Quantum Algorithms for Optimal Power Flow

David Neufeld; Daya Gaur; Md Mohsin Uddin; Robert Benkoczi; Sajad Fathi Hafshejani

REVIEW 4 major objections 4 minor 1 cited by

Quantum Algorithms for Optimal Power Flow

T0 review · 4 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read For the small power-flow problems tested, preconditioned HHL and VQLS reproduce the classical optimal cost and iteration counts.

desk verdict A small, honest empirical study that plausibly shows preconditioned VQLS/HHL can match classical OPF costs on tiny cases, but it omits the exact encoding and conditioning details needed to make the result reproducible or to rule out a semiclassical solve. read the letter →

arxiv 2412.06177 v1 pith:CUMHIO3E submitted 2024-12-09 quant-ph cs.SYeess.SYmath.OC

Sajad Fathi Hafshejani , Md Mohsin Uddin , David Neufeld , Daya Gaur , Robert Benkoczi This is my paper · ORCID

classification quant-phcs.SYeess.SYmath.OC MSC 81P6890C51

keywords optimalpowerflowquantumlinearsystemsHHLalgorithmvariationalsolverinteriorpointmethodspreconditioningAC-OPFDC-OPF

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

The paper asks whether the search-direction solves inside an interior-point optimal power flow solver can be replaced by quantum linear-system algorithms without losing accuracy. On the small direct-current (DC-OPF) cases it could simulate, preconditioned HHL (a quantum linear-system algorithm) and VQLS (a variational quantum linear solver) converge to the same generation cost as the classical solver, matching the classical iteration count on the smallest case and adding one iteration on another. On the one small alternating-current (AC-OPF) case it could run, VQLS again matches the classical cost and iteration count. The authors' conclusion is not that quantum solves are faster, but that they are accurate enough at this scale to justify further work on scaling the hybrid approach.

What carries the argument

The load-bearing object is the KKT linear system (Eq. 2) that an interior-point method solves at every iteration to obtain the Newton search directions ($\Delta X$, $\Delta Z$, $\Delta \lambda$, $\Delta \mu$). The hybrid method replaces the classical solve of this system with a quantum linear-system algorithm: HHL as a direct quantum solver and VQLS as a variational one. Left preconditioning with incomplete LU factorization is applied before the quantum solve, and the paper argues that this controls the condition number whose growth dominates HHL's complexity. The convergence measure gradcond ties the approximate quantum search directions to the classical convergence criteria.

What would settle it

Rerun one of the reported cases while recording the condition number and residual of the preconditioned KKT system before every interior-point step; if the preconditioned matrix is too ill-conditioned for the quantum error bounds to hold, or if the approximate search direction fails to reduce gradcond on a fresh test case, the parity claim is refuted.

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Extended reading notes

Core claim

The paper's central claim is that the KKT linear system arising inside a step-controlled primal-dual interior-point method for optimal power flow can be preconditioned and then solved with HHL or VQLS, and that the resulting hybrid algorithm reaches the same optimal cost as the classical solver on the test cases it could simulate. The reported costs match exactly in every completed run — 746.25 (DC case 3), 2393.31 (case 6ww), 4131.03 (case 9), and 758.21 (AC case 3) — with quantum iteration counts equal or one above the classical counts. The paper presents these runs as a substantial improvement over the earlier load-flow results in [1], and as evidence that the condition-number growth identified in [10] can be contained by ILU preconditioning. It does not claim an asymptotic speedup; the claim is parity and viability at small scale.

Load-bearing premise

The comparison stands on the assumption that an approximate solution read out of HHL or VQLS for the preconditioned KKT system is accurate enough for the interior-point method to keep converging to the true optimum; the paper's own limitations section restricts the tests to three small DC cases and one AC case because of qubit count and numerical instability, and the condition numbers and quantum residuals are never reported.

Editorial extensions

If this is right

If the parity holds beyond the four test cases, a practical division of labor emerges: classical globalization from the interior-point method plus an approximate quantum KKT solve is enough to reach the true optimum.
The preconditioning result isolates the condition number as the manageable bottleneck, so better preconditioners may extend the parity to larger DC-OPF cases and to more AC-OPF cases.
Since iteration counts are equal or within one, the number of quantum solves required is roughly the same as the number of classical solves, so the cost of read-in and read-out, rather than iteration count, becomes the main scaling question.

Reading between the lines

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

The paper never reports condition numbers or the precise oracle encoding of the non-Hermitian KKT matrix; measuring both on the same four cases would separate the value of preconditioning from the value of the quantum solver.
Because VQLS convergence depends on the classical optimizer and the ansatz, the equal costs on tiny cases do not yet reveal how error scales with system size; a controlled test with artificial matrices of known condition number would be the natural next benchmark.
Interior-point methods for other network optimization problems, such as gas or water networks, have the same KKT structure, so the same preconditioned hybrid pattern is a transferable template if the parity result holds.

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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

What's new here is narrow but real: the authors drop preconditioned HHL and VQLS into the search-direction solve of an interior-point OPF loop and report that on three tiny MATPOWER cases the resulting costs match classical SC-PDIPM. The iteration counts are equal or within one of classical, and the gradcond plots show similar convergence. That is a legitimate data point for hybrid quantum-classical optimization, and the paper is honest about its limits: they only did case3, 6ww, 9 for DC-OPF and case3 for AC-OPF, and they say larger cases are out of reach. The comparison is against external classical solutions, so the central result is not circular. Credit where due: the combination of ILU preconditioning with VQLS inside an OPF loop appears to be new, and the empirical agreement, if it holds up, is useful evidence that quantum linear solvers are not automatically useless inside interior-point methods.

Now the soft spots, and one of them is load-bearing. The paper never describes how the KKT matrix from Eq. (2) is encoded for HHL or VQLS. That matrix is symmetric but indefinite, and the [mu/Z] block changes by orders of magnitude as the barrier parameter shrinks. The authors say they use left ILU preconditioning, but M^{-1}A is generally non-symmetric, so it cannot be fed directly to HHL or VQLS as-is. If they embed it in a Hermitian form like [[0, M^{-1}A], [(M^{-1}A)^†, 0]], the dimension doubles and the condition number changes; none of that is reported. No condition numbers, no Pauli-term counts, no VQLS ansatz or optimizer settings, no HHL precision parameters, no error bars. Without those, I cannot tell whether the reported equal costs reflect a genuine approximate quantum solve or a solve that is effectively classical to machine precision. The stress-test note makes this point forcefully, and after reading the paper I think it lands.

The other weaknesses are more minor. The HHL entry for case9 is missing with no explanation. The AC-OPF results cover only one case. And the novelty over the authors' own load-flow paper [1] and prior quantum DC-OPF work is incremental, though the specific OPF-plus-preconditioning combination is new. The complexity theory from [10] is cited but not engaged numerically; the paper does not claim to overturn it, so this is a limitation rather than a flaw.

Who is this for? Readers working on hybrid quantum solvers for power systems will want to know this exists, but they will need the missing details before trusting the table. A serious referee could help extract those details. I would not cite it in my own work yet because the reproducibility bar is not met.

Recommendation: send it to peer review. It is small and incomplete, but the question is timely, the experiment is well-defined, and the missing information is fixable. A good referee should push for the encoding, conditioning data, and error bars, and should also ask for at least one run with a deliberately noisy/crude quantum readout to show the method still behaves like an approximate solver.

Referee Report

4 major / 4 minor

Summary. The paper proposes hybrid quantum-classical solvers for optimal power flow (OPF) by replacing the classical linear-system solve inside a step-controlled primal-dual interior-point method (SC-PDIPM) with HHL or VQLS, using ILU left preconditioning. The authors report simulations on small MATPOWER cases (case3, case6ww, case9 for DC-OPF, and case3 for AC-OPF) and claim that the quantum-assisted methods produce the same optimal costs and similar iteration counts as the classical SC-PDIPM solver, with preconditioning controlling the growth of the condition number.

Significance. If substantiated, the result would be a useful empirical step toward hybrid quantum-classical OPF solvers and would directly address the concern raised by Pareek et al. about the poor asymptotic scalability of HHL for power-flow problems. The comparison against MATPOWER's classical solutions is an external benchmark, so the central empirical claim is not circular. The use of standard test cases and the focus on preconditioning are sensible and relevant to the quantum-optimization community. However, the manuscript currently lacks the essential algorithmic details needed to verify that a quantum linear-system solve actually took place: no Hermitian encoding of the KKT system is described, no condition numbers are reported, and no error tolerances or readout procedures are given. As written, the reported agreement in iteration counts and costs is not reproducible and could be explained by a classical solve of the search-direction system.

major comments (4)

[§V, Algorithm 2 line 9; §IV-B; Eq. (2)] The manuscript never specifies how the KKT linear system in Eq. (2) is encoded as an operator suitable for HHL or VQLS. The KKT matrix is symmetric but indefinite, its (2,2) block [μ/Z] varies by orders of magnitude as the barrier parameter γ changes, and left ILU preconditioning produces the generally non-symmetric matrix M^{-1}A. VQLS, as stated in §IV-B, requires A to be a linear combination of Hermitian unitaries with bounded norm (typically ||A||≤1), and HHL requires a Hermitian or block-encoded operator with bounded norm. The authors must describe the actual encoding used (for example, a doubled Hermitian embedding), the resulting qubit count, the number of Pauli terms, the norm scaling, and the condition number after preconditioning for each test case. Without this information, the equivalence of iteration counts in Table I cannot be attributed to a quantum solve.
[§VI, Table I; Algorithm 2] No stopping criteria, error tolerances, or readout procedures are reported for the inner HHL/VQLS solves. The SC-PDIPM outer loop requires a search direction Δ = (ΔX, ΔZ, Δλ, Δμ); it is never explained how the quantum state |x⟩ is converted into these classical vectors, nor how approximation error in that readout affects the convergence conditions (feascond, gradcond, compcond, costcond). The paper should report the inner-solve tolerance, the number of VQLS outer iterations, the ansatz depth, the optimizer settings, and the HHL phase-estimation precision for each case, together with a sensitivity study showing how perturbing the search direction changes the reported iteration counts.
[§VI, Figures 1–4; §V] The claim that preconditioning leads to 'controlled growth in the condition number' is not supported by any numerical evidence. No condition numbers are given for the original KKT matrix or for the preconditioned system, for any test case. Since the stated complexity of HHL is O(log(N) s^2 κ^2/ε) and VQLS performance also degrades with κ, reporting κ before and after ILU preconditioning is essential to evaluate whether the quantum part is feasible at all. The authors should add a table or figure with these values.
[§VI, Table I and Table II] The empirical basis is very small: only three DC-OPF cases and one AC-OPF case, with HHL missing for case9 and entirely missing for AC-OPF. The conclusion that 'quantum approaches are comparable to classical approaches' is stronger than the data support. Either restrict the conclusion to the tested instances or add more cases and error bars on the reported costs and iteration counts.

minor comments (4)

[Throughout] There are several typographical errors, including 'quadricatically', 'acutal', 'pseduo-code', 'f alse', and 'SC-PDIM' in place of 'SC-PDIPM'. These should be corrected in a revision.
[Table I] The dash for HHL in the case9 row is unexplained; the text in §VI discusses Figure 3 only for classical and VQLS, so the reader cannot tell whether HHL failed, was not run, or was omitted for another reason. Please clarify.
[§VI, Figures 1–4] The figures do not include axis labels or error bars, and the text describing 'slight variations' and 'fluctuations' is qualitative. Adding labeled axes and, where possible, error bars or multiple runs would improve interpretability.
[Abstract and §VI] The abstract states that the results 'substantially improve the results in [1]', but no quantitative comparison with [1] is provided anywhere in the paper. Either include that comparison or soften the claim.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular step; the empirical comparison is anchored to external MATPOWER classical solutions, with only non-load-bearing self-citations to [1].

full rationale

The derivation chain is: OPF model -> Lagrangian (1) -> KKT linear system (2) -> SC-PDIPM with search directions solved by HHL/VQLS -> costs and iterations compared against classical results from MATPOWER (Tables I-II). None of these steps defines an output in terms of the claimed conclusion. The reported optimal costs (e.g., 746.25 for case 3) are compared with an external classical solver, not constructed from the quantum parameters; no fitted constant or uniqueness theorem forces the match. The paper's references to [1] (same research group) are comparative or expository ('this is already an improvement on the simulations in [1]', 'For an explanation of HHL and its use in solving load flow, see [1]') and are not load-bearing: the central OPF comparison stands on the independently defined MATPOWER benchmarks and the equations derived in this paper. The absence of a Hermitian encoding description, condition numbers, Pauli-term counts, and ansatz details is a reproducibility/correctness risk, not a circularity of the derivation. The explicit Limitations paragraph also confines the claims to small DC-OPF and AC-OPF cases, so the paper does not overstate an internally constructed agreement. No circular step can be exhibited with an equation-to-equation reduction, so the appropriate finding is no significant circularity.

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

The paper does not introduce new physical entities. However, it relies on several unstated implementation choices and the critical assumption that the KKT system can be encoded into quantum algorithms with acceptable conditioning.

free parameters (5)

VQLS ansatz layer count L
The number of strongly entangling layers in the ansatz is not reported; it is a hand-chosen hyperparameter that affects accuracy.
VQLS optimizer step size alpha
The step size for the classical optimizer is not specified.
ILU preconditioner tolerances
Dual threshold ILU parameters (drop tolerance, fill-in) are not specified, though they control the condition number of the preconditioned system.
HHL eigenvalue estimation precision
Number of ancilla qubits or phase estimation precision is not given.
SC-PDIPM parameters (xi, sigma, gamma_0)
Step-control and barrier parameters are inherited from [19] but the exact values used are not stated.

assumptions (4)

domain assumption The OPF problem can be formulated as in Section III-A with quadratic cost and standard constraints.
This is the standard OPF model, but its convexity and the validity of the KKT conditions are assumed.
domain assumption SC-PDIPM [19] converges to the OPF optimum for the selected MATPOWER cases when the search direction is solved exactly.
The paper relies on the classical solver's correctness without re-deriving it.
ad hoc to paper The indefinite KKT matrix in Eq. (2) can be embedded as a Hermitian operator suitable for HHL/VQLS, with bounded norm and manageable condition number after preconditioning.
This is a load-bearing assumption that is never demonstrated; the paper does not describe the encoding or report condition numbers.
ad hoc to paper Errors in the approximate quantum search direction do not prevent SC-PDIPM from converging to the same optimum.
The convergence results imply this, but no error analysis ties the quantum approximation error to the IPM convergence.

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Cite this review

Pith. "Pith review of Quantum Algorithms for Optimal Power Flow." pith.science (2026). https://pith.science/paper/CUMHIO3E

@misc{pith2026241206177,
  author       = {Pith},
  title        = {Pith review of: Quantum Algorithms for Optimal Power Flow},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CUMHIO3E}},
  note         = {Machine review of arXiv:2412.06177}
}

read the original abstract

This paper explores the use of quantum computing, specifically the use of HHL and VQLS algorithms, to solve optimal power flow problem in electrical grids. We investigate the effectiveness of these quantum algorithms in comparison to classical methods. The simulation results presented here which substantially improve the results in [1] indicate that quantum approaches yield similar solutions and optimal costs compared to classical methods, suggesting the potential use case of quantum computing for power system optimization.

Figures

Figures reproduced from arXiv: 2412.06177 by the authors.

**Figure 1.** Case 3-DC-OPF: gradcond v/s iterations We study the convergence of the three methods. Values of gradcond close to 0 indicate a locally optimal solution and convergence. Therefore, we plot gradcond values as function of the iteration number for the cases studied here [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗

**Figure 2.** Case 6ww-DC-OPF: gradcond v/s iterations [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Case 9-DC-OPF: gradcond v/s iterations [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Case 3-AC-OPF: gradcond v/s iterations [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Quantum Hardware-in-the-Loop for Optimal Power Flow in Renewable-Integrated Power Systems
eess.SY 2025-05 conditional novelty 5.0 of 10

A hardware-in-the-loop setup running annealer-based power flow and optimal power flow on RTDS matches Newton-Raphson benchmarks on a renewable-integrated nine-bus grid, with OPF solved only by Fujitsu's digital annealer.

Reference graph

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

A hybrid quantum algorithm for load flow

David Neufeld, Sajad Fathi Hafshejani, Daya Gaur, and Robert Benkoczi. A hybrid quantum algorithm for load flow. In Power , Energy and Electrical Engineering , pages 589–600. IOS Press, 2024

work page 2024

[2] [10]

Demystifying quantum power flow: Unveiling the limits of practical quantum advantage

Parikshit Pareek, Abhijith Jayakumar, Carleton Coffrin, and Sidhant Misra. Demystifying quantum power flow: Unveiling the limits of practical quantum advantage. arXiv preprint arXiv:2402.08617 , 2024

arXiv 2024

[3] [2]

Lecture Notes on Optimal Power Flow (OPF)

Spyros Chatzivasileiadis. Lecture notes on optimal power flow (opf). arXiv preprint arXiv:1811.00943 , 2018

work page Pith review arXiv 2018

[4] [3]

Quantum power flow

Fei Feng, Yifan Zhou, and Peng Zhang. Quantum power flow. IEEE Transactions on Power Systems , 36(4):3810–3812, 2021

work page 2021

[5] [4]

Quantum computing for power flow algorithms: Testing on real quantum computers

Brynjar Sævarsson, Spyros Chatzivasileiadis, Hj ¨ortur J ´ohannsson, and Jacob Østergaard. Quantum computing for power flow algorithms: Testing on real quantum computers. In 11th Bulk Power Systems Dynamics and Control Symposium , 2022

work page 2022

[6] [5]

Quantum power flows: From theory to practice

Junyu Liu, Han Zheng, Masanori Hanada, Kanav Setia, and Dan Wu. Quantum power flows: From theory to practice. Quantum Machine Intelligence, 6(55), 2024

work page 2024

[7] [6]

Exploiting quantum power flow in smart grid co-simulation

Dominik Vereno, Amin Khodaei, Christian Neureiter, and Sebastian Lehnhoff. Exploiting quantum power flow in smart grid co-simulation. In 2023 11th Workshop on Modelling and Simulation of Cyber-Physical Energy Systems (MSCPES) , pages 1–6. IEEE, 2023

work page 2023

[8] [7]

Solving dc power flow problems using quantum and hybrid algorithms

Fang Gao, Guojian Wu, Suhang Guo, Wei Dai, and Feng Shuang. Solving dc power flow problems using quantum and hybrid algorithms. Applied Soft Computing , 137:110147, 2023. 6

work page 2023