Impact of Fixing Spins in a Quantum Annealer with Energy Rescaling

Hirotaka Irie; Shu Tanaka; Tadashi Kadowaki; Tomohiro Hattori

REVIEW 3 major objections 5 minor 49 references

Impact of Fixing Spins in a Quantum Annealer with Energy Rescaling

T0 review · 3 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Fixing spins helps or hurts a quantum annealer depending on the energy rescaling.

desk verdict A narrow but genuine result: fixing spins helps or hurts quantum annealing depending on the energy rescaling parameter, demonstrated cleanly for a homogeneous ferromagnetic model, though the practical generalization is limited by the known-ground-state assumption. read the letter →

arxiv 2502.01008 v4 pith:H5RW3OFZ submitted 2025-02-03 cond-mat.stat-mech quant-ph

Tomohiro Hattori , Hirotaka Irie , Tadashi Kadowaki , Shu Tanaka This is my paper · ORCID

classification cond-mat.stat-mechquant-ph

keywords quantumannealingfixingspinsenergyrescalingminimumgapfullyconnectedferromagneticIsingmodelminorembeddingqubitchainsthermodynamiclimit

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper asks whether the fixing-spins size-reduction method, which deletes a subset of variables before annealing, still helps when the hardware forces the problem's energy scale to be rescaled. Working with a homogeneous fully connected ferromagnetic Ising model (all pairs of spins share the same attractive interaction, plus a small magnetic field to remove degeneracy), the paper finds the answer is conditional on the rescaling parameter $r$. At small $r$, fixing spins improves both the lowest energy reached on the annealer and the minimum energy gap; at large $r$, fixing spins shrinks the gap and harms the search. The connecting mechanism is that fixed spins shift the local field to $h' = h + J(N-n)(1-2p_{\mathrm{err}})$, which changes the energy range and hence the rescaling that the hardware applies. The result is supported by experiments on a quantum annealer, by exact diagonalization of the gap, and by a thermodynamic-limit calculation.

What carries the argument

The carrying object is the reduced local field $h' = h + J(N-n)(1-2p_{\mathrm{err}})$, a scalar that converts the whole fixing-spins procedure into one shift of the magnetic field. Rescaling by $r$ turns the problem Hamiltonian into $H''_p = H'_p/r$, so $h'$ sets both the energy range available on the hardware and the effective annealing schedule $B'(s) = B(s)/r$. In the thermodynamic limit the model is reduced by a Holstein-Primakoff transformation (a standard mapping of spin operators to creation and annihilation operators) and then diagonalized by a Bogoliubov transformation, yielding a harmonic oscillator whose frequency, minimized over $s$, is the gap $\Delta_{\min}$. A second mechanism enters on real hardware: qubit chains from minor embedding break when $h'$ is small relative to the fixed chain strength, which happens for large $n$ and large $p_{\mathrm{err}}$ or small $r$.

What would settle it

Reproduce the exact diagonalization of the $N=160$ homogeneous fully connected ferromagnetic model with $p_{\mathrm{err}}=0$: the claimed crossover requires that at $r=0.25$ the minimum gap is larger when fewer spins remain and falls as $n$ grows to 160, while at $r=10$ the gap rises with $n$; if either ordering fails, the central crossover claim is wrong.

Watch

Extended reading notes

Core claim

The central claim is that the benefit of fixing spins cannot be judged on its own: it is controlled by the rescaling parameter $r$. For the homogeneous fully connected ferromagnetic model with $N$ spins, fixing $n$ of them with error probability $p_{\mathrm{err}}$ yields an equivalent reduced model with unchanged coupling $J$ and modified local field $h' = h + J(N-n)(1-2p_{\mathrm{err}})$. Since the hardware rescales the problem parameters by $r$, this shift in $h'$ changes the actual Hamiltonian that is annealed. The paper reports a crossover: with a wide energy range (small $r$), fixing spins lowers the minimum energy and enlarges the minimum gap, while with a narrow range (large $r$) the gap-shrinking effect of rescaling dominates and fixing spins degrades performance. The experiments and exact-diagonalization results agree on this crossover, and the thermodynamic-limit analysis shows the gap increases with $h'$ and has an optimal rescaling value for a fixed parameter range.

Load-bearing premise

The argument presupposes that the ground state is known well enough to define the error probability $p_{\mathrm{err}}$, which is only true for the trivial all-up state of this ferromagnetic model; for real optimization problems the ground state is unknown, so $p_{\mathrm{err}}$ cannot be controlled before annealing and wrongly fixed spins may have effects beyond the single scalar $h'$.

Editorial extensions

If this is right

When the energy scale must be rescaled, the optimal number of fixed spins depends on $r$; for instance, at $r=2.5$ the gap is largest at $n=140$ rather than at the most aggressive reduction.
Fixing spins in the correct direction is doubly useful on real hardware: it both lowers the energy and strengthens $h'$, which protects the embedded qubit chains from breaking.
At error probability $p_{\mathrm{err}}=0.5$, $h'$ returns to the original field $h$, so the energy range is unchanged and fixing spins loses its rescaling-related advantage.
For small $r$ the trade-off favors fixing spins; for large $r$ rescaling penalizes it, so the number of spins to fix and the amount of rescaling should be treated as one tuning decision.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

For inhomogeneous or random Ising problems the scalar $h'$ formula no longer holds, so the $r$-dependent crossover found here may be weaker or reversed; sweeping $r$ while fixing spins on such models would show whether the trade-off is generic.
The same argument suggests a practical protocol: choose the rescaling parameter and the number of fixed spins jointly, since optimizing either alone misses the region where both are favorable.
If a future annealer reduced chain length by adding couplers per qubit, the chain-break penalty at small $r$ would shrink, possibly making aggressive spin fixing useful over a wider rescaling range.
In real optimization the ground state is not known, so $p_{\mathrm{err}}$ cannot be fixed in advance; whether classical preprocessing can keep the effective error low enough to sit in the beneficial small-$r$ regime remains an open question.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The one thing to know: this paper delivers a clean, narrow result on the interplay between spin-fixing size reduction and energy rescaling in a quantum annealer, for a homogeneous fully connected ferromagnetic Ising model. The core finding --- fixing spins helps at small r (wide energy range) and hurts at large r --- is supported by exact diagonalization and a thermodynamic-limit bosonization that agree with no fitted parameters. The hardware data on a D-Wave device is consistent with the trend, though it is presented as the minimum of 100 runs without error bars, so treat the experimental part as qualitative.

What is new: prior work treated fixing spins without rescaling (Ref. 21) and rescaling without fixing (Refs. 35, 36). Combining them exposes a trade-off that matters for parameter tuning: the rescaling parameter controls whether size reduction helps or hurts the gap. Eq. (14) shows the effect is mediated by the single scalar h' = h + J(N-n)(1-2p_err). The thermodynamic-limit calculation is a standard but competently executed Holstein-Primakoff plus Bogoliubov derivation.

Soft spots, in order of importance. First, the central quantity p_err is an input, not an output. The paper says openly that it is easy to calculate because the ground state is trivially known; for real optimization problems you do not know the ground state, and a real preprocessing method produces correlated errors that Eq. (14) cannot capture. So the demonstrated trade-off is for an idealized, ground-state-informed fixing procedure. That is not a fatal flaw for the model studied, but it means the abstract and conclusion overreach when they generalize to “the fixing spins method” without qualification. Second, the hardware experiment uses only the minimum of 100 runs and no variance estimate; chain-break effects are discussed qualitatively but not quantified. Third, the claim in Sec. 3.2 that finite-size results extend to larger sizes rests on the thermodynamic-limit comparison; reasonable, but not a proof of extensivity for the gap.

None of these undercut the core finite-size result for this model. The paper is honestly written, acknowledges the simplest setup, and points to future work on random and constrained models. I would send it to peer review; a referee should push for a scope-limited abstract and a statistical treatment of the hardware runs. A conditional accept is appropriate, not a rejection.

Referee Report

3 major / 5 minor

Summary. The paper studies the interplay between fixing spins (a size-reduction method) and energy rescaling in quantum annealing, using the homogeneous fully connected ferromagnetic Ising model as a testbed. It introduces an error probability p_err for mis-fixed spins, derives the effective local field h' after fixing spins (Eq. 14), and analyzes the minimum energy gap via exact diagonalization for N=160 and via a Holstein-Primakoff/Bogoliubov thermodynamic-limit formula. Hardware experiments on D-Wave Advantage measure the minimum energy over 100 runs as a function of the reduced size n and the rescaling parameter r. The central claim is that fixing spins improves quantum annealing performance at small r (wide energy range) and degrades it at large r, with an optimal n that depends on r and p_err.

Significance. If the results hold, they provide a concrete and largely parameter-free analysis of a nontrivial trade-off: fixing spins increases the energy gap through reduced system size but simultaneously modifies the local fields, and the benefit depends on the rescaling parameter r. The thermodynamic-limit gap formula and the explicit dependence of h' on p_err are useful benchmarks. The paper is transparent about its limitation to a trivial-ground-state model, and the analytic derivations are reproducible. The hardware experiment is illustrative, but it is not statistically characterized and should not be the basis for quantitative claims.

major comments (3)

[3.1, Fig. 2] The experimental claim that fixing spins enhances the quantum annealer's performance is supported only by the minimum energy among 100 stochastic runs, with no error bars, standard deviation, or repeated-measurement statistics. Since D-Wave outputs are stochastic, the differences among cells in Fig. 2 could be noise; please provide a statistical characterization (e.g., error bars, median, or distribution of chain breaks) to justify the claim.
[2 and Conclusion] The error probability p_err is an exogenous parameter that is only well-defined because the ground state is known (all-up state). The paper acknowledges this in Sec. 2, but the Conclusion generalizes to 'a fixing spins method with a low error probability is essential' and proposes hardware design changes. For real optimization problems, p_err is not known a priori and errors are not uniform; Eq. (14) does not capture correlations between fixed-spin errors and local fields. Please scope the conclusions to the homogeneous ferromagnetic model, or add a discussion of what changes when the ground state is unknown.
[3.2, Fig. 4] The statement 'Fixing spins has a positive effect at small r but a negative effect at large r' (Sec. 3.2) is a useful summary but is too coarse for intermediate r: Fig. 4 shows non-monotonic behavior with an optimal n for r around 2–3, and even at p_err = 0.5 the gap varies with n because the system size changes. Please characterize the optimal n as a function of r and p_err, or at least qualify the claim to reflect the non-monotonic structure.

minor comments (5)

[Eq. (15)] Equation (15) is typeset with a brace but no cases; the expression runs across lines without a clear branch structure. Please format it as a single display equation or a proper cases environment.
[3.2, Fig. 5 caption] The phrase 'shows the magnetic process of Δmin' should be 'shows the s-dependence of Δmin' or similar.
[3.2] The claim that 'the properties of the energy gap with fixing spins and rescaling are extensive with respect to size' is not established; the thermodynamic limit is a specific N→∞ limit, not evidence of extensivity. Please rephrase.
[2, Eq. (12)] Equation (12) uses subscript i without distinguishing the reduced-system index i′; this is a minor notation inconsistency.
[Fig. 2 caption] The caption says 'The ground state energy is 0.597' but does not specify the units or the parameter values used to compute this; please clarify.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the central results are computed from the Hamiltonian and benchmarked against exact diagonalization.

full rationale

The paper's derivation chain is self-contained and not circular. The fixing-spins transformation is deterministic: substituting fixed spin values into Eqs. (4)-(7) gives h' = h + J(N-n)(1-2p_err) (Eq. 14) for the homogeneous fully connected ferromagnetic model; this is a definitional consequence of the Hamiltonian, not an assumed conclusion. The central claims about the minimum energy and minimum gap as functions of n, p_err, and r are obtained by exact diagonalization of the full quantum Hamiltonian (Figs. 3-4) and by a thermodynamic-limit bosonization calculation (Eq. 15), neither of which fits parameters to the target quantities or imports a prior result as proof. The same-author citation [21] is used to motivate the definitions of p_err and r, but the new results do not reduce to that citation because the gap is recomputed independently and the r-dependence is the paper's own finding. The admitted limitation that p_err is 'easy to calculate' only for the trivial ferromagnetic ground state is a scope restriction on the method's applicability, not a circular step: the paper honestly states the conditional nature of the conclusions with respect to p_err. No fitted parameter is renamed as a prediction, and no self-citation supplies the load-bearing argument. Therefore the paper exhibits no significant circularity.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central claim rests on a deliberately simple model and on treating the fixing-spins error rate as an input. No data are fitted and no new entities are introduced; the analytic gap calculation relies on standard large-N bosonization.

free parameters (4)

p_err (fixing error probability) = 0.0, 0.1, 0.2, 0.3, 0.4, 0.5
Error probability of fixed spins relative to the known all-up ground state; hand-chosen input parameter, not fitted to data.
rescaling parameter r = 0.25 to 10 (scanned)
Control parameter sweeping the energy scale; not fitted to data.
h, J, N = h=0.1, J=1, N=160
Model constants chosen by hand: h>0 lifts the twofold degeneracy, J sets the scale, N is near the maximum embeddable size on the used D-Wave device.
chain strength = 2.0
Fixed at the maximum permissible value per D-Wave documentation, not fitted.

assumptions (5)

domain assumption The ground state of the homogeneous fully connected ferromagnetic Ising model is the all-up state, so p_err is a well-defined input.
Used throughout Sec. 3 to translate fixed-spin error into the local field h' via Eq. (14).
domain assumption The effect of fixing spins on the remaining problem is fully captured by the modified local fields h' and unchanged couplings J' (Eqs. (6)-(7)).
Equations (6) and (7) define the reduced Hamiltonian; this is exact for the Ising model, but assumes the fixed directions are independent of the remaining spins.
standard math The Holstein-Primakoff bosonization in the thermodynamic limit (N much larger than the occupation number, 1/sqrt(N) to 0) gives the exact energy gap of the fully connected model.
Appendix A, Eqs. (A.4)-(A.20), following Seoane and Nishimori [42].
domain assumption The annealing schedules A(s) and B(s) are such that rescaling only changes the slope of B(s)/r, modifying the effective annealing schedule as in Appendix B.
Used to connect the rescaling parameter r to the gap via Eq. (10) and Fig. B-1.
domain assumption The minimum energy over 100 runs and a fixed chain strength of 2.0 is a valid proxy for quantum annealer performance.
Sec. 3.1; the paper provides no statistical characterization of this proxy.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Impact of Fixing Spins in a Quantum Annealer with Energy Rescaling." pith.science (2026). https://pith.science/paper/H5RW3OFZ

@misc{pith2026250201008,
  author       = {Pith},
  title        = {Pith review of: Impact of Fixing Spins in a Quantum Annealer with Energy Rescaling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/H5RW3OFZ}},
  note         = {Machine review of arXiv:2502.01008}
}

read the original abstract

Quantum annealing is a promising algorithm for solving combinatorial optimization problems. However, various hardware restrictions significantly impede its efficient performance. Size-reduction methods provide an effective approach for addressing large-scale problems but often introduce additional challenges. A notable hardware restriction is the limited number of decision variables quantum annealing can handle compared to the size of the problem. Moreover, when employing size-reduction methods, the interactions and local magnetic fields in the Ising model--used to represent the combinatorial optimization problem--can become excessively large, making them difficult to implement on hardware. Although prior studies suggest that energy rescaling impacts the performance of quantum annealing, its interplay with size-reduction methods remains unexplored. This study examines the relationship between fixing spins, a promising size-reduction method, and the effects of energy rescaling. Numerical simulations and experiments conducted on a quantum annealer demonstrate that the fixing spins method enhances quantum annealing performance while preserving the spin-chain embedding for a homogeneous, fully connected ferromagnetic Ising model.

Figures

Figures reproduced from arXiv: 2502.01008 by the authors.

**Figure 1.** (Color online) Conceptual diagram illustrating the fixing spins method. This method reduces the size of the Ising model while altering the properties of local magnetic fields. The graph on the left represents the original Ising model, where white circles indicate spins determined by quantum annealing, and red circles represent fixed spins. Black arrows denote the local magnetic fields at each vertex, while solid lin… view at source ↗

**Figure 2.** (Color online) The color map shows the distribution of the minimum energy in the n versus r space for the homogeneous fully connected ferromagnetic Ising model with N = 160. The spacing of r values is not necessarily uniform. n = 160 corresponds to the results without fixed spins. The annealing time is set to τ = 2000.0 µs. The ground state energy is 0.597. Each panel represents a different error rate for fixing sp… view at source ↗

**Figure 3.** (Color online) The minimum energy gap ∆min of the Hamiltonian after fixing spins with perr = 0 and rescaling with r for the homogeneous fully connected ferromagnetic Ising model with N = 160. n = 160 corresponds to the results without fixing spins. Solid lines between points are provided as a guide to the eye. the chain length, the weaker local magnetic fields are assigned to each qubit. Consequently, as the spin le… view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: (Color online) Color map of the minimum energy gap ∆min distribution in n versus r space for the homogeneous fully connected ferromagnetic Ising model with N = 160. The spacing of r values is not necessarily uniform. n = 160 represents the results without fixing spins.…

**Figure 5.** Figure 5: (Color online) Minimum energy gap ∆min at the thermodynamic limit for different h ′ . Black lines represent ∆min for the same h range condition with different r. The assumption is that spins are fixed up to the maximum value where the local magnetic fields reach h/r.…

**Figure 4.** Figure 4: The same considerations apply to the energy gap at fi [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

49 extracted references · 45 canonical work pages

[1]

Impact of Fixing Spins in a Quantum Annealer with Energy Rescaling

Introduction Combinatorial optimization problems are ubiquitous across various domains, necessitating high-accuracy solutions to ad- dress their complexity. A critical challenge in these prob- lems is the exponential growth of solution candidates as the problem size increases. With the ongoing advancement of information-driven society, the demand for solv...

work page Pith review arXiv 2025
[2]

The Hamiltonian of the Ising model, representing the problem to be solved, is denoted as Hp

Setup In quantum annealing, solutions to a combinatorial op- timization problem are obtained by mapping them to the ground state of the Ising model. The Hamiltonian of the Ising model, representing the problem to be solved, is denoted as Hp. Quantum fluctuations, which facilitate the exploration of the solution space, are represented by the driver Hamilto...

work page
[3]

Thus, the model generated after fixing spins will not include isolated spins, making it easier to analyze

Results In this study, for simplicity, we focus on a homogeneous fully connected ferromagnetic Ising model, as the Ising model obtained after fixing spins results in another fully connected Ising model. Thus, the model generated after fixing spins will not include isolated spins, making it easier to analyze. The homogeneous fully connected ferromagnetic I...

work page
[4]

Promoting the application of advanced quan- tum technology platforms to social issues

Conclusion We investigated the e ffects of rescaling and fixing spins through experimental hardware and numerical simulations. The experiment clarifies the impact of fixing spins and rescal- ing on a homogeneous fully connected ferromagnetic Ising model. Our results demonstrate that fixing spins e ffectively leads to a low-energy state in quantum annealin...

work page
[5]

Kadowaki and H

T. Kadowaki and H. Nishimori: Phys. Rev. E 58 (1998) 5355

1998
[6]

Finnila, M

A. Finnila, M. Gomez, C. Sebenik, C. Stenson, and J. Doll: Chemical Physics Letters 219 (1994) 343

work page 1994
[7]

Das and B

A. Das and B. K. Chakrabarti: Rev. Mod. Phys. 80 (2008) 1061

2008
[8]

Tanahashi, S

K. Tanahashi, S. Takayanagi, T. Motohashi, and S. Tanaka: Journal of the Physical Society of Japan 88 (2019) 061010

2019

Show all 49 references

[9]

B. K. Chakrabarti, H. Leschke, P. Ray, T. Shirai, and S. Tanaka: Philo- sophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 381 (2023) 20210419

2023
[10]

S. J. Weinberg, F. Sanches, T. Ide, K. Kamiya, and R. Correll: Sci. Rep. 13 (2023) 4770

2023
[11]

Camino, J

B. Camino, J. Buckeridge, P. Warburton, V . Kendon, and S. Woodley: Journal of Applied Physics 133 (2023)

2023
[12]

Honda, K

R. Honda, K. Endo, T. Kaji, Y . Suzuki, Y . Matsuda, S. Tanaka, and M. Muramatsu: Scientific Reports 14 (2024) 13872

2024
[13]

Z. Xu, W. Shang, S. Kim, E. Lee, and T. Luo: npj Computational Ma- terials 11 (2025) 4

2025
[14]

Rosenberg, P

G. Rosenberg, P. Haghnegahdar, P. Goddard, P. Carr, K. Wu, and M. L. De Prado: Proceedings of the 8th workshop on high performance com- putational finance, 2015, pp. 1–7

2015
[15]

Grant, T

E. Grant, T. S. Humble, and B. Stump: Phys. Rev.Appl. 15 (2021) 014012

2021
[16]

Kitai, J

K. Kitai, J. Guo, S. Ju, S. Tanaka, K. Tsuda, J. Shiomi, and R. Tamura: Phys. Rev. Res. 2 (2020) 013319

2020
[17]

A. S. Koshikawa, M. Ohzeki, T. Kadowaki, and K. Tanaka: Journal of the Physical Society of Japan 90 (2021) 064001

2021
[18]

Izawa, K

S. Izawa, K. Kitai, S. Tanaka, R. Tamura, and K. Tsuda: Phys. Rev. Res. 4 (2022) 023062

2022
[19]

Matsumori, M

T. Matsumori, M. Taki, and T. Kadowaki: Scientific Reports 12 (2022) 12143

2022
[20]

K. Nawa, T. Suzuki, K. Masuda, S. Tanaka, and Y . Miura: Phys. Rev. Appl. 20 (2023) 024044

2023
[21]

Inoue, Y

T. Inoue, Y . Seki, S. Tanaka, N. Togawa, K. Ishizaki, and S. Noda: Opt. Express 30 (2022) 43503

2022
[22]

Hirama and M

S. Hirama and M. Ohzeki: Journal of the Physical Society of Japan 92 (2023) 113002

2023
[23]

Kanai, M

H. Kanai, M. Yamashita, K. Tanahashi, and S. Tanaka: IEEE Access 12 (2024) 157669

2024
[24]

Altshuler, H

B. Altshuler, H. Krovi, and J. Roland: Proceedings of the National Academy of Sciences 107 (2010) 12446

2010
[25]

Hattori, H

T. Hattori, H. Irie, T. Kadowaki, and S. Tanaka: Journal of the Physical Society of Japan 94 (2025) 013001

2025
[26]

H. Irie, H. Liang, T. Doi, S. Gongyo, and T. Hatsuda: Sci. Rep. 11 (2021) 8426

2021
[27]

Karimi and G

H. Karimi and G. Rosenberg: Quantum Information Processing 16 (2017) 166

2017
[28]

Karimi, G

H. Karimi, G. Rosenberg, and H. G. Katzgraber: Phys. Rev. E96 (2017) 043312

2017
[29]

Atobe, M

Y . Atobe, M. Tawada, and N. Togawa: IEEE Transactions on Computers 71 (2022) 2606

2022
[30]

Kikuchi, N

S. Kikuchi, N. Togawa, and S. Tanaka: Journal of the Physical Society of Japan 92 (2023) 124002

2023
[31]

P. L. Hammer, P. Hansen, and B. Simeone: Mathematical Programming 28 (1984) 121

1984
[32]

Choi: Quantum Information Processing 7 (2008) 193

V . Choi: Quantum Information Processing 7 (2008) 193

2008
[33]

Choi: Quantum Information Processing 10 (2011) 343

V . Choi: Quantum Information Processing 10 (2011) 343

2011
[34]

P. F. C. McGeoch and K. Boothby: D-Wave Technical Report Series (2022)

2022
[35]

Vinci, T

W. Vinci, T. Albash, G. Paz-Silva, I. Hen, and D. A. Lidar: Phys. Rev. A 92 (2015) 042310

2015
[36]

Vinci, T

W. Vinci, T. Albash, and D. A. Lidar: npj Quantum Information 2 (2016) 1

2016
[37]

Pelofske, G

E. Pelofske, G. Hahn, and H. N. Djidjev: Sci. Rep. 12 (2022) 4499

2022
[38]

Hino and S

K. Hino and S. Tanaka: 2024 IEEE International Conference on Quan- tum Computing and Engineering (QCE), V ol. 02, 2024, pp. 434–435

2024
[39]

M. H. Amin, A. D. King, J. Raymond, R. Harris, W. Bernoudy, A. J. Berkley, K. Boothby, A. Smirnov, F. Altomare, M. Babcock, et al.: arXiv preprint arXiv:2311.01306 (2023)

2023 arXiv
[40]

Braida, S

A. Braida, S. Martiel, and I. Todinca: npj Quantum Information 10 (2024) 40

2024
[41]

Messiah: Quantum Mechanics (1976)

A. Messiah: Quantum Mechanics (1976)

1976
[42]

McGeoch and P

C. McGeoch and P. Farr ´e: D-Wave Technical report (2021)

2021
[43]

D-Wave QPU Architecture Topologies
[44]

Solver Parameters - D-Wave System Documentation
[45]

Tanaka, R

S. Tanaka, R. Tamura, and B. Chakrabarti: Quantum Spin Glasses, An- nealing and Computation (Cambridge University Press, 2017)

2017
[46]

Seoane and H

B. Seoane and H. Nishimori: Journal of Physics A: Mathematical and Theoretical 45 (2012) 435301

2012
[47]

Holstein and H

T. Holstein and H. Primako ff: Phys. Rev. 58 (1940) 1098

1940
[48]

N. N. Bogoljubov: Il Nuovo Cimento (1955-1965) 7 (1958) 794

1958
[49]

Morita: Journal of the Physical Society of Japan 76 (2007) 104001

S. Morita: Journal of the Physical Society of Japan 76 (2007) 104001. 9

2007

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

[1] [1]

Impact of Fixing Spins in a Quantum Annealer with Energy Rescaling

Introduction Combinatorial optimization problems are ubiquitous across various domains, necessitating high-accuracy solutions to ad- dress their complexity. A critical challenge in these prob- lems is the exponential growth of solution candidates as the problem size increases. With the ongoing advancement of information-driven society, the demand for solv...

work page Pith review arXiv 2025

[2] [2]

The Hamiltonian of the Ising model, representing the problem to be solved, is denoted as Hp

Setup In quantum annealing, solutions to a combinatorial op- timization problem are obtained by mapping them to the ground state of the Ising model. The Hamiltonian of the Ising model, representing the problem to be solved, is denoted as Hp. Quantum fluctuations, which facilitate the exploration of the solution space, are represented by the driver Hamilto...

work page

[3] [3]

Thus, the model generated after fixing spins will not include isolated spins, making it easier to analyze

Results In this study, for simplicity, we focus on a homogeneous fully connected ferromagnetic Ising model, as the Ising model obtained after fixing spins results in another fully connected Ising model. Thus, the model generated after fixing spins will not include isolated spins, making it easier to analyze. The homogeneous fully connected ferromagnetic I...

work page

[4] [4]

Promoting the application of advanced quan- tum technology platforms to social issues

Conclusion We investigated the e ffects of rescaling and fixing spins through experimental hardware and numerical simulations. The experiment clarifies the impact of fixing spins and rescal- ing on a homogeneous fully connected ferromagnetic Ising model. Our results demonstrate that fixing spins e ffectively leads to a low-energy state in quantum annealin...

work page

[5] [5]

Kadowaki and H

T. Kadowaki and H. Nishimori: Phys. Rev. E 58 (1998) 5355

1998

[6] [6]

Finnila, M

A. Finnila, M. Gomez, C. Sebenik, C. Stenson, and J. Doll: Chemical Physics Letters 219 (1994) 343

work page 1994

[7] [7]

Das and B

A. Das and B. K. Chakrabarti: Rev. Mod. Phys. 80 (2008) 1061

2008

[8] [8]

Tanahashi, S

K. Tanahashi, S. Takayanagi, T. Motohashi, and S. Tanaka: Journal of the Physical Society of Japan 88 (2019) 061010

2019