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arxiv: 2604.24362 · v1 · submitted 2026-04-27 · 🪐 quant-ph

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Practical lower bounds for hybrid quantum interior point methods in linear programming

Lennart Binkowski
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Pith reviewed 2026-05-08 04:05 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum interior point methodshybrid quantum algorithmslinear programmingquantum linear solversruntime lower boundspractical quantum advantagebenchmarking linear programs
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The pith

Hybrid quantum interior point methods do not provide practical advantages over classical solvers for linear programming.

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

This paper shows that a standard form of hybrid quantum interior point method for solving linear programs cannot achieve practical speedups relative to classical solvers. It derives lower bounds on the quantum runtime by combining the cost of quantum linear system solutions with the number of iterations needed in the interior point loop. These bounds are computed for a wide collection of linear programming problems spanning multiple families and compared against the performance of classical methods. The quantum lower bounds exceed classical runtimes in every case, even when granting the quantum solver highly favorable performance characteristics. Readers should care because this rules out near-term practical utility for this class of hybrid quantum algorithms on problems of realistic size and complexity.

Core claim

Equipping hybrid quantum interior point methods with a Chebyshev-based quantum linear solver and using either modified normal equation or orthogonal subspace formulations for the Newton systems leads to quantum runtime lower bounds that surpass classical solver runtimes on all instances tested from diverse linear programming families, regardless of the quantum cycle duration assumed.

What carries the argument

The exclusion analysis that produces rigorous lower bounds on quantum runtime from the complexity of the quantum linear solver, the dimensions of the linear systems, and the iteration count of the interior point method.

If this is right

  • No practical advantage exists for these hybrid quantum interior point methods on realistic linear programming problems.
  • The conclusion is independent of the specific Newton system formulation used.
  • Significant improvements in quantum linear solver performance would be required to change the outcome.
  • Broad testing on varied problem instances is needed to establish whether quantum methods can be useful in practice.

Where Pith is reading between the lines

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

  • The discrepancy between theoretical asymptotic improvements and practical performance may apply to other hybrid quantum approaches for optimization tasks.
  • Hardware improvements in quantum cycle times alone are unlikely to suffice without advances in algorithm design.
  • Lower bound techniques of this type could be useful for screening other proposed quantum algorithms for practical relevance.
  • Attention might shift toward developing fully quantum methods or different hybridizations for linear programming.

Load-bearing premise

That the quantum linear solver achieves close to its theoretical best-case performance with minimal overhead in the hybrid setting and that the Newton systems can be solved without extra costs beyond the linear system solves.

What would settle it

A measured runtime for the quantum component on a realistic linear programming instance that falls below the calculated lower bound, or a complete hybrid run that finishes faster than classical methods on such an instance.

Figures

Figures reproduced from arXiv: 2604.24362 by Lennart Binkowski.

Figure 1
Figure 1. Figure 1: Pipeline for computing the quantum runtime lower bound. Each dashed arrow marks a benevolent assumption that reduces the bound, making it more view at source ↗
Figure 2
Figure 2. Figure 2: Effective difficulty γ = sκ of the Newton linear systems across all benchmark instances, for the MNES (left) and OSS (right) formulations. The three binary optimisation relaxations—Clique, Independent Set, and Vertex Cover—constitute predominantly easy Newton systems in both cases. In fact, the condition number κ was often found to be exactly 1, so that their difficulties are effectively given by their spa… view at source ↗
Figure 3
Figure 3. Figure 3: Percentage of instances (y-axis), divided into eight problem classes, for which the lower bounds of the two QIPM variants’ runtimes lie below view at source ↗
read the original abstract

Quantum interior point methods (QIPMs) promise polynomial speed-ups over classical solvers for linear programming by outsourcing the solution of Newton linear systems to quantum linear solvers (QLSAs). However, asymptotic speed-ups do not necessarily translate to practical advantages on realistic problem instances. In this work, I evaluate whether practical advantage of a standard hybrid QIPM pipeline can already be excluded relative to the classical open-source solver HiGHS on a broad and diverse collection of LP instances spanning eight problem families, including public benchmark libraries, such as MIPlib, and relaxations of combinatorial optimisation problems. Following the hybrid benchmarking paradigm initiated by Cade et al., I derive rigorous lower bounds on the quantum runtime under a series of highly benevolent assumptions and compare them against classical runtimes. I equip the QIPMs with the best-performing functional QLSA, the Chebyshev-based method, as identified by Lefterovici et al., and evaluate two Newton system formulations proposed by Mohammadisiahroudi et al.: the modified normal equation system and the orthogonal subspace system. The exclusion analysis yields a consistent negative picture: across all instances and for any realistic quantum cycle duration, the quantum runtime lower bounds already exceed the classical runtimes, establishing that these hybrid QIPMs will offer no practical advantage over good classical solvers for realistic linear programming instances.

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.

Referee Report

2 major / 3 minor

Summary. The paper derives rigorous lower bounds on the total quantum runtime of hybrid QIPMs for linear programming, using the Chebyshev-based QLSA applied to the modified normal equation and orthogonal subspace Newton systems. These bounds are computed under a series of highly benevolent assumptions (including optimistic QLSA performance and hardware parameters) and compared against HiGHS runtimes on a broad collection of LP instances from eight families (including MIPLIB and combinatorial relaxations). The central claim is that the quantum lower bounds exceed classical runtimes for any realistic quantum cycle duration, excluding practical advantage for these hybrid methods on realistic instances.

Significance. If the lower-bound derivations hold under the stated assumptions, the work provides a concrete, instance-level exclusion result that bridges asymptotic quantum speed-up claims with practical benchmarking. It strengthens the hybrid benchmarking paradigm of Cade et al. by incorporating the best-performing functional QLSA and two specific Newton formulations, offering falsifiable predictions about when (or whether) such hybrids can compete with mature classical solvers.

major comments (2)
  1. [§4] §4 (QLSA precision and IPM convergence): The lower-bound derivation adopts a fixed or instance-independent precision tolerance ε for the Chebyshev QLSA (e.g., relative error 10^{-10} or similar). Standard IPM analyses require the Newton-step accuracy to scale at least as fast as the current barrier parameter μ or duality gap to preserve centrality and guarantee O(√n log(1/ε)) iterations; a fixed ε independent of iteration progress risks either non-convergence or the need for additional safeguards (adaptive precision or extra iterations), which would strictly increase the true quantum cost above the reported lower bound.
  2. [§3.2] §3.2 and Table 2 (quantum cycle duration and hardware assumptions): The exclusion holds 'for any realistic quantum cycle duration,' yet the manuscript does not provide a quantitative justification or sensitivity analysis for the threshold separating 'realistic' from 'unrealistic' cycle times (e.g., based on current or near-term superconducting or trapped-ion gate times). Because cycle duration is the sole free parameter, an explicit mapping from hardware literature to the chosen range is needed to make the 'no practical advantage' claim load-bearing.
minor comments (3)
  1. [§2.1] §2.1: The eight problem families are mentioned in the abstract but should be enumerated with explicit instance counts and sources in the main text for reproducibility.
  2. [Figure 3] Figure 3 (runtime comparison plots): Axis labels and legends should explicitly state the units (seconds) and the exact quantum cycle durations used in each curve.
  3. Reference list: The citation to Lefterovici et al. for the Chebyshev QLSA should include the arXiv identifier or full bibliographic details.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We address each major comment point by point below, providing clarifications and noting the revisions that will be incorporated.

read point-by-point responses

  1. Referee: [§4] §4 (QLSA precision and IPM convergence): The lower-bound derivation adopts a fixed or instance-independent precision tolerance ε for the Chebyshev QLSA (e.g., relative error 10^{-10} or similar). Standard IPM analyses require the Newton-step accuracy to scale at least as fast as the current barrier parameter μ or duality gap to preserve centrality and guarantee O(√n log(1/ε)) iterations; a fixed ε independent of iteration progress risks either non-convergence or the need for additional safeguards (adaptive precision or extra iterations), which would strictly increase the true quantum cost above the reported lower bound.

    Authors: We appreciate this observation on IPM convergence requirements. Our lower-bound analysis deliberately employs a fixed, conservative precision of 10^{-10} under the benevolent assumptions stated in the manuscript. This value is smaller than the accuracy needed in the final IPM iterations for the instances considered, and we assume convergence holds without extra iterations. If adaptive precision or additional safeguards prove necessary, the actual quantum runtime would exceed our reported lower bound, which would only strengthen the exclusion result. In the revised manuscript we will add a short clarifying paragraph in §4 explaining this choice and its implications for the lower-bound validity. revision: partial

  2. Referee: [§3.2] §3.2 and Table 2 (quantum cycle duration and hardware assumptions): The exclusion holds 'for any realistic quantum cycle duration,' yet the manuscript does not provide a quantitative justification or sensitivity analysis for the threshold separating 'realistic' from 'unrealistic' cycle times (e.g., based on current or near-term superconducting or trapped-ion gate times). Because cycle duration is the sole free parameter, an explicit mapping from hardware literature to the chosen range is needed to make the 'no practical advantage' claim load-bearing.

    Authors: We agree that an explicit mapping and sensitivity analysis would improve transparency. In the revised version we will expand §3.2 with a dedicated paragraph that (i) states the considered range of cycle durations (1 ns to 10 μs), (ii) provides a sensitivity plot or table showing how the exclusion threshold varies across this range, and (iii) cites recent hardware literature on superconducting-qubit and trapped-ion gate times to justify the 'realistic' regime. This will make the claim fully load-bearing without altering the core numerical results. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper derives lower bounds on quantum runtime for hybrid QIPMs from cited external QLSA performance (Chebyshev method) and hardware assumptions, then compares the resulting bounds directly against measured classical runtimes from the independent HiGHS solver across benchmark instances. No load-bearing step reduces the central claim to a self-definition, a fitted parameter renamed as prediction, or a self-citation chain; the cited works on benchmarking paradigms and Newton formulations are external and the comparison uses independent data sources. The exclusion result follows from this non-tautological comparison rather than construction from the target conclusion.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The claim rests on standard linear-algebra assumptions plus optimistic hardware parameters for quantum cycle time; no new entities are introduced.

free parameters (1)
  • quantum cycle duration
    Varied over realistic values to show the bound holds for any such value; not fitted to data but treated as an external parameter.
axioms (1)
  • domain assumption The Chebyshev-based QLSA and the two Newton system formulations achieve their stated asymptotic performance on the chosen instances.
    Invoked when equipping the QIPM with the best-performing QLSA and evaluating the two formulations from prior work.

pith-pipeline@v0.9.0 · 5527 in / 1297 out tokens · 64723 ms · 2026-05-08T04:05:30.037886+00:00 · methodology

discussion (0)

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Reference graph

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