When the initial state $|ψ(0)⟩$ is a computational basis state, there is a simple upper bound on the success probability achievable in closed system reverse annealing: the population of the initial state in the maximum-spin sector (see App.~\ref{append:successbound}). This upper bound explains why the following closed system results have relatively low success probabilities.

We plot in Fig.~\ref{fig:close0001_r1} the simulation results for the total success probability for various annealing rates, subject to the D-Wave annealing schedule shown in Fig.~\ref{fig:schedule}, in the same units. We see that the success probability after a single cycle is non-negligible only when $τ$ is small enough ($τ< 1$ [ns]), in which case diabatic transitions to states with lower energies take place for $s_in< s_Δ$. However, the success probability remains small even in this case, in any case much smaller than in our experimental results where thermal relaxation plays the dominant role.

\begin{figure}[h!]
\subfigure[]{\includegraphics[width=0.49\columnwidth]{ra2/4qubitsunitary_0001_new_largefont.pdf}\label{fig:4qubitsunitary}}
\subfigure[]{\includegraphics[width=0.4939\columnwidth]{ra2/8qubitsunitary_00000001_new_largefont.pdf}\label{fig:8qubitsunitary}}
\caption{Total success probability for closed system with $r=1$ (single cycle) and a single-spin down as the initial state. (a) $N=4$, (b) $N=8$. Here and in the subsequent figures the nanoscecond timescale is set by the energy scale of the D-Wave annealing schedule shown in Fig.~\ref{fig:schedule}.}
\label{fig:close0001_r1}
\end{figure}

\subsubsection{Dependence on initial state and number of iterations}
\label{sec:closed2}

We next choose the initial state $|ψ(0)⟩$ with two spins down $|0011⟩ (m_0=0)$ for $N=4$ and  $|00000011⟩ (m_0=0.5)$ for $N=8$, respectively, i.e., the second excited states of $H_0$ for these respective system sizes.

\begin{figure}[h!]
\subfigure[]{\includegraphics[width=0.4939\columnwidth]{ra2/4qubitsunitary_0011_new_largefont.pdf}\label{fig:4qubitsunitary2}}
\subfigure[]{\includegraphics[width=0.4939\columnwidth]{ra2/8qubitsunitary_00000011_new_largefont.pdf}\label{fig:8qubitsunitary2}}
\subfigure[]{\includegraphics[width=0.4939\columnwidth]{ra2/4qubitsunitary_0011_r2_new_largefont.pdf}\label{fig:4qubitsunitary3}}
\subfigure[]{\includegraphics[width=0.4939\columnwidth]{ra2/8qubitsunitary_00000011_r2_new_largefont.pdf}\label{fig:8qubitsunitary3}}
\caption{Total success probability with two spins down as the initial state. (a) $N=4$ and $r=1$, (b) $N=8$ and $r=1$, (c) $N=4$ and $r=2$, (d) $N=8$ and $r=2$.}
\label{fig:close0011_r1}
\end{figure}

Figure~\ref{fig:close0011_r1} (top row) shows that as expected, for $N=4$ and $r=1$, the success probability is overall lower than the case when the initial state is the first excited state, Fig.~\ref{fig:close0001_r1}. Only for the highest annealing rate (smallest $τ$, most diabatic) is the maximum success probability similar to the case of the first excited state. For $N=8$, the success probability is much smaller, no matter how diabatic the process is. This indirectly confirms the dominant role played by thermal effects in our experimental results.

We plot in Fig.~\ref{fig:close0011_r1} (bottom row) the results for $r=2$ cycles, where we see that the success probability decreases compared to the $r=1$ case. This is consistent with the conclusions of Ref.~\cite{nishimori:reverse-pspin-2}.

Recalling the experimental results reported in the previous section, we may therefore safely conclude that, as expected, the closed-system picture is far from the experimental reality of the D-Wave device.

\subsection{Open quantum system model}
\label{sec:AME}

For an open system, the state after the first cycle is
\begin{equation}
\rho(2t_{\text{inv}}) = V(2t_{\text{inv}},0)\rho(0) \,,
\end{equation}
\begin{equation}
V(t,0) = {\cal T}\exp\left[\int_0^{t}\mathcal{L}(t')dt'\right] \,.
\label{eq:vpropagator}
\end{equation}
$ℒ(t)$ is the time-dependent Liouville superoperator. We use the adiabatic master equation (AME)~\cite{ABLZ:12-SI}, where $ℒ(t)$  at time $t$ satisfies
\begin{align}
\frac{d\rho(t)}{dt}
&= \mathcal{L}(t)\rho(t)\\
&= i \bigl[\rho(t), H(t) + H_\text{LS}(t)\bigr] + \mathcal{D}\bigl[\rho(t)\bigr] \,.
\end{align}
Here, $ H_LS(t) $ is the Lamb shift term and $𝒟 $ is the dissipator. Again, the time dependence of the integrand in Eq.~\eqref{eq:vpropagator} is incorporated into $s(t)$.

In the weak coupling limit, $𝒟$ can be expressed in a diagonal form with Lindblad operators $L_i,ω(t)$:

\begin{eqnarray}
\label{eq:dissdiag}
{\mathcal{D}}_{}  [\rho(t)]  & = &\sum_{i}\sum_{\omega}\gamma_i(\omega)\left(
L_{i,\omega}(t) \rho(t) L^\dagger_{i,\omega}(t)\phantom{\frac{1}{2}} \right. \notag \\
&& \hspace{-0.5cm} \left. \qquad\qquad - \frac{1}{2} \left\{ L_{i,\omega}^{\dagger}(t) L_{i,\omega}(t) , \rho(t) \right\} \right) \,,
\end{eqnarray}
where the summation runs over the qubit index $i$ and the Bohr frequencies $ω$ [all possible differences of the time-dependent eigenvalues of $H(t)$]. This dissipator expresses decoherence in the energy eigenbasis~\cite{Albash:2015nx}: quantum jumps occur only between the eigenstates of $H(t)$~\cite{yip:mcwf}.

At the end of $r$ cyles, the final state $ρ(2rt_inv)$ is expressed as:
\begin{equation}
\rho(2rt_{\text{inv}}) = V(2rt_{\text{inv}}, 0)\rho(0)\,,
\end{equation}
\begin{equation}
V(2rt_{\text{inv}}, 0) = \prod_{i=0}^{r-1}V(2(i+1)t_{\text{inv}}, 2it_{\text{inv}}) \,.
\end{equation}

We consider two different models of system-bath coupling: independent and collective dephasing~\cite{LidarWhaley:03}.

In first case, we assume that the qubit system is coupled to independent, identical bosonic baths, with the bath and interaction Hamiltonians being
\begin{subequations}
\begin{align}
\label{eq:SBm}
H_B &= \sum_{i=1}^N \sum_{k=1}^\infty \omega_k b_{k,i}^\dagger b_{k,i} \ ,  \\
H_{SB}^{\text{ind}} &= g \sum_{i=1}^N
\sigma_i^z \otimes \sum_k \left(b_{k,i}^\dagger + b_{k,i} \right) \ ,
\end{align}
\end{subequations}
where $b_k,i^†$ and $b_k,i$ are, respectively, the raising and lowering operators for the $k$th oscillator mode with natural frequency $ω_k$.  The rates appearing in Eq.~\eqref{eq:dissdiag} are given by
\begin{equation}
\gamma_{i}(\omega) = 2\pi \eta g^2\frac{\omega e^{-|\omega|/\omega_c}}{1-e^{-\beta\omega}}  \ ,
\end{equation}
arising from an Ohmic spectral density, and satisfying the Kubo-Martin Schwinger condition~\cite{kubo1957statistical, martin1959theory} ($γ_i(- ω) = e^-βω γ_i(ω)$, $β= 1/T$ being the inverse temperature).  We use $ηg^2 = 10^-3$, the cutoff frequency $ω_c = 1$ [THz], and the D-Wave device operating temperature $T= 12.1  [mK] = 1.57$ [GHz]. With the independent dephasing assumption, the Lindblad operators are:
\begin{equation}
L_{i,\omega}(t) = \sum_{\epsilon_b-\epsilon_a = \omega} | \varepsilon_a(t) \rangle \langle \varepsilon_a(t) | \sigma^z_{i} | \varepsilon_b(t) \rangle \langle \varepsilon_b(t) | \,,
\end{equation}
corresponding to dephasing in the instantaneous eigenbasis ${|ε_a(t) ⟩}$ of $H(t)$. Similarly to the closed system case, we rotate the density matrix and Lindblad operators into the instantaneous energy eigenbasis at each time step in our numerical simulations. This keeps the matrices sparse, without loss of accuracy. For $N = 8$ we truncate the system size to the lowest $n=18 = 1+1+8+8$ (i.e. total number of degenerate the ground states and first excited states at $s=1$) levels out of $256$.

We also consider the collective dephasing model, where all the qubits are coupled to a collective bath with the same coupling strength $g$, which preserves the spin symmetry. In this case, the interaction Hamiltonian becomes
\begin{align}
H_{SB}^{\text{col}} = g S^z \otimes B,
\end{align}
\begin{align}
S^z = \sum_{i} \sigma_i^z, \quad B = \sum_k \left(b_{k}^\dagger + b_{k} \right).
\end{align}
With this assumption, we can group together the Lindblad operators corresponding to different qubit $i$ into a single one:
\begin{equation}
L_{\omega}(t) = \sum_{\epsilon_b-\epsilon_a = \omega} | \varepsilon_a(t) \rangle \langle \varepsilon_a(t) | S^z | \varepsilon_b(t) \rangle \langle \varepsilon_b(t) | \,.
\end{equation}
The resulting number of Lindblad operators is a factor of $N$ smaller than that of the independent system-bath coupling model.

The total success probability at the end of the $r$ cycles is
\begin{equation}
p(r) = \bra{\text{up}}\rho(2rt_{\text{inv}})\ket{\text{up}} + \bra{\text{down}}\rho(2rt_{\text{inv}})\ket{\text{down}} \,.
\end{equation}

Any relaxation during the reverse annealing dynamics to the global instantaneous ground state or the instantaneous first excited state of $H(t)$ is beneficial, the latter since it becomes degenerate with the ground states ${|up⟩,|down⟩}$ of $H_0$ at the end of the anneal (see App.~\ref{append:spectrum}).

\subsubsection{Dependence on annealing time}
\label{sec:AME-tau}

Figure~\ref{fig:4open_0001_r1} (top row) shows the simulation results for success probability as a function of $s_inv$ with $N=4$, various $τ$, and the initial state $|0001⟩ (m_0=0.5)$, using the independent and collective dephasing models.
\begin{figure}[h!]
\subfigure[]{\includegraphics[width=0.4939\columnwidth]{ra2/4qubitopen_0001_new_largefont.pdf}\label{fig:4qubitopena}}
\subfigure[]{\includegraphics[width=0.4939\columnwidth]{ra2/4qubitopen_0001_collective_new_largefont.pdf}\label{fig:4qubitopenb}}
\subfigure[]{\includegraphics[width=0.4939\columnwidth]{ra2/4qubitopen_0011_new_largefont.pdf}}
\subfigure[]{\includegraphics[width=0.4939\columnwidth]{ra2/4qubitopen_0011_r2_new_largefont.pdf}}
\caption{Top row: Success probability as a function of $s_{\mathrm{inv}}$ with (a) independent and (b) collective dephasing (SB denotes system-bath coupling). Initial state: $\ket{0001}~(m_0=0.5)$ and $r=1$, a single cycle. Bottom row: Success probability with different $\tau$. Initial state: $\ket{0011}~(m_0=0)$. (a) $r=1$, (b) $r=2$.}
\label{fig:4open_0001_r1}
\end{figure}
The independent dephasing model leads to a maximum success probability of $1$ for large $τ$ and small $s_inv$. However, the collective dephasing model leads to a maximum success probability of $1/4$. The reason is the same as in closed system simulations. For the initial state $|0001⟩$, only $1/4$ of its population is in the subspace of maximum quantum spin number.
However, the global (instantaneous) ground state and the first excited state of the annealing Hamiltonian $H(s)$ both belong to the maximum-spin subspace. Since the collective dephasing model preserves the spin symmetry and the dynamics is restricted to each subspace, at most $1/4$ of the initial population can be relaxed to the instantaneous ground state and the first excited state, and reach the correct solutions at the end of the anneal (see App.~\ref{append:successbound} for more details). It is also noteworthy that coherent oscillations are visible for $τ=1$ [ns] (compare with Fig.~\ref{fig:close0001_r1}(a)), but not for the larger values of $τ$ we have simulated. Recall that the experimental timescale in Sec.~\ref{sec:exp} is on the order of a [$μ$s].

Comparing the simulation results in Fig.~\ref{fig:4open_0001_r1} and the experimental results in Fig.~\ref{fig:RA_20_diff_tau}(b), we observe that they share the same main features. Namely, the success probability increases with $τ$; and, as $τ$ increases the maximum success probability can be achieved with a larger inversion point $s_inv$. Most notably, the total success probability also drops to zero for sufficiently large $s_inv$ in our simulations. This is the freeze-out effect that is well captured by the adiabatic master equation (as first reported in Ref.~\cite{ABLZ:12-SI}), since  thermal relaxation is suppressed when the transverse field magnitude becomes so small that the system and system-bath Hamiltonians effectively commute.

Note that the experimental results in Sec.~\ref{sec:exp} have a maximum success probability as high as $1$, while our closed system simulations and open-system simulations with the collective dephasing model have success probabilities upper bounded by some constants $<1$, as already discussed. The high total success probability observed in our experiments is evidence that, as expected, the dynamics in the D-Wave device do not preserve spin symmetry. Collective system-bath coupling cannot explain the experimental results, leaving independent system-bath coupling as the only candidate consistent with the experiments according to our simulations.

\subsubsection{Dependence on initial condition and the number of cycles}
\label{sec:simulations-init}

For the initial state $|0011⟩ (m_0=0)$ and with $r=1$, the independent dephasing model still gives a maximum success probability of $1$ as seen in Fig.~\ref{fig:4open_0001_r1}(c). The collective dephasing model (not shown) has a maximum success probability of $1/6$, since the initial state has $1/42 = 1/6$ of its population in the maximum-spin subspace.

Comparing Fig.~\ref{fig:4open_0001_r1}(a) and (c), the dependence of the success probability on $s_inv$ is  similar to that of the initial state $|0001⟩$. The $τ= 1$ [ns] coherent oscillations visible for the latter are more attenuated for $|0011⟩$, but this was also the case for the closed system simulations (contrast Fig.~\ref{fig:close0001_r1}(a) and Fig.~\ref{fig:close0011_r1}(a) at $τ= 1$ [ns]).
%The only exception is for $\tau = 1$ [ns], where reverse annealing is too fast for the system to experience the open-system effect.
The main feature distinguishing Fig.~\ref{fig:4open_0001_r1}(a) and (c) is the shift to the left of the $τ≥100$ [ns] curves for the initial state $|0011⟩$, i.e., as $m_0$ is reduced from $0.5$ to $0$. This is consistent with our experimental results, as can be seen in Fig.~\ref{fig:RA_20_initial}(b).

In Fig.~\ref{fig:4open_0001_r1}(d), we consider the dependence on the number of cycles $r$, using the independent system-bath model. For $r=2$, we see that the results are similar to those of a single cycle. However, we do see a small improvement in the sense of a slight shift to the right of the curves with $τ≥100$ [ns] compared with $r=1$, which is consistent with the experimental result shown in Fig.~\ref{fig:IRA}(d). We note that this improvement was not observed in the closed system case, as can be seen by contrasting Figs.~\ref{fig:close0011_r1}(a) and (c). Interestingly, there is also a small signature of coherent oscillations for $τ≤100$ [ns].

Finally, we note that compared to the closed systems case, the results depend much less on the initial condition.

\subsubsection{Size dependence and partial success probability}

In Fig.~\ref{fig:4vs8open}, we show the success probability for two different system sizes $N=4$ and $8$. The initial state has a single spin down. We observe that the results do not depend much on the system size, with the total success probabilities slightly larger (shifted to the right) for $N=4$. This is consistent with the experimental results for $N=4$ and $8$ shown in Fig.~\ref{fig:RA_20_size}(b).

\begin{figure}[h!]
\centering
\includegraphics[width=0.45\columnwidth]{ra2/4qubitsvs8qubitsopen_1us_largefont.pdf}
%    \includegraphics[width=0.4939\columnwidth]{8qubitsegplot_open_individual_largefont.pdf}
\caption{Success probability as a function of $s_{\mathrm{inv}}$ for independent dephasing. The initial state has one spin down and $\tau$ is $1 [\mu$s]. The main plot shows the results for $N=4$ and $8$. The inset shows the partial success probabilities for $N=8$.}
\label{fig:4vs8open}
\end{figure}

While the total success probabilities of our open-system simulations with independent dephasing are generally consistent with the experimental total success probabilities, our open-system simulations always produce symmetric partial success probabilities as shown in Fig.~\ref{fig:4vs8open}, in stark contrast with the experiments, where the all-up state is strongly favored over the all-down state in the region around the minimum gap (see all the top panels in the figures in Sec.~\ref{sec:exp}). Clearly, this reflects a significant failure of the AME model.

One reason for this discrepancy is that our simulations do not include a mechanism such as the polarized spin-bath which we believe explains the experimentally observed asymmetry for small $s_inv$ (see Sec.~\ref{sec:spinbathpol}). Moreover, our adiabatic master equation simulations result in a symmetric final all-up and all-down population
since any relaxation event during the anneal (at any $s$) to either the global instantaneous ground state $|ϵ_0(s)⟩$ or the instantaneous first excited state $|ϵ_1(s)⟩$ of $H(s)$, which become degenerate at $s=1$,
eventually populates $|ϵ_0(s=1)⟩ = (|↑⟩^⊗N + |↓⟩^⊗N)/√(2)$ or $|ϵ_1(s=1)⟩ = (|↑⟩^⊗N -
|↓⟩^⊗N)/√(2)$. These two states
have equal all-up and all-down populations.
%Note that even if $\ket{\epsilon_0(s)}$ and $\ket{\epsilon_1(s)}$ have different amplitudes near $s=1$,
Similar observations are also found in Ref.~\cite{albash2015consistency}.

\section{Semi-classical simulations}
\label{sec:classical}

In an effort to understand the asymmetric partial success probability observed in our experiments, we performed semi-classical simulations of the same problem using the spin-vector Monte Carlo (SVMC)~\cite{shin2014quantum} algorithm and a new variant with transverse-field-dependent
updates (SVMC-TF)~\cite{Albash2020ComparingRM}. For the $p$-spin problem, we replace the Hamiltonian of Eq.~\eqref{eq:H} by a classical Hamiltonian:
\begin{equation}
\mathcal{H}(s) = -\frac{A(s)}{2} \left(\sum_{i}^{N} \sin \theta_i\right) - \frac{B(s) N}{2}\left(\frac{1}{N}\sum_{i}^{N} \cos \theta_i\right)^{p}\,.
\label{eq:Hising}
\end{equation}
Each qubit $i$ is replaced by a classical $O(2)$ spin $M⃗_i=(sinθ_i,0,cosθ_i)$, $θ_i ∈[0, π]$. For the purpose of reverse annealing, we again need to specify the $t$ dependence of $s(t)$. The concept of time $t$ is here replaced by the number of Monte Carlo sweeps. we replace $τ$ by a specified number of total sweeps. The total number of sweeps is then $2τ(1-s_inv)$, in analogy to the total annealing time in Eq.~\eqref{eq:t_a}.

To simulate the effect of thermal hopping through this semi-classical landscape with inverse temperature $β$, we perform at each time step a spin update according to the Metropolis rule. In SVMC, a random angle $θ_i^' ∈[0, π]$ is picked for each spin $i$. Updates of the spin angles $θ_i$ to $θ_i^'$ are accepted according to the standard Metropolis rule associated with the change in energy ($ΔE$) of the classical Hamiltonian. For the $p$-spin problem, $ΔE$ cannot be expressed in a simple form as in case of the Ising problem Hamiltonian~\cite{shin2014quantum}.

In SVMC-TF, the random angle $θ_i^' =θ_i + ϵ_i(s)$ is picked in a restricted range
\begin{align}
\epsilon_i(s) \in \left[-\min \left(1,\frac{A(s)}{B(s)}\right) \pi, \min \left(1,\frac{A(s)}{B(s)}\right) \pi\right].
\end{align}
The goal of SVMC-TF is to restrict the angle update for $A(s) < B(s)$, and imitate the freeze-out effect discussed in Sec.~\ref{sec:freezeout}. The full SVMC and SVMC-TF algorithms for reverse annealing are summarized in App.~\ref{append:svmc}, including the expression for $ΔE$.

% A simple intuitive way to visualize the semiclassical dynamics described by the SMVC and SVMC-TF algorithms is to consider the energy landscape defined by $\mathcal{H}(s)$ [Eq.~\eqref{eq:Hising}] when equating all angles ($\theta_i \equiv \theta$). We plot the resulting surface in Fig.~\ref{fig:surface}. Note that the two ground states indeed have all equal angles, namely $\theta = 0$ (up) and $\theta = \pi$ (down). The semiclassical dynamics follow a path that starts from a given excited state in our simulations, hence not on the surface; if a sufficient number of spin flips (corresponding to angle updates) occur so that one of the two ground states is found then the dynamics descend to the surface shown, and remain on it unless another angle update flips a spin so the system reenters an excited state. It is clear that for $s_\mathrm{inv}$ close to the saddle point location at $s=0.334$, any random angle update that takes the system to the ground state will have a high probability of keeping it there.

% \begin{figure}[h!]
% \includegraphics[width = 0.8\columnwidth]{ra2/surface.pdf}
% \caption{The two saddle points are at $s=0.334$ and $\theta = \pi/2\pm 0.309\pi$.}
% \label{fig:surface}
% \end{figure}

\subsection{Total success probability}
We report on SVMC and SVMC-TF simulations for $N=4$ and $8$. We choose the initial condition with a single spin down. In terms of angles, for $N=4$ the initial angles are ${0,0,0,π}$. We again use the temperature $T=12.1$ [mK]. The classical analogue of the annealing time is chosen to be $τ= 10^3$ and $10^4$ sweeps.

In Fig.~\ref{fig:comparealltotal} we display the simulation results for the total success probability using SVMC and SVMC-TF. The number of samples is $10^4$.
\begin{figure}[h!]
% \subfigure[]{\includegraphics[width = 0.4939\columnwidth]{svmcplot_100sweeps.pdf}\label{fig:4qubitsunitarya}}
\subfigure[]{\includegraphics[width = 0.4939\columnwidth]{ra2/svmcplot_1000sweeps_largefont.pdf}}
\subfigure[]{\includegraphics[width = 0.4939\columnwidth]{ra2/svmcplot_10000sweeps_largefont.pdf}}
\caption{Total success probability for SVMC and SVMC-TF. (a) $\tau = 10^3$ sweeps, (b) $\tau = 10^4$ sweeps. Error bars are 2$\sigma$ over $10^4$ samples.}
\label{fig:comparealltotal}
\end{figure}
SVMC gives high total success probabilities even with large inversion points $s_inv$. A large $s_inv$ value means that during the whole reverse annealing process the ratio $A(s)/B(s)$ is small. In the D-Wave device, it is expected that for small $A(s)/B(s)$ the dynamics freezes, which makes it difficult to reach the ground state when the initial state is excited. With the number of sweeps increased from $τ=10^3$ to $10^4$, we observe that even for very large inversion points $s_inv$ the total success probability of SVMC can be as high as $1$. This is because the angle updates in SVMC are completely random and thus with a sufficient number of sweeps, it is possible for the state to flip to the correct solution.

However, in SVMC-TF, the range of angle update is restricted for $A(s)/B(s) < 1$. The restricted angle updates (freeze-out effect) prevent the state from flipping to the correct solutions. Therefore the total success probability for large inversion points $s_inv$ is basically zero in SVMC-TF, regardless of the number of sweeps. This is also what we observe in the experimental data and in the adiabatic master equation simulations, as discussed in Sec.~\ref{sec:AME-tau}.

\subsection{Partial success probability}
\label{sec:PSP-SVMC}

We compare the partial success probability obtained from SVMC and SVMC-TF in Figs.~\ref{fig:compareallpartial}(a) and (b), respectively, for $N=4$ and $τ= 10^3$. The SVMC-TF result for $N=8$ is plotted in Fig.~\ref{fig:compareallpartial}(c). Comparing the experimental data in Fig.~\ref{fig:RA_20_size}(a) and Fig.~\ref{fig:compareallpartial}, we observe a satisfactory agreement with SVMC-TF, in particular the correct trend for the unequal up and down partial success probabilities. Also, for a limited region of $s_inv$, the partial success probabilities of the all up states can be as high as $1$.
Classical SVMC-TF simulations also explain the D-Wave results of random systems in Ref.~\cite{Albash2020ComparingRM}.

\begin{figure}[h!]
\subfigure[]{\includegraphics[width = 0.4939\columnwidth]{ra2/svmcplot_1000sweeps_ge_largefont.pdf}}
\subfigure[]{\includegraphics[width = 0.4939\columnwidth]{ra2/svmctfplot_1000sweeps_n4_ge_largefont.pdf}}
\subfigure[]{\includegraphics[width=0.4939\columnwidth]{ra2/svmctfplot_1000sweeps_n8_ge_largefont.pdf}}
\caption{Partial success probability by (a) SVMC, $N=4 $and (b) SVMC-TF, $N=4$. $\tau = 10^3$ sweeps. (c) SVMC-TF, $N=8$. $\tau = 10^3$ sweeps. Error bars are 2$\sigma$ over $10^4$ samples.}
\label{fig:compareallpartial}
\end{figure}

For $N=8$, we observe a deviation from the experimental data shown in Fig.~\ref{fig:RA_20_size}(a) for $s_inv ≲0.3$, where there exists a small but clear difference in the probabilities of all-up and all-down states, whereas the SVMC-TF data do not show such a trend. As discussed in Sec.~\ref{sec:spinbathpol}, we attribute the difference for $s_inv ≲0.3$ to the spin-bath polarization effect, which is not modeled in our SVMC-TF simulations.

In Fig.~\ref{fig:partial16} we display SVMC-TF reverse annealing simulation results of partial success probabilities for $N=16,32$ with $τ= 10^3,10^4$ sweeps. For both sizes shown, the regime of high partial success probability for the all-up state is shifted slightly to higher $s_inv$ for $τ= 10^4$ sweeps then for $τ= 10^3$. This is consistent with the trend in the experimental results observed in Fig.~\ref{fig:RA_20_diff_tau}(a). However, the trend with system size is inconsistent with the experimental results shown in Fig.~\ref{fig:RA_20_size}(a): the results for $N=16$ and $N=32$ are virtually indistinguishable (apart from statistical fluctuations), which the experimental data shows that $N=16$ is not yet large enough for convergence. This (small) failure of the SVMC-TF model may hint at an interesting way in which to identify a ``quantum signature" in experimental quantum annealing~\cite{Boixo:2014yu,albash2015consistency}.

\begin{figure}[h!]
% \subfigure[]{\includegraphics[width = 0.4939\columnwidth]{svmcplot_100sweeps.pdf}\label{fig:4qubitsunitarya}}
\subfigure[]{\includegraphics[width = 0.4939\columnwidth]{ra2/svmcplot_1000sweeps_n16_ge.pdf}}
\subfigure[]{\includegraphics[width = 0.4939\columnwidth]{ra2/svmcplot_10000sweeps_n16_ge.pdf}}
\subfigure[]{\includegraphics[width = 0.4939\columnwidth]{ra2/svmcplot_1000sweeps_n32_ge.pdf}}
\subfigure[]{\includegraphics[width = 0.4939\columnwidth]{ra2/svmcplot_10000sweeps_n32_ge.pdf}}
\caption{Partial success probability for SVMC-TF. (a) $N=16$, $\tau = 10^3$ sweeps, (b) $N=16$, $\tau = 10^4$ sweeps, (c) $N=32$, $\tau = 10^3$ sweeps, (b) $N=32$, $\tau = 10^4$ sweeps. Error bars are 2$\sigma$ over $10^4$ samples.}
\label{fig:partial16}
\end{figure}

\section{\label{section:Conc}Conclusion of this chapter}
The previous chapter (Ref.~\cite{Passarelli2019}) has reported that the coupling with the environment improves the performance of reverse annealing for the $p$-spin model with $p=3$. In the present study, we have performed reverse annealing on a real device D-Wave 2000Q for the $p$-spin model with $p=2$ and have confirmed that the reverse annealing process yields high success probabilities as in the previous chapter (Ref.~\cite{Passarelli2019}).

We perform closed system simulations, and open-system simulations of quantum adiabatic master equation, which take into account the effect of decoherence in the instantaneous energy eigenbasis. The numerical simulations reveal that thermal relaxation indeed leads to high total success probabilities, which also match well with our experimental results. Experimental data of reverse annealing also in turn helps us understand the noise mechanism in D-Wave machines. Based on the experimental data, we make careful analysis on the simulation methods and the correct choice of system-bath coupling model.  We argue that reverse annealing would fail in a closed system setting or open system simulation with the assumption of collective dephasing.

To further support the experimental results of asymmetrical partial success probabilities, we perform semi-classical simulations of spin-vector Monte Carlo. Our results show that the semi-classical thermal effect also benefits the performance of reverse annealing in real-world quantum annealing devices. We hope that this realization would be a step forward in the understanding and development of devices where thermal effects lead to quantum acceleration.

\section{Acknowledgments of this chapter}
All experimental data is provided by Yuki Bando. The experimental section is written by Yuki Bando. We thank Masayuki Ohzeki for useful comments and Sigma-i Co., Ltd. for providing machine time on D-Wave 2000Q.
The research is based partially upon work supported by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA) and the Defense Advanced Research Projects Agency (DARPA), via the U.S. Army Research Office contract W911NF-17-C-0050. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the ODNI, IARPA, DARPA, or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon. Computation for some of the work described in this chapter was supported by the University of Southern California Center for High-Performance Computing and Communications (hpcc.usc.edu).

\chapter{Adiabatic reverse annealing}
\label{chap: ara}
We now move on to the study of another variant of reverse annealing called adiabatic reverse annealing, originally introduced in~\cite{perdomo:sombrero, nishimori:reverse-pspin, nishimori:reverse-pspin-2}, in an open system setting. We focus on the $p$-spin Hamiltonian with $p=3$.

\section{Introduction and terminology}
The annealing Hamiltonian for adiabatic reverse annealing (ARA) is~\cite{nishimori:reverse-pspin-2, Crosson2020}:
\begin{equation}
\label{eq:H}
H(s, \lambda) = sH_{0} + (1-s)(1- \lambda)H_{\text{init}} + \Gamma (1-s)\lambda V_{\text{TF}} \,.
\end{equation}

The time-dependent parameters $s(t)$ and $λ(t)$ both
change from $0$ to $1$ as time $t$ proceeds from $0$ to $τ$, where
$τ$ is the total anneal time. We focus on the case of a linear function $s(t) = t/τ$.

The notation of Eq.~\eqref{eq:H} is as follows.
\begin{equation}
H_{0} = -N\left(\frac{1}{N}\sum_{i=1}^{N} \sigma^{z}_i\right)^{p}
\end{equation}
is the target problem Hamiltonian, with $p=3$ and
$N$ the number of qubits.
\begin{equation}
H_{\text{init}} = -\sum_{i=1}^{N}\epsilon_i \sigma^{z}_i
\end{equation}
is the initial Hamiltonian, $ϵ_i = ±1$.
\begin{equation}
V_{\text{TF}} = -\sum_{i=1}^{N} \sigma^{x}_i
\end{equation}
is the transverse field Hamiltonian.
% $A(s)$ and $B(s)$ depends on the specific machine. The D-Wave 2000Q NASA processor annealing schedules of $A(s)$ and $B(s)$ are shown in Fig.~\ref{fig: DWavenasaschedule}.
We want to perform (adiabatic) reverse annealing with various initial states $|ψ(s=0)⟩ = |ϵ_1, ϵ_2, …, ϵ_N⟩$ (the ground state of $H_init$).

%\subsection{$\lambda(t) = s(t)$}
% \subsection{Staying in the maximum spin subspace}
% \begin{align}
%   H(s, \lambda) &= sH_{0} + (1-s)(1- \lambda)H_{\text{init}} + \Gamma (1-s)\lambda V_{\text{TF}} &\\
%     &= sH_{0} + (1-s)(1- s)H_{\text{init}} + \Gamma (1-s)s V_{\text{TF}}  \,.
% \end{align}
We further assume that $λ= s$. With this assumption, Eq.~\eqref{eq:H} is simplified as:
\begin{equation}
\label{eq:H_simp}
H(s) = sH_{0} + (1-s)^2 H_{\text{init}} + \Gamma (1-s)s V_{\text{TF}}  \,.
\end{equation}

In the special case that $λ(t) = 1$,
Eq.~\eqref{eq:H} is the quantum annealing (QA)  Hamiltonian.

% \subsection{Schedule}
\section{
Preliminary I: Adiabatic reverse annealing (ARA) Hamiltonian spectrum}
In this section, we want to explore how the spectrum changes by tuning certain parameters of Eq.~\eqref{eq:H_simp}. This is important in the calculations of diabatic transition rates, excitation and relaxation rates, and thus the understanding of open-system behavior of adiabatic reverse annealing (ARA).
The solution state of the $p$-spin problem with $N$ spins is $|↑⟩^⊗N$. Consider the initial state with the number of spin-downs $N_↓= 2$.
% first spin down, i.e. $\epsilon_1=-1$, $\epsilon_i=+1$ for $i \neq 1$.
We plot the spectrum for $N=10$ in Fig.~\ref{fig: ARAspectrum}.

\begin{figure}[h!]
\centering
\includegraphics[width=0.6\linewidth]{ra3/pspin_ARA_spectrum_10.pdf}
\caption{The $16$ lowest lying energies of $10$ qubits, with $H_{\text{init}}$ having $N_\downarrow = 2$.
%such that $\epsilon_1=-1$, $\epsilon_i=+1$ for $i \neq 1$.
\label{fig: ARAspectrum}
\end{figure}

\subsection{Controlling the parameters: $\Gamma$}
The energy gap between instantaneous eigenenergies $ϵ_i(s)$ and $ϵ_j(s)$ is $Δ_ij(s) = ϵ_i(s) - ϵ_j(s)$. The corresponding minimum energy gap is $Δ_ij = min_s Δ_ij(s)$.

In adiabatic reverse annealing, we are interested in the minimum energy gap between the ground state $ϵ_0(s)$ and the $1$st excited state $ϵ_1(s)$. It is denoted by $Δ$ ($=Δ_min)$.
\begin{equation}
\Delta = \Delta_{10} = \min_{s} \Delta_{10}(s) = \min_{s} \epsilon_{1}(s) - \epsilon_{0}(s) \,.
\end{equation}

Consider the initial state with $N_↑=1$. We want to explore how the gap properties change with $Γ$ in Eq.~\eqref{eq:H_simp}.
We calculate and plot in Fig.~\ref{fig: gammaplot} the value of $Δ_10(s)$, for $N=8$ and $Γ= {1, ⋯, 5}$. In the inset we plot the dependence of $Δ$ on $Γ$. In general $Δ$ decreases as $Γ$ decreases. Also, the shape of the gap is sharper with smaller $Γ$. This allows diabatic transitions.

\begin{figure}[h!]
\centering
\includegraphics[width=0.6\linewidth]{ra3/gamma_plot.pdf}
\caption{$\Delta(s)$ for $\Gamma = \{1, \cdots, 5\}$ in units where $h=1$.}
\label{fig: gammaplot}
\end{figure}

\subsection{Controlling the parameters: Initial states $\epsilon_i$ or ($H_{\text{init}}$)}

We want to investigate how the gap properties change with different initial states. In this case we focus on $Γ= 1$.
\begin{figure}[h!]
\centering
\includegraphics[width=0.6\linewidth]{ra3/pspin_p2_spectrum_4_Hinit.pdf}
\caption{$\Delta(s)$ for different $\epsilon_i$. $\Gamma = 1$ in units where $h=1$.}
\label{fig: gammaplot}
\end{figure}

From Fig.~\ref{fig: gammaplot}, we see that in general $Δ$ decreases as the Hamming distance between the initial state and the target state increases.

\subsection{Controlling the parameters: $N$}
We want to investigate how the minimum gap changes with system size $N$. In this case we focus on the initial state of
%$\epsilon_1=-1$, $\epsilon_i=+1$ for $i \neq 1$,
$N_↓=1$ and $Γ= 1$. The value of $Δ$ for each $N$ is plotted in Fig.~\ref{fig: nplot}.
We also plot the $s$ value where the minimum gap is found, i.e.,
\begin{equation}
s_{\mathrm{min}} = \mathrm{argmin} \Delta_{10}(s) \,.
\end{equation}
This is important for the calculation of relaxation rates for reverse annealing.

\begin{figure}[h!]
\centering
\includegraphics[width=0.6\linewidth]{ra3/gaprevealplot.pdf}
\caption{$\Delta$ for $N = \{3, \cdots, 20\}$ in units where $h=1$. Inset: the value of $s_{\mathrm{min}}$ for each $N$.}
\label{fig: nplot}
\end{figure}

\section{Preliminary II: Time evolution}
Let $|ψ(s=0)⟩$ ($|ψ(t=0)⟩$) denote the initial state. The anneal time is $τ$. The final state at the end of the adiabatic reverse annealing (ARA) and the success probability can be expressed as follows.

\subsection{Closed system}
The state after the ARA in a closed system is
\begin{equation}
\ket{\psi(\tau)} = U(\tau,0)\ket{\psi(0)}\,,
\end{equation}
\begin{equation}
U(\tau, 0) = {\cal T}\exp\left[-i\int_0^{\tau}H(t')dt'\right]
\end{equation}
is the unitary operator and $T$ denotes the forward time-ordering. The time-dependence of the Hamiltonian (in the integrand) is given by the $s(t)$ function.

% At the end of $r$ rounds, the final state $\ket{\psi(2rt_{\text{inv}})}$ can be expressed as:
% \begin{equation}
% \ket{\psi(2rt_{\text{inv}})} = U(2rt_{\text{inv}}, 0)\ket{\psi(0)}\,,
% \end{equation}
% with
% \begin{equation}
%     U(2rt_{\text{inv}}, 0) = \prod_{i=0}^{r-1}U(2(i+1)t_{\text{inv}}, 2it_{\text{inv}}) \,.
% \end{equation}

\subsection{Open system: Lindbladian dynamics}
For an open system, the state after ARA is
\begin{equation}
\rho(\tau) = V(\tau,0)\rho(0) \,,
\end{equation}
\begin{equation}
V(\tau,0) = {\cal T}\exp\left[\int_0^{\tau}\mathcal{L}(t')dt'\right] \,.
\label{eq:vpropagator}
\end{equation}
$ℒ(t)$ is the time-dependent Liouville superoperator. For example, in adiabatic master equation, $ℒ(t)$  at time $t$ takes the form of
\begin{align}
\label{eq:ame}
\frac{d\rho(t)}{dt}
&= \mathcal{L}(t)\rho(t)\\
&= \iu \bigl[\rho(t), \ham(t) + \ham\ped{LS}(t)\bigr] + \diss\bigl[\rho(t)\bigr] \,.
\end{align}
$ LS(t) $ is a Lamb shift term and $ $ is the dissipator superoperator. Again, the time-dependence of the integrand in Eq.~\eqref{eq:vpropagator} depends on the $s(t)$ function.

% At the end of $r$ rounds, the final state $\rho(2rt_{\text{inv}})$ can be expressed as:
% \begin{equation}
% \rho(2rt_{\text{inv}}) = V(2rt_{\text{inv}}, 0)\rho(0)\,,
% \end{equation}
% with
% \begin{equation}
%     V(2rt_{\text{inv}}, 0) = \prod_{i=0}^{r-1}V(2(i+1)t_{\text{inv}}, 2it_{\text{inv}}) \,.
% \end{equation}

\section{Preliminary III: Success metrics and classes of problems}
We use two success metrics for ARA: success probability and time to solution (TTS).
\subsection{Success probability}
The success probability refers to the probability of the final state being the target solution state. For the example of $p$-spin ($p=3$) model, the success probability at the end of ARA can be expressed as:

% \begin{equation}
%     \mathsf{Success\,probability} =
%     \begin{cases}
% |\langle\uparrow\uparrow\uparrow\uparrow|\psi(\tau)\rangle|^2, \text{\quad\quad in a closed system.}\\
% \langle\uparrow\uparrow\uparrow\uparrow|\rho(\tau)|\uparrow\uparrow\uparrow\uparrow\rangle , \text{\quad in an open system.}
% \end{cases}
% \end{equation}
\begin{equation}
\mathsf{Success\,probability\,\,} p_g(\tau) =
\begin{cases}
|^{\otimes N}\langle \,\uparrow|\psi(\tau)\rangle|^2, \text{\quad\quad in a closed system.}\\
^{\otimes N}\langle\,\uparrow|\rho(\tau)|\uparrow\,\rangle ^{\otimes N} , \text{\quad in an open system.}
\end{cases}
\end{equation}

\subsection{Time to solution (TTS)}
The time to solution (TTS) is defined as~\cite{ronnow:speedup}:

\begin{equation}
\mathsf{TTS}(\tau, p_d) = \tau \frac{\log (1-p_d)}{\log p_e(\tau)} \,.
\end{equation}

$p_d$ is the desired threshold probability and $p_e(τ) = 1 - p_g(τ)$.

\subsection{Classes of problems}
We want to explore four general classes of problems: 1. Adiabatic reverse annealing in an open system
($𝖠𝖱𝖠_𝖮𝗉𝖾𝗇$), 2. Adiabatic reverse annealing in a closed system ($𝖠𝖱𝖠_𝖢𝗅𝗈𝗌𝖾𝖽$), 3. Standard quantum annealing in an open system ($𝖰𝖠_𝖮𝗉𝖾𝗇$), and 4. Standard quantum annealing in a closed system ($𝖰𝖠_𝖢𝗅𝗈𝗌𝖾𝖽$). The comparison between $𝖠𝖱𝖠_𝖢𝗅𝗈𝗌𝖾𝖽$ and $𝖰𝖠_𝖢𝗅𝗈𝗌𝖾𝖽$ has been made in~\cite{nishimori:reverse-pspin-2} and it was shown that $𝖠𝖱𝖠_𝖢𝗅𝗈𝗌𝖾𝖽$ can have an advantage over $𝖰𝖠_𝖢𝗅𝗈𝗌𝖾𝖽$. Since there are many studies (e.g.~\cite{passarelli:pspin} and~\cite{Passarelli2019}) showing that dissipation can give quantum enhancement, we want to investigate further if and under what conditions $𝖠𝖱𝖠_𝖮𝗉𝖾𝗇$ performs better than  $𝖠𝖱𝖠_𝖢𝗅𝗈𝗌𝖾𝖽$. On an equal setting, we want also to compare if $𝖠𝖱𝖠_𝖮𝗉𝖾𝗇$ has any computational advantage over $𝖰𝖠_𝖮𝗉𝖾𝗇$, in order to support the use of the formal protocol in a realistic experimental setting. Any advantage/enhancement is defined according to the two success metrics mentioned before.

We have already studied how the spectrum depends on $Γ$, $H_init$ and $N$. We want to evaluate how the performance of $𝖠𝖱𝖠$ and $𝖰𝖠$ depends on these parameters. For each set of such parameters, we performed simulations in a range of anneal times $τ$.

As noted in~\cite{Crosson2020}, diabatic transition to higher excited states may be a shortcut of the route to the final solution. Therefore, we include in our studies anneal times $τ$ shorter than the one set by adiabatic condition~\cite{albash:review-aqc}. We also explore the cases with small $Γ$ and large Hamming distance between the initial state and the target state, which result in a very small and sharp gap and thus diabatic transitions to higher excited states.

\section{Illustrative examples}
We perform numerical experiments for both closed and open system. We use the full Hamiltonian without sector decomposition. The coupling is $η= 1e^-3$ and the cutoff frequency is  $ω_c = 1$THz. We use an independent dephasing bath $ind = g ∑_i σ_i^z ⊗B_i$ and a collective dephasing bath  $col = g ∑_i σ_i^z ⊗B$, and perform simulations of the adiabatic master equation Eq.~\eqref{eq:ame}. We begin by studying three examples in the following subsections (Sec.~\ref{subsectiona}-~\ref{subsectionc}) and illustrate the dynamics of ground state population during the anneal.

%\subsection{Introductory examples}
\subsection{Results of $8$-qubit simulation: $\tau = 40$ns. $\Gamma = 1$.}
\label{subsectiona}
We simulate the closed and open-system evolution with different initial states. In this example we have $N=8$, $τ= 40$ns. $Γ= 1$. The results are plotted in Fig.~\ref{fig:8ame_727}. The solid lines are open-system simulation results while the dotted lines are closed-system simulation results.
\begin{figure}[h!]
\centering
\includegraphics[width = 12cm]{ra3/ame_plot_2_727.pdf}\label{fig:8qubitsame}
\caption{Open-system and close-system adiabatic reverse annealing (ARA) simulation results. $\tau = 40$ns. $\Gamma = 1$.}
\label{fig:8ame_727}
\end{figure}

\textit{Conclusion: }
From Fig.~\ref{fig:8ame_727}, we see that open-system effect (thermal excitation) is detrimental in this case, i.e., for all initial states specified, the solid lines are below the dotted lines along the evolution and at the end of the evolution.

\subsection{Results of $4$-qubit simulation: $\tau = 4$ns. $\Gamma = 1$.}
In this example we have $N=4$, $τ= 4$ns, and $Γ= 1$. The results are plotted in Fig.~\ref{fig:4ame_723}. The solid lines are open-system simulation results while the dotted lines are closed-system simulation results.

\begin{figure}[h!]
\centering
\includegraphics[width = 12cm]{ra3/ame_plot_2_723.pdf}\label{fig:8qubitsame}
\caption{Open-system and close-system adiabatic reverse annealing (ARA) simulation results. $\tau = 4$ns. $\Gamma = 1$.}
\label{fig:4ame_723}
\end{figure}

\textit{Conclusion: }
From Fig.~\ref{fig:4ame_723}, we see that open-system effect makes no difference, i.e. for all initial states specified, the solid lines almost overlap with the dotted lines along the evolution and at the end of the evolution. The total anneal time $τ= 4$ns is too short for the environment (bath) to come into effect. The environment causes little difference to the success probabilities. This short anneal time also leads to observable diabatic transitions to higher excited states.

\subsection{Results of $4$-qubit simulation: $\tau = 400$ns. $\Gamma = 0.3$.}
\label{subsectionc}
In this example we have $N=4$, $τ= 400$ns. $Γ= 0.3$. The results are plotted in Fig.~\ref{fig:4ame_724_2}.

\begin{figure}[h!]
\centering
\includegraphics[width = 12cm]{ra3/ame_plot_2_724_2.pdf}\label{fig:8qubitsame}
\caption{Open-system and close-system adiabatic reverse annealing (ARA) simulation results. $\tau = 400$ns. $\Gamma = 0.3$.}
\label{fig:4ame_724_2}
\end{figure}
\textit{Conclusion: }
From Fig.~\ref{fig:4ame_724_2}, we observe sudden diabatic transitions, since $Γ= 0.3$ results in very small gaps for most of the $H_init$. Meanwhile, a relatively long anneal time of $τ= 400$ns allows open-system relaxation mechanism to increase the (instantaneous and final success probability) ground state population for some of the initial states.
% \begin{figure}[h!]
% \subfigure[]{\includegraphics[width = 8cm]{ame_plot_2_724_2.pdf}\label{fig:4qubitsunitarya}}
% \subfigure[]{\includegraphics[width = 8cm]{ame_plot_2_724_2.pdf}\label{fig:8qubitsunitary}}
% \caption{Open-system and close-system adiabatic reverse annealing (ARA) simulation results. $\tau = 400$ns. $\Gamma = 0.3$.}
% \label{fig:compareall}
% \end{figure}
% \subsubsection{Results of $4$-qubit simulation: $\tau = 400$ns. $\Gamma = 5$.}

\section{ARA: Random initial state $c=0.5$.}
As in previous studies, the success of ARA depends on whether our initial state is close to the correct solution or not. In this section, we study the most general case where the initial state is a random guess ($c=0.5$, where $c=N_↑/N$) and want to see if $𝖠𝖱𝖠_𝖮𝗉𝖾𝗇$ can give an enhancement over $𝖠𝖱𝖠_𝖢𝗅𝗈𝗌𝖾𝖽$. Instead of looking at ground state population along the anneal, we now focus only on the results of ARA at the end of anneal. A summary of the results as a function of $N$ and $τ$ is shown in Fig.~\ref{fig:compareall}.

\begin{figure}[h!]
\subfigure[]{\includegraphics[width = 8cm]{ra3/annealsuccessprob_closed_randomguess.pdf}\label{fig:4qubitsunitarya}}
\subfigure[]{\includegraphics[width = 8cm]{ra3/annealsuccessprob_open_randomguess.pdf}\label{fig:8qubitsunitary}}
\subfigure[]{\includegraphics[width = 8cm]{ra3/annealsuccessprob_col_randomguess.pdf}\label{fig:8qubitsunitary}}
\subfigure[]{\includegraphics[width = 8cm]{ra3/annealsuccessprob_qa_randomguess_TTS_compare_10.pdf}\label{fig:compareall}}
\caption{ARA performance for (a) closed system, (b) open system with independent system-bath coupling, (c) open system with collective system-bath coupling, (d) comparison of the 3 situations in terms of TTS ($N=10$).}
% \label{fig:compareall}
\end{figure}

First of all, we observe that success probabilities go up monotonically with anneal time $τ$ for the case of $𝖠𝖱𝖠_𝖢𝗅𝗈𝗌𝖾𝖽$. As expected, the longer the anneal time $τ$, the more adiabatic the evolution and the higher the success probability.
For $𝖠𝖱𝖠_𝖮𝗉𝖾𝗇$, however, we observe that there can exist an intermediate anneal time $τ$ that gives a locally maximal success probability, and this is true for both SB coupling models. Such non-monotonic dependence of success probabilities in terms of anneal times $τ$ is due to the environmental induced relaxation after diabatic transitions to higher excited states.

Secondly, at long anneal times, for the case of $𝖠𝖱𝖠_𝖢𝗅𝗈𝗌𝖾𝖽$ the ground state is reached with with certainty (success probability $= 1$); while for the case of $𝖠𝖱𝖠_𝖮𝗉𝖾𝗇$, the Gibbs state is reached. Therefore, in all situations $𝖠𝖱𝖠_𝖢𝗅𝗈𝗌𝖾𝖽$ always outperforms $𝖠𝖱𝖠_𝖮𝗉𝖾𝗇$ for long enough anneal times $τ$. However, as shown in later section~\ref{sec:modelcompare}, the steady state reached by the two different system bath coupling model is different.

To see if relaxation can provide any quantum enhancement, we do observe that for some $N$ $𝖠𝖱𝖠_𝖮𝗉𝖾𝗇$ performs better than $𝖠𝖱𝖠_𝖢𝗅𝗈𝗌𝖾𝖽$ in the cases of intermediate anneal times $τ$. We demonstrate in Fig.~\ref{fig:compareall} for the case of $N=10$ that $𝖠𝖱𝖠$ performs better in an open system settings in terms of shorter $𝖳𝖳𝖲$. For both  $𝖠𝖱𝖠_𝖢𝗅𝗈𝗌𝖾𝖽$ and $𝖠𝖱𝖠_𝖮𝗉𝖾𝗇$ we observe optimal TTS. To avoid the misinterpretation of TTS, we avoid the calculation of $𝖳𝖳𝖲(τ)$ where the success probability $p_g(τ)$ is very close to $0$ and $1$. Lastly, we can see that in general the performance of both $𝖠𝖱𝖠_𝖢𝗅𝗈𝗌𝖾𝖽$ and  $𝖠𝖱𝖠_𝖮𝗉𝖾𝗇$ worsens as $N$ increases, as expected due to the closure of the gap with increasing system size.

% We draw several conclusions:
% 1. $\mathsf{ARA_\text{Open}}$ has optimal stopping times. But $\mathsf{ARA_\text{Closed}}$ does not.
% 2. Whether $\mathsf{ARA_\text{Open}}$ can beat $\mathsf{ARA_\text{Closed}}$ is case by case scenario
% Attached plot where it beats
% 3.
% The conclusion of optimal stopping times is clearer when we look at the TTS.
% We made the conclusion that instead of

\subsection{QA}
\begin{figure}[h!]
% \subfigure[]{\includegraphics[width = 8cm]{annealsuccessprob_qa_closed_randomguess.pdf}\label{fig:4qubitsunitarya}}
\subfigure[]{\includegraphics[width = 8cm]{ra3/annealsuccessprob_qa_open_randomguess.pdf}\label{fig:8qubitsunitary}}
\subfigure[]{\includegraphics[width = 8cm]{ra3/annealsuccessprob_qa_col_randomguess.pdf}\label{fig:8qubitsunitary}}
\caption{Standard QA in an open system with two different system-bath coupling model.}
\label{fig:qaopen}
\end{figure}

We show in Fig.~\ref{fig:qaopen} the case of quantum annealing in open system $𝖠𝖱𝖠_𝖮𝗉𝖾𝗇$. Note that the initial state of QA is the equal superposition  $|+⟩^⊗N$. Comparing Fig.~\ref{fig:qaopen} with Fig.~\ref{fig:compareall}, we can see easily that $𝖰𝖠_𝖮𝗉𝖾𝗇$ achieves higher success probabilities for all $N$ and $τ$. Therefore, for random initial states $c=0.5$,  $𝖠𝖱𝖠_𝖮𝗉𝖾𝗇$ cannot outperform $𝖰𝖠_𝖮𝗉𝖾𝗇$.

\section{ARA performance in terms of $\Gamma$ and $H_{\mathrm{init}}$}
We now want to study the dependence of ARA performance in terms of $H_init$ and $Γ$. For this purpose, we fix the number of qubits $N$ in our studies in this section.
\subsection{Min. Gap vs $H_{\mathrm{init}}$ vs $\Gamma$. $N=4$.}
\begin{figure}[h!]
\centering
\includegraphics[width=0.8\textwidth]{ra3/pspin_ARA_spectrum_3d.pdf}
\caption{The value of minimum gap as a function of $H_{\mathrm{init}}$ and $\Gamma$. $N=4$.}
\label{fig:spectrum_3d}
\end{figure}

We first plot in Fig.~\ref{fig:spectrum_3d} the value of minimum gap of ARA ($N=4$) as a function of initial state $H_init$ and $Γ$. Except for initial states that are close to the correct solutions, in general the larger the transverse field $Γ$ the larger the minimum gap $Δ_min$.

\subsection{Different Anneal times + Different $\Gamma = \{1,2,5\}$. $N=4$. $\mathsf{ARA_\text{Open}}$}
% \begin{figure}[h!]
% \begin{subfigure}[b]{0.49\textwidth}
% \includegraphics[width=3.33in]{annealsuccessprob_89_trial.pdf}
% \caption{}
% \label{fig:sub1}
% \end{subfigure}
% \begin{subfigure}[b]{0.49\textwidth}
% \includegraphics[width=3.33in]{annealsuccessprob_825_trial.pdf}
% \caption{}
% \label{fig:sub3}
% \end{subfigure}
% \begin{subfigure}[b]{0.49\textwidth}
% \includegraphics[width=3.33in]{annealsuccessprob_810_trial.pdf}
% \caption{}
% \label{fig:sub3}
% \end{subfigure}
% \caption[]{}
% \label{fig: 3atoms}
% \end{figure}
\begin{figure}[h!]
\subfigure[]{\includegraphics[width = 8cm]{ra3/annealsuccessprob_89_trial.pdf}\label{fig:4qubitsunitarya}}
\subfigure[]{\includegraphics[width = 8cm]{ra3/annealsuccessprob_825_trial.pdf}\label{fig:8qubitsunitary}}
\subfigure[]{\includegraphics[width = 8cm]{ra3/annealsuccessprob_810_trial.pdf}\label{fig:8qubitsunitary}}
\caption{$\mathsf{ARA_{\text{open}}}$ results (a) $\Gamma=1$, (b) $\Gamma=2$, (c) $\Gamma=5$. Independent system-bath coupling.}
\label{fig:ARAopen4qubits}
\end{figure}

We simulate the performance of $𝖠𝖱𝖠_𝖮𝗉𝖾𝗇$ as a function of initial state $H_init$ and $Γ$, and plot the results in Fig.~\ref{fig:ARAopen4qubits} . We focus on the assumption of Independent system-bath coupling. In general the closer the initial state to the correct solution, the higher the success probabilities of $𝖠𝖱𝖠_𝖮𝗉𝖾𝗇$. Except for initial states that are close to the correct solutions, in general we want to increase the transverse field $Γ$ to enhance ARA performance because it results in a larger gap, in agreement with the observation in Fig.~\ref{fig:spectrum_3d}. We also plot this dependence on $Γ$ in Fig.~\ref{fig:ARAopen4qubitsuudd} in terms of success probability and $𝖳𝖳𝖲$. $𝖳𝖳𝖲$ is also shorter for larger $Γ$ and we observe an optimal $𝖳𝖳𝖲$ for all three values of $Γ$.
% \subsection{Different Anneal times + $\Gamma = 1$. $4$ qubits. Closed vs Open.}
% We want to explore different anneal times $\tau$. One observation about the TTS is that

% \begin{figure}[h!]
% \includegraphics[width=0.8\textwidth]{success_duuu_close_open.pdf}\label{fig:4qubitscloseame}
% \label{fig:4ame_724_2}
% \caption{Black: open, Blue: closed.}
% \end{figure}

\begin{figure}[h!]
\subfigure[]{\includegraphics[width = 8cm]{ra3/annealsuccessprob_89_810_825.pdf}\label{fig:4qubitsunitarya}}
\subfigure[]{\includegraphics[width = 8cm]{ra3/TTS_89_810_825.pdf}\label{fig:8qubitsunitary}}
\caption{$\mathsf{ARA_{\text{open}}}$ results, with initial state being $\ket{\downarrow\downarrow\uparrow\uparrow}$. (a) success probability, (b) $\mathsf{TTS}$. Independent system-bath coupling.}
\label{fig:ARAopen4qubitsuudd}
\end{figure}

% \begin{figure}[h!]
% \includegraphics[width=0.8\textwidth]{annealsuccessprob_89_810_825.pdf}\label{fig:8qubitsame}
% \label{fig:4ame_724_2}
% \end{figure}
\subsection{Different Anneal times + Different  $\Gamma<1$. $N=4$. $\mathsf{ARA_\text{Closed}}$ and $\mathsf{ARA_\text{Open}}$}
We study the case of $Γ<1$, which would results in a smaller gap (Fig.~\ref{fig:spectrum_3d}) except for the case where the initial state is equal to the correct solution. We plot in Fig.~\ref{fig:gammasmaller1_closed} and Fig.~\ref{fig:gammasmaller1_open} the simulated performance of $𝖠𝖱𝖠_𝖢𝗅𝗈𝗌𝖾𝖽$ and $𝖠𝖱𝖠_𝖮𝗉𝖾𝗇$ respectively as a function of initial state $H_init$ and $Γ$. Again we focus on the assumption of independent system-bath coupling.

\begin{figure}[h!]
\subfigure[]{\includegraphics[width = 8cm]{ra3/annealsuccessprob_82_Gamma01}\label{fig:4qubitsunitarya}}
\subfigure[]{\includegraphics[width = 8cm]{ra3/annealsuccessprob_82_Gamma05}\label{fig:8qubitsunitary}}
\caption{$\mathsf{ARA_\text{Closed}}$  (a) $\Gamma = 0.1$, (b) $\Gamma = 0.5$. $N=4$.}
\label{fig:gammasmaller1_closed}
\end{figure}

\begin{figure}[h!]
\subfigure[]{\includegraphics[width = 8cm]{ra3/annealsuccessprob_82_open_Gamma01}\label{fig:4qubitsunitarya}}
\subfigure[]{\includegraphics[width = 8cm]{ra3/annealsuccessprob_82_open_Gamma05}\label{fig:8qubitsunitary}}
\caption{$\mathsf{ARA_\text{Open}}$. (a) $\Gamma = 0.1$, (b) $\Gamma = 0.5$. $N=4$. Independent system-bath coupling.}
\label{fig:gammasmaller1_open}
\end{figure}

Comparing the results of $𝖠𝖱𝖠_𝖢𝗅𝗈𝗌𝖾𝖽$ (Fig.~\ref{fig:gammasmaller1_closed}) and $𝖠𝖱𝖠_𝖮𝗉𝖾𝗇$ (Fig.~\ref{fig:gammasmaller1_open}), we again observe the non-monotonic dependence of success probabilities in terms of anneal times $τ$ exclusively for $𝖠𝖱𝖠_𝖮𝗉𝖾𝗇$. We also see noticeable improvements of success probabilities due to open-system effect at intermediate anneal times $τ$, especially if the initial states are far away from the correct solution. For both $𝖠𝖱𝖠_𝖢𝗅𝗈𝗌𝖾𝖽$ and $𝖠𝖱𝖠_𝖮𝗉𝖾𝗇$, the larger $Γ$ the higher the success probabilities, except for the case where the initial state is equal to the correct solution.
% \clearpage
% \subsection{Different Anneal times + $\Gamma = 0.5$. $4$ qubits. Closed vs Open}
% \begin{figure}[h!]
% \includegraphics[width=0.8\textwidth]{annealsuccessprob_82_Gamma05_dddd_openvsclose.pdf}\label{fig:4qubitscloseame}
% \label{fig:4ame_724_2}
% \caption{Black: open, Blue: closed.}
% \end{figure}

% \clearpage
% \section{Pausing}
% \subsection{Different Anneal times + $\Gamma = 1$. $4$ qubits. Pausing for a duration $\tau$ at $s=0.3$. Closed vs Open}
% Closed:
% \begin{figure}[h!]
% \includegraphics[width=0.65\textwidth]{annealsuccessprob_Gamma1_pausing}\label{fig:4qubitscloseame}
% \label{fig:4ame_724_2}
% \caption{}
% \end{figure}

% \clearpage
% \subsection{Different Anneal times + $\Gamma = 1$. $4$ qubits. Pausing for a duration $\tau$ at $s=0.3$. Closed vs Open}
% Open:
% \begin{figure}[h!]
% \includegraphics[width=0.65\textwidth]{annealsuccessprob_open_Gamma1_pausing}\label{fig:4qubitscloseame}
% \label{fig:4ame_724_2}
% \caption{}
% \end{figure}

% \clearpage
% \subsection{Different Anneal times + $\Gamma = 1$. $4$ qubits. Pausing for a duration $\tau$ at $s=0.3$. Closed vs Open}
% \begin{figure}[h!]
% \includegraphics[width=0.8\textwidth]{annealsuccessprob_Gamma1_dddd_openvsclose_pausing}\label{fig:4qubitscloseame}
% \label{fig:4ame_724_2}
% \caption{Black: open, Blue: closed.}
% \end{figure}

% \clearpage
% \section{$8$, $10$-qubit example}
% \subsection{Different Anneal times. $8$ qubits. Two spins down. $\Gamma = 1$.}
% \begin{figure}[h!]
% \includegraphics[width=0.8\textwidth]{annealsuccessprob_dduuuuuu.pdf}\label{fig:8qubitsame}
% \caption{Black: open, Blue: closed.}
% \label{fig:4ame_724_2}
% \end{figure}

% \clearpage
% \subsection{Different Anneal times. $8$ qubits. Two spins down. $\Gamma = 1$.}
% \begin{figure}[h!]
% \includegraphics[width=0.8\textwidth]{TTS_dduuuuuu.pdf}\label{fig:8qubitsame}
% \caption{Black: open, Blue: closed.}
% \label{fig:4ame_724_2}
% \end{figure}

% \clearpage
% \subsection{Different Anneal times. $10$ qubits. Two spins down. $\Gamma = 1$.}
% \begin{figure}[h!]
% \includegraphics[width=0.8\textwidth]{annealsuccessprob_dduuuuuuuu.pdf}\label{fig:8qubitsame}
% \caption{Black: open, Blue: closed.}
% \label{fig:4ame_724_2}
% \end{figure}

% \clearpage
% \subsection{Different Anneal times. $10$ qubits. Two spins down. $\Gamma = 1$.}
% \begin{figure}[h!]
% \includegraphics[width=0.8\textwidth]{annealTTS_dduuuuuuuu.pdf}\label{fig:8qubitsame}
% \caption{Black: open, Blue: closed.}
% \label{fig:4ame_724_2}
% \end{figure}

% \clearpage
% \subsection{Different Anneal times. $8$ qubits. Two spins down. $\Gamma = 0.1$.}
% \begin{figure}[h!]
% \includegraphics[width=0.8\textwidth]{annealsuccessprob_dduuuuuu_2.pdf}\label{fig:8qubitsame}
% \caption{Black: open, Blue: closed.}
% \label{fig:4ame_724_2}
% \end{figure}

% \clearpage
% \subsection{Different Anneal times. $8$ qubits. Two spins down. $\Gamma = 0.1$.}
% \begin{figure}[h!]
% \includegraphics[width=0.8\textwidth]{annealTTS_dduuuuuu_2.pdf}\label{fig:8qubitsame}
% \caption{Black: open, Blue: closed.}
% \label{fig:4ame_724_2}
% \end{figure}

% \clearpage
% \subsection{Different Anneal times. $10$ qubits. Two spins down. $\Gamma = 0.1$.}
% \begin{figure}[h!]
% \includegraphics[width=0.8\textwidth]{annealsuccessprob_dduuuuuuuu_2.pdf}\label{fig:8qubitsame}
% \caption{Black: open, Blue: closed.}
% \label{fig:4ame_724_2}
% \end{figure}

\subsection{Different Anneal times. $\Gamma = 0.1$, $N_{\downarrow}=2$. $N=\{8,10\}$. $\mathsf{ARA_\text{Open}}$.}
We study the extreme case of $Γ= 0.1$ for larger $N$. Note that for such small $Γ= 0.1$, the transverse field is very weak and the Hamiltonian is dominated by the problem Hamiltonian $H_0$ and $H_init$. The resulting gap is very small and sharp and there is a large degree of diabatic transition to higher excited states.

We plot in Fig.~\ref{fig:8_10_gamma01} the $𝖳𝖳𝖲$ dependence of anneal times $τ$, for $N={8,10}$ both with $N_↓=2$. Due to the significant diabatic transition to higher excited states, $𝖠𝖱𝖠_𝖮𝗉𝖾𝗇$ tends to perform better than $𝖠𝖱𝖠_𝖢𝗅𝗈𝗌𝖾𝖽$ because of the relaxation to the ground state.
\begin{figure}[h!]
\subfigure[]{\includegraphics[width = 8cm]{ra3/annealTTS_dduuuuuu_2.pdf}\label{fig:4qubitsunitarya}}
\subfigure[]{\includegraphics[width = 8cm]{ra3/annealTTS_dduuuuuuuu_2.pdf}\label{fig:8qubitsunitary}}
\caption{(a) $N=8$, (b) $N=10$. $\Gamma = 0.1$ and $N_{\downarrow}=2$. Independent system-bath coupling.}
\label{fig:8_10_gamma01}
\end{figure}

\section{Independent vs collective SB coupling}
\label{sec:modelcompare}
We investigate the difference between the  independent and collective system-bath coupling assumption.

\subsection{Long anneal times $\tau$}
As observed in Fig.~\ref{fig:compareall}, for long anneal times $τ$ the success probabilities given by simulation with the collective SB coupling assumption are higher than those given by independent SB coupling.
In fact, the steady state solution $ρ(s = 1,τ→∞)$ of the Lindbladian master equation in the weak coupling limit is the Gibbs state set by the problem Hamiltonian.
\begin{equation}
\rho(s = 1,\tau \rightarrow \infty) = \frac{e^{- \beta H_S(1)}}{\mathcal{Z}} = \frac{e^{- \beta H_0}}{\mathcal{Z}}\ .
\label{eq:rhoS0}
\end{equation}
where $𝒵 = Tr(exp(- βH_0 ))$.

Recall that the permutation symmetry allows us to write the Hamiltonian in terms of only a set of total spin operators of two subsystems,
\begin{equation}
\hat S^{x,z}_1 \equiv \frac{1}{2}\sum_{i=1}^{|N_{\uparrow}|}\hat\sigma^{x,z}_i,\quad
\hat S^{x,z}_2 \equiv \frac{1}{2}\sum_{i=|N_{\uparrow}|+1}^{N}\hat\sigma^{x,z}_i \,.
\label{ra:eq:s1s2}
\end{equation}
% as
% \begin{align}
%  \hat H_0 &= -N(\frac{2}{N}(\hat S^z_1 + \hat S^z_2))^p \\
%  \hat H_{\text{init}} &= -2(\hat S^z_1 - \hat S^z_2),
%  \hat V_{\text{TF}}=-(\hat S^x_1 + \hat S^x_2).
%  \label{eq:H_blocked}
% \end{align}

% Since \((\hat{\bf{S}}_1)^2\) and \((\hat{\bf{S}}_2)^2\) commute with the total Hamiltonian, the system stays in the subspace with the largest values of \((\hat{\bf{S}}_1)^2\) and \((\hat{\bf{S}}_2)^2\) because the initial state lies in this subspace.

For collective SB coupling model, the dynamics occurs entirely in the two subspaces with maximum eigenvalues
of $S_1^2$ and $S^2_2$. The Lindbladian is defined according to the projection of $H(s)$ to such  subspaces.
% therefore we can restrict numerical simulations to the subspace
% given by their tensor product, having dimension $D = (\lfloor Nc \rfloor + 1)(N - \lfloor Nc\rfloor +
% 1)$, quadratic rather than exponential in $N$. In the two subspaces, suitable
% bases are given by the simultaneous eigenstates of $\{ S_j^2, S_j^z \}$, denoted $\left|j; w\right>$ and
% such that $S_j^z \left|j; w\right> = (S^z_{j,\text{max}} - w)\left|j; w\right>$, with $S^z_{1,\text{max}} = \lfloor Nc \rfloor /2$ and $S^z_{2,\text{max}} =
% (N - \lfloor Nc \rfloor)/2$. In this representation, the initial state is
% $\left|\psi(0)\right> = \left|1;0 \right> \otimes \left|2;N - \lfloor Nc \rfloor\right>$.
For collective SB coupling model, the steady state is given by:
\begin{equation}
\rho'(s = 1,\tau \rightarrow \infty) = \frac{e^{- \beta H_0'}}{\mathcal{Z'}}
\label{eq:rhoS0}
\end{equation}
where $𝒵' = Tr(exp(- βH_0' ))$ and $H_0' = PH_0P$. $P = ∑_α |ψ_α⟩⟨ψ_α|$ and $|ψ_α⟩ = |1;w_1> ⊗|2;w_2>$ is the basis that has the maximum eigenvalues of $S_1^2$ and $S_2^2$.

A simple proof below shows that for long anneal times the success probability given by collective SB coupling model is always higher than the one given by independent SB coupling model.
\begin{proof}
Since $H_0'$ is the projection of $H_0$ in the subspace with the largest values of \((\hat{\bf{S}}_1)^2\) and \((\hat{\bf{S}}_2)^2\), $\mathcal{Z'}<\mathcal{Z}$. The success (ground state) probability $\langle g | \rho'(s = 1, \tau \rightarrow \infty) | g \rangle = \frac{e^{- \beta e_g}}{\mathcal{Z'}} > \frac{e^{- \beta e_g}}{\mathcal{Z}} = \langle g | \rho(s = 1, \tau \rightarrow \infty) | g \rangle$, where $e_g$ is the ground state energy.
\end{proof}

\subsection{Intermediate anneal times $\tau$}
In general, since the assumption of independent SB coupling allows decoherence to many more eigenstates of $H(s)$ outside the maximal spin subspaces, it gives a lower success probability than the independent SB model for almost all anneal times $τ$. For the example of $c=0.5$ and $τ= 10$ns, we see in Fig.~\ref{fig:indvscol_n} that the difference of the success probability set by the two models diverges as system size $N$ increases. This is expected since the dimension reduction due to the restriction to maximal spin subspaces also scales exponentially in $N$.
% \begin{figure}[h!]
% \includegraphics[width=0.8\textwidth]{annealsuccessprob_n.pdf}\label{fig:8qubitsame}
% \label{fig:4ame_724_2}
% \end{figure}
\begin{figure}[h!]
\centering
\includegraphics[width=0.8\textwidth]{ra3/annealsuccessprob_vscol.pdf}
\caption{Dependence of success probability as a function of $N$ due to the two different SB coupling model. $c=0.5$, $\Gamma=1$ and $\tau = 10$ns.}
\label{fig:indvscol_n}
\end{figure}

% \clearpage
% \subsection{Different $N$.}
% We want to explore different $N$.

% \textit{Conclusion: }
% From Fig.~\ref{fig:4ame_724_2}, we observe sudden diabatic transitions, since $\Gamma = 0.3$ results in very small gaps for most of the $H_{\text{init}}$. Meanwhile, a relatively long anneal time of $\tau = 400$ns allows open-system relaxation mechanism to increase the (instantaneous and final success probability) ground state population for some of the initial states.

% \section{Conditions where open-system ARA is better than unitary evolution}
% In general, as $\Gamma$ is smaller, $c$ is smaller, and $N$ is larger,
% \begin{enumerate}
%   \item .
%   \item Intermediate anneal times
% \end{enumerate}

\section{Conclusion of this chapter}
We studied the performance of ARA as a function of transverse field $Γ$, initial state $H_init$ and system size $N$. In general, except for the cases where the initial state is equal to the correct solution, ARA performs better with larger $Γ$.

We observe non-monotonic dependence of success probabilities in terms of the anneal times $τ$ for $𝖠𝖱𝖠_𝖮𝗉𝖾𝗇$. For long anneal times, $𝖠𝖱𝖠_𝖢𝗅𝗈𝗌𝖾𝖽$ reaches the solution with certainty, while $𝖠𝖱𝖠_𝖮𝗉𝖾𝗇$ reaches the Gibbs state. For $𝖠𝖱𝖠_𝖮𝗉𝖾𝗇$, the difference due to independent system-bath coupling assumption and collective system-bath coupling assumption is studied for both long anneal times $τ$ and intermediate anneal times $τ$. In general, the former case always results in a lower success probability (or longer TTS).

$𝖠𝖱𝖠_𝖮𝗉𝖾𝗇$ tends for outperform $𝖠𝖱𝖠_𝖢𝗅𝗈𝗌𝖾𝖽$ in the following conditions:
\begin{enumerate}
\item Small $\Gamma$.
\item Initial state far away from the correct solution.
\item Large $N$.
\item Intermediate anneal times $\tau$.
\end{enumerate}
The first three conditions result in a small and sharp gap for diabatic transitions for intermediate anneal times $τ$. Because of the environmentally-induced relaxation back to the ground state after diabatic transitions to higher excited states, $𝖠𝖱𝖠_𝖮𝗉𝖾𝗇$ performs better than $𝖠𝖱𝖠_𝖢𝗅𝗈𝗌𝖾𝖽$ with the above conditions, in agreement with the prospects of quantum enhancement for diabatic annealing~\cite{Crosson2020}.

% Research Topic 1

% More Research Topics
% \include{ResearchTopic2}

% Conclusion and ongoing work
\include{Conclusion}

% Using single-space for reference list.
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% Bibliography
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\field{journaltitle}{Science}
\field{number}{5564}
\field{title}{Theory of Quantum Annealing of an {I}sing Spin Glass}
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\field{year}{2002}
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\field{journaltitle}{Rev. Mod. Phys.}
\field{month}{9}
\field{title}{\textit{Colloquium}: Quantum annealing and analog quantum computation}
\field{volume}{80}
\field{year}{2008}
\field{pages}{1061\bibrangedash 1081}
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\field{journaltitle}{Journal of Mathematical Physics}
\field{number}{12}
\field{title}{Mathematical foundation of quantum annealing}
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\field{year}{2008}
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\entry{Hauke2020}{article}{}
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given={William\bibnamedelima D.},
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\field{issn}{13616633}
\field{journaltitle}{Rep. Prog. Phys.}
\field{number}{5}
\field{title}{{Perspectives of quantum annealing: Methods and implementations}}
\field{volume}{83}
\field{year}{2020}
\field{pages}{054401}
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\entry{novikov2018exploring}{inproceedings}{}
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given={Daniel\bibnamedelima A},
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given={Kenneth\bibnamedelima M},
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\field{booktitle}{2018 IEEE International Conference on Rebooting Computing (ICRC)}
\field{title}{Exploring more-coherent quantum annealing}
\field{year}{2018}
\field{pages}{1\bibrangedash 7}
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\endentry
\entry{amin_decoherence_2009}{article}{}
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given={M.\bibnamedelimi H.\bibnamedelimi S.},
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given={Dmitri\bibnamedelima V.},
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given={James\bibnamedelima A.},
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\field{abstract}{We have studied the decoherence properties of adiabatic quantum computation (AQC) in the presence of in general non-Markovian, e.g., low-frequency, noise. The developed description of the incoherent Landau-Zener transitions shows that the global AQC maintains its properties even for decoherence larger than the minimum gap at the anticrossing of the two lowest-energy levels. The more efficient local AQC, however, does not improve scaling of the computation time with the number of qubits n as in the decoherence-free case. The scaling improvement requires phase coherence throughout the computation, limiting the computation time and the problem size n.}
\field{journaltitle}{Phys. Rev. A}
\field{month}{2}
\field{number}{2}
\field{title}{Decoherence in adiabatic quantum computation}
\field{urlday}{16}
\field{urlmonth}{3}
\field{urlyear}{2014}
\field{volume}{79}
\field{year}{2009}
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\entry{smirnov2018theory}{article}{}
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{IOP Publishing}%
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\field{journaltitle}{New Journal of Physics}
\field{number}{10}
\field{title}{Theory of open quantum dynamics with hybrid noise}
\field{volume}{20}
\field{year}{2018}
\field{pages}{103037}
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\entry{davies1978open}{article}{}
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\field{journaltitle}{Journal of Statistical Physics}
\field{number}{5}
\field{title}{Open quantum systems with time-dependent Hamiltonians and their linear response}
\field{volume}{19}
\field{year}{1978}
\field{pages}{511\bibrangedash 523}
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\entry{Albash:2015nx}{article}{}
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\field{journaltitle}{Physical Review A}
\field{number}{6}
\field{title}{Decoherence in adiabatic quantum computation}
\field{volume}{91}
\field{year}{2015}
\field{pages}{062320}
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\field{journaltitle}{New J. of Phys.}
\field{number}{12}
\field{title}{{Quantum adiabatic Markovian master equations}}
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\field{number}{19}
\field{title}{Macroscopic resonant tunneling in the presence of low frequency noise}
\field{volume}{100}
\field{year}{2008}
\field{pages}{197001}
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\field{journaltitle}{arXiv e-prints}
\field{title}{Relaxation vs. adiabatic quantum steady state preparation: which wins?}
\field{year}{2016}
\field{pages}{arXiv\bibrangedash 1612}
\range{pages}{-1}
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\field{journaltitle}{Quantum Information Processing}
\field{month}{2}
\field{number}{1}
\field{title}{A study of heuristic guesses for adiabatic quantum computation}
\field{volume}{10}
\field{year}{2011}
\field{pages}{33\bibrangedash 52}
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\entry{Albash-Lidar:RMP}{article}{}
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\field{journaltitle}{arXiv:1611.04471}
\field{title}{Adiabatic Quantum Computing}
\field{year}{2016}
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\verb http://arXiv.org/abs/1611.04471
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\field{journaltitle}{Physical Review A}
\field{number}{2}
\field{title}{Diagonal catalysts in quantum adiabatic optimization}
\field{volume}{103}
\field{year}{2021}
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\field{journaltitle}{New Journal of Physics}
\field{month}{2}
\field{number}{2}
\field{title}{Modernizing quantum annealing using local searches}
\field{volume}{19}
\field{year}{2017}
\field{pages}{023024}
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\entry{King:2018aa}{article}{}
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given={Andrew\bibnamedelima D.},
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given={Paul\bibnamedelima I.},
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familyi={P\bibinithyphendelim L\bibinitperiod},
given={Anatoly\bibnamedelima Yu.},
giveni={A\bibinitperiod\bibinitdelim Y\bibinitperiod}}}%
given={Loren\bibnamedelima J.},
giveni={L\bibinitperiod\bibinitdelim J\bibinitperiod}}}%
given={Mark\bibnamedelima H.},
giveni={M\bibinitperiod\bibinitdelim H\bibinitperiod}}}%
given={Mark\bibnamedelima W.},
giveni={M\bibinitperiod\bibinitdelim W\bibinitperiod}}}%
given={Mohammad\bibnamedelima H.},
giveni={M\bibinitperiod\bibinitdelim H\bibinitperiod}}}%
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\field{abstract}{The work of Berezinskii, Kosterlitz and Thouless in the 1970s1,2 revealed exotic phases of matter governed by the topological properties of low-dimensional materials such as thin films of superfluids and superconductors. A hallmark of this phenomenon is the appearance and interaction of vortices and antivortices in an angular degree of freedom---typified by the classical XY model---owing to thermal fluctuations. In the two-dimensional Ising model this angular degree of freedom is absent in the classical case, but with the addition of a transverse field it can emerge from the interplay between frustration and quantum fluctuations. Consequently, a Kosterlitz--Thouless phase transition has been predicted in the quantum system---the two-dimensional transverse-field Ising model---by theory and simulation3--5. Here we demonstrate a large-scale quantum simulation of this phenomenon in a network of 1,800 in situ programmable superconducting niobium flux qubits whose pairwise couplings are arranged in a fully frustrated square-octagonal lattice. Essential to the critical behaviour, we observe the emergence of a complex order parameter with continuous rotational symmetry, and the onset of quasi-long-range order as the system approaches a critical temperature. We describe and use a simple approach to statistical estimation with an annealing-based quantum processor that performs Monte Carlo sampling in a chain of reverse quantum annealing protocols. Observations are consistent with classical simulations across a range of Hamiltonian parameters. We anticipate that our approach of using a quantum processor as a programmable magnetic lattice will find widespread use in the simulation and development of exotic materials.}
\field{isbn}{1476-4687}
\field{journaltitle}{Nature}
\field{number}{7719}
\field{title}{Observation of topological phenomena in a programmable lattice of 1,800 qubits}
\field{volume}{560}
\field{year}{2018}
\field{pages}{456\bibrangedash 460}
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