This function is extremely well fit by a simple exponential in $N$, with a
coefficient that depends on $\kappa$. Empirically, it can be well approximated
as,

$\displaystyle
w\left(N,\kappa\right)^{2}\propto\left(1+a\kappa^{b}\right)^{N},$ (38)

where fitting a range of $\kappa$ values from $0.4$ to $1.6$ gives $a=0.066$
and $b=2.25$, in fairly good agreement with the result
$\left(1+\kappa^{2}/16\right)^{M}$ one can derive from simple second order
perturbation theory.

Of note is that Eq. 37 is both of similar quantitative accuracy as the results
plotted in FIG. 10, and it predicts that the decay exponent smoothly
approaches zero as $\kappa\to 2$, at which point the minimum gap decays
inverse polynomially. Similar behavior–a crossover to polynomial scaling when
theory predicts the decay exponent must vanish–was observed both analytically
and numerically in the 1d transverse field Ising chain in Grattan et al.
(2023). We see this as lending further indirect support to our polynomial time
approximation hardness prediction in Eq. IV.4–though the situation is fairly
different all these calculations identify a crossover to polynomial scaling by
the point at which a predicted decay exponent vanishes in careful many-body
theory.

##  Appendix B Band structure considerations for linear spectral folding

In the simulations in section VI, we noticed that the quadratic AQC
formulation of spectrally folded quantum optimization performed similarly to
the rough prediction in section IV, returning a worst case approximation ratio
of $q_{a}=0.7$ compared to the approximate theoretical estimate 0.68. In
contrast, the linear trial minimum annealing versions more notably
outperformed the analyitical prediction (returning $q_{a}=0.75$ as compared to
0.6), and in numerical experiments the best choice for the lowering parameter
$C\left(t\right)$ in Eq. 8 was 2-3 times that one would naively expect. We
suspect that this has to do with the band structure of the dressed eigenstates
in $\mathcal{T}$ (the band of classical states at or very near the fold
energy) when the transverse field $\kappa$ is nonzero, and in this subsection
argue why this might be the case. We present these arguments as suggestions
and not a rigorous proof, and think more work on this issue could be valuable
for shedding light on the detailed structure of these optimization algorithms.

Let $\left\\{\left|T_{j}\right>\right\\}$ be the set of all $N_{\mathcal{T}}$
bitstring states in $\mathcal{T}$; states in $\mathcal{T}$ all have $E\simeq
AE_{\mathrm{GS}}$ before any corrections from the lowering Hamiltonian.
Further let $\left|\psi_{k}\right>$ be a dressed eigenstate whose spectral
weight is concentrated in $\mathcal{T}$. Thus

$\displaystyle\left|\psi_{k}\right>\simeq\sum_{j\in\mathcal{T}}^{N_{\mathcal{T}}}c_{jk}\left|T_{j}\right>,\;\;\left<\left|c_{jk}^{2}\right|\right>\simeq\frac{1}{N_{\mathcal{T}}}.$
(39)

Finally, let $\Omega_{Lj}$ be the $M$-spin tunneling matrix element into a
bitstring state $\left|T_{j}\right>$ from $\left|L_{D}\right>$, where
$\left<\Omega_{Lj}\right>_{j}=\Omega_{0}$ as calculated in Eq. IV.3.

We choose our gauge and basis so that all transverse field terms are negative
and $H$ is real, which is straightforward since $H$ is stoquastic. In this
gauge we can assume all the $\Omega_{Lj}$ matrix elements are negative, though
their magnitudes can of course vary substantially over $\mathcal{T}$ (but we
expect are fairly well-correlated locally, when considering states in
$\mathcal{T}$ only a few flips apart). Because the states in $\mathcal{T}$ are
all near-zero energy, the local transverse field-induced “hopping” matrix
elements connecting them are either direct (if a given pair of states in
$\mathcal{T}$ are one flip apart), or the result of short ranged, few-flip
tunneling through local excited states. In either case, the resulting matrix
element is negative, and the band of $\left|\psi_{k}\right>$ states are
(approximately) the eigenstates of a hopping-like model on a sparse,
disordered graph of $N_{\mathcal{T}}$ sites, where most hopping matrix
elements are negative and there is local potential disorder. With all these
quantities defined, we now want to estimate the tunneling matrix element
$\tilde{\Omega}_{Lk}$ from $\left|L_{D}\right>$ into $\left|\psi_{k}\right>$.

We first consider states $\left|\psi_{k}\right>$ near the band center, e.g.
where the energy shifts from the transverse field are solely due to the second
order local processes captured in section IV and the “hopping energy” is
nearly zero. For such states, we can assume the signs of the amplitudes
$c_{jk}$ are randomized. Then by the law of large numbers:

$\displaystyle\left<\tilde{\Omega}_{Lk}^{2}\right>_{k}$ $\displaystyle\simeq$
$\displaystyle\left<\left(\sum_{j\in\mathcal{T}}^{N_{\mathcal{T}}}c_{jk}\Omega_{Lj}\right)^{2}\right>_{k}$
$\displaystyle\propto$ $\displaystyle
N_{\mathcal{T}}\left<c_{jk}^{2}\Omega_{Lj}^{2}\right>_{j}\propto\left<\Omega_{Lj}^{2}\right>.$

Thus, the average squared matrix element to tunnel into a state in the band
center has the same scaling as one obtains from a more naive calculation that
ignores the band structure entirely. Since the majority of states in the band
are near the center in this respect, this calculation shows that ignoring the
local band structure in $\mathcal{T}$ is a decent approximation for obtaining
the average collective tunneling rate that enters into equations like IV.4.

But what happens when we consider states that are more extremal, e.g. near the
top or bottom of the band? Near the top of the band, we can assume significant
local alternation in the signs/phases of the $c_{jk}$ coefficients, and thus
the assumption of randomization is still fairly good and Eq. B likely
accurately captures the scaling. For states near the bottom of the band,
however, the situation changes. First, these states will be lower in energy by
an $O\left(N\right)$ factor, which means $C\left(t\right)$ must be further
reduced for $\left|L_{D}\right>$ to cross them, and the individual tunneling
matrix elements $\Omega_{Lj}$ into such states may scale differently as a
result. We label these matrix elements $\Omega_{Lj}^{\prime}$ and do not
attempt to predict what, if any, changes to scaling may arise.

Second, to minimize the energy the signs/phases of the $c_{jk}$ of nearby (in
Hamming distance) configurations will be synchronized given that the matrix
elements that couple them are real and (mostly) negative (again, in this gauge
choice). And this synchronization can dramatically enhance tunneling rates.
Let us guess, for example, that in the band ground state
$\left|\psi_{0}\right>$, _most_ $c_{j0}$ are positive. Then:

$\displaystyle\left|\tilde{\Omega}_{L0}^{2}\right|\simeq\left<\left(\sum_{j\in\mathcal{T}}^{N_{\mathcal{T}}}c_{j0}\Omega_{Lj}^{\prime}\right)^{2}\right>\propto
N_{\mathcal{T}}\left<\left(\Omega_{Lj}^{\prime}\right)^{2}\right>.$ (41)

Since $N_{\mathcal{T}}$ is exponentially large in $N$ (Eq. 31), the tunneling
matrix element into the band ground state can be exponentially larger than the
average matrix element for tunneling into the band center, though if
$\Omega_{Lj}^{\prime}$ decays more quickly with $N$ that may reduce or erase
this effect. We expect that collective tunneling into low-lying states in the
band could be similarly enhanced even if there is more variation in the signs
of the $c_{jk}$ terms, as the matrix elements $\Omega_{Lj}$ are locally
correlated in magnitude. And though there are exponentially fewer states near
the band bottom compared to the band center, if the tunneling matrix elements
are exponentially larger the weighted average over all states in the band can
be substantially increased–we saw this effect already in averaging over the
relative distance to $\left|L\right>$ of different states in $\mathcal{T}$
(Eq. 32). It is thus quite plausible that band structure effects could further
increase collective tunneling rates and be responsible for the relative
overperformance of TMA spectral folding in our simulations, as compared to the
analytical prediction.

All that said, we did not include this analysis in the main text because we do
not presently have the mathematical tools to quantitatively predict the band
structure and nature of the dressed eigenstates, nor can we predict the
potential scaling changes in the collective tunneling matrix elements
$\Omega_{Lj}^{\prime}$ to states near the bottom of the band. So we present
these results to justify our claim that ignoring band structure is likely to
underestimate the achievable approximation ratio, and to suggest interesting
directions for further research.

Figure 11: Probabilities of returning states within Hamming distance
$D_{H}=\left\\{2/5,1/3,1/4,1/5,1/8,0\right\\}N$ flips from the ground state
$G$ for QAOA (top to bottom curves), for
$N_{C}=\left\\{2,4,6,3\sqrt{N}/2\right\\}N$ (clockwise from top left) and
application parameters in text. As seen in the figures, the per-shot
probability of finding the ground state is essentially independent of
constraint density and tracks the prediction $N2^{-0.28N}$ in Eq. 15. On the
other hand, the probabilities of returning states at various extensive
fractional Hamming distances, e.g. $N/4$ or fewer flips, decay much more
slowly, and the task of finding states comparatively close to $G$ becomes
easier as $N_{C}/N$ increases. That said, these probabilities all decay
exponentially with $N$ in all cases, consistent with the worsening
approximation ratios observed in figure 6.

Figure 12: Probabilities of returning states within Hamming distance
$D_{H}=\left\\{2/5,1/3,1/4,1/5,1/8\right\\}N$ flips from the ground state $G$
for the AQC formulation of spectral folding in figure 7 (top to bottom
curves), for $N_{C}=\left\\{2,4,6,3\sqrt{N}/2\right\\}N$ (clockwise from top
left) and application parameters in text. For $N_{C}=2N$ some probabilities
decay slowly with system size, due to competition with other minima (though
non-monotonicity makes the fitting somewhat ambiguous here); for all other
cases they are constant or increase toward some large $N$ saturation value,
indicating that this variation is finding states near the global minimum with
constant probability. The probability of finding $G$ is not plotted, as for
spectrally folded optimization it’s essentially zero.

Figure 13: Probabilities of returning states within Hamming distance
$D_{H}=\left\\{2/5,1/3,1/4,1/5,1/8\right\\}N$ flips from the ground state $G$
for the TMA formulation of spectral folding in figure 8 (top to bottom
curves), for $N_{C}=\left\\{2,4,6,3\sqrt{N}/2\right\\}N$ (clockwise from top
left) and application parameters in text. For $N_{C}=2N$ the probabilities
decay slowly with system size, due to competition with other minima; for all
other cases they are constant or increase toward some large $N$ saturation
value, indicating that this variation is finding states near the global
minimum with constant probability. The probability of finding $G$ is not
plotted, as for spectrally folded optimization it’s essentially zero.

Figure 14: Performance quadratic AQC spectral folding, for
$N_{C}=\left\\{2,4,6,3\sqrt{N}/2\right\\}N$ (clockwise from top left) with
increased approximation target $A=0.85$ and all other parameters equal. In
comparison to the $A=0.75$ results in figure 7, increasing $A$ improves
prefactors but does not improve scaling and in the case of $N_{C}/N=6$,
modestly reduces $q_{a}$. This suggests we have reached the performance
ceiling for this variation on the tested PPSP classes.

## References

  * Korte et al. (2011) B. H. Korte, J. Vygen, B. Korte, and J. Vygen, _Combinatorial optimization_ , vol. 1 (Springer, 2011).
  * Garey et al. (1974) M. R. Garey, D. S. Johnson, and L. Stockmeyer, in _Proceedings of the sixth annual ACM symposium on Theory of computing_ (1974), pp. 47–63.
  * Ullman (1975) J. D. Ullman, Journal of Computer and System sciences 10, 384 (1975).
  * Grest et al. (1986) G. S. Grest, C. Soukoulis, and K. Levin, Physical Review Letters 56, 1148 (1986).
  * Crescenzi et al. (1995) P. Crescenzi, V. Kann, and M. Halldórsson, _A compendium of np optimization problems_ (1995).
  * Hochba (1997) D. S. Hochba, ACM Sigact News 28, 40 (1997).
  * Monasson (1998) R. Monasson, Journal of Physics A: Mathematical and General 31, 513 (1998).
  * Woeginger (2003) G. J. Woeginger, in _Combinatorial optimization: eureka, you shrink!_ (Springer, 2003), pp. 185–207.
  * Bapst et al. (2013) V. Bapst, L. Foini, F. Krzakala, G. Semerjian, and F. Zamponi, Physics Reports 523, 127 (2013).
  * Hillar and Lim (2013) C. J. Hillar and L.-H. Lim, Journal of the ACM (JACM) 60, 1 (2013).
  * Venturelli et al. (2015) D. Venturelli, S. Mandrà, S. Knysh, B. Gorman, R. Biswas, and V. Smelyanskiy, Physical Review X 5, 031040 (2015).
  * Bellitti et al. (2021) M. Bellitti, F. Ricci-Tersenghi, and A. Scardicchio, _Entropic barriers as a reason for hardness in both classical and quantum algorithms_ (2021), eprint 2102.00182.
  * Jones et al. (2022) C. Jones, K. Marwaha, J. S. Sandhu, and J. Shi, arXiv preprint arXiv:2210.03006 (2022).
  * Arora and Barak (2009) S. Arora and B. Barak, _Computational complexity: a modern approach_ (Cambridge University Press, 2009).
  * Finnila et al. (1994) A. Finnila, M. Gomez, C. Sebenik, C. Stenson, and J. Doll, Chemical physics letters 219, 343 (1994).
  * Kadowaki and Nishimori (1998) T. Kadowaki and H. Nishimori, Physical Review E 58, 5355 (1998).
  * Das and Chakrabarti (2008) A. Das and B. K. Chakrabarti, Reviews of Modern Physics 80, 1061 (2008).
  * Johnson et al. (2011) M. W. Johnson, M. H. Amin, S. Gildert, T. Lanting, F. Hamze, N. Dickson, R. Harris, A. J. Berkley, J. Johansson, P. Bunyk, et al., Nature 473, 194 (2011).
  * 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, Nature Physics 10, 218 (2014).
  * Hauke et al. (2019) P. Hauke, H. G. Katzgraber, W. Lechner, H. Nishimori, and W. D. Oliver, arXiv preprint arXiv:1903.06559 (2019).
  * King et al. (2023) A. D. King, J. Raymond, T. Lanting, R. Harris, A. Zucca, F. Altomare, A. J. Berkley, K. Boothby, S. Ejtemaee, C. Enderud, et al., Nature pp. 1–6 (2023).
  * Farhi et al. (2000) E. Farhi, J. Goldstone, S. Gutmann, and M. Sipser, arXiv:quant-ph/0001106 (2000).
  * Albash and Lidar (2017) T. Albash and D. A. Lidar, arXiv:1611.04471 (2017).
  * Farhi et al. (2014) E. Farhi, J. Goldstone, and S. Gutmann, arXiv preprint arXiv:1411.4028 (2014).
  * Grattan et al. (2023) G. Grattan, B. A. Barton, S. Feeney, G. Mossi, P. Patnaik, J. C. Sagal, L. D. Carr, V. Oganesyan, and E. Kapit, _Exponential acceleration of macroscopic quantum tunneling in a floquet ising model_ (2023), eprint 2311.17814.
  * Albash and Lidar (2018) T. Albash and D. A. Lidar, Physical Review X 8, 031016 (2018).
  * King et al. (2018) A. D. King, J. Carrasquilla, J. Raymond, I. Ozfidan, E. Andriyash, A. Berkley, M. Reis, T. Lanting, R. Harris, F. Altomare, et al., Nature 560, 456 (2018).
  * Ebadi et al. (2022) S. Ebadi, A. Keesling, M. Cain, T. T. Wang, H. Levine, D. Bluvstein, G. Semeghini, A. Omran, J.-G. Liu, R. Samajdar, et al., Science 376, 1209 (2022).
  * Babbush et al. (2021) R. Babbush, J. R. McClean, M. Newman, C. Gidney, S. Boixo, and H. Neven, PRX Quantum 2, 010103 (2021).
  * Parisi (1979) G. Parisi, Physical Review Letters 43, 1754 (1979).
  * Earl and Deem (2005) D. J. Earl and M. W. Deem, Physical Chemistry Chemical Physics 7, 3910 (2005).
  * Shaydulin et al. (2023) R. Shaydulin, C. Li, S. Chakrabarti, M. DeCross, D. Herman, N. Kumar, J. Larson, D. Lykov, P. Minssen, Y. Sun, et al., arXiv preprint arXiv:2308.02342 (2023).
  * Baldwin and Laumann (2018) C. Baldwin and C. Laumann, Physical Review B 97, 224201 (2018).
  * Smelyanskiy et al. (2018) V. N. Smelyanskiy, K. Kechedzhi, S. Boixo, S. V. Isakov, H. Neven, and B. Altshuler, arXiv preprint arXiv:1802.09542 (2018).
  * Kechedzhi et al. (2018) K. Kechedzhi, V. Smelyanskiy, J. R. McClean, V. S. Denchev, M. Mohseni, S. Isakov, S. Boixo, B. Altshuler, and H. Neven, arXiv preprint arXiv:1807.04792 (2018).
  * Smelyanskiy et al. (2019) V. N. Smelyanskiy, K. Kechedzhi, S. Boixo, H. Neven, and B. Altshuler, arXiv preprint arXiv:1907.01609 (2019).
  * Susa et al. (2018) Y. Susa, Y. Yamashiro, M. Yamamoto, I. Hen, D. A. Lidar, and H. Nishimori, Phys. Rev. A 98, 042326 (2018), URL https://link.aps.org/doi/10.1103/PhysRevA.98.042326.
  * Graß (2019) T. Graß, Phys. Rev. Lett. 123, 120501 (2019), URL https://link.aps.org/doi/10.1103/PhysRevLett.123.120501.
  * Sels and Polkovnikov (2017) D. Sels and A. Polkovnikov, Proceedings of the National Academy of Sciences 114, E3909 (2017).
  * Bao et al. (2018) S. Bao, S. Kleer, R. Wang, and A. Rahmani, Physical Review A 97, 062343 (2018).
  * Yang et al. (2017) Z.-C. Yang, A. Rahmani, A. Shabani, H. Neven, and C. Chamon, Physical Review X 7, 021027 (2017).
  * Čepaitė et al. (2023) I. Čepaitė, A. Polkovnikov, A. J. Daley, and C. W. Duncan, PRX Quantum 4, 010312 (2023), URL https://link.aps.org/doi/10.1103/PRXQuantum.4.010312.
  * Montanari (2021) A. Montanari, SIAM Journal on Computing pp. FOCS19–1 (2021).
  * Farhi et al. (2022) E. Farhi, J. Goldstone, S. Gutmann, and L. Zhou, Quantum 6, 759 (2022).
  * Boulebnane and Montanaro (2021) S. Boulebnane and A. Montanaro, arXiv preprint arXiv:2110.10685 (2021).
  * Basso et al. (2021) J. Basso, E. Farhi, K. Marwaha, B. Villalonga, and L. Zhou, arXiv preprint arXiv:2110.14206 (2021).
  * Altshuler et al. (2010) B. Altshuler, H. Krovi, and J. Roland, Proceedings of the National Academy of Sciences 107, 12446 (2010).
  * Knysh (2016) S. Knysh, Nature communications 7 (2016).
  * Tang and Kapit (2021) Z. Tang and E. Kapit, Physical Review A 103, 032612 (2021).
  * Farhi et al. (2010) E. Farhi, J. Goldstone, D. Gosset, S. Gutmann, and P. Shor, arXiv preprint arXiv:1010.0009 (2010).
  * Sahni and Gonzalez (1976) S. Sahni and T. Gonzalez, Journal of the ACM (JACM) 23, 555 (1976).
  * Håstad (2001) J. Håstad, Journal of the ACM (JACM) 48, 798 (2001).
  * Mézard et al. (2002) M. Mézard, G. Parisi, and R. Zecchina, Science 297, 812 (2002).
  * Mézard et al. (2005) M. Mézard, T. Mora, and R. Zecchina, Physical Review Letters 94, 197205 (2005).
  * Hartmann and Weigt (2006) A. K. Hartmann and M. Weigt, _Phase transitions in combinatorial optimization problems: basics, algorithms and statistical mechanics_ (John Wiley & Sons, 2006).
  * Krzakala and Kurchan (2007) F. Krzakala and J. Kurchan, Physical Review E 76, 021122 (2007).
  * Altarelli et al. (2008) F. Altarelli, R. Monasson, and F. Zamponi, in _Journal of Physics: Conference Series_ (IOP Publishing, 2008), vol. 95, p. 012013.
  * Ibrahimi et al. (2012) M. Ibrahimi, Y. Kanoria, M. Kraning, and A. Montanari, in _Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms_ (SIAM, 2012), pp. 760–779.
  * Basso et al. (2022) J. Basso, D. Gamarnik, S. Mei, and L. Zhou, in _2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)_ (IEEE, 2022), pp. 335–343.
  * Boulebnane and Montanaro (2022) S. Boulebnane and A. Montanaro, arXiv preprint arXiv:2208.06909 (2022).
  * Anshu and Metger (2023) A. Anshu and T. Metger, Quantum 7, 999 (2023).
  * Benchasattabuse et al. (2023) N. Benchasattabuse, A. Bärtschi, L. P. García-Pintos, J. Golden, N. Lemons, and S. Eidenbenz, arXiv preprint arXiv:2308.15442 (2023).
  * Roland and Cerf (2002) J. Roland and N. J. Cerf, Phys. Rev. A 65, 042308 (2002).
  * Dubois and Mandler (2002) O. Dubois and J. Mandler, Comptes Rendus Mathematique 335, 963 (2002).
  * Håstad and Venkatesh (2002) J. Håstad and S. Venkatesh, in _Proceedings of the thiry-fourth annual ACM symposium on Theory of computing_ (2002), pp. 43–52.
  * Kowalsky et al. (2021) M. Kowalsky, T. Albash, I. Hen, and D. A. Lidar, arXiv preprint arXiv:2103.08464 (2021).
  * Zhu et al. (2022) L. Zhu, H. L. Tang, G. S. Barron, F. Calderon-Vargas, N. J. Mayhall, E. Barnes, and S. E. Economou, Physical Review Research 4, 033029 (2022).
  * Kapit and Oganesyan (2021) E. Kapit and V. Oganesyan, Quantum Science and Technology 6, 025013 (2021).
  * Barak et al. (2015) B. Barak, A. Moitra, R. O’Donnell, P. Raghavendra, O. Regev, D. Steurer, L. Trevisan, A. Vijayaraghavan, D. Witmer, and J. Wright, arXiv preprint arXiv:1505.03424 (2015).
  * Anshu et al. (2021) A. Anshu, D. Gosset, K. J. M. Korol, and M. Soleimanifar, Physical Review Letters 127, 250502 (2021).
  * Marwaha and Hadfield (2022) K. Marwaha and S. Hadfield, Quantum 6, 757 (2022).
  * Wang and Zunger (1994a) L. W. Wang and A. Zunger, The Journal of Physical Chemistry 98, 2158 (1994a).
  * Wang and Zunger (1994b) L.-W. Wang and A. Zunger, The Journal of Chemical Physics 100, 2394 (1994b).
  * McClean et al. (2016) J. R. McClean, J. Romero, R. Babbush, and A. Aspuru-Guzik, New Journal of Physics 18, 023023 (2016).
  * Santagati et al. (2018) R. Santagati, J. Wang, A. A. Gentile, S. Paesani, N. Wiebe, J. R. McClean, S. Morley-Short, P. J. Shadbolt, D. Bonneau, J. W. Silverstone, et al., Science advances 4, eaap9646 (2018).
  * Zhang et al. (2021) F. Zhang, N. Gomes, Y. Yao, P. P. Orth, and T. Iadecola, Physical Review B 104, 075159 (2021).
  * Tilly et al. (2022) J. Tilly, H. Chen, S. Cao, D. Picozzi, K. Setia, Y. Li, E. Grant, L. Wossnig, I. Rungger, G. H. Booth, et al., Physics Reports 986, 1 (2022).
  * Tazi and Thom (2023) L. C. Tazi and A. J. Thom, arXiv preprint arXiv:2305.04783 (2023).
  * Kapit (2021a) E. Kapit, _Systems and methods for accelerated quantum optimization_ (2021a), uS Patent App. 17/027,146.
  * Bravyi et al. (2011) S. Bravyi, D. P. DiVincenzo, and D. Loss, Annals of physics 326, 2793 (2011).
  * Pietracaprina et al. (2016) F. Pietracaprina, V. Ros, and A. Scardicchio, Physical Review B 93, 054201 (2016).
  * Baldwin et al. (2016) C. Baldwin, C. Laumann, A. Pal, and A. Scardicchio, Physical Review B 93, 024202 (2016).
  * Baldwin et al. (2017) C. Baldwin, C. Laumann, A. Pal, and A. Scardicchio, Physical Review Letters 118, 127201 (2017).
  * Scardicchio and Thiery (2017) A. Scardicchio and T. Thiery, arXiv preprint arXiv:1710.01234 (2017).
  * Derrida (1980) B. Derrida, Physical Review Letters 45, 79 (1980).
  * Jörg et al. (2010) T. Jörg, F. Krzakala, J. Kurchan, A. C. Maggs, and J. Pujos, EPL (Europhysics Letters) 89, 40004 (2010).
  * Schiffer et al. (2023) B. F. Schiffer, D. S. Wild, N. Maskara, M. Cain, M. D. Lukin, and R. Samajdar, arXiv preprint arXiv:2306.13131 (2023).
  * Kapit (2021b) E. Kapit, _Systems and methods for passive quantum error correction_ (2021b), uS Patent 10,956,267.
  * Mossi et al. (2023) G. Mossi, V. Oganesyan, and E. Kapit, arXiv preprint arXiv:2306.10632 (2023).
  * Suzuki et al. (2021) Y. Suzuki, Y. Kawase, Y. Masumura, Y. Hiraga, M. Nakadai, J. Chen, K. M. Nakanishi, K. Mitarai, R. Imai, S. Tamiya, et al., Quantum 5, 559 (2021).
  * Isakov et al. (2016) S. V. Isakov, G. Mazzola, V. N. Smelyanskiy, Z. Jiang, S. Boixo, H. Neven, and M. Troyer, Physical review letters 117, 180402 (2016).
  * Andriyash and Amin (2017) E. Andriyash and M. H. Amin, arXiv preprint arXiv:1703.09277 (2017).
  * Jiang et al. (2017a) Z. Jiang, V. N. Smelyanskiy, S. V. Isakov, S. Boixo, G. Mazzola, M. Troyer, and H. Neven, Physical Review A 95, 012322 (2017a).
  * Jiang et al. (2017b) Z. Jiang, V. N. Smelyanskiy, S. Boixo, and H. Neven, Physical Review A 96, 042330 (2017b).
  * King et al. (2019) A. D. King, J. Raymond, T. Lanting, S. V. Isakov, M. Mohseni, G. Poulin-Lamarre, S. Ejtemaee, W. Bernoudy, I. Ozfidan, A. Y. Smirnov, et al., arXiv preprint arXiv:1911.03446 (2019).
  * Heim et al. (2015) B. Heim, T. F. Rønnow, S. V. Isakov, and M. Troyer, Science 348, 215 (2015).