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arxiv: 2604.11825 · v1 · submitted 2026-04-11 · 🪐 quant-ph · math-ph· math.MP

Quantum algorithms for Young measures: applications to nonlinear partial differential equations

Shi Jin , Nana Liu , Maria Lukacova-Medvidova , Yuhuan Yuan This is my paper

Pith reviewed 2026-05-10 16:01 UTC · model grok-4.3

classification 🪐 quant-ph math-phmath.MP
keywords quantum algorithmsYoung measuresnonlinear PDEslinear programmingmeasure-valued solutionsdissipative solutionsquantum speeduprandom PDEs
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The pith

Quantum linear programming algorithms achieve polynomial speedups for Young measures in random nonlinear PDEs but none for expected values alone.

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

Nonlinear PDEs often produce oscillatory or unstable solutions that standard numerical methods struggle to capture accurately, leading researchers to use Young measures as a way to describe the average behavior over possible fine-scale oscillations. The measure-valued formulation converts the PDE into a linear programming problem whose size grows rapidly with dimension, creating a computational bottleneck on classical machines. The paper applies quantum linear programming algorithms to this optimization task and finds that methods such as the quantum central path algorithm can deliver polynomial improvements over classical interior-point solvers in some regimes. When only the expected values of the Young measure are required, quantum methods show no advantage over direct classical PDE solvers. For random PDEs, however, recovering the full Young measure yields a polynomial speedup over standard classical approaches while supplying a more complete picture of the solution.

Core claim

The central claim is that quantum linear programming algorithms, including the quantum central path method, solve the linear programs arising from dissipative measure-valued formulations of nonlinear PDEs with up to polynomial advantage over classical interior point methods. For dissipative weak solutions given by the expected values of the Young measure, the quantum approach offers no advantage relative to direct classical solvers. For random PDEs, the same quantum methods recover the full Young measure with polynomial speedup compared with direct classical PDE solvers, and this measure supplies a richer description of solution behavior than averages alone.

What carries the argument

The linear programming problem obtained by reformulating a nonlinear PDE in terms of its dissipative measure-valued solution, solved with quantum linear programming algorithms such as the quantum central path algorithm.

Load-bearing premise

The derived linear program is large enough and structured so that the asymptotic quantum speedups are not overwhelmed by the costs of encoding the problem or correcting errors.

What would settle it

A concrete runtime benchmark on a random nonlinear PDE where the total wall-clock time of the quantum algorithm, including all state preparation and error correction, exceeds that of a direct classical PDE solver for the same accuracy on the Young measure.

Figures

Figures reproduced from arXiv: 2604.11825 by Maria Lukacova-Medvidova, Nana Liu, Shi Jin, Yuhuan Yuan.

Figure 1
Figure 1. Figure 1: Experiment 1 for the inviscid Burgers equation: the measure characterized by F(T, x, ξ), solution u(T, x), energy Eˆ(T, x), the evolution of total energy R Ω Edx ˆ and the total energy defects R Ω Edx with T = 1 and (Nt , Nx, Nξ) = (150, 200, 200) [PITH_FULL_IMAGE:figures/full_fig_p030_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: shows the measure characterized by F(T, x, ξ), solution u(T, x), energy Eˆ(T, x), the evolution of total energy R Ω Edx ˆ and the total energy defects R Ω Edx obtained with (Nt , Nx, Nξ) = (150, 200, 200). Further, the L p -errors and corresponding rates on consecu￾tively refined meshes are presented in [PITH_FULL_IMAGE:figures/full_fig_p031_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: shows the measure characterized by F(T, x, ξ), solution u(T, x), energy Eˆ(T, x), the evolution of total energy R Ω Edx ˆ and the total energy defects R Ω Edx obtained with (Nt , Nx, Nf ) = (300, 400, 401) [PITH_FULL_IMAGE:figures/full_fig_p032_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: presents the density ϱ, momentum m, the evolution of total energy R Ω Edx ˆ and energy defect R Ω Edx obtained with (Nt , Nx, Nϱ, Nm) = (200, 300, 151, 151) [PITH_FULL_IMAGE:figures/full_fig_p033_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Experiment 2 for the Euler system: density ϱ, momentum m, the evolution of total energy R Ω Edx ˆ and energy defect R Ω Edx obtained with (Nt , Nx, Nϱ, Nm) = (50, 100, 51, 201). Example A.6 (Riemann problem) The computational domain is taken to be Ω = [0, 1] and the outflow boundary conditions are used. The initial conditions are chosen to be as follows ϱ0(x) = ( 3 if x < 0.5, 1 if x ≥ 0.5, m0(x) = 0. (A.6… view at source ↗
Figure 6
Figure 6. Figure 6: shows density ϱ, momentum m, the evolution of total energy R Ω Edx ˆ and energy defect R Ω Edx obtained with (Nt , Nx, Nϱ, Nm) = (180, 200, 201, 201) [PITH_FULL_IMAGE:figures/full_fig_p034_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Experiment 1 for the Allen-Cahn equation (Double interfaces): the measure char￾acterized by F(T, x, ξ), solution u(T, x), the evolution of total energy R Ω Edx ˆ , the defect of total energy R Ω Edx as well as the defect of total regularized energy R Ω E REdx obtained with (Nt , Nx, NU ) = (100, 50, 200) [PITH_FULL_IMAGE:figures/full_fig_p037_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Experiment 2 for the Allen-Cahn equation: the measure characterized by F(T, x, ξ), solution u(T, x), the evolution of total energy R Ω Edx ˆ , the defect of total energy R Ω Edx as well as the defect of total regularized energy R Ω E REdx obtained with (Nt , Nx, NU ) = (150, 80, 100). 37 [PITH_FULL_IMAGE:figures/full_fig_p037_8.png] view at source ↗
read the original abstract

Many nonlinear PDEs have singular or oscillatory solutions or may exhibit physical instabilities or uncertainties. This requires a suitable concept of physically relevant generalized solutions. Dissipative measure-valued solutions have been an effective analytical tool to characterize PDE behavior in such singular regimes. They have also been used to characterize limits of standard numerical schemes on classical computers. The measure-valued formulation of a nonlinear PDE yields an optimization problem with a linear cost functional and linear constraints, which can be formulated as a linear programming problem. However, this linear programming problem can suffer from the curse of dimensionality. In this article, we propose solving it using quantum linear programming (QLP) algorithms and discuss whether this approach can reduce costs compared to classical algorithms. We show that some QLP algorithms, such as the quantum central path algorithm, have up to polynomial advantage over the classical interior point method. For problems where one is interested in the dissipative weak solution, namely the expected values of the Young measure, we show that the QLP algorithms offer no advantage over direct classical solvers. Moreover, for random PDEs, there can be up to polynomial advantage in obtaining the Young measure over direct classical PDE solvers. This is a significant advantage over standard PDE solvers, since the Young measure provides a more detailed description of the solution. We also propose some open questions for future development in this direction.

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 / 1 minor

Summary. The paper proposes reformulating dissipative measure-valued solutions of nonlinear PDEs as linear programming problems and solving them with quantum linear programming algorithms such as the quantum central-path method. It asserts that these QLP algorithms can yield up to polynomial advantage over classical interior-point methods for the full LP, while offering no advantage over direct classical solvers when only expected values of the Young measure are required; for random PDEs, however, polynomial advantage is claimed for recovering the entire Young measure, which supplies a more detailed description than standard weak solutions.

Significance. If the claimed polynomial speedups survive realistic encoding and oracle-construction costs, the work would open a quantum route to computing detailed measure-valued descriptions of singular or uncertain nonlinear PDEs, where classical methods are limited by dimensionality. The explicit separation between expected-value and full-measure regimes is a useful clarification that could guide future quantum-PDE research.

major comments (2)
  1. [Abstract] Abstract and the section on QLP advantages: the statement that 'some QLP algorithms, such as the quantum central path algorithm, have up to polynomial advantage over the classical interior point method' is made without the concrete complexity analysis or the precise conditions (e.g., sparsity, conditioning, or oracle model) under which the advantage holds for the Young-measure LP.
  2. [Discussion of advantages for random PDEs] Section discussing advantages for random PDEs: the claimed polynomial gain in recovering the full Young measure presupposes that the dense constraint matrix (arising from the weak form plus moment/entropy conditions) can be presented to the quantum algorithm via oracles whose construction cost is negligible compared with the LP solve; the manuscript provides no bound on this preprocessing cost, which scales at least linearly with the total number of nonzero entries O(N M K) for a spatial grid of size N, measure-space binning of size M, and moment order K.
minor comments (1)
  1. The notation for the Young-measure variables and the precise mapping from the weak form to the LP constraint matrix could be stated more explicitly, perhaps with a small illustrative example, to help readers verify the claimed linearity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments, which help clarify the scope of our claims. We address the major comments point by point below and will revise the manuscript to incorporate the suggested qualifications and additional discussion.

read point-by-point responses

  1. Referee: [Abstract] Abstract and the section on QLP advantages: the statement that 'some QLP algorithms, such as the quantum central path algorithm, have up to polynomial advantage over the classical interior point method' is made without the concrete complexity analysis or the precise conditions (e.g., sparsity, conditioning, or oracle model) under which the advantage holds for the Young-measure LP.

    Authors: We agree that the statement in the abstract would benefit from greater precision. The polynomial advantage is drawn from established results on the quantum central-path method in the QLP literature, which yields a polynomial improvement in the number of iterations (and thus overall complexity) relative to classical interior-point methods when the LP data can be accessed via efficient oracles and the instance satisfies standard assumptions on conditioning and sparsity. However, we did not supply a tailored complexity analysis for the specific Young-measure LP arising from the weak form plus moment/entropy constraints. In the revision we will add a short subsection that recalls the relevant QLP complexity bounds, states the oracle and conditioning assumptions under which they apply, and discusses how the structure of our constraint matrix (block-sparse contributions from the PDE weak form combined with denser moment blocks) interacts with those assumptions. revision: yes

  2. Referee: [Discussion of advantages for random PDEs] Section discussing advantages for random PDEs: the claimed polynomial gain in recovering the full Young measure presupposes that the dense constraint matrix (arising from the weak form plus moment/entropy conditions) can be presented to the quantum algorithm via oracles whose construction cost is negligible compared with the LP solve; the manuscript provides no bound on this preprocessing cost, which scales at least linearly with the total number of nonzero entries O(N M K) for a spatial grid of size N, measure-space binning of size M, and moment order K.

    Authors: The referee correctly notes that the constraint matrix is dense and that the manuscript does not quantify the cost of building the oracles that supply matrix entries to the quantum algorithm. Under the standard quantum LP model, oracle construction is assumed to be provided “for free” once the classical data are known; our claim of polynomial advantage for recovering the full Young measure in random PDEs is made inside that model. We acknowledge that, for a generic dense discretization, the classical preprocessing cost O(N M K) could be comparable to or larger than the quantum solve time and would therefore need to be included in any end-to-end comparison. In the revised manuscript we will (i) explicitly state the oracle-model assumption, (ii) note that the advantage holds only when this preprocessing is sub-dominant, and (iii) give examples of structured random PDEs (e.g., those admitting low-rank or sparse-in-measure representations) where oracle construction can be performed more efficiently. A complete, discretization-specific bound on preprocessing is beyond the scope of the present work and will be flagged as an open question for future investigation; hence the revision is partial. revision: partial

Circularity Check

0 steps flagged

No significant circularity in the quantum advantage claims for Young-measure LPs

full rationale

The paper starts from the standard measure-valued formulation of nonlinear PDEs (dissipative weak solutions via Young measures), which produces an LP with linear objective and linear constraints by direct transcription of the weak form plus moment/entropy conditions. It then composes this LP with externally established quantum linear programming algorithms (quantum central-path method and similar) whose polynomial complexity improvements are taken from the independent quantum-computing literature. The distinction between recovering only expected values (no quantum advantage) versus the full Young measure for random PDEs follows immediately from the structure of the output functional and the scaling of the discretized measure space; no fitted parameter is renamed as a prediction, no self-citation supplies a load-bearing uniqueness theorem, and no ansatz is smuggled in. The derivation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim relies on the domain assumption that nonlinear PDEs admit dissipative measure-valued solutions expressible as LPs, and on the complexity results of quantum LP algorithms from prior literature.

axioms (1)
  • domain assumption The measure-valued formulation of nonlinear PDEs can be cast as a linear programming problem with linear cost and constraints.
    This is the foundational step stated in the abstract.

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