# Determination of Optimal Chain Coupling made by Embedding in D-Wave Quantum
Annealer

Hayun Park  Department of Liberal Studies, Kangwon National University,
Samcheok, 25913, Republic of Korea  Hunpyo Lee<EMAIL_ADDRESS>Department
of Liberal Studies, Kangwon National University, Samcheok, 25913, Republic of
Korea  Quantum Sub Inc., Samcheok, 25913, Republic of Korea

###### Abstract

The qubits in a D-wave quantum annealer (D-wave QA) are designed on a Pegasus
graph that is different from structure of a combinatorial optimization
problem. This situation requires embedding with the chains connected by
ferromagnetic (FM) coupling $J_{c}$ between the qubits. Weak and strong
$J_{c}$ values induce chain breaking and enforcement of chain energy, which
reduce the accuracy of quantum annealing (QA) measurements, respectively. In
addition, we confirmed that even though the D-Wave Ocean package provides a
default coupling $J_{c}^{\text{default}}$, it is not an optimal coupling
$J_{c}^{\text{optimal}}$ that maximizes the possible correct rate of QA
measurements. In this paper, we present an algorithm how
$J_{c}^{\text{optimal}}$ with the maximum probability $p$ for observing the
possible lowest energy is determined. Finally, we confirm that the extracted
$J_{c}^{\text{optimal}}$ show much better $p$ than $J_{c}^{\text{default}}$ in
QA measurements of various parameters of frustrated and fully connected
combinatorial optimization problems. The open code is available in
https://github.com/HunpyoLee/OptimizeChainStrength.

††preprint: AIP/123-QED

##  I Introduction

The recent progress in quantum technology has led to the emergence of quantum
machines. In addition, they are being developed and built more frequently than
before Preskill (2018). In particular, a D-Wave quantum annealer (D-Wave QA)
is an example of such a quantum machine Johnson  _et al._ (2011). Unlike gate-
type quantum machines based on circuits, quantum annealing (QA) is implemented
in the parameterized Hamiltonian of a transverse-field Ising model containing
binary superconducting qubits Kadowaki and Nishimori (1998); King  _et al._
(2022); Das and Chakrabarti (2008). The primary advantage of this architecture
quantum machine is that qubits are added much easier than gate-type quantum
computers with maintaining the accuracy of the results, due to short quantum
coherence time. Thus, the D-Wave QA with rapid increases of qubit capacity
approaches computational speed of classical digit machine. In addition,
studying the dynamic behaviors observed in real materials is a very
interesting research topic, as it is not easy to study using numerical
simulation methods. Ca3Co2O6 compound is an example with unconventional
dynamics that has remained a puzzle Nekrashevich  _et al._ (2022). Recent QA
experiment on frustrated spin system reported that it can be used as a
programmable quantum simulator for both the dynamic and equilibrium behaviors
observed in Ca3Co2O6 compound King  _et al._ (2021a). Consequently, this
quantum machine has been not only employed in combinatorial optimization
problems requiring annealing process for the possible lowest energy and for
the research of Ising model, which exhibits the phase transition between
unconventional phases at zero temperature, but also used as programmable
quantum simulator for exploring dynamic behaviors shown in real materials King
_et al._ (2021a); Amin  _et al._ (2018); Isakov  _et al._ (2016); Mazzola,
Smelyanskiy, and Troyer (2017); Inoue  _et al._ (2021); Kairys  _et al._
(2020); King  _et al._ (2021b); Irie  _et al._ (2021); Park and Lee (2022);
Rønnow  _et al._ (2014); Albash and Lidar (2018); Boixo  _et al._ (2014).

However, the D-Wave QA cannot technically describe the exact couplings between
long-distance qubits, and it retains qubits designed on the specific
architectures such as Pegasus and Kimera graphs that depend on D-Wave QA
hardware types. These situations require physical embedding that the
architecture of the original problem topologically matches that of D-Wave QA
Lanthaler and Lechner (2021); Konz  _et al._ (2021). The auxiliary chains with
ferromagnetic (FM) coupling $J_{c}$ between several qubits, represented as one
variable in the architecture of the original problem, are required. Weak and
strong $J_{c}$ induce chain breaking and reinforcement of chain energy to
sustain FM order of qubits in chains, respectively, lowering the accuracy of
the results measured by D-Wave QA. In addition, even though the D-Wave QA
Ocean package provides a default coupling $J_{c}^{\text{default}}$, it is not
an optimal coupling $J_{c}^{\text{optimal}}$. Therefore, determining
$J_{c}^{\text{optimal}}$ is highly desirable for accurate QA measurements.

Figure 1:  (Color online) (a) Unit cell of $2\times 2$ qubits on a two-
dimensional (2D) square structure with the nearest-neighbor coupling $J_{1}$
and diagonal-neighbor coupling $J_{2}$ as a combinatorial optimization
problem. (b) Embedding on Pegasus graph for a unit cell of $2\times 2$ qubits
on the 2D square structure. The chains with coupling $J_{c}$ to sustain
ferromagnetic order of qubits are marked as dotted lines.

In this study, we first analyze the full energy spectrum on an embedding of a
combinatorial optimization problem of small size to estimate
$J_{c}^{\text{optimal}}$, through an exact approach that visits all energy
states. We confirm that $J_{c}^{\text{optimal}}$ would be slightly larger than
the critical chain coupling $J_{\text{c}}^{*}$ between chain breaking and
unbroken state without any chain breaking. Next, we explain an algorithm by
which $J_{c}^{*}$ is extracted. We confirmed that the maximum probability
$p^{\text{max}}$ with the possible lowest energy in QA measurements of various
combinatorial optimization problems appears at a chain coupling value that is
$1.2$ times larger than $J_{c}^{*}$.

As examples of the combinatorial optimization problems to demonstrate the
usefulness of $J_{c}^{\text{optimal}}$ extracted by our algorithm, we consider
a $J_{1}-J_{2}$ Ising system on a two-dimensional (2D) $L\times L$ square
lattice and fully connected Ising system with all couplings determined by
randomness on $N$ sites. We present $p$ with the possible lowest energy in
many QA shots of both the combinatorial optimization problems. Finally, it is
confirmed that the extracted $J_{c}^{\text{optimal}}$ showed better $p$ than
$J_{c}^{\text{default}}$ in both the combinatorial optimization problems.

The remainder of this paper is organized as follows. Section II describes the
Hamiltonian made by embedding with chains of FM order between qubits. In
Section III, we analyze the full energy spectrum on the combinatorial
optimization problem of the 2D $4\times 4$ qubits via exact approach. $9$
qubits are artificially added in the chains. We roughly estimate
$J_{c}^{\text{optimal}}$ from analysis of the exact full energy spectrum
result. We also account for the method how $J_{c}^{\text{optimal}}$ is
extracted. Section IV shows that $J_{c}^{\text{optimal}}$ is better $p$ than
$J_{c}^{\text{default}}$ in QA measurements of $J_{1}-J_{2}$ and fully
connected combinatorial optimization problems. Finally, conclusions are
presented in Section V.

##  II Hamiltonian in Embedding

The Hamiltonian of the D-Wave QA is given by

$H=H_{\text{cop}}+H_{\text{chain}},$ (1)

where $H_{\text{cop}}$ and $H_{\text{chain}}$ are the parts of the
combinatorial optimization problem and of the chain, respectively. Here,
$H_{\text{chain}}$ is expressed as follows:

$H_{\text{chain}}=-\sum_{i_{\text{chain}}}\sum_{<k,k^{\prime}>}^{n_{i_{\text{chain}}}}J_{c,i_{\text{chain}}}\sigma_{i_{\text{chain}},k}^{z}\sigma_{i_{\text{chain}},k^{\prime}}^{z},$
(2)

where $J_{c,i_{\text{chain}}}$ is the coupling of the FM order between $k$ and
$k^{\prime}$ qubits at the chain site $i_{\text{chain}}$, and
$n_{i_{\text{chain}}}$ is the number of qubits. The total number of qubits
$N_{\text{chain}}$ in all chains is given by
$N_{\text{chain}}=\sum_{i_{\text{chain}}}n_{i_{\text{chain}}}$.

##  III Determination of the optimal chain coupling

###  III.1 Hamiltonian

Figure 2:  (Color online) Interval $\Delta_{c}/J_{1}$ between the ground and
the first excited energies on the 2D $4\times 4$ qubits of the combinatorial
optimization problem with $9$ qubits in the chains as a function of
$J_{c}/J_{1}$ for $J_{2}/J_{1}=0.42$, $0.46$, and $0.48$. $\Delta_{c}/J_{1}$
are computed by the exact method with visiting all energy states. Note that if
any chain is broken, we set that $\Delta_{c}/J_{1}$ is $0$.

As an example to show an algorithm how $J_{c}^{\text{optimal}}$ is determined,
we considered a $J_{1}-J_{2}$ combinatorial optimization problem on a 2D
$L\times L$ square lattice. The Hamiltonian $H_{\text{cop}}$ of the
$J_{1}-J_{2}$ combination optimization problem is defined as follows:

$H_{\text{cop}}=-J_{1}\sum_{<i,j>}\sigma_{i}^{z}\sigma_{j}^{z}+J_{2}\sum_{<<i,j^{\prime}>>}\sigma_{i}^{z}\sigma_{j^{\prime}}^{z},$
(3)

where the nearest- and diagonal-neighbors are denoted by $<i,j>$ and
$<<i,j^{\prime}>>$, respectively Jin, Sen, and Sandvik (2012); Jin  _et al._
(2013). $J_{1}$ and $J_{2}$ mean the nearest-neighbor and diagonal-neighbor
couplings respectively, as shown in Fig. 1 (a). Here, the reason we chose this
$J_{1}-J_{2}$ combinatorial optimization problem as the example is because the
interval $\Delta$ between the energies of the ground and of the first-excited
states is systematically adjusted by $J_{2}/J_{1}$ value Jin, Sen, and Sandvik
(2012); Jin  _et al._ (2013). $\Delta$ is an important variable that affects
the accuracy of QA measurements, because the plausible local minima close to
global minimum in objective function of the combinatorial optimization problem
increase with decreasing $\Delta$. In addition, another energy gap
$\Delta_{c}$ between the energies of the ground and of the first-excited
states in Eq. (1) of an embedded combinatorial optimization problem on Pegasus
graph is occurred by chains with $J_{c}$. Figs. 1(a) and (b) illustrate the
unit cell of $2\times 2$ qubits on the 2D square structure with the nearest-
neighbor coupling $J_{1}$ and diagonal-neighbor coupling $J_{2}$ as a
combinatorial optimization problem and its embedding on Pegasus graph,
respectively. The chains with the coupling $J_{c}$ to keep FM order of qubits
are marked as dotted lines in Fig. 1(b).

###  III.2 Rough estimation of optimal chain coupling by analysis of full
energy spectrum

We analyze the full energy spectrum of $J_{1}-J_{2}$ combinatorial
optimization problem on a 2D $4\times 4$ square lattice in the embedding,
using the exact method with visiting all energy states. $9$ qubits are also
added in the chains. We find that when all chains are broken in weak
$J_{c}/J_{1}$, a stable state occurs. As $J_{c}/J_{1}$ increases, the stable
state with all chain breakings changes into a state where chain breaking
disappears individually, depending on the chain connecting shape between the
qubits of the combinatorial optimization problem and the chain. This implies
that the critical coupling for chain breaking at each chain depends on the
chain connecting shapes. After $J_{c}^{*}/J_{1}$, the state in which any chain
is not broken becomes a stable state.

Figure 3:  (Color online) Chain coupling $J_{c}$ as a function of
$\Delta/J_{1}$ on the bottom label (and $J_{2}/J_{1}$ on top label) in
$J_{1}-J_{2}$ combinatorial optimization problem on a 2D $8\times 8$ square
lattice. Here, $\Delta/J_{1}$ in the bottom label are approximately estimated
by interval between the lowest energy and the second lowest energy in many
quantum annealing (QA) shots. The top label is $J_{2}/J_{1}$, matching
$\Delta/J_{1}$.

Fig. 2 shows $\Delta_{c}/J_{1}$ as a function of $J_{c}/J_{1}$ for
$J_{2}/J_{1}=0.42$, $0.46$, and $0.48$. Here, we assume that when all chains
are unbroken, the most stable state occurs to prevent deformation of the
combinatorial optimization problem. So, if any chain is broken, we set
$\Delta_{c}/J_{1}$ to $0$. A stable state without any chain breaking appears
above the critical chain couplings $J_{c}^{*}/J_{1}=1.68$, $1.84$, and $1.92$
for $J_{2}/J_{1}=4.2$, $4.6$, and $4.8$, respectively. $\Delta_{c}/J_{1}$
increases as $J_{c}/J_{1}$ increases from $J_{c}^{*}/J_{1}$ until kink
behavior is observed at $J_{c}^{**}/J_{1}=2.96$, $2.48$, and $2.24$ for
$J_{2}/J_{1}=4.2$, $4.6$, and $4.8$, respectively. After $J_{c}^{**}/J_{1}$
for all $J_{2}/J_{1}$, $\Delta_{c}/J_{1}$ converges to $\Delta/J_{1}$. We
believe that the chain breaking and dominant chain energy states that sustain
the FM order of the qubits in the chain would appear below $J_{c}^{*}/J_{1}$
and above $J_{c}^{**}/J_{1}$, respectively. The range between $J^{*}$ and
$J^{**}$ decreases with increasing $J_{2}/J_{1}$ (or decreasing
$\Delta/J_{1}$). Finally, we inferred three possibilities from exact result in
Fig. 2: (i) $J_{c}^{\text{optimal}}/J_{1}$ might be between $J_{c}^{*}/J_{1}$
and $J_{c}^{**}/J_{1}$. (ii) When $\Delta/J_{1}$ is tiny, it may be difficult
to determine $J_{c}^{\text{optimal}}/J_{1}$. (iii) The accuracy of QA
measurements will be poor with decreasing $\Delta/J_{1}$.

###  III.3 Procedure for extracting the optimal chain coupling $J_{c}$ value

The D-Wave Ocean package provides the default coupling
$J_{c}^{\text{default}}$ given as

$J_{c}^{\text{default}}=1.41*(\frac{1}{M_{c}}\sum_{<i,j>}J_{ij}^{2})^{\frac{1}{2}}*(\frac{1}{N_{c}}\sum_{i=1}\text{deg}(i))^{\frac{1}{2}},$
(4)

where $M_{c}$, $N_{c}$ and $\text{deg}(i)$ are the number of total
connectivities between qubits, the number of total qubits and the coordination
number at $i$-chain, respectively. Even though $J_{c}^{\text{default}}$ gives
not bad results in accuracy of QA measurements, we confirmed that
$J_{c}^{\text{default}}$ is not $J_{c}^{\text{optimal}}$ from estimation of
the exact result in Fig. 2 and from various results measured by D-Wave QA.

Here, we design more sophisticated algorithm to obtain
$J_{\text{c}}^{\text{optimal}}$ that provides better results than
$J_{c}^{\text{default}}$. The procedure of the algorithm is followed as: (i)
In the first step, we set to initial chain coupling $J_{c}$ given as
$J_{c}=0.1\times J_{c}^{\text{default}}$ of weak chain coupling. The stable
state at initial $J_{c}$ is when almost all chains are broken. (ii) In the
second step, we increase $J_{c}$ by $10$ percent of $J_{c}^{\text{default}}$
and perform QA measurements. The chain breaking gradually disappears with
increasing $J_{c}$. (iii) In the third step, we set the critical chain
coupling $J_{c}^{*}$, when the chain is no longer broken. (iv) Finally, we set
$J_{c}^{*}$ multiplied by $1.2$ as $J_{c}^{\text{optimal}}$. The factor $1.2$
is obtained empirically from the best results of QA measurements in various
combinatorial optimization problems on D-Wave Advantage machine. In open code
we set annealing time to $1.0\times 10^{-3}$ second in QA measurements, and
perform a hundred of annealing shots to find $J_{c}^{*}$ and
$J_{c}^{\text{optimal}}$.

Figure 4:  (Color online) Probability $p$ for observing the possible lowest
energies measured by $5000$ times QA shots with $J_{c}^{\text{default}}$ and
$J_{c}^{\text{optimal}}$ on $J_{1}-J_{2}$ combinatorial optimization problem
on 2D $8\times 8$ square lattice.

##  IV results

###  IV.1 $J_{1}-J_{2}$ combinatorial optimization problem

In order to demonstrate the superiority of the results in
$J_{c}^{\text{optimal}}$ extracted by our algorithm, we first consider
$J_{1}-J_{2}$ combinatorial optimization problem on 2D $8\times 8$ square
lattice expressed by Hamiltonian of Eq. (3). We compare
$J_{c}^{\text{default}}/J_{1}$ computed by Eq. (4) to
$J_{c}^{\text{optimal}}/J_{1}$ determined by our algorithm. Fig. 3 shows
$J_{c}^{\text{default}}$ and $J_{c}^{\text{optimal}}$ values as a function of
$\Delta/J_{1}$ in bottom label (and $J_{2}/J_{1}$ in top label). Here,
$\Delta/J_{1}$ values in bottom label are approximately estimated by interval
between the lowest energy and the second lowest energy in many QA shots. The
top label is $J_{2}/J_{1}$, matching $\Delta/J_{1}$. We observe that
$J_{c}^{\text{default}}/J_{1}$ values are larger than
$J_{c}^{\text{optimal}}/J_{1}$ in all $\Delta/J_{1}$.
$J_{c}^{\text{default}}/J_{1}$ increases slowly, while
$J_{c}^{\text{optimal}}/J_{1}$ increases quickly as $\Delta/J_{1}$ decreases.

In addition, we plot $p$ for observing the possible lowest energies measured
by QA shots of $5000$ times with both $J_{c}^{\text{default}}/J_{1}$ and
$J_{c}^{\text{optimal}}/J_{1}$ in Fig. 4. $p$ (or accuracy of QA measurements)
decrease with decreasing $\Delta/J_{1}$ in both $J_{c}^{\text{default}}/J_{1}$
and $J_{c}^{\text{optimal}}/J_{1}$, because many plausible local minimum that
are similar to the global minimum are created in tiny $\Delta/J_{1}$. As a
main result, we find that $p$ values measured in
$J_{c}^{\text{optimal}}/J_{1}$ are higher than those in
$J_{c}^{\text{default}}/J_{1}$. We think that $J_{c}^{\text{optimal}}/J_{1}$
provides the optimal compromise between chain breaking and chain energy, while
$J_{c}^{\text{default}}/J_{1}$ lowers the accuracy of QA measurements by the
strong chain energy.

Figure 5:  (Color online) $J_{c}$ as a function of $N$ on bottom label (and
$N_{T}$ on top label) in fully connected combinatorial optimization problem.
Here, $N$ on bottom label and $N_{T}$ on top label are the number of qubits in
fully connected combinatorial optimization problem and the number of qubits in
embedding, respectively.

###  IV.2 Fully connected combinatorial optimization problem

Finally, in order to confirm a more general case we select a fully connected
combinatorial optimization problem, where obtaining accurate results of QA
measurements is more difficult than $J_{1}-J_{2}$ combinatorial optimization
problem because it produces more plausible local minima than $J_{1}-J_{2}$
combinatorial optimization problem, due to strong degeneracy in energy states.
The Hamiltonian $H_{\text{cop}}$ is given as

$H_{\text{cop}}=-\sum_{<i,j>}^{N}J_{<i,j>}\sigma_{i}^{z}\sigma_{j}^{z},$ (5)

where $<i,j>$ and $N$ mean all couplings between qubits and the number of
qubits in the combination optimization problem, respectively. $J_{<i,j>}$
values are either $-1.0$, $0.0$ or $1.0$ determined by random number
generator.

Same as $J_{1}-J_{2}$ combinatorial optimization problem, we plot $J_{c}$
values and $p$ as a function of $N$ on bottom label in Fig. 5 and Fig. 6,
respectively. $p$ are measured by QA shots of $5000$ times with
$J_{c}^{\text{default}}$ and $J_{c}^{\text{optimal}}$.
$J_{c}^{\text{default}}$ values are always bigger than
$J_{c}^{\text{optimal}}$ values in all $N$ of Fig. 5. The number of total
qubits $N_{T}$ on top label in embedding is increasing with $N$ in Fig 5. It
is also confirmed that $J_{c}^{\text{optimal}}$ gives more accurate results
than $J_{c}^{\text{default}}$ in QA measurements of Fig. 6.

Figure 6:  (Color online) $J_{c}$ as a function of $N$ in fully connected
combinatorial optimization problem. $p$ are measured by QA shots of $5000$
times with $J_{c}^{\text{default}}$ and $J_{c}^{\text{optimal}}$.

##  V Conclusion

The D-Wave QA displays significant potential due to the recent rapid increase
in qubit capacity. Moreover, the D-Wave QA has been extensively applied to
solving combinatorial optimization problems. However, as a limitation,
physical embedding with FM chains of several qubits is required to consider
the exact long-range coupling between qubits ignored in D-Wave QA. The
separation of FM qubits in the chains or the dominant enhancement of the chain
energy to sustain FM order of qubits in chains lowers the accuracy of the
results measured by D-Wave QA. In addition, we confirm that even though the
D-Wave Ocean package provides the default coupling $J_{c}^{\text{default}}$,
it is not an optimal coupling $J_{c}^{\text{optimal}}$. Therefore, determining
the optimal coupling $J_{c}^{\text{optimal}}$ for maintaining the appropriate
FM ordered qubits in the chains is required to improve accuracy.

In this paper, we present the algorithm how $J_{c}^{\text{optimal}}$ is
determined. We confirm that the extracted $J_{c}^{\text{optimal}}$ shows much
better $p$ than $J_{c}^{\text{default}}$ in various parameters of
$J_{1}-J_{2}$ and fully connected combinatorial optimization problems.
Finally, We believe that our method provides practical assistance for D-Wave
QA users to search for $J_{c}^{\text{optimal}}$ with decreased essential
errors. The open code is available in https://github.com/HunpyoLee/
OptimizeChainStrength.

##  VI Acknowledgments

This work was supported by Institute of Information and communications
Technology Planning Evaluation (IITP) grant funded by the Korean government
(MSIT) (No. RS-2023-0022952422282052750001), Ministry of Science through
NRF-2021R1111A2057259 and by the quantum computing technology development
program of the National Research Foundation of Korea (NRF) funded by the
Korean government (Ministry of Science and ICT(MSIT)) (No. 2020M3H3A1110365).

##  VII Data Availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

##  VIII Conflict of interest

The authors have no conflicts to disclose.

## References

  * Preskill (2018) J. Preskill, “Quantum Computing in the NISQ era and beyond,” Quantum 2, 79 (2018).
  * Johnson  _et al._ (2011) M. W. Johnson, M. H. S. Amin, S. Gildert, T. Lanting, F. Hamze, N. Dickson, R. Harris, A. J. Berkley, J. Johansson, P. Bunyk, E. M. Chapple, C. Enderud, J. P. Hilton, K. Karimi, E. Ladizinsky, N. Ladizinsky, T. Oh, I. Perminov, C. Rich, M. C. Thom, E. Tolkacheva, C. J. S. Truncik, S. Uchaikin, J. Wang, B. Wilson, and G. Rose, “Quantum annealing with manufactured spins,” Nature 473, 194–198 (2011).
  * Kadowaki and Nishimori (1998) T. Kadowaki and H. Nishimori, “Quantum annealing in the transverse ising model,” Phys. Rev. E 58, 5355–5363 (1998).
  * King  _et al._ (2022) A. D. King, S. Suzuki, J. Raymond, A. Zucca, T. Lanting, F. Altomare, A. J. Berkley, S. Ejtemaee, E. Hoskinson, S. Huang, E. Ladizinsky, A. J. R. MacDonald, G. Marsden, T. Oh, G. Poulin-Lamarre, M. Reis, C. Rich, Y. Sato, J. D. Whittaker, J. Yao, R. Harris, D. A. Lidar, H. Nishimori, and M. H. Amin, “Coherent quantum annealing in a programmable 2,000 qubit ising chain,” Nature Physics 18, 1324–1328 (2022).
  * Das and Chakrabarti (2008) A. Das and B. K. Chakrabarti, “Colloquium: Quantum annealing and analog quantum computation,” Rev. Mod. Phys. 80, 1061–1081 (2008).
  * Nekrashevich  _et al._ (2022) I. Nekrashevich, X. Ding, F. Balakirev, H. T. Yi, S.-W. Cheong, L. Civale, Y. Kamiya, and V. S. Zapf, “Reaching the equilibrium state of the frustrated triangular ising magnet ${\mathrm{ca}}_{3}{\mathrm{co}}_{2}{\mathrm{o}}_{6}$,” Phys. Rev. B 105, 024426 (2022).
  * King  _et al._ (2021a) A. D. King, C. D. Batista, J. Raymond, T. Lanting, I. Ozfidan, G. Poulin-Lamarre, H. Zhang, and M. H. Amin, “Quantum annealing simulation of out-of-equilibrium magnetization in a spin-chain compound,” PRX Quantum 2, 030317 (2021a).
  * Amin  _et al._ (2018) M. H. Amin, E. Andriyash, J. Rolfe, B. Kulchytskyy, and R. Melko, “Quantum boltzmann machine,” Phys. Rev. X 8, 021050 (2018).
  * Isakov  _et al._ (2016) S. V. Isakov, G. Mazzola, V. N. Smelyanskiy, Z. Jiang, S. Boixo, H. Neven, and M. Troyer, “Understanding quantum tunneling through quantum monte carlo simulations,” Phys. Rev. Lett. 117, 180402 (2016).
  * Mazzola, Smelyanskiy, and Troyer (2017) G. Mazzola, V. N. Smelyanskiy, and M. Troyer, “Quantum monte carlo tunneling from quantum chemistry to quantum annealing,” Phys. Rev. B 96, 134305 (2017).
  * Inoue  _et al._ (2021) D. Inoue, A. Okada, T. Matsumori, K. Aihara, and H. Yoshida, “Traffic signal optimization on a square lattice with quantum annealing,” Scientific Reports 11, 3303 (2021).
  * Kairys  _et al._ (2020) P. Kairys, A. D. King, I. Ozfidan, K. Boothby, J. Raymond, A. Banerjee, and T. S. Humble, “Simulating the shastry-sutherland ising model using quantum annealing,” PRX Quantum 1, 020320 (2020).
  * King  _et al._ (2021b) A. D. King, J. Raymond, T. Lanting, S. V. Isakov, M. Mohseni, G. Poulin-Lamarre, S. Ejtemaee, W. Bernoudy, I. Ozfidan, A. Y. Smirnov, M. Reis, F. Altomare, M. Babcock, C. Baron, A. J. Berkley, K. Boothby, P. I. Bunyk, H. Christiani, C. Enderud, B. Evert, R. Harris, E. Hoskinson, S. Huang, K. Jooya, A. Khodabandelou, N. Ladizinsky, R. Li, P. A. Lott, A. J. R. MacDonald, D. Marsden, G. Marsden, T. Medina, R. Molavi, R. Neufeld, M. Norouzpour, T. Oh, I. Pavlov, I. Perminov, T. Prescott, C. Rich, Y. Sato, B. Sheldan, G. Sterling, L. J. Swenson, N. Tsai, M. H. Volkmann, J. D. Whittaker, W. Wilkinson, J. Yao, H. Neven, J. P. Hilton, E. Ladizinsky, M. W. Johnson, and M. H. Amin, “Scaling advantage over path-integral monte carlo in quantum simulation of geometrically frustrated magnets,” Nature Communications 12, 1113 (2021b).
  * Irie  _et al._ (2021) H. Irie, H. Liang, T. Doi, S. Gongyo, and T. Hatsuda, “Hybrid quantum annealing via molecular dynamics,” Scientific Reports 11, 8426 (2021).
  * Park and Lee (2022) H. Park and H. Lee, “Frustrated ising model on d-wave quantum annealing machine,” Journal of the Physical Society of Japan 91 (2022), https://doi.org/10.7566/JPSJ.91.074001.
  * Rønnow  _et al._ (2014) T. F. Rønnow, Z. Wang, J. Job, S. Boixo, S. V. Isakov, D. Wecker, J. M. Martinis, D. A. Lidar, and M. Troyer, “Defining and detecting quantum speedup,” Science 345, 420–424 (2014).
  * Albash and Lidar (2018) T. Albash and D. A. Lidar, “Demonstration of a scaling advantage for a quantum annealer over simulated annealing,” Phys. Rev. X 8, 031016 (2018).
  * Boixo  _et al._ (2014) S. Boixo, T. F. Rønnow, S. V. Isakov, Z. Wang, D. Wecker, D. A. Lidar, J. M. Martinis, and M. Troyer, “Evidence for quantum annealing with more than one hundred qubits,” Nature Physics 10, 218–224 (2014).
  * Lanthaler and Lechner (2021) M. Lanthaler and W. Lechner, “Minimal constraints in the parity formulation of optimization problems,” New Journal of Physics 23, 083039 (2021).
  * Konz  _et al._ (2021) M. S. Konz, W. Lechner, H. G. Katzgraber, and M. Troyer, “Embedding overhead scaling of optimization problems in quantum annealing,” PRX Quantum 2, 040322 (2021).
  * Jin, Sen, and Sandvik (2012) S. Jin, A. Sen, and A. W. Sandvik, “Ashkin-teller criticality and pseudo-first-order behavior in a frustrated ising model on the square lattice,” Phys. Rev. Lett. 108, 045702 (2012).
  * Jin  _et al._ (2013) S. Jin, A. Sen, W. Guo, and A. W. Sandvik, “Phase transitions in the frustrated ising model on the square lattice,” Phys. Rev. B 87, 144406 (2013).