+\ket{g}\bra{e})+\frac{\hbar\Omega^{^{\prime}}}{2}(\ket{e}\bra{u}+\ket{u}\bra{e}),$
(3.30)

where electric-field-dependent energies $\omega_{\mu=e,g,u}$ of atomic states
are derived as $\omega_{e}=-7.81-(E-500)\times 7.3^{-4}$ THz,
$\omega_{g}=-7.92-(E-500)\times 5.5^{-4}$ THz and
$\omega_{e}=-7.88-(E-500)\times 6.3^{-4}$ THz [141]. The atom-resonator
interaction is expressed as,

$V_{a}=\frac{\hbar
g_{a}}{2}(b^{\dagger}\ket{g}\bra{e}+\ket{e}\bra{g}b)+\frac{\hbar
g^{^{\prime}}_{a}}{2}(b^{\dagger}\ket{u}\bra{e}+\ket{e}\bra{u}b),$ (3.31)

with coupling strengths $g_{a}=|\bra{e}D\ket{g}|\varepsilon/\hbar$ and
$|\bra{e}D\ket{u}|\varepsilon/\hbar=2\pi\times 0.5$ GHz. The atom resonantly
interacts with the resonator at $E^{(2)}$ $(\omega_{e}-\omega_{u}=\omega_{0})$
and at $E^{(4)}$ $(\omega_{e}-\omega_{g}=\omega_{0})$ [139, 141].

The three-JJ flux qubit is coupled to the LC resonator through means of mutual
induction with inductance $M$. $M$ is chosen to be equal to $M=27pH$ to obtain
a flux strong enough for coupling, furthermore, the flux qubit is biased by an
external magnetic flux $\Phi$ to tune the frequency spacing
$\varepsilon=2I_{p}\Phi_{0}\gamma_{q}/\hbar$ with phase bias parameter
$\gamma_{q}=\Phi/\Phi_{0}-1/2$ (values introduced in Section 3.2.1) [27, 141].
The qubit is tuned between the states $\ket{R}$ and $\ket{L}$. Thus, the
Hamiltonian for the flux-qubit is similar to Eq. (3.19) [140, 141],

$H_{f}=-\frac{\hbar\varepsilon}{2}\sigma_{f,z}-\frac{\hbar\Delta}{2}\sigma_{f,x},$
(3.32)

but with the definitions of the Pauli matrices being,
$\sigma_{f,z}=\ket{L}\bra{L}-\ket{R}\bra{R}$,
$\sigma_{f,x}=\sigma^{\dagger}_{f,-}+\sigma_{f,-}$ and
$\sigma_{f,-}=\ket{R}\bra{L}$. The tunnel splitting is denoted as $\Delta$
with the flux qubit-resonator interaction potential given by,

$V_{f}=-\hbar g_{f}(b^{\dagger}+b)\sigma_{f,z},$ (3.33)

with the coupling strength given by
$g_{f}=(MI_{p}/\hbar)\sqrt{\hbar\omega_{0}/(2L)}$ [141]. Fig. 3.10(c) shows
the varying $\gamma_{q}$ against the energy spectrum of the flux qubit-
resonator interface [141],

$(H_{LC}+H_{f}+V_{f})\ket{\psi_{f-LC}}=\hbar\omega_{f-LC}\ket{\psi_{f-LC}},$
(3.34)

where $\omega_{f-LC}$ is the eigenvalue and the eigenstate $\ket{\psi_{f-LC}}$
is spanned in the basis $\\{\ket{o,n_{p}},o=L,;n_{p}=0,1,2,...\\}$ where
$n_{p}$ is microwave photon number. In the Hybrid system, the microwave photon
number is between $0,1$. The energy separation at the avoided crossing between
the states $\ket{R,1}$ and $\ket{L,0}$ at $\gamma_{q}^{(2)}$ is
$\delta=\frac{2V_{f}\Delta}{\hbar\sqrt{\varepsilon^{2}+\Delta^{2}}}$ (in Fig.
3.10(d)).

The final system Hamiltonian is,

$H=H_{LC}+H_{a}+H_{f}+V_{a}+V_{f}.$ (3.35)

This will be used to model the formation of the GHZ state used in the $E_{2}$
gate in Section 3.4.

###  3.4 Forming the Non-Local GHZ State

This section underlines the steps necessary to create the GHZ state. The GHZ
state is prepared using the methodology seen in [141]. The GHZ state
describing the flux-resonator-atom is,

$\ket{GHZ}=\frac{1}{\sqrt{2}}(\ket{L,1,e}+\ket{R,0,g}).$ (3.36)

The methodology used to form the $\ket{GHZ}$ in [141] follows a five step
process. Initially, the electric field and the phase bias parameter are set to
$E^{(1)}$ and $\gamma_{q}^{(1)}$, respectively. Here, both the atom and the
flux qubit are far-off-resonantly coupled to the resonator, thus the hybrid
system is in the $\ket{R,0,e}$ state. the external electric field, $E$, is
increased rapidly to $E^{(5)}$ for a time duration of $\pi/(2\Omega)$ which
allows the atom to transit to the $(\ket{e}+\ket{g})/\sqrt{2}$. The density
matrix of the result of the simulated transition is visualized in Fig.
3.11(a). The graph indicates the amplitude (probability) of the density matrix
($\rho=\ket{\psi}\bra{\psi}$) where the matrix elements are denoted on the
axis. The trace of the density matrix
$\bra{e}\rho\ket{e}+\bra{g}\rho\ket{g}+\bra{u}\rho\ket{u}$ is equal to 1,
where $\bra{u}\rho\ket{u}\approx 0$, $\bra{e}\rho\ket{e}\approx 0.45$ and
$\bra{g}\rho\ket{g}\approx 0.55$. The off-diagonal elements denoting the
superposition state $(\ket{e}+\ket{g})/\sqrt{2}$, are seen to have a
probability of 50%. This density matrix corresponds to a fidelity of 0.99
relative to the ideal $(\ket{e}+\ket{g})/\sqrt{2}$ state.

By switching on the resonant interaction between the microwave resonator and
the atomic $\ket{e}-\ket{u}$ transition, $E$ is reduced to $E^{(2)}$ for a
time of $\pi/g^{^{\prime}}_{a}$ before going back to $E^{(1)}$ which evolves
the $\ket{e}$ state to the auxiliary $\ket{u}$ state in order for the hybrid
system to transition to state $\frac{1}{\sqrt{2}}(\ket{R,1,u}+\ket{R,0,g})$.
The results of this transition is seen depicted in Fig. 3.11(b) where the atom
and resonator state are combined, thus,
$\frac{1}{\sqrt{2}}(\ket{1,u}+\ket{0,g})$. The diagonal terms show
$\bra{g,0}\rho\ket{g,0}\approx 0.9$ and $\bra{u,1}\rho\ket{u,1}\approx 0.1$
while the off-diagonal terms show an approximate 30% probability for the
$\bra{u,1}\rho\ket{g,0}$ and $\bra{g,0}\rho\ket{u,1}$ states. This corresponds
to a 0.95 fidelity relative to the
$\frac{1}{\sqrt{2}}(\ket{R,1,u}+\ket{R,0,g})$ state.

Moving to the flux qubit side, $\gamma_{q}$ is then switched to
$\gamma_{q}^{(2)}$ to resonate with the resonator for a time of $\pi/\delta$
which completes the transfer from the resonator to the flux qubit wherein the
flux qubit experiences a state flip. Thus, the system state is now at
$\frac{1}{\sqrt{2}}(\ket{L,0,u}+\ket{R,0,g})$. Then, $E$ is set to $E^{(3)}$
for a time length of $\pi/\Omega^{^{\prime}}$ consequently evolving the
$\ket{u}$ state to the $\ket{e}$. The resulting state is shown in Fig.
3.11(c), whereby only the atom system is shown. The diagonal terms show
$\bra{u}\rho\ket{u}\approx 0.7$ and $\bra{g}\rho\ket{g}\approx 0.3$ while the
off-diagonal terms show an approximate 40% probability for the
$\bra{u}\rho\ket{g}$ and $\bra{g}\rho\ket{u}$ states. This corresponds to a
0.95 fidelity relative to the $\frac{1}{\sqrt{2}}(\ket{L,0,u}+\ket{R,0,g})$
state.

Finally by repeating the nonadiabatic reduction to $E^{(2)}$ for a time length
of $\pi/g^{^{\prime}}_{a}$ and then tuning to $E^{(3)}$ for a time length of
$\pi/\Omega^{^{\prime}}$, the system reaches the GHZ state shown in Fig.
3.11(d). The diagonal terms show $\bra{L,1,e}\rho\ket{L,1,e}\approx 0.05$ and
$\bra{R,0,g}\rho\ket{R,0,g}\approx 0.8$. It is noticeable that an extra state
$\bra{L,1,g}\rho\ket{L,1,g}$ is measured to be around 15%. The off-diagonal
terms show an approximate 15% probability for the $\bra{L,1,e}\rho\ket{R,0,g}$
and $\bra{R,0,g}\rho\ket{L,1,e}$ states. The states,
$\bra{L,1,g}\rho\ket{R,0,g}$ and $\bra{R,0,g}\rho\ket{L,1,g}$, also show an
approximate 35% chance of being in a superposition. This corresponds to a 0.93
fidelity relative to the $\frac{1}{\sqrt{2}}(\ket{L,1,e}+\ket{R,0,g})$ state.

The GHZ state in [141] was reported to be completed in 8 ns with a fidelity
0.977. Due to differing methods of solving the master equation and using non-
dynamic time steps, the optimized fidelity reached was 0.93 in approximately
17 ns which is still in accordance with the timescale in [141]. The code for
the numerical simulation can be seen in Appendix C.2. The numerical simulation
begins by initializing the Hamiltonians of each system as well as the
interactions seen in the functions: rydberg_hamiltonian,
rydberg_resonator_hamiltonian, flux_hamiltonian, flux_resonator_hamiltonian,
and resonator_hamiltonian. The function solve_master_equation takes these
Hamiltonians and evolves them over a given time inputted by the user. The
output of this function is an array of evolved states at each time step. An
additional function, find_optimal_time, takes this array and finds the state
with the highest fidelity. Thus, it evaluates the fidelity of the vector state
at each time step with the ideal final state and finds the time step with the
best fidelity [76]. The five steps are defined as non-value returning
functions, step_1, step_2, step_3, step_4, and step_5. The timescale in the
range of nanoseconds is of importance due to the fact that the preparation
time is much shorter than the decoherence time, which will be seen in Section
4.3. From this, it means that it is indeed possible to establish a feasible
$E_{2}$ gate between the Rydberg atom computer and the flux qubit computer.

Figure 3.11: The probability (denoted as a color) of the density matrix of the
specific systems $\rho=\ket{\psi}\bra{\psi}$ at different steps of the GHZ
implementation. (a) Density state of the atom system after increasing $E$ from
$E^{(1)}$ to $E^{(5)}$ where the system transitions to the
$\frac{1}{\sqrt{2}}(\ket{R,0,e}+\ket{R,0,g})$ with 0.99 fidelity. (b) Density
state of the atom-resonator system after reducing $E$ from $E^{(5)}$ to
$E^{(2)}$ nonadiabatically. After this step, the system is at
$\frac{1}{\sqrt{2}}(\ket{R,1,u}+\ket{R,0,g})$ with 0.95 fidelity. (c) Density
state of atom system after $E$ is increased to $E^{(3)}$ quickly wherein the
final system state is $\frac{1}{\sqrt{2}}(\ket{L,0,e}+\ket{R,0,g})$ with 0.97
fidelity. (d) The final GHZ density state of the system with 0.93 fidelity.

##  Chapter 4 Modeling Noise and Decoherence in Quantum Systems

In order to properly simulate the Hybrid system evolving through the quantum
phase algorithm, the model requires noise models. The noise models will be
dependent on each system, which, like Chapter 3, will be split into three
sections. Section 4.1 will derive the noise model encountered in the Rydberg
atom computer. Section 4.2 will derive the noise fluctuations encountered in
the flux qubit computer. Section 4.3 describes the noise from the flux-
resonator-atom apparatus responsible for realizing the connection between both
quantum computers, which follows the theory described in [141]. The goal of
this chapter is to acquire the noise models of the systems for their
implementation in the execution of the quantum phase estimation discussed in
Chapter 5.

###  4.1 Rydberg Atom Noise

The Rydberg atom system encounters noise from two main sources, the laser-atom
interaction and the energy level dissipation. The goal of this section will be
to obtain the Lindbladians describing the Rydberg noise. A Lindbladian
describes open quantum systems by generalizing the Schrödinger equation to a
system that is in contact with its environment, which can be seen explained in
greater detail in Appendix B.2 [124], that will be used to simulate the
Rydberg system dynamics. The Lindbladian describing the noise dynamics with a
given initial density matrix, $\rho$, is seen below,

$\mathcal{L}[\rho]=i\sum_{k}\gamma_{k}\bigg{(}A_{k}\rho
A_{k}^{\dagger}-\frac{1}{2}\Big{\\{}A_{k}^{\dagger}A_{k},\rho\Big{\\}}\bigg{)},$
(4.1)

where $\\{A_{k}\\}$ are the Lindblad operators representing the coupling of
the system to the environment and thus the dissipative process [57, 74, 77,
89].

In order to derive the Linbaldian for the local phase noise, it is first
necessary to derive the full Rydberg interaction Hamiltonian which includes
the interaction of the atoms interacting with each other as well as the atoms
interacting with the laser. For simplicity, it will be assumed that the atom
contains two main energy states, the ground state $\ket{g}$ and the excited
state $\ket{e}$. The distinction of the hyperfine states will be ignored
because the noise and dissipation of both hyperfine levels are very similar.

The laser field denoted by $\mathcal{\boldsymbol{E}}(\boldsymbol{r},t)$, can
be described as [2],

$\mathcal{\boldsymbol{E}}(\boldsymbol{r},t)=\boldsymbol{E}_{0}\mathrm{cos}(\omega_{L}t+\phi(t)-\boldsymbol{k}\cdot\boldsymbol{r})=\frac{1}{2}\boldsymbol{\mathcal{E}}_{0}\Big{(}e^{i(\omega_{L}t+\phi(t)-\boldsymbol{k}\cdot\boldsymbol{r}})+e^{-i(\omega_{L}t+\phi(t)-\boldsymbol{k}\cdot\boldsymbol{r})}\Big{)}.$
(4.2)

The amplitude of the field is denoted by $\boldsymbol{\mathcal{E}}_{0}$ where
$t$ denotes the time, $\omega_{L}$ and $\phi$ are the field frequency and the
phase noise, respectively. The Hamiltonian of the Rydberg is considered to
have an atomic Hamiltonian whereby the atomic interactions are expressed and a
light interaction Hamiltonian. This is of the form,

$\mathcal{H}=H_{atom}+H_{int}=\hbar\omega_{eg}\sigma_{ee}+q\boldsymbol{r}\cdot\boldsymbol{E}(\boldsymbol{r},t),$
(4.3)

where $q$ is the electron charge, $q=-e$, and $\sigma_{ij}=\ket{i}\bra{j}$
where $i,j$ denote the energy states. The full interaction Hamiltonian can be
obtained by applying the unitary transformation to the Hamiltonian described
in Eq. (4.3),

$U=e^{-i\hbar\omega_{ge}t\sigma_{ee}}.$ (4.4)

Here, $\omega_{eg}$ is the frequency between both energy levels,
$\omega_{eg}=(E_{e}-E_{g})/\hbar$. This leads to,

$\mathcal{H}^{I}=U^{\dagger}\mathcal{H}U-\hbar\Delta\sigma_{ee}.$ (4.5)

After using the rotating wave approximation and cancelling out negligible
terms, the equation becomes,

$\mathcal{H}^{I}=-\hbar\Delta\sigma_{ee}+\frac{\hbar\Omega}{2}\bigg{(}\sigma_{ge}e^{i(\phi(t)-\phi(t^{^{\prime}}))}+\sigma_{eg}e^{-i(\phi(t)-\phi(t^{^{\prime}}))}\bigg{)},$
(4.6)

where $\Omega=\boldsymbol{E}_{0}\cdot\mu/\hbar$ describes the Rabi frequency
where $\mu=e\bra{g}\boldsymbol{r}\ket{e}$ is the dipole matrix and
$\Delta=\omega_{L}-\omega_{eg}$. The instantaneous change in phase noise is
described as $e^{i\phi(t-t^{^{\prime}})}$. The instantaneous change of phase
noise is the so-called frequency noise which corresponds to the energetic
shifts of the Rydberg state and is characterized as [2],

$\varepsilon(t)=\dot{\phi}=\frac{\mathrm{d}\phi}{\mathrm{d}t}.$ (4.7)

A final transformation $U=e^{-i\hbar\varepsilon(t)\sigma_{ee}}$ can be applied
to Eq. (4.6) to remove the time-dependence of the frequency noise in the
exponent [83] leaving,