Annealing-based approach to solving partial differential equations

Kazue Kudo

arxiv: 2406.17364 · v5 · submitted 2024-06-25 · 🧮 math.NA · cond-mat.dis-nn· cond-mat.stat-mech· cs.NA· quant-ph

Annealing-based approach to solving partial differential equations

Kazue Kudo This is my paper

Pith reviewed 2026-05-24 00:25 UTC · model grok-4.3

classification 🧮 math.NA cond-mat.dis-nncond-mat.stat-mechcs.NAquant-ph

keywords annealingpartial differential equationseigenvalue problemsRayleigh quotientsimulated annealingnumerical methodsdiscretization

0 comments

The pith

An iterative annealing algorithm solves discretized PDEs by computing eigenvectors to arbitrary precision without adding variables.

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

The paper reformulates discretized partial differential equations as generalized eigenvalue problems that are then turned into optimization tasks using a generalized Rayleigh quotient. An algorithm is proposed that applies annealing iteratively to refine the eigenvector solutions. This iteration enables the target precision to be reached without enlarging the variable count. Simulated annealing experiments track how the required iteration count grows with system size and annealing duration. Overall performance is shown to depend on system size, annealing time, and the specific PDE characteristics.

Core claim

The central claim is that the proposed iterative algorithm enables efficient annealing-based computation of eigenvectors to arbitrary precision without increasing the number of variables, by repeatedly optimizing the generalized Rayleigh quotient for the eigenvalue problem that arises from PDE discretization.

What carries the argument

Iterative annealing optimization of the generalized Rayleigh quotient to refine eigenvector solutions for the discretized generalized eigenvalue problem.

If this is right

Eigenvectors for PDE-derived eigenvalue problems can be obtained to any desired accuracy by repeated annealing steps rather than by enlarging the formulation.
The number of iterations needed grows with system size and annealing time in a manner that can be measured directly via simulated annealing.
Computational cost is governed by the interplay of system size, annealing schedule, and the concrete PDE being solved.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same iterative refinement might apply to other linear-algebra problems that can be cast as Rayleigh-quotient optimization.
If the observed scaling holds for quantum annealing devices, the method could become viable for very large sparse systems where classical eigensolvers become expensive.
Direct comparison against standard iterative solvers such as Lanczos on the same discretized operators would quantify any practical advantage.

Load-bearing premise

The iterative computations converge reliably to the target precision for the discretized systems arising from PDEs, with scaling that remains practical as system size grows.

What would settle it

A test on the discretized Poisson equation in which the eigenvector error fails to fall below a fixed threshold no matter how many iterations or how long the annealing time is increased.

Figures

Figures reproduced from arXiv: 2406.17364 by Kazue Kudo.

**Figure 2.** Figure 2: The number of precision updates required for solving the one-dimensional symmetric Poisson [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** Figure 3: The number of iterations at the iterative descent stage required for solving (a) the one-dimensional [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

**Figure 4.** Figure 4: The average number of iterations at the iterative descent stage for solving the one-dimensional [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗

**Figure 4.** Figure 4: However, the coefficient α in the exponential function e αn is relatively small, even for the asymmetric Poisson equation. This suggests the proposed method, together with using an Ising machine, can outperform conventional methods for certain classes of simple problems. It is important to note that this analysis focuses on the number of iterations rather than the actual annealing time. In the simulated a… view at source ↗

read the original abstract

Solving partial differential equations (PDEs) using an annealing-based approach involves solving generalized eigenvalue problems. Discretizing a PDE yields a system of linear equations (SLE). Solving an SLE can be formulated as a general eigenvalue problem, which can be transformed into an optimization problem with an objective function given by a generalized Rayleigh quotient. The proposed algorithm requires iterative computations. However, it enables efficient annealing-based computation of eigenvectors to arbitrary precision without increasing the number of variables. Investigations using simulated annealing demonstrate how the number of iterations scales with system size and annealing time. Computational performance depends on system size, annealing time, and problem characteristics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper recasts discretized PDEs as iterative annealing optimization of the generalized Rayleigh quotient but supplies almost no error numbers or solver comparisons.

read the letter

The main takeaway is that the authors recast discretized PDEs as generalized eigenvalue problems and then solve them by iteratively optimizing the generalized Rayleigh quotient with annealing. This keeps the problem size fixed while using multiple runs to improve accuracy. The abstract states that the algorithm enables computation to arbitrary precision without increasing variables, which is achieved through the iterations. The simulated annealing investigations show the scaling, which is useful information even if it's on a classical simulator rather than actual annealing hardware. What the paper does is lay out the steps from discretization to the optimization formulation and then report scaling behavior from simulated annealing runs. The claim that arbitrary precision is reachable without adding variables comes directly from the iterative nature of the procedure. The scaling investigations with respect to system size and annealing time are the most tangible results provided. The mapping itself is not revolutionary; it builds on known ways to turn linear systems into Rayleigh quotient minimizations. The value, if any, lies in showing that annealing can be applied in this iterative fashion for PDE-derived problems. The soft spot is the absence of concrete accuracy measures. The abstract mentions the investigations but does not include error values, convergence rates, or head-to-head tests against standard methods such as finite element solvers or iterative linear solvers. This makes it difficult to assess whether the approach is competitive or reliable for typical PDE applications. The work is aimed at researchers exploring alternative computing paradigms for numerical mathematics, particularly those with access to or interest in annealing hardware. A reader focused on classical numerical PDE methods would likely find it too preliminary. I think it deserves peer review. The idea is internally consistent and the scaling data provides a starting point for discussion. Referees could usefully push for the quantitative validation that is missing from the current version.

Referee Report

2 major / 0 minor

Summary. The paper proposes an annealing-based method for solving PDEs: discretize the PDE to obtain a system of linear equations, recast this as a generalized eigenvalue problem, and convert the latter into an optimization problem whose objective is the generalized Rayleigh quotient. This optimization is solved by simulated annealing; the algorithm performs iterative computations to reach arbitrary precision while keeping the number of variables fixed. Scaling behavior of iteration count with system size and annealing time is investigated via simulated annealing.

Significance. If the iterative annealing procedure can be shown to converge reliably with practical scaling, the approach would supply a new route to PDE solution that maps directly onto annealing hardware without enlarging the variable count at each iteration. The reported scaling investigations are a positive step, but the absence of quantitative error metrics, convergence rates, or baseline comparisons leaves the practical significance difficult to evaluate.

major comments (2)

[Abstract] Abstract: the central claim that eigenvectors are obtained 'to arbitrary precision' via iterative annealing is unsupported by any reported error norms, residual values, or convergence data; only the scaling of iteration count is mentioned.
[Abstract] Abstract: the statement that the method 'enables efficient annealing-based computation ... without increasing the number of variables' is load-bearing for the contribution, yet no explicit comparison of problem size before and after iteration, nor any demonstration that iteration count remains sub-linear in system size, is supplied.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on the abstract claims. We address each major comment below and will revise the manuscript to provide the requested supporting evidence.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim that eigenvectors are obtained 'to arbitrary precision' via iterative annealing is unsupported by any reported error norms, residual values, or convergence data; only the scaling of iteration count is mentioned.

Authors: We agree that explicit quantitative support for the 'arbitrary precision' claim is needed. The iterative procedure refines the solution by repeated annealing runs on the fixed-size generalized Rayleigh quotient, but the current version reports only iteration scaling. In the revised manuscript we will add residual norm values, error metrics, and convergence plots versus iteration count to substantiate the claim. revision: yes
Referee: [Abstract] Abstract: the statement that the method 'enables efficient annealing-based computation ... without increasing the number of variables' is load-bearing for the contribution, yet no explicit comparison of problem size before and after iteration, nor any demonstration that iteration count remains sub-linear in system size, is supplied.

Authors: The discretization determines a fixed number of variables that is unchanged by the iterative annealing steps; each iteration optimizes the same Rayleigh quotient without expanding the variable count. The manuscript already examines how iteration count scales with system size, but we acknowledge that an explicit before/after comparison and discussion of the scaling regime (including whether it remains sub-linear) would strengthen the claim. We will add this clarification and supporting analysis in the revision. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on standard equivalences

full rationale

The paper's chain proceeds from PDE discretization to a system of linear equations, reformulated as a generalized eigenvalue problem, then cast as minimization of the generalized Rayleigh quotient—an optimization problem solved iteratively by annealing. These steps invoke standard mathematical identities (Rayleigh quotient properties for eigenvectors) without redefining any quantity in terms of the target output or fitting parameters to the same data that is later 'predicted.' No self-citations appear as load-bearing premises, no uniqueness theorems are imported from prior author work, and no ansatz is smuggled via citation. Scaling investigations are performed empirically with simulated annealing on the resulting optimization problem, leaving the central claim independent of its own fitted values.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The approach rests on standard numerical discretization of PDEs and the equivalence between linear systems and generalized eigenvalue problems; annealing time and iteration count function as tunable parameters whose scaling is investigated but not derived from first principles.

free parameters (2)

annealing time
Abstract states that computational performance depends on annealing time and reports scaling investigations with it.
number of iterations
Abstract notes that the algorithm requires iterative computations whose number scales with system size and annealing time.

axioms (2)

domain assumption Discretizing a PDE yields a system of linear equations that can be cast as a generalized eigenvalue problem.
Explicitly stated as the starting point in the abstract.
standard math The generalized Rayleigh quotient serves as a valid objective function whose optimization yields the desired eigenvectors.
Invoked without further justification in the abstract's description of the optimization formulation.

pith-pipeline@v0.9.0 · 5633 in / 1416 out tokens · 31068 ms · 2026-05-24T00:25:02.163593+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The proposed algorithm requires iterative computations. However, it enables efficient annealing-based computation of eigenvectors to arbitrary precision without increasing the number of variables.
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Solving an SLE can be formulated as a general eigenvalue problem, which can be transformed into an optimization problem with an objective function given by a generalized Rayleigh quotient.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

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[1] [1]

Y. Cao, A. Papageorgiou, I. Petras, J. Traub, S. Kais, Quantum algorithm and circuit design solving the Poisson equation, New J. Phys. 15 (1) (2013) 013021

work page 2013

[2] [2]

A. M. Childs, J.-P. Liu, A. Ostrander, High-precision quantum algorithms for partial differential equa- tions, Quantum 5 (2021) 574. doi:https://doi.org/10.22331/q-2021-11-10-574

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Physical Review A33(5), 2913–2927 (1986)

H.-L. Liu, Y.-S. Wu, L.-C. Wan, S.-J. Pan, S.-J. Qin, F. Gao, Q.-Y. Wen, Variational quantum algorithm for the Poisson equation, Phys. Rev. A 104 (2021) 022418. doi:https://doi.org/10.1103/PhysRevA. 104.022418

work page doi:10.1103/physreva 2021

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Y. Sato, R. Kondo, S. Koide, H. Takamatsu, N. Imoto, Variational quantum algorithm based on the minimum potential energy for solving the Poisson equation, Phys. Rev. A 104 (2021) 052409. doi:https://doi.org/10.1103/PhysRevA.104.052409

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R. Demirdjian, D. Gunlycke, C. A. Reynolds, J. D. Doyle, S. Tafur, Variational quantum solutions to the advection–diffusion equation for applications in fluid dynamics, Quantum Inf. Proc. 21 (9) (2022)

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Bravo-Prieto, R

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work page doi:10.1103/physreva.108 2023

[10] [10]

Garai, S

A. Teplukhin, B. K. Kendrick, D. Babikov, Calculation of Molecular Vibrational Spectra on a Quantum Annealer, J. Chem. Theor. Comp. 15 (8) (2019) 4555–4563. doi:https://doi.org/10.1021/acs. jctc.9b00402

work page doi:10.1021/acs 2019

[11] [11]

Teplukhin, B

A. Teplukhin, B. K. Kendrick, S. Tretiak, P. A. Dub, Electronic structure with direct diagonalization on a D-wave quantum annealer, Sci. Rep. 10 (1) (2020) 20753. doi:10.1038/s41598-020-77315-4

work page doi:10.1038/s41598-020-77315-4 2020

[12] [12]

Krakoff, S

B. Krakoff, S. M. Mniszewski, C. F. A. Negre, Controlled precision QUBO-based algorithm to compute eigenvectors of symmetric matrices, PLOS ONE 17 (5) (2022) e0267954. doi:https://doi.org/10. 1371/journal.pone.0267954

work page 2022

[13] [13]

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, G. Rose, Quantum annealing with manufactur...

work page doi:10.1038/nature10012 2011

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Yamaoka, C

M. Yamaoka, C. Yoshimura, M. Hayashi, T. Okuyama, H. Aoki, H. Mizuno, A 20k-spin Ising chip to solve combinatorial optimization problems with CMOS annealing, IEEE J. Solid-State Circ. 51 (1) (2016) 303–309. doi:https://doi.org/10.1109/JSSC.2015.2498601

work page doi:10.1109/jssc.2015.2498601 2016

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