$\mathcal{H}^{I}=-\hbar\Delta\sigma_{ee}-\hbar\varepsilon(t)\sigma_{ee}+\frac{\hbar\Omega}{2}(\sigma_{ge}+\sigma_{eg}).$
(4.8)

The full derivation of the interaction Hamiltonian can be seen in Appendix D.6
[19, 34, 95]. In a many atom system, the final Hamiltonian can be formulated
to be similar to that seen in Eq. (3.7) with an extra noise term,

$\mathcal{H}^{I}=\frac{\Omega}{2}\sum_{i}\Big{(}\sigma_{eg}^{(i)}+\sigma^{(i)}_{ge}\Big{)}+V_{0}\sum_{i<j}\frac{\sigma_{ee}^{(i)}\sigma_{ee}^{(j)}}{|i-j|^{6}}-\sum_{i}\varepsilon^{(i)}(t)\sigma_{ee}^{(i)},$
(4.9)

where $\hbar=1$ and the interatomic distance between two atoms is defined as
$r_{ij}\equiv|i-j|$. It is important to note that the interaction is possible
when $\omega_{eg}=\omega_{L}$ thus, $\Delta=0$.

The frequency noise $\varepsilon(t)$ is dependent on the phase noise $\phi(t)$
which causes fast and slow fluctuations in the laser electric field with a
correlation time $\tau_{c}$ [2]. The phase noise acts as a time-dependent
detuning and shares many similarities with Brownian motion which is described
by the Langevin equation [83],

$\ddot{\phi}=-\gamma\dot{\phi}+F(t),$ (4.10)

where $\gamma$ is the inverse correlation time $\tau_{c}=1/\gamma$ and $F(t)$
is a Gaussian distribution denoting a rapidly fluctuating force with zero
ensemble average $\overline{F(t)}=0$ and $\overline{F(t)^{2}}\neq 0$. It is
assumed that $F(t)$ has a very short correlation time compared to the other
characteristic time scales of the system, and thus approximately [2, 59],

$\overline{F(t)F(t^{^{\prime}})}=2D\delta(t-t^{^{\prime}}),$ (4.11)

where $D$ is the magnitude of the fluctuations and $\delta(t-t^{^{\prime}})$
is the the Dirac delta function. Together with $\gamma$, $D$ characterizes the
spectral width $\Gamma$ of the Lorentzian line shape [83],

$\Gamma=\frac{2D}{\gamma^{2}}.$ (4.12)

The spectral line width, $\Gamma$ is often referred to as the dephasing rate.
To dynamically model the dephasing rate using the system Hamiltonian derived
in Eq. (4.9), one must model the Langevin as a Lindbladian (Eq. (4.1)). This
follows from the derivation of the Born-Markov master equation [19],

$\hat{\dot{\rho}}=-\int^{t}_{-\infty}\mathrm{d}t^{^{\prime}}\mathrm{tr}_{B}\Big{[}\sum_{i}\varepsilon^{(i)}(t)\sigma_{ee}^{(i)},\Big{[}\sum_{i}\varepsilon^{(i)}(t)\sigma_{ee}^{(i)},\hat{\rho}(t)\Big{]}\Big{]},$
(4.13)

where $\mathrm{tr}_{B}[x]$ is equivalent to the average $\langle x\rangle$.
Expanding equation Eq. (4.13) gives,

$\hat{\dot{\rho}}=-\int^{t}_{-\infty}\mathrm{d}t^{\prime}\sum_{i,j}\mathrm{Tr}_{B}\Big{\\{}\varepsilon^{(i)}(t)\varepsilon^{(j)}(t^{\prime})\sigma_{ee}^{(i)}\sigma_{ee}^{(j)}\hat{\rho}+h.c.-\varepsilon^{(i)}(t)\sigma_{ee}^{(i)}\hat{\rho}\varepsilon^{(j)}+h.c.\Big{\\}}=-\sum_{i,j}\int^{t}_{-\infty}\mathrm{d}t^{\prime}\mathrm{Tr}_{B}\Big{\\{}\varepsilon^{(i)}(t)\varepsilon^{(j)}(t^{^{\prime}})\Big{\\}}\Big{[}\sigma_{ee}^{(i)}\hat{\rho}\sigma_{ee}^{(j)}-\frac{1}{2}\\{\sigma_{ee}^{(i)}\sigma_{ee}^{(j)},\hat{\rho}\\}\Big{]}=\mathcal{L}[\hat{\rho}].$
(4.14)

A local laser is used when addressing qubits in the quantum systems since, as
observed in section 3.1.3, individual qubits are addressed per operation.
Thus, for the local noise, the correlation between the phase noise experienced
by atoms $i$ and $j$ at two different times is
$\langle\varepsilon^{(i)}(t)\varepsilon^{(j)}(t^{\prime})\rangle=(\Gamma/2)\delta_{ij}\delta(t-t^{\prime})$
thus finding the local phase noise superoperator [2, 46],

$\mathcal{L}[\hat{\rho}]^{l}=\Gamma\sum_{i}\bigg{[}\sigma_{ee}^{(i)}\hat{\rho}\sigma_{ee}^{(i)}-\frac{1}{2}\\{\sigma_{ee}^{(i)}\sigma_{ee}^{(i)},\hat{\rho}\\}\bigg{]}.$
(4.15)

The final decoherence effect from the system comes from energy decay or the
rate at which the population from an excited state $\ket{e}$ is transferred to
a lower lying level $\ket{g}$ via photon emission into the vacuum. Spontaneous
emission can be derived through Weisskopf-Wigner theory assuming complete
coupling with closely spaced cavity modes with the emission spectrum centered
at the atomic transition frequency. Weisskopf-Wigner theory also highlights
that phenomenologically, the excited state is only capable of emitting to the
ground state but not vice versa [111]. The final superoperator that describes
the spontaneous decay from the state $\ket{e}$ reads,

$\mathcal{L}[\hat{\rho}]^{e}=\gamma_{e}\sum_{i}\Big{[}\sigma_{ge}^{(i)}\hat{\rho}\sigma_{eg}^{(i)}-\frac{1}{2}\\{\sigma_{eg}^{(i)}\sigma_{eg}^{(i)},\hat{\rho}\\}\Big{]},$
(4.16)

where $\gamma_{e}$ is the spontaneous emission rate from the excited state.
Using ARC [115], the spontaneous decay time for the $\ket{e}=\ket{70S_{1/2}}$
is $375\mu$s. The Linbladians observed in Eq. (4.15) and (4.16) will be used
to model the noise. The implementation will be discussed in Chapter 5.

###  4.2 Flux Qubit Noise

Figure 4.1: Potential noise sources that can arrise in superconducting qubits.
Inspired by [98].

The goal of this section is to describe the flux qubit noise in Linbladian
form. Flux qubit noise arises from the general noise in a superconducting
circuit which can be seen in figure 4.1. There exists two categories. The
first category of noise is the noise generated from effects on the circuit
such as decoherence from qubit coupling effects [94]. The second group of
noise is the non-circuit related noise such as fluctuations in the magnetic
field or the Josephson junction itself.

The noises generated from the circuit (often from defects in the dielectrics)
are typically said to cause charge fluctuations, while the noise generated by
the magnetic field and the stray fields originating from the Josephson
junctions are said to cause flux fluctuations. Flux fluctuations can be
described by introducing a perturbation around the magnetic frustration
$(\delta f)$ to the anharmonic system described in Eq. (3.18). The Hamiltonian
of the flux fluctuations is thus a first order perturbation of the potential
energy in terms of the magnetic frustration. Using the definition for
$f_{\varepsilon}$ and $\varphi_{-}$ seen in Section 3.2, as well as, Eq.
(3.18), the flux fluctuation Hamiltonian can be analytically modelled as,

$\delta\mathcal{H}=\delta U=-2\pi\alpha E_{J}\mathrm{sin}(2\pi
f_{\varepsilon}+2\varphi_{-})\delta f=-2\pi\alpha
E_{J}(\mathrm{sin}\phi\mathrm{cos}(2\varphi_{-^{\prime}})+\mathrm{cos}\phi\mathrm{sin}(2\varphi_{-^{\prime}}))\delta
f\approx-2\pi\alpha
E_{J}(\mathrm{sin}\phi(1-2\varphi_{z}^{2}(\hat{a}+\hat{a}^{\dagger}))\delta
f,$ (4.17)

where $\varphi_{-^{\prime}}=\varphi_{-}\varphi_{-}^{*}$, where
$\varphi_{-}^{*}$ is the $\varphi$ corresponding to the minimum of the
potential well where $\ket{L}$ resides, seen in figure 3.8(d), and
$\varphi_{z}=(E_{C,-}/4E_{J,-})^{1/4}$. Furthermore, $E_{C,-}$ is the
effective charging energy for the negative minimum and $E_{J,-}$ is the
effective Josephson energy in the negative minimum and
$\phi=\phi(f_{\varepsilon})=2\pi
f_{\varepsilon}+2\varphi_{-}^{*}(f_{\varepsilon})$.

Fluctuations near the frequency $\Omega$ are considered because they have the
possibility of inducing transitions between the $\ket{L}$ and $\ket{R}$
states. Thus, in the case $\delta f\propto\mathrm{cos}(\Omega t)$, the system
simplifies to two levels,

$\mathcal{H}=\frac{1}{2}(\hbar\omega_{q}\sigma_{z}+I_{m}\Phi_{0}\delta
f\sigma_{x})$ (4.18)

where $\omega_{q}=\omega_{q}(f_{\varepsilon})=\Omega$ and $I_{m}$, the
$f_{\varepsilon}$-dependent current difference between the parametrised
circulating-current states, is defined as
$I_{m}=I_{m}(f_{\varepsilon})=-8\pi\alpha\sigma_{z}\mathrm{cos}\phi
E_{J}/\Phi_{0}$, where the parameters are described in Section 3.2.2.

The next step is to derive the charge fluctuation noise on the Hamiltonian.
Investigating the charge fluctuations ($\delta q_{a}$,$\delta q_{b}$) requires
invoking a perturbation via the kinetic energy $T$. Perturbing the kinetic
energy given by Eq. (3.22) yields,

$\delta
T=\delta\boldsymbol{q}^{\intercal}\boldsymbol{C}^{-1}\boldsymbol{Q}=-\frac{e}{C(2\zeta+2\alpha+1)}[-in_{z}(\hat{a}-\hat{a}^{\dagger})](\delta
q_{A}-\delta q_{B}),$ (4.19)

where $n_{z}=(E_{J,-}/4E_{C,-})^{1/4}$ is the quantum ground-state uncertainty
in Cooper-pair number and $\zeta$ is called the shunt factor which relates the
capacitance of the smaller junction ($C_{3}$ seen in figure 3.8(c)) to the
capacitance of the shunt capacitor through $C_{sh}=\zeta C_{3}$ [136]. This
factor is considered to satisfy $\zeta\gg 1$. The charge noise is orthogonal
to the flux noise, thus it couples to the Hamiltonian in the $\sigma_{y}$
basis. The perturbation depends only on the differential mode of the induced
charges between the two islands (described by nodes $A$ and $B$ seen in figure
3.8(c)) [136]. Thus, placing the charge fluctuation back into Eq. (4.18),
gives,

$\mathcal{H}=\frac{1}{2}(\hbar\omega_{q}\sigma_{z}+I_{m}\Phi_{0}\delta
f\sigma_{x}+n_{z}E_{C,-}\delta n_{-}\sigma_{y}),$ (4.20)

where $\delta n_{-}=(\delta q_{A}-\delta q_{B})/(-e)$ is the differential
electron number fluctuation [136].

The importance of the C-shunted flux design is observed in the noise
sensitivity. Because of this design, the charge noise sensitivity diminishes
with larger $\zeta$ due to the relation, $n_{z}E_{C,-}\propto
E^{3/4}_{C,-}\propto\zeta^{-3/4}$. Each one of these fluctuation terms can be
applied as a Linbladian to the Hamiltonian seen in Eq. (3.21). The
Lindbladians modelling the noise due to the fluctuations in flux and charge
noise in the $\sigma_{x},\sigma_{y},\sigma_{z}$ Pauli bases, are thus [136],

$\mathcal{L}[\hat{\rho}]_{z}=\frac{1}{2}\omega_{q}\Big{[}\sigma_{z}\hat{\rho}\sigma_{z}^{\dagger}-\frac{1}{2}\\{\sigma_{z}^{\dagger}\sigma_{z},\hat{\rho}\\}\Big{]},$
(4.21) $\mathcal{L}[\hat{\rho}]_{x}=\frac{1}{2\hbar}I_{m}\Phi_{0}\delta
f\Big{[}\sigma_{x}\hat{\rho}\sigma_{x}^{\dagger}-\frac{1}{2}\\{\sigma_{x}^{\dagger}\sigma_{x},\hat{\rho}\\}\Big{]},$
(4.22) $\mathcal{L}[\hat{\rho}]_{y}=\frac{1}{2\hbar}n_{z}E_{C,-}\delta
n_{-}\Big{[}\sigma_{y}\hat{\rho}\sigma_{y}^{\dagger}-\frac{1}{2}\\{\sigma_{y}^{\dagger}\sigma_{y},\hat{\rho}\\}\Big{]}.$
(4.23)

The final non-circuit noise source comes from the CPW coupled to the flux
qubit. The CPW has been simplified to operate similarly to a resonant cavity.
Due to this, the system encounters noise in the form of the Purcell effect.
The Purcell effect is the enhancement of a quantum system’s spontaneous
emission rate due to its environment. In this case, the CPW radiates a wave
which has been reflected from the environment which consequently excites the
resonance out of phase. This is difficult to analytically model due to the
vast sources of electromagnetic waves possible, thus, using experimental
results [135], the rate at which the resonator encounters spontaneous emission
is modelled by,

$\frac{1}{\tau_{photon}}=\frac{\sqrt{A\mathrm{ln}2}}{\hbar}\Big{|}\frac{\delta\omega_{LR}}{\delta
f_{\varepsilon}}\Big{|},$ (4.24)

where $\omega_{LR}$ is the transition frequency between the states $\ket{L}$
and $\ket{R}$, and $A$ is a fit parameter. According to [135],
$1/\tau_{photon}$ was measured to be around 9.19 MHz for
$0.5<f_{\varepsilon}<0.55$. The Lindbladian can, thus, be described as,

$\mathcal{L}[\hat{\rho}]_{purcell}=\frac{1}{\tau_{photon}}\Big{[}\sigma_{y}\hat{\rho}\sigma_{y}^{\dagger}-\frac{1}{2}\\{\sigma_{y}^{\dagger}\sigma_{y},\hat{\rho}\\}\Big{]}.$
(4.25)

The Lindbladians observed in Eq. (4.21), (4.22), (4.23) and (4.25) will be
used to model the noise. The implementation will be discussed in Chapter 5.

###  4.3 Hybrid System Noise

The goal of this section is to describe the noise (in Linbladian form) of the
flux-resonator-atom qubit connection responsible for the construction of the
$E_{2}$ gate seen in Section 3.3. Due to the short time scale taken to build
the $E_{2}$ gate, the noise of this system is different to the noise
encountered in Sections 4.1 and 4.2. It follows the approximations and
simplified noise factors seen in [141].

Following Yu et al., there are four relevant sources of noise. The flux qubit
has been simplified to contain two noise sources, qubit dephasing, given by a
dephasing rate, as well as a relaxation loss of a qubit decohering from
$\ket{R}$ state to a $\ket{L}$ state (seen in Section 3.3), given by a
relaxation rate [141]. The dephasing rate and relaxation rate are denoted by
$\gamma_{relax}$ and $\gamma_{\phi}$, respectively. The main noise source
observed in the hybrid system, following [141], is the spontaneous decay rate
seen in Section 4.1. The decay is observed for the states $\mu=e,u$ to the
ground state $\ket{g}$. The spontaneous emission rate is assumed to be equal
for all $\mu$ because the Rydberg state is close to the capacitor surface
which induces extra noise from stray fields and thus drastically reduces the
lifetime of the states [141]. Finally, the noise of the resonator is
characterized by $\kappa=\omega_{0}/Q$ (assuming a Q-factor value
$Q=\mathrm{10^{5}}$) as the loss rate due to the Purcell effect, as described
in Section 4.2. All the values can be seen on table F.1.

The Hybrid system, thus, possesses the following Lindbladians,

$\mathcal{L}^{f}_{relax}=\gamma_{relax}\Big{[}\sigma_{f,-}\rho\sigma_{f,-}^{\dagger}-\frac{1}{2}\\{\sigma_{f,-}^{\dagger}\sigma_{f,-},\rho\\}\Big{]}$
(4.26)
$\mathcal{L}^{f}_{dephase}=\frac{\gamma_{\phi}}{2}\Big{[}\sigma_{f,z}\rho\sigma_{f,z}^{\dagger}-\frac{1}{2}\\{\sigma_{f,z}^{\dagger}\sigma_{f,z},\rho\\}\Big{]}$
(4.27) $\mathcal{L}^{r}=\kappa\Big{[}b\rho
b^{\dagger}-\frac{1}{2}\\{b^{\dagger}b,\rho\\}\Big{]}$ (4.28)
$\mathcal{L}^{a}=\gamma_{e}\sum_{\mu=e,u}\Big{[}\sigma_{\mu g}\rho\sigma_{\mu
g}^{\dagger}-\frac{1}{2}\\{\sigma_{\mu g}^{\dagger}\sigma_{\mu
g},\rho\\}\Big{]}$ (4.29)

where $\sigma_{g\mu}=\bra{g}\ket{\mu}$ and the other parameters can be seen in
Section 3.3 [141]. The final fidelity with noise is very similar to the
fidelity without noise at the same time scale. This is due to the fact that
the decoherence timescale is a factor of $\mathrm{10^{3}}$ larger than the
creation of the $E_{2}$ gate.

##  Chapter 5 Quantum Algorithm Simulation and Analysis with Gradient Ascent
Pulse Engineering

This chapter will detail the simulation of the distributed phase estimation
algorithm using the Hamiltonians and noise sources for both the flux and
Rydberg qubit computers as seen in Chapters 3 and 4. As seen in Chapter 2, the
distributed phase estimation algorithm, when using four counting qubits, has a
circuit depth of around 50 gates. In the case of numerous gates, it becomes
imperative that each gate maintains a high level of accuracy since errors in
individual gates tend to compound or multiply as they are applied
successively. To accomplish accurate gates, the gates will be optimised using
the GRAPE (Gradient Ascent Pulse Engineering) algorithm explained in Section
5.1. The accuracy of the gates will then be observed using process tomography,
a method of experimentally measuring the fidelity of a gate, in Section 5.2.
Finally, the implementation of the gates as well as the GRAPE algorithm in the
distributed phase estimation algorithm will be elaborated upon in Section 5.3.

###  5.1 Introduction to GRAPE with BFGS

Simulating quantum algorithms that require many quantum gates or depth on a
noisy open system is unachievable because the system decoheres far too fast.
Knowing the noise contributions, one could try analytically solving the master
equation and adding the shifts necessary to accomplish the creation of a gate
for a certain time. However, this requires solving the Schrödinger equation at
each step of the evolution which is far too difficult, especially for a system
of many qubits.

In the current NISQ era there is a large push towards correcting and
optimizing qubit gates through optimal control [65]. Quantum optimal control
theory (QOCT) refers to a set of methods used to implement shapes of external
electromagnetic fields capable of manipulating the quantum dynamical processes
at the atomic level in the most efficient way possible. Gradient Ascent Pulse
Engineering (GRAPE) algorithm is one of many optimal control methods that was
first developed by Khaneja et al. [62]. GRAPE has garnered attention among
other algorithms because it utilizes direct analytical expressions for the
gradient as opposed to difference methods. This means that it can efficiently
find suitable solutions in the parameter space with fast convergence speed
[26, 62]. This section will aim to introduce the GRAPE algorithm, which will
then be implemented to optimise gates in Sections 5.2 and 5.3.

To fully grasp GRAPE, it is crucial to first characterize the system. The
system is modelled using the Hamiltonian described in Eq. (B.12) where the
Hamiltonian is a linear combination of a drift Hamiltonian and a control
Hamiltonian. To model the system into an open system, one must first utilize
the density operator $\rho(t)$ and characterize the equation of motion as a
Liouville-von Neumann equation ( defined in Eq. (B.15)),

$\dot{\rho}(t)=-\frac{i}{\hbar}\bigg{[}\Big{(}\mathcal{H}_{0}+\sum^{m}_{k=1}u_{k}(t)\mathcal{H}_{k}\Big{)},\rho(t)\bigg{]},$
(5.1)

where $\mathcal{H}_{0}$ is the drift Hamiltonian, $\mathcal{H}_{k}$ are the
radiofrequency Hamiltonians that correspond to the available control fields
and $u(t)=(u_{1}(t),u_{2}(t),\cdots,u_{m}(t))$ is the control vector or a set
of vector amplitudes that can be changed [62]. The goal is to find the optimal
amplitudes $u_{k}(t)$ of the RF fields capable of steering a given density
operator in initial state $\rho(0)=\rho_{0}$ in a total time $T$ to a density
operator $\rho(T)$ with maximum overlap to some desired target $C$ measured by
the standard inner product,

$\braket{C}{\rho(T)}=\mathrm{Tr}\\{C^{\dagger}\rho(T)\\}.$ (5.2)

In the case of open quantum systems, the dynamics is described by the
Markovian master equation seen in Eq. (4.1). The formal solution to such
equation is [15],

$\rho(t)=\mathrm{exp}\Big{\\{}\int^{t}_{0}\mathcal{L}(t^{^{\prime}})dt^{^{\prime}}\Big{\\}}\rho(0).$
(5.3)

To iterate through time, the transfer time $T$ is discretised in $N$ equal
steps of duration $\Delta t=T/N$. It is assumed that at each time step, the
control amplitudes $u_{k}(t)$ are constant. This means that at an arbitrary
time step $l$, the amplitude $u_{k}(t)$ of the $k$th control Hamiltonian is
given by $u_{k}(l)$. Using the Lindbladian formulation described Eq. (4.1),
the master equation is,

$\dot{\rho}=-i[H,\rho]+\mathcal{L}[\rho],$ (5.4)

where $\mathcal{L}$ acts as an operator on a density matrix, $\rho$. Using
this definition means that one can define a discretised unitary propagator
during a time step $l$,

$U_{l}\boldsymbol{\cdot}=\mathrm{exp}\bigg{\\{}-i\Delta
t([H_{l},\boldsymbol{\cdot}]+\mathcal{L}[\boldsymbol{\cdot}]\bigg{\\}}.$ (5.5)

A performance index is required for the algorithm to optimise a gate, which
can be expressed as the overlap between a final state $\rho(T)$ and the target
state, $C$. The performance index is of the form,

$\Phi_{0}=\mathrm{Tr}\Big{\\{}CU_{N}\cdots U_{1}\rho(0)\Big{\\}}.$ (5.6)

The derivative of the performance index takes the form [15],

$\frac{\partial\Phi_{0}}{\partial
u_{k}(l)}=\mathrm{Tr}\bigg{\\{}\lambda_{l}(C)\frac{\partial U_{l}}{\partial
u_{k}(l)}\rho_{l-1}\bigg{\\}},$ (5.7)

where,

$\rho_{l}=U_{l}\cdots U_{1}\rho(0),$ (5.8)

is the forward in time evolved density matrix and,

$\lambda_{j}(C)=U_{l+1}^{\dagger}\cdots U_{N}^{\dagger}C,$ (5.9)

is the backwards in time evolution from the final target state.The first order
derivative of the $l^{th}$ time-evolution operator in terms of $\Delta t$ is
[15, 1],

$\frac{\partial U_{l}\cdot}{\partial u_{l}(l)}\approx-i\Delta
t[H_{k},(U_{l}\boldsymbol{\cdot})].$ (5.10)

The derivative of the performance index is thus,

$\frac{\partial\Phi_{0}}{\partial u_{k}(l)}=-i\Delta
t\mathrm{Tr}\\{\lambda_{l}(C)[H_{k},\rho_{l}]\\}.$ (5.11)

The result seen in Eq. (5.11) is the core of the GRAPE algorithm [1, 15].

Figure 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
$\Phi_{0}$.

Fig. 5.1 exhibits how an arbitrary $u_{k}(t)$ evolves over time. The
efficiency of the algorithm is dependent on maximizing the performance
gradient $\partial\Phi_{0}/\partial u_{k}(l)$ denoted by arrows on Fig. 5.1.
The performance change of the control amplitude is acquired by allowing [26],

$u_{k}(l)\rightarrow u_{k}(l)+\epsilon\ \frac{\delta\Phi_{0}}{\delta
u_{k}(l)},$ (5.12)

so that the performance function $\Phi_{0}$ increases when $\epsilon$ is a
small enough step size. Tailoring the GRAPE algorithm for qubit optimal
control requires modifying the definition of the performance function
$\Phi_{0}$. In the field of qubit optimal control the performance function
described by (5.2) can be referred to as the fitness function, $f$ [78]. This
means that the fitness function is the metric that controls the modification
such that $\varepsilon=1-f$. The closer the algorithm gets to reaching the
desired fidelity, the smaller the modification becomes. In a full evolution,
there will be $N\times M$ variables where $M$ are the amount of control
Hamiltonians. The general algorithm for GRAPE can be seen in Alg. 1.

Algorithm 1 The general GRAPE algorithm.

Get minimum fidelity error $x_{min}$

Guess initial controls $u_{k}(l)$

$\rho\leftarrow\rho_{0}$

$done\leftarrow\mathrm{False}$

while not $done$ do

$l\leftarrow 0$

while $l\leq N$ do

$Iterator_{f}\leftarrow 0$

while $Iterator_{f}\leq l$ do

$\rho\leftarrow U_{Iterator_{f}}\rho$

$Iterator_{f}\leftarrow Iterator_{f}+1$

end while

$\lambda\leftarrow C$

$Iterator_{b}\leftarrow N$

while $Iterator_{b}\leq l+1$ do

$\lambda\leftarrow U^{\dagger}_{Iterator_{b}}$

$Iterator_{b}=Iterator_{b}+1$

end while

end while

Evaluate $\partial\Phi_{0}/\partial u_{k}(l)$ (Eq. (5.11)), $\varepsilon=1-f$
and update $m\times N$ control amplitudes $u_{k}(l)$ according to Eq. (5.12)

if $\varepsilon\leq x_{min}$ then

$done\leftarrow\mathrm{True}$

end if

end while

The main challenge the GRAPE algorithm as well as other optimization
algorithms encounter is that the methods have a tendency to get stuck at a
local minima. For a system that possesses many control fields
$\mathcal{H}_{k}$, this is usually not a problem since one can define a
fidelity with sufficient precision to reach. On the other hand, it is very
difficult (or impossible) to know the highest possible fidelity for a system
that lacks the required control fields. This consequently forces a trade-off
between the number of iterations required to reach the minima and the
possibility to overstep a minima.

An approach one can take to counteract this is to translate the local problem
landscape to a parabola which in turn makes it possible to step into the
minima without having to guess the step size. For this reason, GRAPE utilizes
a quasi-Newtonian gradient descent algorithm called Broyden-Fletcher-Goldfarb-
Shanno (BFGS) algorithm which utilizes Hessian based approximation techniques
[22, 80]. This algorithm efficiently computes the second-order differentials
of Hessian matrices that would otherwise pose a challenging calculation. This
algorithm is seen explained in Appendix E.5 [22, 32, 42, 131].

###  5.2 Quantum Process Tomography and Implementation of GRAPE Optimised
Gates

The goal of this section is to analyze the fidelity of the gates using GRAPE.
The implementation technique will thus focus on single operations at a time,
thus decoherence of atomic states will not be taken into account when the
qubit is not being processed. This constraint is reasonably valid since the
most noise originates from detuning, magnetic fluctuations and laser effects
which are all consequences of qubit state manipulation. Furthermore, when
conducting optimal control on hardware, one cannot access the quantum state in
the middle of a running execution. For this reason, optimal control is used to
create final states from initial states using modelled system noise that has
been measured from previous runs [99].

The initial parameters to consider are those from Eq. (5.1). The Pauli
matrices, $\sigma_{x}$, $\sigma_{y}$ and $\sigma_{z}$ will be used as the
controls. For the Flux system, the drift Hamiltonian will be Eq. (3.21) with
the superoperators Eq. (4.21) - (4.25). For the Rydberg atom system, the drift
Hamiltonian will use Eq. (3.7) with the superoperators Eq. (4.15) and (4.16).
These master equations will be fed into the GRAPE algorithm that already forms
part of the QuTiP optimal control package.

The GRAPE algorithm has many different values that can be modified. This study
will focus on varying two, the number of time steps in a specified time $T$
and the number of iterations the algorithm is allowed to run through before
reaching the minimum fidelity error ($\mathrm{10^{-10}}$). Allowing the
algorithm to run many times for a pulse sequence, allows the GRAPE algorithm
to choose different guesses each time in an attempt to lower the fidelity
error. The time $T$ chosen was 50 $\mu$s because it is at a small enough
timescale to reasonably neglect decoherence effects from state lifetimes. The
time steps will vary from 50 to 200 steps which means that the lowest amount
of time per step will be 250ns which, according to PASQAL [116], is enough
time to pulse a full rotation gate on a Rydberg atom system. The order of
magnitude of the timescale required to form the hybrid GHZ state (in the order
of a few nanoseconds) is far too small for GRAPE to be feasible because it
would require the ability of pulsing control fields at the picosecond scale.
Thus, GRAPE will not be utilized in the creation of the hybrid GHZ gate.

An additional restriction to be imposed on the algorithm pertains to the
allotted CPU time for optimizing a time step before proceeding to the
subsequent iteration. This is referred to as wall time or wall clock time
because it is counted in seconds and is used as a timeout if the algorithm
encounters a local minimum and cannot minimise the error [33]. This is chosen
to be at 200 seconds.

Appendix C.3 shows the code detailing the construction of the GRAPE optimised
noisy qubit gates. The function single_qubit_hamiltonian builds the master
equation for the single qubit interaction both the flux and Rydberg atom
system depending on the argument input. This is also the case for the double
qubit interaction using the function double_qubit_hamiltonian. The parameter
values can be seen in table C.1. This section will analyze the purple section
of the simulation schematic seen in figure C.1. As can be seen in the
schematic, the gate optimisation is dependent on the iterating values for both
the GRAPE iteration amount and the time step amount.

An initial examination of how the GRAPE algorithm fine-tunes and constructs
both Rydberg and flux quantum gates can be observed through the application of
quantum process tomography. Quantum process tomography is the process by which
known quantum states are used to probe a quantum process to find how the
process or gate is described [89]. Quantum process tomography stems from
quantum state tomography which is the procedure of experimentally determining
an unknown quantum state. Since it is not possible to distinguish the
superposition states of an unknown state $\rho$ with a single measurement,
quantum state tomography allows one to estimate $\rho$ if many copies of
$\rho$ are available. This is the case for a quantum gate where the process
can be executed many times on a known initial state to generate many copies of
$\rho$.

The goal is to determine a set of operation elements $\\{E_{i}\\}$ for
$\varepsilon$,

$\varepsilon(\rho)=\sum_{i}E_{i}\rho E_{i}^{\dagger}.$ (5.13)

Experimental results however, do not involve operators, they involve numbers.
Thus, it is convenient to rewrite $E_{i}$ to [89],

$E_{i}=\sum_{m}e_{im}\tilde{E}_{m}$ (5.14)

where $\tilde{E}$ is a fixed set of operators and $e_{i}$ is a set of complex
numbers. This is often described as the chi matrix representation since
$\varepsilon$ can be completely described by a complex number matrix $\chi$
defined as, $\chi_{mn}\equiv\sum_{i}e_{im}e_{in}^{*}$ given that $\rho$
remains Hermitian with trace one which follows
$\sum_{i}E^{\dagger}_{i}E_{i}=I$ [89].

The input states and measurement projectors must each form a basis for the set
of $n$-qubit density matrices, requiring $d^{2}=2^{2n}$ in each set [92]. This
means $\varepsilon$ is,

$\varepsilon(\rho)=\sum^{d^{2}-1}_{m,n=0}\tilde{E}_{m}\rho\tilde{E}^{\dagger}_{n}\chi_{mn}.$
(5.15)

Matrix $\chi$ is the transformation matrix that describes how much
$\tilde{E}_{m}\rho\tilde{E}^{\dagger}_{n}$ contributes to the final state
$\epsilon(\rho)$, making it the most important value to analyze. Using $\chi$,
the fidelity is then calculated by,

$F(\chi_{th},\chi_{re})=\frac{\mathrm{Tr}(\chi_{re}\chi_{th}^{\dagger})}{\sqrt{\mathrm{Tr}(\chi_{th}^{\dagger}\chi_{th})}\sqrt{\mathrm{Tr}(\chi_{re}^{\dagger}\chi_{re})}},$
(5.16)

where $\chi_{th}$ and $\chi_{re}$ are the theoretical and resulting (measured)
$\chi$ matrices. The two gates that will be preliminary investigated against
the GRAPE iteration number and the time step number are the Hadamard gate and
the CNOT gate with the purpose primarily being to analyze both single and
multi-qubit interactions. The $\chi_{re}$ matrix is seen plotted in figures
5.2 and 5.4.

Figure 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. Figure 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. Figure 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 $I\otimes I$.

Hadamard gate can be intuitively understood through the contribution of the
Pauli bases $\sigma_{x}$ and $\sigma_{z}$ as
$U^{Hadamard}=\frac{1}{\sqrt{2}}(\sigma_{x}+\sigma_{z})$ [10, 114]. A more
formal derivation of the $\chi$ matrix for the ideal Hadamard gate is seen in
Appendix B.3. Thus, when calculating the process tomography, the two matrices
that have to be measured are the $\sigma_{x}$ and $\sigma_{z}$ contributions.
This can be seen in figure 5.2 for the 500 iteration step and 50 time step
case for both the flux and Rydberg systems where the fidelity is approximately
equal to 1. The color bar on the left hand side of each graph shows the phase,
$\theta$, of the phase factor $e^{i\theta}$ multiplied to each $\chi$ element,
$\chi_{mn}$ (seen in Appendix B.3). A reduction of the time steps or the
iteration steps however, produces fidelity errors in the flux system that are
more noticeable than those in the Rydberg system. Furthermore, observing the
flux system, it is possible to note that the number of iterations affects the
system’s fidelity more than the number of steps when reducing each number by a
factor of five.

The GRAPE algorithm’s optimisation using the process tomography theory can be
visualized in time for a better understanding. Figure 5.3(a) depicts a 3D
scatter plot with the axes showing the expected contributions of the
$\sigma_{x},\sigma_{y}$ and $\sigma_{z}$ matrices (depicted as $X$, $Y$, and
$Z$, respectively). The two sets of points show the evolution over time for
the propagation using GRAPE optimisation in purple, using 100 time steps and
500 GRAPE iterations, and the propagation without GRAPE optimisation is in
blue. The ideal Hadamard gate is depicted with the red star. Figure 5.3(b)
shows the distance from the ideal Hadamard gate over time. As seen from figure
5.3(a), the GRAPE algorithm utilizes the controls in order to minimize the
fidelity error towards the target. With the amount of time steps used, it is
possible to see that the GRAPE algorithm has sufficient control at each point
in time in order to counter act the system noise, which is seen pulling the
system towards a $\sigma_{y}$ contribution. The fidelity of the Hadamard gate
officially calculated by Eq.(5.16), can be intuitively thought as the distance
away from the ideal gate as seen in figure 5.3(b).

The CNOT gate on the other hand is constructed using the Pauli bases
$\sigma_{x}$ and $\sigma_{z}$ as well as the identity $I$ as
$U^{CNOT}=\frac{1}{2}(I\otimes I+I\otimes\sigma_{x}+\sigma_{z}\otimes
I-Z\otimes X)$. This is seen in 5.4 where at fidelity 1 (seen in the final
Rydberg graph), the contributions are indeed $I$, $\sigma_{x}$, and
$\sigma_{z}$ where the double grouped indices represent a tensor product, e.g
$II=I\otimes I$. As expected, the contributions $\sigma_{x}\otimes\sigma_{z}$
when paired with $I$ or $\sigma_{x}$ contain a factor $e^{-i\pi}$ where
$\theta=\pi$. The Rydberg system once more produces less fidelity error than
the flux system when the parameters are changed. The flux system surprisingly
does not converge to an almost perfect fidelity as seen in the single qubit
gate example as seen from the incorrect Pauli basis artefacts after 1000 GRAPE
iterations and 200 time steps.

It can be seen that the flux system suffers greater decoherence effects than
the Rydberg system. This can be seen analytically from the noise fluctuation
Hamiltonian, Eq. (4.20) whereby the dynamics are governed by
$\sigma_{x},\sigma_{z}$ and, when considering interactions between flux
qubits, $\sigma_{y}$. The magnitude of the noise parameters make it difficult
for the GRAPE algorithm to optimise the gates. This is commonly the issue with
superconducting qubits, both the charge and the fluctuation noise impact the
decoherence heavily. At $E_{J}/E_{C}\geq 50$, optimised superconducting
C-shunted qubits have a lifetime of 50$\mu$s - 100 $\mu$s [64]. The flux qubit
used in this study is closely based on the flux qubit used in [136] where
$E_{J}/E_{C}=65$. On the other hand, Rydberg qubits in the non-excited state
have very long coherence times compared to superconducting qubits because
atomic states with $L\leq 2$, have 100$\mu$s - 1ms lifetimes, which means that
they do not suffer as much from spontaneous emission [29].

In conclusion, it has been seen that high fidelity single qubit gates are
possible for both the flux and Rydberg atom systems but only the Rydberg
qubits observe high fidelity in the double qubit gates. The impact of these
gates on the phase estimation algorithm will be seen in Section 5.3, where the
algorithm will be executed using differing time step and GRAPE iteration
amounts. Because the flux qubit system has been observed to perform poorly,
differing values for the shunt factor, $\zeta$, will be used in the simulation
of the distributed phase estimation algorithm in order to obtain gates with
higher fidelities.

###  5.3 GRAPE Optimised Distributed Phase Estimation

Section 5.2 successfully demonstrated the GRAPE algorithm optimising the
single and double qubit gate fidelities for both the flux and Rydberg atom
systems. This section will use the optimized gates in order to execute the
distributed phase estimation algorithm. Additionally, Section 5.2 showed that
the flux qubit system performs poorly when compared to the Rydberg atom
system. This section will introduce the modification of the shunt factor
$\zeta$, previously described in Sections 3.2.2 and 4.2, as a variable in
order to analyse potential improvements in the accuracy of the distributed
phase estimation algorithm.

The diagram depicting the simulation can be seen in figure C.1. Section 5.2
showed the creation of the gates, which are placed in a gate dictionary
(purple area), and Section 2.4 detailed the distributed phase estimation
algorithm, which is seen depicted in orange. As one can see, there is an extra
iteration that occurs (depicted as $k$). This is because, during the
distributed phase estimation algorithm, one party needs to measure their ebit
in order to communicate them to the other party. In the analysis, the
expectation value of a specific final state gives insight in the accuracy of
the phase estimation algorithm. However, as part of the wavefunction is
collapsed by intermediate measurements, ten simulation runs are conducted to
calculate the expected outcome. Thus, the orange section of the diagram in
figure C.1 is executed ten times and the mean outcomes are recorded.

Firstly, the gates observed in Section 5.2 will be used in the simulation
whereby, $\zeta=10$. Although the single qubit gates demonstrate a very high
fidelity for the flux qubit, it is important to analyze the impact of the
noisy multi-qubit gate seen in figure 5.4. To do so, the phase to estimate
will be chosen to be $\varphi=3/16$ (or 0011 in the computational basis).
Thus, the accuracy will depend on the mean probability (since the algorithm
will be run 10 times) of achieving this outcome for a variable GRAPE iteration
number and a time step number.

It is important to note that the GRAPE algorithm will not be run on the
communication $E_{2}$ gate because the GRAPE algorithm is not efficient in
these timescales (order of nanoseconds). Because none of the error correction
techniques described in this investigation can be employed on the $E_{2}$
gate, it will be assumed to possess perfect fidelity. This will allow errors
arising from GRAPE to be observed more easily. Following figures 5.2 and 5.4,
the ranges of interest chosen were 50-200 time steps and 100-800 GRAPE
iterations due to the computational resources available. This leads to a
simulation time of approximately 22 hours on a desktop computer.

The flux qubit computer was chosen to possess three qubits, one communication
qubit, one counting qubit and a qubit to hold the phase to estimate. The
Rydberg atom computer was chosen to have four qubits in which, three were
counting qubits and one was a communication qubit. The visualization of the
entire simulation of a distributed phase estimation algorithm involving this
architecture for ranging time steps (50-200) and ranging GRAPE iterations
(100-800) can be seen in figure 5.6, where $\zeta=10$. One can observe that
the accuracy of the system at this shunt value, is mainly dependent on the
amount of GRAPE iterations. This follows that of the conclusion obtained in
Section 5.2, where figure 5.4 showed a better relative performance when
changing the GRAPE iteration number for the flux qubit. The highest accuracy
obtained for estimating $\varphi=3/16$ was around 40%.

Experimentally, the factor $\zeta$ has been modified to be between 1-1000
between different superconducting qubit technologies [12, 136]. Thus, the next
step is to analyse the performance of the architecture when $\zeta$ is
increased by a factor of 10, thus $\zeta=100$. This is seen in figure 5.7. It
can be observed that, although the accuracy does not improve significantly to
that of the $\zeta=10$ case for large GRAPE iteration numbers and time step
numbers, it does saturate to a probability of 40% in less GRAPE iterations and
time step numbers. Overall, at this effective shunt factor, it is still not
able to accurately estimate the phase for the given GRAPE iteration and time
step number range.

A final shunt factor of $\zeta=1000$ is simulated. The simulation results can
be seen in figure 5.8. It can be observed that for approximately 190 time
steps and approximately 700 GRAPE iterations, the algorithm results in an
accuracy of around 95%. The accuracy makes a steep jump from 40% to 95% after
an increase of only 5 time steps (180 to 185 time steps). Observing the range
of very low probability ($<10\%$) between 700-800 GRAPE iterations and 100-160
time steps, it is noticeable that there is a trade off between the time step
number and GRAPE iteration number occurring.

To describe this trade off, the evolution of flux qubit system aiming to
create a Hadamard gate can be observed using two different GRAPE
configurations. One configuration is set to use 100 GRAPE iterations while the
other configuration is set to use 500 GRAPE iterations. This can be seen in
figure 5.5. In the 100 GRAPE iteration case, the fidelity reached at the end
of the evolution is 0.97 while for the 500 GRAPE iteration case, the fidelity
reached is 0.92. The graph shows that, although the final fidelity for the 500
GRAPE iteration case was lower than the 100 GRAPE iteration case at the end of
the run, the 500 grape iteration evolution manages to achieve fidelities
higher than the 100 GRAPE iteration evolution in the total run. Observing the
500 GRAPE iteration evolution after 30 $\mu$s, one can see that the evolution
holds the fidelity around 0.98 but then dips to 0.92. This dip is most likely
due to the fact that the system encounters a control landscape whereby the
time step required to modify the control amplitudes is not small enough to
converge. Thus, at these time steps, a minimum local fidelity is reached until
the landscape changes over time as seen at the 40 $\mu$s mark. This concept is
further explained in [72]. Observing figure 5.8, one can see that at 750 GRAPE
iterations, the optimisation goes from poor convergence in the control
landscape at 170 time steps to proper convergence at 185 time steps.

In conclusion, performing a distributed phase estimation algorithm using the
architecture described in Section 3.3 is shown to achieve accurate results
using GRAPE optimisation configured with a 700-800 GRAPE iteration range and
180-200 time step range (>90%) when the flux qubit observes a high C-shunt
value.

Figure 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. Figure 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. Figure 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. Figure 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.

##  Chapter 6 Final Insights and Concluding Remarks

###  6.1 Conclusion

This investigation aimed to determine whether or not two quantum devices with
distinct physical architectures can collaborate to solve a distributed
computational task. Three sub-questions were proposed in order to answer this
question.

The first sub-question, which explored the feasibility of distributing a
quantum algorithm, was addressed by examining the quantum phase estimation
algorithm in Chapter 2. Section 2.3 demonstrated that, with the use of an
$E_{2}$ gate, a distributed version of the phase estimation algorithm is
possible. Section 2.4 outlined the circuits to distribute the phase estimation
algorithm over two quantum computers, one comprising of three qubits and the
other comprising of four qubits. The visualization of the QuTiP simulation
without noise was presented at the conclusion of the chapter, wherein
variations were made in both the number of counting qubits and the phase to be
estimated (seen in figure 2.13). The simulation proved that the distributed
version of the phase estimation algorithm was as accurate as the local
version.

The second sub-question exploring the feasibility of connecting two quantum
systems was addressed in Chapter 3. The quantum hybrid system was broken down
into three sub-system, the Rydberg atom system seen in Section 3.1, the flux
qubit system seen in Section 3.2 and the flux-resonator-atom system used to
couple both architectures inspired by [141], seen in Section 3.3.

Section 3.1.3 details the Rydberg atom configuration where the Rydberg atom
system described in Section 3.1.2 links to the configuration described by Yu
et al.. The outlined method consists of an atom placed near a resonator (a co-
planar waveguide) with an electric field plate and spherical capacitors
capable of tuning the atom between a $22D_{5/2}$ state and an arbitrary
$\Phi_{n=20}$ state, which can couple and uncouple from the resonator.
Furthermore, an explanation and a diagram was given on how this state could
then couple to the other qubits to enable quantum state transfer between the
atom and the flux qubit. It was concluded that for the Rydberg atom
configuration to operate with the entire hybrid system, the three chosen qubit
states were two computational states in the $\ket{0}=5S_{1/2},F=1$ and
$\ket{1}=5S_{1/2},F=2$ as well as the excited state $\ket{r}=70S_{1/2}$. Using
ARC, this gave a dipole-dipole coefficient $C_{3}=32.45$ GHz
$\mu\mathrm{m}^{3}$ and a van der Waals coefficient of $C_{6}=801.98$ GHz
$\mu\mathrm{m}^{6}$ consequently leading to the chosen Rydberg blockade
distance and the single gate operating distance to be $R_{b}=$ 3.5 $\mu$m and
$R=$ 9 $\mu$m, respectively.

Section 3.2.1 introduces the key elements describing a flux qubit. The
Hamiltonian of the system is shown as well as the anharmoncity of the states
depending on the magnetic frustration and the $\alpha$ component. Section
3.2.2 uses this to describe the flux qubit in the context of the hybrid
system, thus, its coupling to the resonator. The Hamiltonian from section
3.2.1 is modified due to the introduction of the C-shunt as well as by adding
the interaction Hamiltonian between the resonator and flux qubit. The primary
insight was that a trade-off exists between the $\alpha$ value, where lower
$\alpha$ values effectively decrease flux fluctuation noise, while higher
values increase the anharmonicity, a factor enabling the system to interact
with the resonator. The value was ultimately chosen to be $\alpha=0.8$ because
of its high anharmonicity and its use in literature [136].

Section 3.3 explained the flux-resonator-atom connection using the co-planar
waveguide. Following the steps described in [141], Section 3.4 depicted the
system’s ability to evolve into a GHZ state with 0.93 fidelity in 17
nanoseconds as seen in figure 3.11. Chapter 3, thus proved that connecting two
quantum computers is feasible.

In order to address the the third sub-question regarding the possibility of a
distributed system distributing an algorithm with accuracy, a noise model had
to be created on top of the driving Hamiltonians previously described in
Chapter 3. Chapter 4 addresses the noise, in terms of Lindbladian equations,
for the three different systems, the Rydberg system seen in Section 4.1, the
flux system noise seen in Section 4.2 and the flux-resonator-atom coupling
device seen in Section 4.3. The Linbladians described in Sections 4.1 and 4.2
are then used in Chapter 5 to model the noise of the flux and Rydberg atom
computers, however, the Linbladian to establish the connection or GHZ gate for
the flux-resonator-atom system was shown to have no effect in Section 4.3.
Thus, it was concluded that the noise would be neglected for the flux-
resonator-atom system.

Answering the third sub-question requires the use of GRAPE optimised gates,
which were introduced in Section 5.1. Section 5.2 described the implementation
of the master equations involving the driving Hamiltonians (derived in Chapter
3) and the Lindbladians (derived in Chapter 4) for use in the GRAPE algorithm.
This section showed that both the Rydberg and flux systems were able to create
single-qubit high fidelity ($F\approx 1$) gates when 500 GRAPE iterations and
50 time steps were used. For the double-qubit gates, the Rydberg atom managed
to reach a high fidelity ($F\approx 1$) CNOT gate with 1000 GRAPE iterations
and 200 time steps, but the flux qubit gate only managed to reach a fidelity
of $F=0.90$. These optimised gates were then used in Section 5.3 to run the
distributed phase estimation algorithm. The initial result (figure 5.6) showed
that the flux system was too noisy to complete a distributed phase estimation
algorithm. However, changing the shunt effectiveness to $\zeta=1000$, resulted
in an estimated phase with an accuracy of over $90\%$ already at 190 time
steps and 700 GRAPE iterations (configured to be $\varphi=3/16$ or
$\ket{0011}$ in the computational basis). The chapter concluded that a
distributed quantum system is capable of executing a distributed quantum
algorithm with high accuracy.

Overall, with the results having shown the possibility of answering the three
sub-questions posed, the investigation has successfully concluded that two
quantum devices with distinct physical architectures can collaborate to solve
a distributed computational task.

###  6.2 Discussion on Limitations and Further Research

The simulations of this investigation mainly used QuTiP. QuTiP proved to be a
powerful tool due to the package’s ability of easily simulating open quantum
systems, more specifically, open quantum gates with configurable Lindbladians.
However, due to the fact that QuTiP is not yet equipped to handle non-local
processes, the construction of the non-local gates were built using local
assumptions. A consequence of non-locality in distributed quantum computing is
asynchronicity which occurs when two operations are not accomplished
sequentially. The $E_{2}$ gate was simply assumed to be two sequential
operations with an immediate response from both parties over a classical
network. Experimentally, this would not work using current classical
guidelines. Both van Brandwijk and Parakh have shown that the TCP/IP structure
of classical networks would require extra quantum operations from both parties
[123]. This would far exceed the nanosecond timescale gate creation time set
in section 3.4, and would ultimately lead to the noise described in Section
4.3 to be non-negligible [141].

Investigating the TCP/IP effect on the $E_{2}$ gate requires the use of a
different framework for the simulations. QooSim [123] addresses the TCP/IP
issue, but cannot simulate noisy circuits. On the other hand, different
methods of establishing quantum networks are being studied. For example, Wang
et al. proposed a protocol in which an additional virtual path is established
between entangled qubits or nodes connected to repeaters that are responsible
for optimising the fidelity by iterating through links between the qubits
[128]. Further research must evaluate the decoherence loss on hardware due to
classical networks and whether or not a different strategy for classical
communication between quantum computers is worth implementing.

The proposed quantum devices to couple in this investigation were Rydberg atom
computers and superconducting computers. Overall, it is unlikely that a
specific qubit will be standardised throughout quantum computing [37]. Thus,
the selection of both quantum devices will likely remain valid for the
foreseeable future. However, implementing a coupling between both of these
devices can be done in a variety of different forms. In hybrid qubit coupling,
there exists two main methods of entangling two hybrid qubits, direct or
indirect coupling [134]. Direct coupling refers to a coupling in which the
magnetic and electric fields resonate at similar resonances meaning that two
qubits physically next to each other can couple or entangle without the need
of an apparatus in the middle. Indirect coupling is when an apparatus
mitigates the resonances of both qubits, this is seen in this study by the co-
planar waveguide. Direct coupling leads to less decoherence and is, thus, more
desired but is sometimes impossible.

Focusing on the system proposed in this investigation, some limitations of the
system need to be further investigated. The indirect coupling utilized comes
from [141] whereby the $22D_{5/2}$ state was researched numerically as the
coupling state between a flux qubit-resonator-atom system. Other system
architectures may couple better than the system introduced in this
investigation, for example, Thiele et al. experimentally showed that a cloud
of meta-stable helium atoms are potential candidates for coupling to
superconducting surfaces [121] because they are not affected by stray fields
generated by the superconducting surface. Although [141] theorized that the
hybrid entanglement would not be affected by a single flux qubit’s stray
field, an experiment would need to be conducted to verify if that remains
valid.

On top of this, for distributed quantum computation to work in its fullest
extent, the quantum computers should also be able to communicate over longer
distances. The current scenario requires both computers to be approximately in
the same vicinity. It has been shown that qubits can be transferred via fibre
optic cables by indirectly opto-mechanically coupling phonons to photons [41,
71]. This is already partially the case within the co-planar wave guide used
in the investigation, however, further optimisation for photon trapping within
the resonator can be accomplished [61]. To do so, each system would need a
single qubit to be attached to an opto-mechanical resonator and these
resonators have to be attached to a transmission or optical fiber cable.

ARC was utilized to calculate the parameters in the Rydberg atom qubit
configuration. However, ARC does not take into consideration the calculation
of hyperfine states. As seen in figure 3.4(a), it was assumed that the RF
transition between a $\Phi_{n=20}$ state and a $5S_{1/2}F=1$ state was
possible. This would need to be further researched experimentally. It may be
the case that an auxiliary state needs to be provided, which would lead to an
increase in decoherence.

The flux qubit was shown to suffer more decoherence than the Rydberg atom
qubit. However, because the hybrid system from [141] was chosen, it put
further strain on the flux qubit. For the system to work, the anharmonic
energy levels in figure 3.8(d) are required in order for the system to
properly couple to the resonator without affecting the rest of the flux qubit
system. C-shunt flux qubits operate best (with an approximate decoherence of
100MHz) when $\alpha<0.5$. Unfortunately, this cannot be done on the system
used because there is a large drop in anharmonicity when $\alpha$ drops below
0.5 as seen in figure 3.7(b). This would mean that the $\ket{R}$ and $\ket{L}$
states would disappear and the flux qubit system would constantly be coupled
to the resonator, thus it would generalize the states to a $\ket{0}$ and
$\ket{1}$ energy state, which forbids the construction of the GHZ state seen
Section 3.4.

In order to show the effect a C-shunt flux qubit with $\alpha=0.49$ has on
noise, the process tomography for the construction of a CNOT gate is
visualized in figure 6.1. The gate constructed using GRAPE with 1000 GRAPE
iterations and 200 time steps has a 0.09 higher fidelity than the gate
obtained when $\alpha=0.8$, depicted in figure 5.4. Utilizing the optimised
gates with a flux qubit $\alpha=0.49$ in the distributed phase algorithm is
seen in figure 6.2. As observed, the distributed phase estimation algorithm,
estimates the correct state with 90% accuracy, at the 400 iteration mark with
120 time steps and at the 500 iteration mark with 60 time steps. This occurs
in less time steps/GRAPE iterations when compared to the $\alpha=0.8$ case
(seen in figure 5.8). Using $\alpha=0.49$, however, is not theoretically
feasible since it would introduce further noise effects caused by lack of
anharmonicity. Thus, a method in which the connection does not rely on the
qubit’s anharmonicity must be researched in order to utilize C-shunt flux
qubits with $\alpha<0.5$ values.

Although the GRAPE algorithm was mainly used in this investigation, it is
worth noting that other algorithms for optimal control exist. The two notable
ones are Krotov [47] and CRAB [38]. Krotov was initially tested but it was far
too computationally expensive when optimising pulses for a multi-qubit gate.
The main strength Krotov possesses is its ability to converge to small
fidelity errors when optimising systems comprised of many qubits [47]. CRAB on
the other hand, did not converge at the given fidelity error rate at all
because it is a gradient-free optimisation and not robust enough to minimise
the fidelity using the controls. Goerz et al. states that GRAPE must be used
when the control parameters are discrete and the derivative of error rate with
respect to the control parameters are known and Hermitian while Krotov is used
when the control parameters are near-continuous without the restriction of
Hermitian Hamiltonians [47]. In the case used in this research, it is apparent
that the best choice was indeed GRAPE.

Instead of using optimal quantum control, one can use variational quantum
algorithms (VQAs) to optimise quantum circuits. Instead of pulsing a quantum
gate at pulse level, one could replace this with a gate or series of gates
capable of manipulating the quantum state to match the process of a perfect
quantum gate. Although GRAPE cannot be utilized in VQAs, other algorithms or
machine learning techniques offer a less intrusive method of engineering gates
[81]. This would defeat the need to offer custom control parameters assuming
both systems can generate Pauli basis states. Variational quantum algorithms
and quantum optimal control can be used in parallel, offering an interesting
expansion of this study [81]. Integrating VQAs through machine learning could
simulate circuit noise and subsequently rectify it, thus enhancing the
examination of the original assumption, which solely introduced noise during
qubit manipulation.

Figure 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 $I\otimes I$. Figure 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.

## References

  * Abdelhafez et al. [2019] Mohamed Abdelhafez, David I. Schuster, and Jens Koch.  Gradient-based optimal control of open quantum systems using quantum trajectories and automatic differentiation.  _Physical Review A_ , 99(5), may 2019.  doi: 10.1103/physreva.99.052327.  URL https://doi.org/10.1103%2Fphysreva.99.052327.
  * Abdussalam [2017] Wildan Abdussalam.  _Dynamics of Rydberg atom lattices in the presence of noise and dissipation_.  Phd thesis, Technische Universität Dresden, Dresden, Germany, August 2017\.  Available at https://nbn-resolving.org/urn:nbn:de:bsz:14-qucosa-227485.
  * Abramowitz and Stegun [2013] Milton Abramowitz and Irene A. Stegun.  _Handbook of Mathematical Functions with formulas, graphs, and mathematical tables_.  Dover Publ, 2013.
  * Ackermann [2023] Rebecca Ackermann.  Black mirror’s big ai episode has the wrong villain, 2023.  URL https://www.vox.com/culture/2023/6/23/23770239/black-mirror-joan-is-awful-review-ai-annie-murphy-salma-hayek.
  * Ansel et al. [2022] Quentin Ansel, Jonas Fischer, Dominique Sugny, and Bruno Bellomo.  Optimal control and selectivity of qubits in contact with a structured environment.  _Physical Review A_ , 106(4), oct 2022.  doi: 10.1103/physreva.106.043702.  URL https://doi.org/10.1103%2Fphysreva.106.043702.
  * Bates and Damgaard [1949] D. R. Bates and Agnete Damgaard.  The calculation of the absolute strengths of spectral lines.  _Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences_ , 242(842):101–122, 1949.  ISSN 00804614.  URL http://www.jstor.org/stable/91450.
  * Benenti et al. [2018] Giuliano Benenti, Giulio Casati, Davide Rossini, and Giuliano Strini.  _Principles of Quantum Computation and Information_.  WORLD SCIENTIFIC, 2018.  doi: 10.1142/10909.  URL https://www.worldscientific.com/doi/abs/10.1142/10909.
  * Benioff [1980] Paul Benioff.  The computer as a physical system: A microscopic quantum mechanical hamiltonian model of computers as represented by turing machines.  _Journal of Statistical Physics_ , 22:563–591, 05 1980\.  doi: 10.1007/BF01011339.
  * Bernstein and Vazirani [1993] Ethan Bernstein and Umesh Vazirani.  Quantum complexity theory.  In _Proceedings of the Twenty-Fifth Annual ACM Symposium on Theory of Computing_ , STOC ’93, page 11–20, New York, NY, USA, 1993. Association for Computing Machinery.  ISBN 0897915917.  doi: 10.1145/167088.167097.  URL https://doi.org/10.1145/167088.167097.
  * Beterov et al. [2016] I I Beterov, M Saffman, E A Yakshina, D B Tretyakov, V M Entin, G N Hamzina, and I I Ryabtsev.  Simulated quantum process tomography of quantum gates with rydberg superatoms.  _Journal of Physics B: Atomic, Molecular and Optical Physics_ , 49(11):114007, may 2016.  doi: 10.1088/0953-4075/49/11/114007.  URL https://doi.org/10.1088%2F0953-4075%2F49%2F11%2F114007.
  * Beterov et al. [2018] I. I. Beterov, G. N. Hamzina, E. A. Yakshina, D. B. Tretyakov, V. M. Entin, and I. I. Ryabtsev.  Adiabatic passage of radio-frequency-assisted förster resonances in rydberg atoms for two-qubit gates and the generation of bell states.  _Physical Review A_ , 97(3), mar 2018.  doi: 10.1103/physreva.97.032701.  URL https://doi.org/10.1103%2Fphysreva.97.032701.
  * Birenbaum [2014] Jeffrey Birenbaum.  The c-shunt flux qubit: A new generation of superconducting flux qubit.  2014\.  URL https://api.semanticscholar.org/CorpusID:124189962.
  * Blatt and Wineland [2008] R. Blatt and David Wineland.  Entangled states of trapped atomic ions.  _Nature_ , 453:1008–15, 07 2008.  doi: 10.1038/nature07125.
  * Bohr [1913] Niels Bohr.  On the constitution of atoms and molecules, part i.  _Philosophical Magazine_ , 26:1–25, 1913.
  * Boutin et al. [2017] Samuel Boutin, Christian Kraglund Andersen, Jayameenakshi Venkatraman, Andrew J. Ferris, and Alexandre Blais.  Resonator reset in circuit qed by optimal control for large open quantum systems.  _Physical Review A_ , 96(4), oct 2017.  doi: 10.1103/physreva.96.042315.  URL https://doi.org/10.1103%2Fphysreva.96.042315.
  * Brassard et al. [1998] Gilles Brassard, Peter HØyer, and Alain Tapp.  Quantum counting.  In Kim G. Larsen, Sven Skyum, and Glynn Winskel, editors, _Automata, Languages and Programming_ , pages 820–831, Berlin, Heidelberg, 1998. Springer Berlin Heidelberg.  ISBN 978-3-540-68681-1.
  * Brennen [2003] Gavin K. Brennen.  An observable measure of entanglement for pure states of multi-qubit systems.  2003\.  doi: 10.48550/ARXIV.QUANT-PH/0305094.  URL https://arxiv.org/abs/quant-ph/0305094.
  * Breuer and Petruccione [2006] Heinz-Peter Breuer and Francesco Petruccione.  _The Theory of Open Quantum Systems_.  01 2006.  ISBN 9780199213900.  doi: 10.1093/acprof:oso/9780199213900.001.0001.
  * Breuer and Petruccione [2007] Heinz-Peter Breuer and Francesco Petruccione.  _The Theory of Open Quantum Systems_.  Oxford University Press, 01 2007.  ISBN 9780199213900.  doi: 10.1093/acprof:oso/9780199213900.001.0001.  URL https://doi.org/10.1093/acprof:oso/9780199213900.001.0001.
  * Briand and Kleppner [1987] Jean-Pierre Briand and Daniel Kleppner.  _Introduction to Rydberg Atoms_ , page 57–75.  Plenum Press, 1987.
  * Brooks [2019] Michael Brooks.  Beyond quantum supremacy: the hunt for useful quantum computers.  _Nature_ , 574:19 – 21, 2019.  URL https://api.semanticscholar.org/CorpusID:203626094.
  * Byrd et al. [1995] Richard H. Byrd, Peihuang Lu, Jorge Nocedal, and Ciyou Zhu.  A limited memory algorithm for bound constrained optimization.  _SIAM Journal on Scientific Computing_ , 16(5):1190–1208, 1995.  doi: 10.1137/0916069.  URL https://doi.org/10.1137/0916069.
  * Caleffi et al. [2022] Marcello Caleffi, Michele Amoretti, Davide Ferrari, Daniele Cuomo, Jessica Illiano, Antonio Manzalini, and Angela Sara Cacciapuoti.  Distributed quantum computing: a survey, 2022.
  * Cappellaro [2012] Paola Cappellaro.  Quantum theory of radiation interactions, Fall 2012.
  * Chakraborty et al. [2023] Chiranjib Chakraborty, Manojit Bhattacharya, Kuldeep Dhama, and Sang-Soo Lee.  Quantum computing on nucleic acid research: Approaching towards next-generation computing.  _Molecular Therapy - Nucleic Acids_ , 33:53–56, 2023.  ISSN 2162-2531.  doi: https://doi.org/10.1016/j.omtn.2023.06.007.  URL https://www.sciencedirect.com/science/article/pii/S2162253123001488.
  * Chen et al. [2022] Yuquan Chen, Yajie Hao, Ze Wu, Bi-Ying Wang, Ran Liu, Yanjun Hou, Jiangyu Cui, Man-Hong Yung, and Xinhua Peng.  Iterative gradient ascent pulse engineering algorithm for quantum optimal control, 2022.
  * Chiorescu et al. [2003] I. Chiorescu, Y. Nakamura, C. J. P. M. Harmans, and J. E. Mooij.  Coherent quantum dynamics of a superconducting flux qubit.  _Science_ , 299(5614):1869–1871, mar 2003.  doi: 10.1126/science.1081045.  URL https://doi.org/10.1126%2Fscience.1081045.
  * Cleve et al. [1998] R. Cleve, A. Ekert, C. Macchiavello, and M. Mosca.  Quantum algorithms revisited.  _Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences_ , 454(1969):339–354, jan 1998.  doi: 10.1098/rspa.1998.0164.  URL https://doi.org/10.1098%2Frspa.1998.0164.
  * Cohen and Thompson [2021] Sam R. Cohen and Jeff D. Thompson.  Quantum computing with circular rydberg atoms.  _PRX Quantum_ , 2:030322, Aug 2021.  doi: 10.1103/PRXQuantum.2.030322.  URL https://link.aps.org/doi/10.1103/PRXQuantum.2.030322.
  * Coppersmith [2002] D. Coppersmith.  An approximate fourier transform useful in quantum factoring, 2002.  URL https://arxiv.org/abs/quant-ph/0201067.
  * Cutress [2022] Dr. Ian Cutress.  Intel’s process roadmap to 2025: with 4nm, 3nm, 20a and 18a?!, 2022.  URL https://www.anandtech.com/show/16823/intel-accelerated-offensive-process-roadmap-updates-to-10nm-7nm-4nm-3nm-20a-18a-packaging-foundry-emib-foveros.
  * Dalgaard et al. [2020] Mogens Dalgaard, Felix Motzoi, Jesper Hasseriis Mohr Jensen, and Jacob Sherson.  Hessian-based optimization of constrained quantum control.  _Physical Review A_ , 102(4), oct 2020.  doi: 10.1103/physreva.102.042612.  URL https://doi.org/10.1103%2Fphysreva.102.042612.
  * de Fouquieres et al. [2011] P. de Fouquieres, S.G. Schirmer, S.J. Glaser, and Ilya Kuprov.  Second order gradient ascent pulse engineering.  _Journal of Magnetic Resonance_ , 212(2):412–417, oct 2011.  doi: 10.1016/j.jmr.2011.07.023.  URL https://doi.org/10.1016%2Fj.jmr.2011.07.023.
  * de Vega and Alonso [2017] Inés de Vega and Daniel Alonso.  Dynamics of non-markovian open quantum systems.  _Rev. Mod. Phys._ , 89:015001, Jan 2017.  doi: 10.1103/RevModPhys.89.015001.  URL https://link.aps.org/doi/10.1103/RevModPhys.89.015001.
  * Deutsch [1985] David Deutsch.  Quantum theory, the church–turing principle and the universal quantum computer.  _Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences_ , 400(1818):97–117, 1985.  doi: 10.1098/rspa.1985.0070.
  * Ditzhuijzen [2009] C.S.E van Ditzhuijzen.  _Dipole-dipole interaction between cold Rydberg atoms_.  PhD thesis, s.n., 2009.
  * DiVincenzo [2000] David P. DiVincenzo.  The physical implementation of quantum computation.  _Fortschritte der Physik_ , 48(9-11):771–783, sep 2000.  doi: 10.1002/1521-3978(200009)48:9/11<771::aid-prop771>3.0.co;2-e.  URL https://doi.org/10.1002%2F1521-3978%28200009%2948%3A9%2F11%3C771%3A%3Aaid-prop771%3E3.0.co%3B2-e.
  * Doria et al. [2011] Patrick Doria, Tommaso Calarco, and Simone Montangero.  Optimal control technique for many-body quantum dynamics.  _Physical Review Letters_ , 106(19), may 2011.  doi: 10.1103/physrevlett.106.190501.  URL https://doi.org/10.1103%2Fphysrevlett.106.190501.
  * Eindhoven [2023] Technische Universiteit Eindhoven.  Rydberg atom quantum computing and simulation, 2023.  URL https://quantumdelta.nl/catalyst-programs/quantum-computing-simulation.
  * Eisert et al. [2000] J. Eisert, K. Jacobs, P. Papadopoulos, and M. B. Plenio.  Optimal local implementation of nonlocal quantum gates.  _Physical Review A_ , 62(5), oct 2000.  doi: 10.1103/physreva.62.052317.  URL https://doi.org/10.1103%2Fphysreva.62.052317.
  * Fiaschi et al. [2021] Niccolò Fiaschi, Bas Hensen, Andreas Wallucks, Rodrigo Benevides, Jie Li, Thiago P. Mayer Alegre, and Simon Gröblacher.  Optomechanical quantum teleportation.  _Nature Photonics_ , 15(11):817–821, oct 2021\.  doi: 10.1038/s41566-021-00866-z.  URL https://doi.org/10.1038%2Fs41566-021-00866-z.
  * Fletcher [2000] R. Fletcher.  _Newton-Like Methods_ , chapter 3, pages 44–79.  John Wiley & Sons, Ltd, 2000.  ISBN 9781118723203.  doi: https://doi.org/10.1002/9781118723203.ch3.  URL https://onlinelibrary.wiley.com/doi/abs/10.1002/9781118723203.ch3.
  * Fowler and Hollenberg [2004] Austin G. Fowler and Lloyd C. L. Hollenberg.  Scalability of shor’s algorithm with a limited set of rotation gates.  _Physical Review A_ , 70(3), sep 2004.  doi: 10.1103/physreva.70.032329.  URL https://doi.org/10.1103%2Fphysreva.70.032329.
  * Friedman et al. [2000] Jonathan Friedman, Vijay Patel, Wei Chen, Sergey Tolpygo, and JE Lukens.  Quantum superposition of distinct macroscopic states.  _Nature_ , 406:43–6, 08 2000.  doi: 10.1038/35017505.
  * Gambetta et al. [2017] Jay M. Gambetta, Jerry M. Chow, and Matthias Steffen.  Building logical qubits in a superconducting quantum computing system.  _npj Quantum Information_ , 3(1), jan 2017.  doi: 10.1038/s41534-016-0004-0.  URL https://doi.org/10.1038%2Fs41534-016-0004-0.
  * Gea-Banacloche et al. [1995] Julio Gea-Banacloche, Yong-qing Li, Shao-zheng Jin, and Min Xiao.  Electromagnetically induced transparency in ladder-type inhomogeneously broadened media: Theory and experiment.  _Phys. Rev. A_ , 51:576–584, Jan 1995.  doi: 10.1103/PhysRevA.51.576.  URL https://link.aps.org/doi/10.1103/PhysRevA.51.576.
  * Goerz et al. [2019] Michael H. Goerz, Daniel Basilewitsch, Fernando Gago-Encinas, Matthias G. Krauss, Karl P. Horn, Daniel M. Reich, and Christiane P. Koch.  Krotov: A python implementation of krotov’s method for quantum optimal control.  _SciPost Phys._ , 7:080, 2019.  doi: 10.21468/SciPostPhys.7.6.080.  URL https://scipost.org/10.21468/SciPostPhys.7.6.080.
  * Goerz [2015] Michael Hartmut Goerz.  _Optimizing Robust Quantum Gates in Open Quantum Systems_.  PhD thesis, Kassel, Universität Kassel, Fachbereich Mathematik und Naturwissenschaften, 05 2015.
  * Guccione et al. [2020] Giovanni Guccione, Tom Darras, Hanna Le Jeannic, Varun B. Verma, Sae Woo Nam, Adrien Cavaillès, and Julien Laurat.  Connecting heterogeneous quantum networks by hybrid entanglement swapping.  _Science Advances_ , 6(22):eaba4508, 2020.  doi: 10.1126/sciadv.aba4508.  URL https://www.science.org/doi/abs/10.1126/sciadv.aba4508.
  * Harris et al. [2007] R. Harris, A. J. Berkley, M. W. Johnson, P. Bunyk, S. Govorkov, M. C. Thom, S. Uchaikin, A. B. Wilson, J. Chung, E. Holtham, J. D. Biamonte, A. Yu. Smirnov, M. H. S. Amin, and Alec Maassen van den Brink.  Sign- and magnitude-tunable coupler for superconducting flux qubits.  _Physical Review Letters_ , 98(17), apr 2007.  doi: 10.1103/physrevlett.98.177001.  URL https://doi.org/10.1103%2Fphysrevlett.98.177001.
  * Harrow et al. [2009] Aram W. Harrow, Avinatan Hassidim, and Seth Lloyd.  Quantum algorithm for linear systems of equations.  _Phys. Rev. Lett._ , 103:150502, Oct 2009.  doi: 10.1103/PhysRevLett.103.150502.  URL https://link.aps.org/doi/10.1103/PhysRevLett.103.150502.
  * Heinz et al. [2023] Irina Heinz, Adam R. Mills, Jason R. Petta, and Guido Burkard.  Analysis and mitigation of residual exchange coupling in linear spin qubit arrays.  2023\.  doi: 10.48550/arXiv.2308.11308.  URL https://doi.org/10.48550/arXiv.2308.11308.
  * Hita-Pérez et al. [2022] María Hita-Pérez, Gabriel Jaumà, Manuel Pino, and Juan José García-Ripoll.  Ultrastrong capacitive coupling of flux qubits.  _Phys. Rev. Appl._ , 17:014028, Jan 2022.  doi: 10.1103/PhysRevApplied.17.014028.  URL https://link.aps.org/doi/10.1103/PhysRevApplied.17.014028.
  * Huang et al. [2023] Cupjin Huang, Tenghui Wang, Feng Wu, Dawei Ding, Qi Ye, Linghang Kong, Fang Zhang, Xiaotong Ni, Zhijun Song, Yaoyun Shi, Hui-Hai Zhao, Chunqing Deng, and Jianxin Chen.  Quantum instruction set design for performance.  _Physical Review Letters_ , 130(7), feb 2023.  doi: 10.1103/physrevlett.130.070601.  URL https://doi.org/10.1103%2Fphysrevlett.130.070601.
  * Hughes [1967] J W B Hughes.  Stark states and o(4) symmetry of hydrogenic atoms.  _Proceedings of the Physical Society_ , 91(4):810, aug 1967.  doi: 10.1088/0370-1328/91/4/306.  URL https://dx.doi.org/10.1088/0370-1328/91/4/306.
  * Jaksch et al. [2000] D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Côté, and M. D. Lukin.  Fast quantum gates for neutral atoms.  _Physical Review Letters_ , 85(10):2208–2211, sep 2000.  doi: 10.1103/physrevlett.85.2208.  URL https://doi.org/10.1103%2Fphysrevlett.85.2208.
  * Joachain [1983] C. J. Joachain.  _Quantum collision theory_.  North-Holland, 1983.
  * Johansson et al. [2013] J.R. Johansson, P.D. Nation, and Franco Nori.  Qutip 2: A python framework for the dynamics of open quantum systems.  _Computer Physics Communications_ , 184(4):1234–1240, 2013.  ISSN 0010-4655.  doi: https://doi.org/10.1016/j.cpc.2012.11.019.  URL https://www.sciencedirect.com/science/article/pii/S0010465512003955.
  * Jr. [1974] Murray Sargent III; Marlan O. Scully; Willis E. Jr.  _Laser Physics_.  CRC Press, 1974.  ISBN 9780201069037.
  * Kaiser et al. [2021] Manuel Kaiser, Conny Glaser, Li Yuan Ley, Jens Grimmel, Helge Hattermann, Daniel Bothner, Dieter Koelle, Reinhold Kleiner, David Petrosyan, Andreas Günther, and József Fortágh.  Cavity driven rabi oscillations between rydberg states of atoms trapped on a superconducting atom chip, 2021.
  * Kellner et al. [2023] Nicole Kellner, Niklas Hüttner, Marco Will, Pertti Hakonen, and Andreas K. Hüttel.  Stepwise fabrication and optimization of coplanar waveguide resonator hybrid devices.  _physica status solidi (b)_ , jun 2023.  doi: 10.1002/pssb.202300187.  URL https://doi.org/10.1002%2Fpssb.202300187.
  * Khaneja et al. [2005] Navin Khaneja, Timo Reiss, Cindie Kehlet, Thomas Schulte-Herbrüggen, and Steffen J. Glaser.  Optimal control of coupled spin dynamics: design of nmr pulse sequences by gradient ascent algorithms.  _Journal of Magnetic Resonance_ , 172(2):296–305, 2005.  ISSN 1090-7807.  doi: https://doi.org/10.1016/j.jmr.2004.11.004.  URL https://www.sciencedirect.com/science/article/pii/S1090780704003696.
  * Kitaev [1995] A. Yu. Kitaev.  Quantum measurements and the abelian stabilizer problem, 1995.  URL https://arxiv.org/abs/quant-ph/9511026.
  * Kjaergaard et al. [2020] Morten Kjaergaard, Mollie E. Schwartz, Jochen Braumüller, Philip Krantz, Joel I.-J. Wang, Simon Gustavsson, and William D. Oliver.  Superconducting qubits: Current state of play.  _Annual Review of Condensed Matter Physics_ , 11(1):369–395, 2020.  doi: 10.1146/annurev-conmatphys-031119-050605.  URL https://doi.org/10.1146/annurev-conmatphys-031119-050605.
  * Koch et al. [2022] Christiane P. Koch, Ugo Boscain, Tommaso Calarco, Gunther Dirr, Stefan Filipp, Steffen J. Glaser, Ronnie Kosloff, Simone Montangero, Thomas Schulte-Herbrüggen, Dominique Sugny, and Frank K. Wilhelm.  Quantum optimal control in quantum technologies. strategic report on current status, visions and goals for research in europe.  _EPJ Quantum Technology_ , 9(1), jul 2022.  doi: 10.1140/epjqt/s40507-022-00138-x.  URL https://doi.org/10.1140%2Fepjqt%2Fs40507-022-00138-x.
  * Kostelecký and Nieto [1985] V. Alan Kostelecký and Michael Martin Nieto.  Analytical wave functions for atomic quantum-defect theory.  _Phys. Rev. A_ , 32:3243–3246, Dec 1985.  doi: 10.1103/PhysRevA.32.3243.  URL https://link.aps.org/doi/10.1103/PhysRevA.32.3243.
  * Kounalakis [2019] M. Kounalakis.  _Nonlinear couplings for quantum control of superconducting qubits and electrical/mechanical resonators_.  Phd thesis, Technische Universiteit Delft, Delft, The Netherlands, Dec 2019.  Available at https://doi.org/10.4233/uuid:7ffe6dfa-2a24-44d1-991f-7e1bbe2397d6.
  * Kouwenhoven and Mooij [2005] Leo Kouwenhoven and J. Mooij.  Quantum information processing and communication: Strategic report on current status, visions and goals for research in europe.  01 2005.
  * Labuhn et al. [2016] Henning Labuhn, Daniel Barredo, Sylvain Ravets, Sylvain de Léséleuc, Tommaso Macrì, Thierry Lahaye, and Antoine Browaeys.  Tunable two-dimensional arrays of single rydberg atoms for realizing quantum ising models.  _Nature_ , 534(7609):667–670, 2016.  doi: 10.1038/nature18274.  URL https://hal.science/hal-01366953.
  * Lachance-Quirion et al. [2019] Dany Lachance-Quirion, Yutaka Tabuchi, Arnaud Gloppe, Koji Usami, and Yasunobu Nakamura.  Hybrid quantum systems based on magnonics.  _Applied Physics Express_ , 12(7):070101, jun 2019\.  doi: 10.7567/1882-0786/ab248d.  URL https://doi.org/10.7567%2F1882-0786%2Fab248d.
  * Lago-Rivera et al. [2023] Dario Lago-Rivera, Jelena V. Rakonjac, Samuele Grandi, and Hugues de Riedmatten.  Long distance multiplexed quantum teleportation from a telecom photon to a solid-state qubit.  _Nature Communications_ , 14(1), apr 2023.  doi: 10.1038/s41467-023-37518-5.  URL https://doi.org/10.1038%2Fs41467-023-37518-5.
  * Larocca and Wisniacki [2021] Martín Larocca and Diego Wisniacki.  Krylov-subspace approach for the efficient control of quantum many-body dynamics.  _Physical Review A_ , 103(2), feb 2021.  doi: 10.1103/physreva.103.023107.  URL https://doi.org/10.1103%2Fphysreva.103.023107.
  * Leclerc and Henriet [2022] Lucas Leclerc and Loïc Henriet.  Pusho, a pulse-level variational quantum algorithm for rydberg atom platforms.  In _2022 IEEE International Conference on Quantum Computing and Engineering (QCE)_ , pages 839–841, 2022.  doi: 10.1109/QCE53715.2022.00138.
  * [74] Nelson Leung.  Quantum computing with distributed modules.  page 184.  doi: https://doi.org/10.6082/uchicago.1700.  URL http://knowledge.uchicago.edu/record/1700.
  * Levine et al. [2019] Harry Levine, Alexander Keesling, Giulia Semeghini, Ahmed Omran, Tout T. Wang, Sepehr Ebadi, Hannes Bernien, Markus Greiner, Vladan Vuletić, Hannes Pichler, and Mikhail D. Lukin.  Parallel implementation of high-fidelity multiqubit gates with neutral atoms.  _Physical Review Letters_ , 123(17), oct 2019.  doi: 10.1103/physrevlett.123.170503.  URL https://doi.org/10.1103%2Fphysrevlett.123.170503.
  * Li et al. [2022] Boxi Li, Shahnawaz Ahmed, Sidhant Saraogi, Neill Lambert, Franco Nori, Alexander Pitchford, and Nathan Shammah.  Pulse-level noisy quantum circuits with qutip.  _Quantum_ , 6:630, jan 2022.  doi: 10.22331/q-2022-01-24-630.  URL https://doi.org/10.22331%2Fq-2022-01-24-630.
  * Lindblad [1976] G. Lindblad.  On the generators of quantum dynamical semigroups.  _Communications in Mathematical Physics_ , 48(2):119–130, 1976.  doi: 10.1007/bf01608499.
  * Lu et al. [2017] Dawei Lu, Keren Li, Jun Li, Hemant Katiyar, Annie Jihyun Park, Guanru Feng, Tao Xin, Hang Li, Guilu Long, Aharon Brodutch, Jonathan Baugh, Bei Zeng, and Raymond Laflamme.  Enhancing quantum control by bootstrapping a quantum processor of 12 qubits.  _npj Quantum Information_ , 3(1), oct 2017.  doi: 10.1038/s41534-017-0045-z.  URL https://doi.org/10.1038%2Fs41534-017-0045-z.
  * Lukin et al. [2001] M. D. Lukin, M. Fleischhauer, R. Cote, L. M. Duan, D. Jaksch, J. I. Cirac, and P. Zoller.  Dipole blockade and quantum information processing in mesoscopic atomic ensembles.  _Physical Review Letters_ , 87(3), jun 2001.  doi: 10.1103/physrevlett.87.037901.  URL https://doi.org/10.1103%2Fphysrevlett.87.037901.
  * Machnes et al. [2011] S. Machnes, U. Sander, S. J. Glaser, P. de Fouquières, A. Gruslys, S. Schirmer, and T. Schulte-Herbrüggen.  Comparing, optimizing, and benchmarking quantum-control algorithms in a unifying programming framework.  _Phys. Rev. A_ , 84:022305, Aug 2011.  doi: 10.1103/PhysRevA.84.022305.  URL https://link.aps.org/doi/10.1103/PhysRevA.84.022305.
  * Magann et al. [2021] Alicia B. Magann, Christian Arenz, Matthew D. Grace, Tak-San Ho, Robert L. Kosut, Jarrod R. McClean, Herschel A. Rabitz, and Mohan Sarovar.  From pulses to circuits and back again: A quantum optimal control perspective on variational quantum algorithms.  _PRX Quantum_ , 2(1), jan 2021.  doi: 10.1103/prxquantum.2.010101.  URL https://doi.org/10.1103%2Fprxquantum.2.010101.
  * McKay et al. [2017] David C. McKay, Christopher J. Wood, Sarah Sheldon, Jerry M. Chow, and Jay M. Gambetta.  Efficient z-gates for quantum computing.  _Physical Review A_ , 96(2), aug 2017.  doi: 10.1103/physreva.96.022330.  URL https://doi.org/10.1103%2Fphysreva.96.022330.
  * Meschede et al. [1985] D. Meschede, H. Walther, and G. Müller.  One-atom maser.  _Phys. Rev. Lett._ , 54:551–554, Feb 1985.  doi: 10.1103/PhysRevLett.54.551.  URL https://link.aps.org/doi/10.1103/PhysRevLett.54.551.
  * Moore [2006] Gordon E. Moore.  Cramming more components onto integrated circuits, reprinted from electronics, volume 38, number 8, april 19, 1965, pp.114 ff.  _IEEE Solid-State Circuits Society Newsletter_ , 11(3):33–35, 2006.  doi: 10.1109/N-SSC.2006.4785860.
  * Morgado and Whitlock [2021] M. Morgado and S. Whitlock.  Quantum simulation and computing with rydberg-interacting qubits.  _AVS Quantum Science_ , 3(2):023501, jun 2021\.  doi: 10.1116/5.0036562.  URL https://doi.org/10.1116%2F5.0036562.
  * Morgan and Hogan [2020] A. A. Morgan and S. D. Hogan.  Coupling rydberg atoms to microwave fields in a superconducting coplanar waveguide resonator.  _Phys. Rev. Lett._ , 124:193604, May 2020.  doi: 10.1103/PhysRevLett.124.193604.  URL https://link.aps.org/doi/10.1103/PhysRevLett.124.193604.
  * Neumann et al. [2020] Niels M. P. Neumann, Roy van Houte, and Thomas Attema.  Imperfect distributed quantum phase estimation.  In Valeria V. Krzhizhanovskaya, Gábor Závodszky, Michael H. Lees, Jack J. Dongarra, Peter M. A. Sloot, Sérgio Brissos, and João Teixeira, editors, _Computational Science – ICCS 2020_ , pages 605–615, Cham, 2020. Springer International Publishing.  ISBN 978-3-030-50433-5.
  * Newrock et al. [2000] R.S. Newrock, C.J. Lobb, U. Geigenmüller, and M. Octavio.  The two-dimensional physics of josephson junction arrays.  volume 54 of _Solid State Physics_ , pages 263–512. Academic Press, 2000.  doi: https://doi.org/10.1016/S0081-1947(08)60250-7.  URL https://www.sciencedirect.com/science/article/pii/S0081194708602507.
  * Nielsen and Chuang [2021] Michael A. Nielsen and Isaac L. Chuang.  _Quantum Computation and Quantum Information_.  Cambridge University Press, 2021.
  * NL [2023] Quantum Delta NL.  Realising the unlimited possibilities of quantum probabilities, 2023.  URL https://quantumdelta.nl/catalyst-programs/quantum-computing-simulation.
  * Noel et al. [2022] Crystal Noel, Pradeep Niroula, Daiwei Zhu, Andrew Risinger, Laird Egan, Debopriyo Biswas, Marko Cetina, Alexey V. Gorshkov, Michael J. Gullans, David A. Huse, and Christopher Monroe.  Measurement-induced quantum phases realized in a trapped-ion quantum computer.  _Nature Physics_ , 18(7):760–764, jun 2022.  doi: 10.1038/s41567-022-01619-7.  URL https://doi.org/10.1038%2Fs41567-022-01619-7.
  * O'Brien et al. [2004] J. L. O'Brien, G. J. Pryde, A. Gilchrist, D. F. V. James, N. K. Langford, T. C. Ralph, and A. G. White.  Quantum process tomography of a controlled-not gate.  _Physical Review Letters_ , 93(8), aug 2004.  doi: 10.1103/physrevlett.93.080502.  URL https://doi.org/10.1103%2Fphysrevlett.93.080502.
  * Orlando et al. [1999] T. P. Orlando, J. E. Mooij, Lin Tian, Caspar H. van der Wal, L. S. Levitov, Seth Lloyd, and J. J. Mazo.  Superconducting persistent-current qubit.  _Phys. Rev. B_ , 60:15398–15413, Dec 1999.  doi: 10.1103/PhysRevB.60.15398.  URL https://link.aps.org/doi/10.1103/PhysRevB.60.15398.
  * Paauw [2009] Floor Paauw.  Superconducting flux qubits: Quantum chains and tunable qubits.  pages 11–97, 2009.
  * Patsch [2022] Sabrina Patsch.  _Control of Rydberg atoms for quantum technologies_.  Dissertation, 2022.  URL http://dx.doi.org/10.17169/refubium-34581.
  * Petrini et al. [2018] Michela Petrini, Gianfranco Pradisi, and Alberto Zaffaroni.  A guide to mathematical methods for physicists.  _Advanced Textbooks in Physics_ , 2018.  doi: 10.1142/q0162.
  * Ph.D. [1890] J. R. Rydberg Ph.D.  Xxxiv. on the structure of the line-spectra of the chemical elements.  _The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science_ , 29(179):331–337, 1890.  doi: 10.1080/14786449008619945.  URL https://doi.org/10.1080/14786449008619945.
  * Plantenberg [2007] J.H. Plantenberg.  _Coupled superconducting flux qubits_.  Phd thesis, Technische Universiteit Delft, Delft, The Netherlands, May 2007.  Available at http://resolver.tudelft.nl/uuid:5a893612-654a-4faa-83b3-235332d7194e.
  * Porotti et al. [2023] Riccardo Porotti, Vittorio Peano, and Florian Marquardt.  Gradient-ascent pulse engineering with feedback.  _PRX Quantum_ , 4, 07 2023.  doi: 10.1103/PRXQuantum.4.030305.
  * Preskill [2018] John Preskill.  Quantum computing in the nisq era and beyond.  _Quantum_ , 2:79, aug 2018.  doi: 10.22331/q-2018-08-06-79.  URL https://doi.org/10.22331%2Fq-2018-08-06-79.
  * Preskill [2023] John Preskill.  Quantum computing 40 years later, 2023.
  * Psaroudaki and Panagopoulos [2021] Christina Psaroudaki and Christos Panagopoulos.  Skyrmion qubits: A new class of quantum logic elements based on nanoscale magnetization.  _Physical Review Letters_ , 127(6), aug 2021.  doi: 10.1103/physrevlett.127.067201.  URL https://doi.org/10.1103%2Fphysrevlett.127.067201.
  * Radzihovsky et al. [2021] Matthew P Radzihovsky, Zachary I Espinosa, and Yewon Gim.  netquil: A quantum playground for distributed quantum computing simulations.  _AT &T Foundry_, page 1–6, Sep 2021.
  * Ravets et al. [2015] Sylvain Ravets, Henning Labuhn, Daniel Barredo, Thierry Lahaye, and Antoine Browaeys.  Measurement of the angular dependence of the dipole-dipole interaction between two individual rydberg atoms at a förster resonance.  _Phys. Rev. A_ , 92:020701, Aug 2015.  doi: 10.1103/PhysRevA.92.020701.  URL https://link.aps.org/doi/10.1103/PhysRevA.92.020701.
  * Rivas and Huelga [2012] Angel Rivas and Susana F. Huelga.  _Open Quantum Systems_.  Springer Berlin Heidelberg, 2012.  doi: 10.1007/978-3-642-23354-8.  URL https://doi.org/10.1007%2F978-3-642-23354-8.
  * Ryabtsev et al. [2016] I I Ryabtsev, I I Beterov, D B Tretyakov, V M Èntin, and E A Yakshina.  Spectroscopy of cold rubidium rydberg atoms for applications in quantum information.  _Physics-Uspekhi_ , 59(2):196–208, feb 2016.  doi: 10.3367/ufne.0186.201602k.0206.  URL https://doi.org/10.3367%2Fufne.0186.201602k.0206.
  * Saffman [2016] M Saffman.  Quantum computing with atomic qubits and rydberg interactions: progress and challenges.  _Journal of Physics B: Atomic, Molecular and Optical Physics_ , 49(20):202001, oct 2016.  doi: 10.1088/0953-4075/49/20/202001.  URL https://doi.org/10.1088%2F0953-4075%2F49%2F20%2F202001.
  * Sakurai and Napolitano [2021] J. J. Sakurai and Jim Napolitano.  _Approximation Methods_.  Cambridge University Press, 2 edition, 2021.
  * Samsung [2022] Samsung.  Samsung begins chip production using 3nm process technology with gaa architecture, 2022.  URL https://news.samsung.com/global/samsung-begins-chip-production-using-3nm-process-technology-with-gaa-architecture.
  * Schumacher [1995] Benjamin Schumacher.  Quantum coding.  _Phys. Rev. A_ , 51:2738–2747, Apr 1995.  doi: 10.1103/PhysRevA.51.2738.  URL https://link.aps.org/doi/10.1103/PhysRevA.51.2738.
  * Scully and Zubairy [1997] Marlan O. Scully and M. Suhail Zubairy.  _Quantum Optics_.  Cambridge University Press, 1997.  doi: 10.1017/CBO9780511813993.
  * Shende et al. [2006] V.V. Shende, S.S. Bullock, and I.L. Markov.  Synthesis of quantum-logic circuits.  _IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems_ , 25(6):1000–1010, jun 2006.  doi: 10.1109/tcad.2005.855930.  URL https://doi.org/10.1109%2Ftcad.2005.855930.
  * Shor [1994] P.W. Shor.  Algorithms for quantum computation: Discrete logarithms and factoring.  _Proceedings 35th Annual Symposium on Foundations of Computer Science_ , 1994.  doi: 10.1109/sfcs.1994.365700.
  * Shukla et al. [2018] Abhishek Shukla, Mitali Sisodia, and Anirban Pathak.  Complete characterization of the single-qubit quantum gates used in the ibm quantum processors, 05 2018.
  * Šibalić et al. [2017] N. Šibalić, J.D. Pritchard, C.S. Adams, and K.J. Weatherill.  Arc: An open-source library for calculating properties of alkali rydberg atoms.  _Computer Physics Communications_ , 220:319–331, nov 2017\.  doi: 10.1016/j.cpc.2017.06.015.  URL https://doi.org/10.1016%2Fj.cpc.2017.06.015.
  * Silvério et al. [2022] Henrique Silvério, Sebastián Grijalva, Constantin Dalyac, Lucas Leclerc, Peter J. Karalekas, Nathan Shammah, Mourad Beji, Louis-Paul Henry, and Loïc Henriet.  Pulser: An open-source package for the design of pulse sequences in programmable neutral-atom arrays.  _Quantum_ , 6:629, jan 2022.  doi: 10.22331/q-2022-01-24-629.  URL https://doi.org/10.22331%2Fq-2022-01-24-629.
  * Simon [1994] D.R. Simon.  On the power of quantum computation.  In _Proceedings 35th Annual Symposium on Foundations of Computer Science_ , pages 116–123, 1994.  doi: 10.1109/SFCS.1994.365701.
  * Stone [2000] Marshall H. Stone.  _Linear transformations in Hilbert space and their applications to analysis_.  American Mathematical Society, 2000.
  * Struck and Burkard [2016] Philipp R. Struck and Guido Burkard.  _Spin Quantum Computing_ , pages 71–103.  Springer Netherlands, Dordrecht, 2016.  ISBN 978-94-007-6892-5.  doi: 10.1007/978-94-007-6892-5_5.  URL https://doi.org/10.1007/978-94-007-6892-5_5.
  * Szmytkowski [2018] Radoslaw Szmytkowski.  Second-order stark effect and polarizability of a relativistic two-dimensional hydrogenlike atom in the ground state.  _Physical Review A_ , 98(4), oct 2018.  doi: 10.1103/physreva.98.042507.  URL https://doi.org/10.1103%2Fphysreva.98.042507.
  * Thiele et al. [2014] T. Thiele, S. Filipp, J. A. Agner, H. Schmutz, J. Deiglmayr, M. Stammeier, P. Allmendinger, F. Merkt, and A. Wallraff.  Manipulating rydberg atoms close to surfaces at cryogenic temperatures.  _Physical Review A_ , 90(1), jul 2014.  doi: 10.1103/physreva.90.013414.  URL https://doi.org/10.1103%2Fphysreva.90.013414.
  * Thompson and Salpeter [1977] John Thompson and Edwin E. Salpeter.  _Quantum mechanics of one and two electron atoms_ , volume 1.  Springer, 1 edition, 1977.
  * van Brandwijk and Parakh [2016] J. Joel van Brandwijk and Abhishek Parakh.  Asynchronous operations on qubits in distributed simulation environments using qoosim, 2016.
  * Viamontes et al. [2002] George F. Viamontes, Manoj Rajagopalan, Igor L. Markov, and John P. Hayes.  Gate-level simulation of quantum circuits, 2002.  URL https://arxiv.org/abs/quant-ph/0208003.
  * Walker and Saffman [2008] Thad G. Walker and M. Saffman.  Consequences of zeeman degeneracy for the van der waals blockade between rydberg atoms.  _Phys. Rev. A_ , 77:032723, Mar 2008.  doi: 10.1103/PhysRevA.77.032723.  URL https://link.aps.org/doi/10.1103/PhysRevA.77.032723.
  * Wallquist et al. [2009] M Wallquist, K Hammerer, P Rabl, M Lukin, and P Zoller.  Hybrid quantum devices and quantum engineering.  _Physica Scripta_ , T137:014001, dec 2009.  doi: 10.1088/0031-8949/2009/t137/014001.  URL https://doi.org/10.1088%2F0031-8949%2F2009%2Ft137%2F014001.
  * Wang et al. [2022a] Haoxiang Wang, Maurice Weber, Josh Izaac, and Cedric Yen-Yu Lin.  Predicting properties of quantum systems with conditional generative models, 2022a.
  * Wang et al. [2022b] Zhaoying Wang, Jian Li, Kaiping Xue, Shaoyin Cheng, Nenghai Yu, Qibin Sun, and Jun Lu.  An asynchronous entanglement distribution protocol for quantum networks.  _IEEE Network_ , 36(5):40–47, 2022b.  doi: 10.1109/MNET.001.2200177.
  * Weber et al. [2017] Sebastian Weber, Christoph Tresp, Henri Menke, Alban Urvoy, Ofer Firstenberg, Hans Peter Büchler, and Sebastian Hofferberth.  Calculation of rydberg interaction potentials.  _Journal of Physics B: Atomic, Molecular and Optical Physics_ , 50(13):133001, jun 2017.  doi: 10.1088/1361-6455/aa743a.  URL https://doi.org/10.1088%2F1361-6455%2Faa743a.
  * Weisstein [2005] Eric W. Weisstein.  Hilbert space.  https://mathworld.wolfram.com/HilbertSpace.html, 2005.  Accessed: 2023-03-10.
  * Wolfe [1969] Philip Wolfe.  Convergence conditions for ascent methods.  _SIAM Review_ , 11(2):226–235, 1969.  doi: 10.1137/1011036.  URL https://doi.org/10.1137/1011036.
  * Wonfor et al. [2021] Adrian Wonfor, Catherine White, Andrew Lord, Reza Nejabati, Timothy P. Spiller, James F. Dynes, Andrew J. Shields, and Richard V. Penty.  Quantum networks in the uk.  In Atul K. Srivastava, Madeleine Glick, and Youichi Akasaka, editors, _Metro and Data Center Optical Networks and Short-Reach Links IV_ , volume 11712, page 1171207. International Society for Optics and Photonics, SPIE, 2021.  doi: 10.1117/12.2578598.  URL https://doi.org/10.1117/12.2578598.
  * Wu et al. [2021] Xiaoling Wu, Xinhui Liang, Yaoqi Tian, Fan Yang, Cheng Chen, Yong-Chun Liu, Meng Khoon Tey, and Li You.  A concise review of rydberg atom based quantum computation and quantum simulation.  _Chinese Physics B_ , 30(2):020305, Feb 2021.  doi: 10.1088/1674-1056/abd76f.  URL https://doi.org/10.1088%2F1674-1056%2Fabd76f.
  * Xiang et al. [2013] Ze-Liang Xiang, Sahel Ashhab, J. Q. You, and Franco Nori.  Hybrid quantum circuits: Superconducting circuits interacting with other quantum systems.  _Reviews of Modern Physics_ , 85(2):623–653, apr 2013.  doi: 10.1103/revmodphys.85.623.  URL https://doi.org/10.1103%2Frevmodphys.85.623.
  * Yamamoto et al. [2014] T Yamamoto, K Inomata, K Koshino, P-M Billangeon, Y Nakamura, and J S Tsai.  Superconducting flux qubit capacitively coupled to an lc resonator.  _New Journal of Physics_ , 16(1):015017, jan 2014\.  doi: 10.1088/1367-2630/16/1/015017.  URL https://dx.doi.org/10.1088/1367-2630/16/1/015017.
  * Yan et al. [2016] Fei Yan, Simon Gustavsson, Archana Kamal, Jeffrey Birenbaum, Adam P Sears, David Hover, Ted J. Gudmundsen, Danna Rosenberg, Gabriel Samach, S Weber, Jonilyn L. Yoder, Terry P. Orlando, John Clarke, Andrew J. Kerman, and William D. Oliver.  The flux qubit revisited to enhance coherence and reproducibility.  _Nature Communications_ , 7(1), nov 2016.  doi: 10.1038/ncomms12964.  URL https://doi.org/10.1038%2Fncomms12964.
  * Yimsiriwattana and Lomonaco [2004a] Anocha Yimsiriwattana and Samuel J. Lomonaco.  Generalized ghz states and distributed quantum computing, 2004a.  URL https://arxiv.org/abs/quant-ph/0402148.
  * Yimsiriwattana and Lomonaco [2004b] Anocha Yimsiriwattana and Samuel J. Lomonaco.  Distributed quantum computing: a distributed shor algorithm.  In Eric Donkor, Andrew R. Pirich, and Howard E. Brandt, editors, _SPIE Proceedings_. SPIE, aug 2004b.  doi: 10.1117/12.546504.  URL https://doi.org/10.1117%2F12.546504.
  * Yu et al. [2016a] Deshui Yu, Alessandro Landra, Marí"a Martínez Valado, Christoph Hufnagel, Leong Chuan Kwek, Luigi Amico, and Rainer Dumke.  Superconducting resonator and rydberg atom hybrid system in the strong coupling regime.  _Physical Review A_ , 94(6), dec 2016a.  doi: 10.1103/physreva.94.062301.  URL https://doi.org/10.1103%2Fphysreva.94.062301.
  * Yu et al. [2016b] Deshui Yu, María Martínez Valado, Christoph Hufnagel, Leong Chuan Kwek, Luigi Amico, and Rainer Dumke.  Charge-qubit-atom hybrid.  _Physical Review A_ , 93(4), apr 2016b.  doi: 10.1103/physreva.93.042329.  URL https://doi.org/10.1103%2Fphysreva.93.042329.
  * Yu et al. [2017] Deshui Yu, Leong Chuan Kwek, Luigi Amico, and Rainer Dumke.  Superconducting qubit-resonator-atom hybrid system.  _Quantum Science and Technology_ , 2(3):035005, jul 2017.  doi: 10.1088/2058-9565/aa7c50.  URL https://doi.org/10.1088%2F2058-9565%2Faa7c50.
  * Zhang and Mitroy [2007] J.-Y. Zhang and J. Mitroy.  Long-range dispersion interactions. i. formalism for two heteronuclear atoms.  _Phys. Rev. A_ , 76:022705, Aug 2007.  doi: 10.1103/PhysRevA.76.022705.  URL https://link.aps.org/doi/10.1103/PhysRevA.76.022705.

##  Appendix A Quantum Logic Gates

###  A.1 Single Qubit Gates

###  A.2 Multiple Qubit Gates

##  Appendix B Proofs and Concepts in Quantum Computation

###  B.1 Quantum Fourier Transform Product Representation

The equivalence of the product representation seen in Eq. 2.13 and the formal
definition seen in Eq. 2.12 follows from,

$\displaystyle\ket{j}$
$\displaystyle\rightarrow\frac{1}{2^{n/2}}\sum^{2^{n}-1}_{k=0}e^{2\pi
ijk/2^{n}}\ket{k}$ (B.1)
$\displaystyle=\frac{1}{2^{n/2}}\sum^{1}_{k_{1}=0}\cdots\sum^{1}_{k_{n}=0}e^{2\pi
ij\big{(}\sum^{n}_{l=1}k_{l}2^{-l})}\ket{k_{1}\cdots k_{n}}$ (B.2)
$\displaystyle=\frac{1}{2^{n/2}}\sum^{1}_{k_{1}=0}\cdots\sum^{1}_{k_{n}=0}\otimes^{n}_{l=\mathbb{1}}e^{2\pi
ijk_{l}2^{-l}}\ket{k_{l}}$ (B.3)
$\displaystyle=\frac{1}{2^{n/2}}\otimes^{n}_{l=1}\sum^{1}_{k_{l}=0}e^{2\pi
ijk_{l}2^{-}l}\ket{k_{l}}$ (B.4)
$\displaystyle=\frac{1}{2^{n/2}}\otimes^{n}_{l=1}\Big{[}\ket{0}+e^{2\pi
ij2^{-l}}\ket{1}\Big{]}$ (B.5) $\displaystyle=\frac{\big{(}\ket{0}+e^{2\pi
i0.j_{n}}\ket{1}\big{)}\big{(}\ket{0}+e^{2\pi
i0.j_{n-1}j_{n}}\ket{1}\big{)}\cdots\big{(}\ket{0}+e^{2\pi i0.j_{1}j_{2}\cdots
j_{n}}\ket{1}\big{)}}{2^{n/2}}.$ (B.6)

###  B.2 Introduction to Open Quantum Systems and the Lindbladian Derivation

The first postulated quantum equation was introduced by Erwan Schrödinger
which describes the behaviour of an isolated or closed system. By definition,
a closed system is one that does not exchange information with another system.
Assuming the system is in a pure state $\ket{\psi(t)}\in\mathcal{H}$ at time
$t$ where $\mathcal{H}$ describes the Hilbert space of the system, the time
evolution is,

$\frac{d}{dt}\ket{\psi(t)}=-\frac{i}{\hbar}H(t)\ket{\psi(t)}$ (B.7)

where $H(t)$ is the Hamiltonian operator of the system. An important property
of of the Schrödinger equation is that it does not change the norm of the
states,

$\frac{d}{dt}\bra{\psi(t)}\ket{\psi(t)}=\bigg{(}\frac{d\bra{\psi(t)}}{dt}\bigg{)}\ket{\psi(t)}+\bra{\psi(t)}\bigg{(}\frac{d\ket{\psi(t)}}{dt}\bigg{)}=i\bra{\psi(t)}H^{{\dagger}}(t)\ket{\psi(t)}-i\bra{\psi(t)}H(t)\ket{\psi(t)}=0.$
(B.8)

This is due to the Hamiltonian being self-adjoint, $H(t)=H^{\dagger}(t)$
[105]. Exploiting the linearity of the Schrödinger equation, one can describe
its solution using an evolution family,
$\ket{\psi(t)}=U(t-t_{0})\ket{\psi(t_{0})}$, in which
$U(t_{0},t_{0})=\mathbb{1}$. Equation B.8 implies that the evolution operator
is isometric. Thus, if the system possesses finite dimensions, the system can
be considered a unitary operator [105]. The specific form of the evolution
operator relies on the properties of the system Hamiltonian. In the simplest
case, whereby the Hamiltonian is time independent, the solution to
Schrödinger’s equation is obtained using,

$U(t,t_{o})=e^{-\frac{i}{\hbar}(t-t_{0})H},$ (B.9)

however, for a time-dependent system, the unitary evolves as a Dyson expansion
[57],

$U(t,t_{0})=\mathcal{T}e^{\int^{t}_{t_{0}}H(t^{\prime})dt^{\prime}},$ (B.10)

where $\mathcal{T}$ is the time ordering operator. From this one can deduce
that due to the self-adjoint nature $H(t)$ possesses, $-H(t)$ is also self
adjoint suggesting that the every evolution has a physical inverse.
Furthermore, the unitary also possesses an inverse, for example in the time-
independent case, $U(t_{1},t_{0})=U(t-t_{0})$. This can be generalized to
$U^{-1}(\tau)=U(-\tau)=U^{\dagger}(\tau)$ such that
$U^{-1}(\tau)U(\tau)=U(\tau)U^{-1}(\tau)=\mathbb{1}$. For time dependent
Hamiltonians, the subtlety is confined within the evolution operator’s
definition, $U(t,s)$. The inverse is shown to be unitary by
$U^{-1}(s,t)=U(s,t)=U^{\dagger}(s,t)$ [105].

Inherited from classical computing, a quantum circuit is an abstracted model
of a set of operations performed on a single or many two level states. The
execution of a circuit on a quantum state is given by,

$\ket{\psi_{f}}=U_{k}U_{K-1}\cdots U_{2}U_{1}\ket{\psi_{i}},$ (B.11)

where $\ket{\psi_{f}}$ and $\ket{\psi_{i}}$ are the final and initial state
respectively and $U_{k}$ with $k\in{1,2,\cdots,K}$ the quantum gates.
Simulating quantum circuits is implemented by representing each the gates and
states of the system as complex matrices and vectors. This is referred to as a
gate level quantum circuit description because it is a quantum algorithm at an
abstract level [124]. The set of unitaries described in eq. (B.11) can be
thought of a process multiplied to a state, which in turn applies the process
to the state.

In order to understand how unitaries evolve a system it is first important to
understand time dependent Hamiltonians. One can define a Hamiltonian as a
function of time $t$ as a set of two Hamiltonians. A non-controllable drift
Hamiltonian $H_{d}$, which describes the system evolution over time and a
control Hamiltonian $H_{c}$, which can control the evolution of the system in
time [74]. This may be depicted as,

$H(t)=H_{d}(t)+H_{c}(t)=H_{d}(t)+\sum_{j}c_{j}(t)H_{j}$ (B.12)

where $H_{j}$ describes the effects of physical controls on the system
modulated by time-dependent control coefficients $c_{j}(t)$ used to realize
desired unitary gates [58, 74]. The unitary is a solution of the Schrödinger
equation,

$i\hbar\frac{\partial U(t)}{\partial t}=H(t)U(t).$ (B.13)

Implementing the desired unitaries can be obtained by choosing $H(t)$,
equally, the choice of the solver depends on the parametrization of the
control coefficients $c_{j}(t)$. Parameters within $c_{j}(t)$ can be
determined through theoretical models or through cost optimisation [58].

In order to evolve the Hamiltonian through the unitaries one needs to define
the respective density matrix. A density matrix is a generalization of a wave
function which describes the physical state of a quantum system. Density
formalism allows a mathematical representation of decoherence on a state
vector $\ket{\psi}$. A density matrix is defined as,

$\rho=\sum_{j}p_{j}\ket{\rho_{j}}\bra{\rho_{j}}.$ (B.14)

A density matrix is said to describe a pure state if the sum has a single
term. If not, it is called a mixed state which describes a classical ensemble
of quantum states. The evolution of $\rho$ in time $t$ can be depicted using
unitaries, $\rho(t_{1})=U(t_{1},t_{0})\rho(t_{0})U^{\dagger}(t_{1},t_{0})$
[105]. For a pure state, the density matrix is equivalent to the state space
vector in Hilbert Space, i.e $\rho=\ket{\psi}\bra{\psi}$ and can be
substituted into the Schrödinger equation, forming the so called Liouville-von
Nuemann equation [48],

$i\hbar\frac{\partial}{\partial t}\rho=[H,\rho].$ (B.15)

The density matrix is also a key parameter to the analysis of quantum systems.
For example, considering a larger system composed by two quantum subsystems in
Hilbert space $\mathcal{H}=\mathcal{H_{A}}+\mathcal{H_{B}}$, the density
matrix of subsystem A is given by [89, 105],

$\rho_{A}=\mathrm{Tr}_{B}(\rho).$ (B.16)

Provided that the total evolution is not factorized
$U(t_{1},t_{0})=U_{A}(t_{1},t_{0})\otimes U_{B}(t_{1},t_{0})$, both quantum
systems $A$ and $B$ are interchanging information, ergo converting the system
to an open system [105]. Even in such system, the density matrix of an
independent subsystem at time $t$ can still be acquired, for example, for
system $A$,

$\rho_{A}(t_{1})=\mathrm{Tr}_{B}[U(t_{1},t_{0})\rho(t_{0})U^{\dagger}(t_{1},t_{0})].$
(B.17)

This is a so called dynamical map [48, 18]. In order to fully establish a true
open quantum system, the environment and the quantum system must entangle over
time. Allowing $\rho_{S}$ and $\rho_{E}$ to depict the qubit and the
environment density matrices, respectively, the full quantum system can be
initialized as a separable state $\rho_{S}\otimes\rho_{E}$ that entangles over
later time $t$ [48]. Knowing that both the time evolution of in this composite
Hilbert space is unitary with eq. B.15 and that the state of the qubit can be
obtained by taking a partial trace over the environment, the equation of
motion can be derived,

$i\hbar\frac{\partial}{\partial t}\rho_{S}=\mathrm{Tr}_{E}[H,\rho].$ (B.18)

The solution of $\rho_{S}(t)$ is thus the dynamical map [48],

$\rho_{S}(t)=\mathcal{E}(t,0)\rho_{S}(0)=\mathrm{Tr}_{E}\Big{[}U(t,0)\rho_{SE}(0)U^{\dagger}(t,0)\Big{]}.$
(B.19)

Fig. B.1 details the relationship between the unitary evolution and the
dynamical map.

Figure 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].

The operation of a dynamical map is considered independent of any prior
evolution in the total Hilbert space when the separation or distinguishability
between two initial states is assumed to not grow during its evolution [18].
If this is satisfied, a dynamical map can be split into two time intervals
$t_{1}$ and $t_{2}$ as,

$\mathcal{E}(t_{1}+t_{2},0)=\mathcal{E}(t_{2},0)\mathcal{E}(t_{1},0).$ (B.20)

Eq. B.20 can be rewritten into the following form [18],

$\mathcal{E}(t,0)=e^{-\frac{i}{\hbar}\mathcal{L}t},$ (B.21)

where $\frac{i}{\hbar}$ has been factored out of $\mathcal{L}$, a generator
that follows $\mathcal{L}(\cdot)=-i[H(t),\cdot]$ [105]. Substituting this
relation into Eq. B.18 leads to [48],

$i\hbar\frac{\partial}{\partial
t}\rho_{S}(t)=\mathcal{L}[\rho_{S}(t)]\equiv\lim_{t\to
0}\frac{1}{t}(\mathcal{E}(t,0)\rho_{S}-\rho_{S}).$ (B.22)

Choosing a complete basis $\\{R_{i}\\}$ of $N_{S}^{2}$ operators of dimensions
$N_{S}$ in Hilbert space such that $\Big{[}R_{N^{2}_{S}}\Big{]}=1$ and
$\mathrm{Tr}[R_{i}]=0$ [48]. Using the expansion of Kraus operators [18], the
equation of motion can then be reduced to,

$\mathcal{L}[\rho_{S}]=[\mathrm{H},\rho_{S}]+i\sum^{N_{S}^{2}-1}_{i,j=1}a_{ij}\bigg{(}R_{i}\rho_{S}R_{j}^{\dagger}-\frac{1}{2}\Big{\\{}R_{j}^{\dagger}R_{i},\rho_{S}\Big{\\}}\bigg{)},$
(B.23)

where $\\{\cdot,\cdot\\}$ denotes an anti-commutator, $\mathrm{H}$ is a
Hermitian operator constructed from $\\{R_{i}\\}$ and $a_{ij}$ is a given
coefficient matrix. Rotating the basis to a new basis $\\{A_{k}\\}$ such that
the coefficient matrix is diagonalized with eigenvalues $\gamma_{k}$, one
arrives at the master equation in Lindblad form [77],

$\mathcal{L}[\rho_{S}]=[\mathrm{H},\rho_{S}]+i\sum_{k}\gamma_{k}\bigg{(}A_{k}\rho_{S}A_{k}^{\dagger}-\frac{1}{2}\Big{\\{}A_{k}^{\dagger}A_{k},\rho_{S}\Big{\\}}\bigg{)},$
(B.24)

where $\\{A_{k}\\}$ are the Lindblad operators representing the coupling of
the system to the environment and thus the dissipative process [89, 77]. The
Lindbladian is not a unitary, however, it does still satisfy the property of
being trace-preserving and positive for any initial condition. In this
investigation, the $[H,\rho_{S}]$ term will be omitted to only describe the
noise dynamics of a system, thus,

$\mathcal{L}[\rho_{S}]=i\sum_{k}\gamma_{k}\bigg{(}A_{k}\rho_{S}A_{k}^{\dagger}-\frac{1}{2}\Big{\\{}A_{k}^{\dagger}A_{k},\rho_{S}\Big{\\}}\bigg{)},$
(B.25)

###  B.3 Measuring Expected Values with the $\chi$ Matrix for a Hadamard Gate

The final state, $\varepsilon(t)$, can be expressed as,

$\varepsilon(t)=\rho_{out}\sum_{m,n=0}\tilde{E}_{m}\rho\tilde{E}_{n}^{\dagger}\chi_{mn}=U\rho_{in},$
(B.26)

where $U$ is a propagator for density matrix in superoperator form. One can
then identify,

$U=\sum_{mn}\chi_{mn}\tilde{E}_{m}\tilde{E}_{n}^{\dagger}$ (B.27)

The interest lies in forming the $\chi_{mn}$ matrix originating from a
process, which is a $N^{4}\times N^{4}$ matrix, where $N$ is the amount of
qubits involved. The product $\tilde{E}_{m}\tilde{E}_{n}^{\dagger}$ can be
written as a $N^{4}\times N^{4}$ matrix called $M$. This means,

$U_{I}=\sum^{N^{4}}_{J}M_{IJ}\chi_{J},$ (B.28)

with the solution for $\chi$,

$\chi=M^{-1}U.$ (B.29)

For an ideal Hadamard gate, the $U$ superoperator is $U=(G_{H}\otimes
I)\times(I\otimes G_{H}^{\dagger})$ where $G_{H}$ is the Hadamard gate seen
below,

$\frac{1}{\sqrt{(}2)}\begin{bmatrix}1&1\\\ 1&-1\end{bmatrix}.$ (B.30)

Next, the list of operators that define the basis $\\{\tilde{E}_{i}\\}$ are
defined in the form of list operators for each composite system. For the
Hadamard gate, the Pauli gates are used to define the basis, thus one can
define a matrix $\Pi$, such that,

$\Pi=\begin{bmatrix}I&X&Y&Z\end{bmatrix}\otimes\begin{bmatrix}I\\\ X\\\ Y\\\
Z\end{bmatrix}=\begin{bmatrix}II&IX&IY&IZ\\\ XI&XX&XY&XZ\\\ YI&YX&YY&YZ\\\
ZI&ZX&ZY&ZZ\\\ \end{bmatrix}.$ (B.31)

The $\chi$ matrix is computed by,

$\chi=\frac{1}{N^{2}}\begin{bmatrix}\mathrm{Tr}(II\times
U)&&\mathrm{Tr}(IX\times U)&&\mathrm{Tr}(IY\times U)&&\mathrm{Tr}(IZ\times
U)\\\ \mathrm{Tr}(XI\times U)&&\mathrm{Tr}(XX\times U)&&\mathrm{Tr}(XY\times
U)&&\mathrm{Tr}(XZ\times U)\\\ \mathrm{Tr}(YI\times U)&&\mathrm{Tr}(YX\times
U)&&\mathrm{Tr}(YY\times U)&&\mathrm{Tr}(YZ\times U)\\\ \mathrm{Tr}(ZI\times
U)&&\mathrm{Tr}(ZX\times U)&&\mathrm{Tr}(ZY\times U)&&\mathrm{Tr}(ZZ\times
U)\end{bmatrix}.$ (B.32)

For the ideal Hadamard gate this equals,

$\chi_{Hadamard}=\begin{bmatrix}0&0&0&0\\\ 0&0.5\cdot e^{i\cdot 0}&0&0.5\cdot
e^{i\cdot 0}\\\ 0&0&0&0\\\ 0&0,5\cdot e^{i\cdot 0}&0&0.5\cdot e^{i\cdot 0}\\\
\end{bmatrix},$ (B.33)

which is seen plotted in figure 5.2 for the $F\approx 1$ cases. The colour in
of the elements depicts the phase, $\theta$, of the phase factor
($e^{i\theta}$) multiplied to the $\chi_{mn}$ element.

##  Appendix C Code & Functions

###  C.1 Distributed Phase Estimation

Apply initial hadamard gates function ` `

Establish the E2 gate (GHZ state) between the two systems ` `

Implementation of the CPHASE gate ` `

Performing a measurement on a specific qubit ` `

Applying the non-local controlled phase gates seen in figure 2.11 ` `

Applying the DCP circuit seen in figure 2.11 ` `

Performing the non-local section of the inverse Fourier transform circuit seen
in figure 2.12 ` `

Execute the inverse Fourier transform circuit in figure 2.12 ` `

Distributed phase estimation algorithm for estimating 16 different phases ` `

###  C.2 Flux-Resonator-Atom Hybrid Numerical Simulation

Constructor of the Rydberg hamiltonian ` `

Constructor of the Rydberg-Resonator Hamiltonian ` `

Constructor of the Flux Hamiltonian ` `

Constructor of the Flux Resonator Hamiltonian ` `

Constructor of the Resonator Hamiltonian ` `

Function to simulate a hamiltonian for a given time. ` `

Function to maximize the fidelity depending on evolution time. ` `

First step of the Flux-Resonator-Atom system ` `

Second step of the Flux-Resonator-Atom system ` `

Third step of the Flux-Resonator-Atom system ` `

Fourth step of the Flux-Resonator-Atom system ` `

Fifth step of the Flux-Resonator-Atom system ` `

Executing the E2 (GHZ) connection. ` `

###  C.3 Creation of GRAPE optimised gates

Fabrication of the RZ gate. ` `

Fabrication of the CNOT gate. ` `

Function to establish the drift and controls for the single qubit case. ` `

Function to establish the drift and controls for the double qubit case. ` `

Function to create single qubit optimised gates. ` `

Function to create double qubit optimised gates. ` `

Creating the optimised gates using GRAPE. ` `

Function executing the GRAPE optimiser. ` `

###  C.4 Values for Parameters

Parameters | Values
---|---
$E_{J}$ (GHz) | 65
$E_{C}$ (GHz) | 1
$\Delta$ (GHz) | 0.82
$\alpha$ | 0.8
$\varepsilon$ (GHz) | 6.7
$f_{\varepsilon}$ | 0.53
$\zeta$ | 10,100,1000
$g$ (GHz) | 2
$\delta n_{m}$ | 1
$\Omega$ (GHz) | $2\pi\times 6.8$
$\Gamma$ (MHz) | $2\pi\times 470$
$\gamma_{e}$ (MHz) | $2\times\pi 0.03$
Table C.1: List of physical parameters used for the master equations.

###  C.5 Diagram of Simulation Processes

Figure C.1: Detailed schematic of the steps involved when simulating the
distributed phase estimation with GRAPE optimised gates.

##  Appendix D Quantum Principles in Rydberg Systems

###  D.1 Wavefunctions for Alkali Atoms

Alkali atoms are the names given to the chemical elements Li, Na, K, Rb and
Cs. These atoms are found in the first group in the periodic table along with
Hydrogen. Although alkali atoms obviously differ from hydrogen, both hydrogen
and alkali metals have an outermost electron in an s-orbital. Thus it possible
to derive the wavefunction for a hydrogen atom to derive the analytical
wavefunction for alkali atoms. In order to do so, one needs to solve the time-
independent Schrödinger equation,

$-\frac{1}{2}\Delta^{2}\psi(\boldsymbol{r})-\frac{1}{r}\psi(\boldsymbol{r})=E\psi(\boldsymbol{r}),$
(D.1)

where the Coulomb potential is exhibited by $V=-\frac{1}{r}$ where $r$ is the
distance of the electron to the nucleus. Due to the spherical symmetric nature
of an atom, one can separate the Schrödinger equation into a radial and
angular part using spherical coordinates,

$\psi(\boldsymbol{r})=R(r)Y_{lm}(\theta,\phi).$ (D.2)

The angular part $Y_{lm}(\theta,\phi)$ has a solution [122],

$Y_{lm}(\theta,\phi)=\sqrt{\frac{(2l+1)}{4\pi}\frac{(l-m)!}{(l+m)!}}P_{l}^{m}(\mathrm{cos}\theta)e^{im\phi}$
(D.3)

where $P^{m}_{l}$ the associated Legendre function [3]. $l$ is the orbital
angular momentum quantum number and $m$ is the magnetic quantum number.
Solving the radial part of the Schrödinger equation requires one to rewrite
$u(r)=rR(r)$ to obtain,

$\bigg{(}-\frac{1}{2}\frac{\mathrm{d}^{2}}{\mathrm{d}r^{2}}+\frac{l(l+1)}{2r^{2}}-\frac{1}{r}\bigg{)}u(r)=Eu(r).$
(D.4)

The solution to $R(r)$ can be generalized to [122],

$R_{nl}(r)=\sqrt{\bigg{(}\frac{2}{n}\bigg{)}^{3}\frac{(n-l-1)!}{2n(n+l)!}}e^{-r/n}\bigg{(}\frac{2r}{n}\bigg{)}^{l}L^{2l+1}_{n-l-1}\bigg{(}\frac{2r}{n}\bigg{)},$
(D.5)

with energy,

$E_{n}=-\frac{1}{2n^{2}},$ (D.6)

where $n$ is the principal quantum number, $l$ is the angular momentum quantum
number which cannot be greater than $n-1$. Eq. (D.6) can be adjusted for
alkali atoms by adding a term to the orbital quantum number known as a quantum
defect $\delta_{nlj}$ [36] where,

$\delta_{nlj}=\delta_{0}+\frac{\delta_{2}}{(n-\delta_{0})^{2}}.$ (D.7)

In this formulation, both $\delta_{0}$ and $\delta_{2}$ are dependent on $l$
but the spin orbit coupling extends the dependence to an additional spin orbit
coupling term $j$. The discrepancy in energy between the energy of a hydrogen
atom and an alkali metal arises from the potential near the core which in turn
changes the energy. Hence, the energy can be rewritten as,

$E_{nlj}=-\frac{1}{2(n-\delta_{nlj})^{2}}.$ (D.8)

This defect was originally measured and postulated by Johannes Rydberg in 1980
[97]. In order to solve the radial wavefunction for alkali metals, equation
(D.5) must be rewritten using the new definitions for the principal quantum
number $n$ and the orbital momentum quantum number $l$. The new principal
quantum number can simply be rewritten as $n^{*}=n-\delta_{nlj}$. For the
orbital quantum number however, $l*=l-\delta_{nlj}+I(l)$ where $I(l)$ is an
integer responsible for fixing the number of radial nodes as seen in [6, 66].
The restriction applied to $I(l)$ is that $\delta_{nlj}-l-\frac{1}{2}<I(l)\leq
n_{min}-l-1$ where $n_{min}$ is the principal quantum number of the ground
state. Setting $I(l)=\delta_{nlj}$ establishes a satisfactory agreement for
the lifetime values 5s-5p [36].

$R_{n^{*}l^{*}}(r)=\sqrt{\bigg{(}\frac{2}{n^{*}}\bigg{)}^{2}\frac{(n^{*}-l^{*}-1)!}{2n^{*}(n^{*}+l^{*})!}}e^{-r/n^{*}}\bigg{(}\frac{2r}{n^{*}}\bigg{)}^{l^{*}}L^{2l^{*}+1}_{n^{*}-l^{*}-1}\bigg{(}\frac{2r}{n^{*}}\bigg{)}.$
(D.9)

Fig. D.1 shows the absolute value of Eq. (D.9) multiplied by $r$ given in
$a_{0}$ squared as a function of $r$ for the state $\ket{22D,j=5/2,s=1/2}$ for
$\mathrm{Rb^{87}}$. This shows the probability of finding an electron from the
nucleus at distance $r$.

Figure 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.

###  D.2 Non-Degenerate Perturbation Theory

Solving physical problems in large Hilbert spaces, much like analyzing the
interaction of radiation with matter, is analytically too complex to solve
exactly. A method of solving such complex problems can be solved using an
approximate method.

Assume a Hamiltonian in the form,

$\mathcal{H}=\mathcal{H}_{0}-\varepsilon V$ (D.10)

where the eigenvalue for the unperturbed Hamiltonian $\mathcal{H}_{0}$ is
known and $\varepsilon$ is a value varying between 0 and 1 responsible for the
expansion. The solution for $\varepsilon\rightarrow 1$ is the desired solution
to (D.10). Knowing the eigenvalues and eigenkets of the uperturbed
Hamiltonian, $\mathcal{H}_{0}\ket{k}=E_{k}^{(0)}\ket{k}$ allows one to rewrite
the complete Hamiltonian as,

$(\mathcal{H}_{0}+\varepsilon
V)\ket{\varphi_{k}}_{\varepsilon}=E_{k}(\varepsilon)\ket{\phi_{k}}_{\varepsilon},$
(D.11)

this is assuming that at first that the energy spectrum is not degenerate
(that is, all $E_{k}^{(0)}$ are different). Expanding the eigenket where
$\ket{\phi^{(0)}_{k}}=\ket{k}$,

$\ket{\phi_{k}}=\ket{\phi_{k}^{(0)}}+\varepsilon\ket{\phi_{k}^{(1)}}+\varepsilon^{2}\ket{\phi_{k}^{(2)}}+\dots,$
(D.12)

and

$E_{k}=E_{k}^{(0)}+\varepsilon E^{(1)}+\varepsilon^{2}E_{k}^{(2)}+\dots.$
(D.13)

When $\varepsilon$ is small, the energy shift due to the perturbation is given
by $\Delta_{k}=E_{k}=E_{k}^{(0)}$. This means that equation (D.11) can be
rewritten to $(E^{(0)}_{k}-\mathcal{H}_{0})\ket{\varphi_{k}}=(\varepsilon
V-\Delta_{k})\ket{\varphi_{k}}$. Then, projecting this onto $\bra{k}$,
$\bra{k}(E^{(0)}_{k}-\mathcal{H}_{0})\ket{\varphi_{k}}=\bra{k}(\varepsilon
V-\Delta_{k})\ket{\varphi_{k}}$ leaves [24, 108],

$\Delta_{k}=\varepsilon\frac{\bra{k}V\ket{\varphi_{k}}}{\braket{k}{\varphi_{k}}},$
(D.14)

since
$\bra{k}\mathcal{H}_{0}\ket{\varphi_{k}}=\bra{k}E_{k}^{(0)}\ket{\varphi_{k}}$
and $\bra{k}\varepsilon V-\Delta_{k}\ket{\varphi_{k}}=0$. It is possible to
introduce the relation $\braket{k}{\varphi_{k}}=1$, however this implies that
the perturbed state $\ket{\varphi_{k}}$ is not normalized. It can be shown
that by defining,

$\ket{\psi_{k}}=\frac{\ket{\varphi_{k}}}{\sqrt{\braket{\varphi_{k}}{\varphi_{k}}}}$
(D.15)

such that $\braket{k}{\psi_{k}}=1/\sqrt{\braket{\varphi_{k}}{\varphi_{k}}}$,
$\braket{k}{\phi_{k}}$ can be approximated to 1 [24]. Calculating the
perturbed normalization factor,

$\braket{\varphi_{k}}{\varphi_{k}}=\braket{k+\varepsilon\varphi_{k}^{(1)}+\dots}{k+\varepsilon\varphi_{k}^{(1)}+\dots}=1+\varepsilon\braket{k}{\varphi^{(1)}_{k}}+\dots,$
(D.16)

where $\varepsilon\braket{k}{\varphi^{(1)}_{k}}=0$. Thus,
$\braket{\varphi_{k}}{\varphi_{k}}\approx 1$ meaning that
$\braket{k}{\phi_{k}}\approx 1$.

Using the approximation $\braket{k}{\phi_{k}}\approx 1$, it follows that
$\Delta_{k}=\varepsilon\braket{k}{V}{\varphi_{k}}$. Replacing, $\Delta_{k}$ by
$\varepsilon E^{1}_{k}+\varepsilon^{2}E^{2}_{k}+\dots$ and $\ket{\varphi_{k}}$
by it’s expansion and equating terms of the same order in $\epsilon$, it is
possible to obtain,

$E^{n}_{k}=\braket{k}{V\varphi_{k}^{(n-1)}}.$ (D.17)

This result shows that by knowing only the eigenstates of lower orders, one
can find the energy of all orders. The question is now finding the value of
$\varphi_{k}^{(n-1)}$. Due to the fact that
$(E^{(0)}_{k}-\mathcal{H}_{0})^{-1}\ket{k}$ possesses a singularity, it is not
possible to invert the operator in the expansion
$\ket{\varphi_{k}}=(E^{(0)}_{k}-\mathcal{H}_{0})^{-1}(\varepsilon
V-\Delta_{k})\ket{\varphi_{k}}$. To ensure
$(E^{(0)}_{k}-\mathcal{H}_{0})^{-1}$ is never applied to eigenstates of the
unperturbed Hamiltonian, $\ket{\psi_{k}}$ needs to equal $(\varepsilon
V-\Delta_{k})\ket{\varphi_{k}}\neq\ket{k}$ for any $\ket{\varphi_{k}}$ [24].
By defining a projector $P_{k}=\mathds{1}-\ket{k}\bra{k}=\sum_{h\neq
k}\ket{k}\bra{k}$, one ensures that $\forall\ket{\psi}$ the projected state is
$\ket{\psi}^{\prime}=P_{k}\ket{\psi}$ is such that
$\braket{k}{\psi^{\prime}}=0$ since
$\bra{k}P_{k}\ket{\psi}=\braket{k}{\psi}-\braket{k}{k}\braket{k}{\psi}=0$
[108, 24]. Using the projector, $(E_{k}^{(0)}-\mathcal{H_{0}})$ can be
multiplied by $P_{k}\ket{\psi}$, thus giving,

$P_{k}(E_{k}^{(0)}-\mathcal{H}_{0})\ket{\varphi_{k}},$ (D.18)

which is now solvable since $(E_{k}^{(0)}-\mathcal{H_{0}})^{-1}\ket{P_{k}}$ is
no longer ill defined. Thus, taking
$(E_{k}^{(0)}-\mathcal{H}_{0})\ket{\varphi_{k}}=(\varepsilon
V-\Delta_{k})\ket{\varphi_{k}}$, multiplying it by $P_{k}$ and rearranging for
$P_{k}\ket{\varphi_{k}}$ gives,

$P_{k}\ket{\varphi_{k}}=(E_{k}^{(0)}-\mathcal{H}_{0})^{-1}P_{k}(\varepsilon
V-\Delta_{k})\ket{\varphi_{k}}$ (D.19)

Noting that
$P_{k}\ket{\varphi_{k}}=\ket{\varphi_{k}}-\ket{k}\braket{k}{\varphi_{k}}=\ket{\varphi_{k}}-\ket{k}$,
one can obtain [108, 24],

$\ket{\varphi_{k}}=\ket{k}+(E^{(0)}_{k}-\mathcal{H}_{0})^{-1}P_{k}(\varepsilon
V-\Delta_{k})\ket{\varphi_{k}}.$ (D.20)

In order to simplify the expression, it is possible to define an operator
$F_{k}$ as,

$F_{k}=(E_{k}^{(0)}-\mathcal{H}_{0})^{-1}P_{k}=\sum_{h\neq
k}\frac{\ket{h}\bra{h}}{E^{0}_{k}-E^{0}_{h}}.$ (D.21)

By using the expansion,

$\ket{k}+\varepsilon\ket{\varphi^{(1)}_{k}}+\dots=\ket{k}+F_{k}\varepsilon(V-E_{k}^{1}-\varepsilon
E^{2}_{k}-\dots)(\ket{k}+\varepsilon\ket{\varphi_{k}^{(1)}}+\dots)$ (D.22)

one can solve to obtain the first order term,

$\ket{\varphi_{k}^{(1)}}=F_{k}(V-E_{k}^{1})\ket{k}=F_{k}(V-\bra{k}V\ket{k})\ket{k}=F_{k}V\ket{k}$
(D.23)

and the second order term,

$E^{2}_{k}=\bra{k}V\ket{\varphi_{k}^{(1)}}=\bra{k}VF_{k}V\ket{k}=\bra{k}V\bigg{(}\sum_{h\neq
k}\frac{\ket{h}\bra{h}}{E^{0}_{k}-E^{0}_{h}}\bigg{)}V\ket{k}$ (D.24)

which is explicitly equal to [108, 24],

$E^{2}_{k}=\sum_{h\neq k}\frac{|V_{kh}|^{2}}{E^{0}_{k}-E^{0}_{h}}.$ (D.25)

###  D.3 Stark Effect

Suppose the atom under study has a Hamiltonian in the form
$\mathcal{H}=\mathcal{H}_{0}+\mathcal{H}_{1}$ where $\mathcal{H}_{0}$ is the
unperturbed part of the Hamiltonian and $\mathcal{H}_{1}=e|\boldsymbol{E}|z$
is the perturbed part of the Hamiltonian. This problem can be studied using
non-degenerate perturbation theory assuming that the unperturbed states are
non-degenerative. A change in energy of the eigenstates characterized by
quantum numbers $n$,$l$, and $m$ in the presence of a small electric field is
given by,

$\Delta
E_{nlm}=e|\boldsymbol{E}|\bra{n,l,m}z\ket{n,l,m}+e^{2}|\boldsymbol{E}|^{2}\sum_{n^{\prime},l^{\prime},m^{\prime}=n,l,m}\frac{|\bra{n,l,m}z\ket{n^{\prime},l^{\prime},m^{\prime}}|^{2}}{E_{nlm}-E_{n^{\prime}l^{\prime}m^{\prime}}}$
(D.26)

Trivially, it is possible to see that in almost all cases,
$\bra{n,l,m}z\ket{n^{\prime},l^{\prime},m^{\prime}}$ is zero. Thus, it is
important to determine a set of rules for these matrix elements to be non-
zero. These rules are referred to as selection rules. Knowing that the z
orbital angular momentum operator commutes with z ($[L_{z},z]=1$), it follows
that [108, 24],

$\bra{n,l,m}[L_{z},z]\ket{n^{\prime},l^{\prime},m^{\prime}}=\bra{n,l,m}L_{z}z-zL_{z}\ket{n^{\prime},l^{\prime},m^{\prime}}=\hbar(m-m^{\prime})\bra{n,l,m}z\ket{n^{\prime},l^{\prime},m^{\prime}}=0.$
(D.27)

This shows two clear selection rules. The first being that $m^{\prime}=m$ has
to hold in order for $\bra{n,l,m}z\ket{n^{\prime},l^{\prime},m^{\prime}}$ to
be non-zero. The second being that $l^{\prime}=l\pm 1$ which can be calculated
by using the commutation of the total angular momentum $L^{2}$ with $z$ [24].

$\Delta E_{nlm}=e^{2}|\boldsymbol{E}|^{2}\sum_{n^{\prime},l^{\prime}=l\pm
1}\frac{|\bra{n,l,m}z\ket{n^{\prime},l^{\prime},m^{\prime}}|^{2}}{E_{nlm}-E_{n^{\prime}l^{\prime}m}}.$
(D.28)

This is thus the definition of the quadratic Stark effect. Since, $\Delta
E=-\frac{1}{2}\alpha|\boldsymbol{E}|^{2}$, it follows that,

$\alpha_{nlm}=2e^{2}\sum_{n^{\prime},l^{\prime}=l\pm
1}\frac{|\bra{n,l,m}z\ket{n^{\prime},l^{\prime},m^{\prime}}|^{2}}{E_{nlm}-E_{n^{\prime}l^{\prime}m}}.$
(D.29)

For states that are degenerate at zero field, the Stark effect will cause a
splitting of the states. This means that at these states, the energy shift is
linear meaning that they have a permanent dipole moment [36]. This occurs in
alkalis for all higher angular momentum states with equal $n$. Following [55],
it is convenient to convert the quantum number $l$ to $k$, the parabolic
quantum number, using the parabolic coordinate system. The state $\ket{nkm}$
is thus expressed as [36, 55],

$\ket{nkm}=\sum_{l}C_{kl}\ket{nlm},$ (D.30)

where the Clebsch-Gordan coefficient is,

$C_{kl}=(-1)^{m}\sqrt{2l+1}\begin{pmatrix}\frac{1}{2}(n-1)&\frac{1}{2}(n-1)&l\\\
\frac{1}{2}(m+k)&\frac{1}{2}(m-k)&m\end{pmatrix}.$ (D.31)

The parabolic quantum number $k$ is restricted to $n-|m|-1<k<-n+|m|+1$ in
steps of 2. The Stark shift resulting in [122],

$\Delta E=\frac{3}{2}nkE$ (D.32)

###  D.4 Anti-Crossing

Assume a two level state with levels $h$ and $k$ with energies $E^{0}_{k}$ and
$E^{0}_{h}$ which is perturbed by an applied perturbation $V$ connecting the
two states ($\bra{l}V\ket{j}=0$) while still different to zero for the
transition from $h$ to $k$ ($\bra{h}V\ket{k}\neq 0$). The perturbed state
energies, assuming the perturbation is small, are [24],

$E_{k}^{(2)}=\sum_{j\neq
k}\frac{|V_{kj}|^{2}}{E_{k}^{0}-E^{0}_{j}}=\frac{|V_{kh}|^{2}}{E_{k}^{0}-E^{0}_{h}},$
(D.33)

and,

$E_{k}^{(2)}=\sum_{j\neq
h}\frac{|V_{hj}|^{2}}{E_{h}^{0}-E^{0}_{j}}=\frac{|V_{kh}|^{2}}{E_{h}^{0}-E^{0}_{k}}=-E_{k}^{(2)}.$
(D.34)

In the absence of a perturbation, these two energy levels would cross when
$E_{k}^{0}=E^{0}_{h}$. However, with an added perturbation, both these energy
levels repel each other with opposite energy shifts. This phenomenon is called
an anti-crossing and is more noticeable when the energies $E_{k}^{0}$ and
$E_{h}^{0}$ are closer to each other [108].

###  D.5 Förster Energy Defect

Consider the electric dipole-dipole interaction between an atom A and an atom