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arxiv: 2606.28752 · v1 · pith:QTOUUWQSnew · submitted 2026-06-27 · 🧮 math.NA · cs.NA

Schr\"odingerization based quantum algorithms for regularized Wasserstein proximal operators

Shi Jin , Nana Liu , Yue Yu This is my paper

Pith reviewed 2026-06-30 09:12 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords quantum algorithmsWasserstein proximal operatorSchrödingerizationCole-Hopf transformationoptimal transportFokker-Planck equationHamilton-Jacobi equationmean-field games
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The pith

Quantum algorithm for regularized Wasserstein proximal operators achieves O(d N_x T log²(1/ε)) query complexity via Schrödingerization.

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

The paper develops a quantum algorithm for the regularized Wasserstein proximal operator by first adding a diffusive regularization to the Benamou-Brenier continuity equation, producing a coupled Fokker-Planck and viscous Hamilton-Jacobi system. The Cole-Hopf transformation converts both equations into forward heat equations, after which Schrödingerization solves the heat equations while Hadamard division and product steps are performed through matrix-vector multiplications. The resulting procedure prepares an ε-approximation to the terminal density whose query cost scales linearly with dimension d and grid points N_x per dimension, in contrast to classical grid methods whose cost per step scales as N_x^d. A reader would care because this linear scaling suggests quantum computers could address high-dimensional optimal transport and mean-field game problems that are currently intractable classically.

Core claim

By applying the Cole-Hopf transformation to the regularized forward-backward PDE system, both the Fokker-Planck and viscous Hamilton-Jacobi equations become forward heat equations. Schrödingerization solves these heat equations, while the required Hadamard division for initial data and Hadamard product for terminal density recovery are realized as matrix-vector multiplications. The complete quantum algorithm therefore prepares an ε-approximation of the terminal density state with O(d N_x T log²(1/ε)) query complexity, up to constants that depend on the potential and initial density but remain independent of d and N_x.

What carries the argument

Schrödingerization method applied to the pair of heat equations obtained from the Cole-Hopf transformation, together with matrix representations of the Hadamard division and product operations.

If this is right

  • The terminal density state can be approximated with only logarithmic dependence on the error tolerance ε.
  • The query cost grows linearly with spatial dimension d and with the number of grid points N_x per dimension.
  • The algorithm supplies an exponential advantage over classical methods whose per-step cost scales as N_x^d.
  • The approach directly applies to the proximal step arising in mean-field games and related variational problems.

Where Pith is reading between the lines

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

  • If the stated polylog overhead is achieved, the same transformation pipeline could be reused for other regularized transport problems whose governing equations admit Cole-Hopf linearization.
  • The matrix representations of Hadamard operations may combine with other quantum linear-system solvers to address broader classes of nonlinear PDEs.
  • Practical verification on small quantum devices would test whether the hidden constants remain modest when d and N_x are moderate.

Load-bearing premise

The Cole-Hopf transformation together with Schrödingerization and the matrix realizations of Hadamard division and product can be implemented with only polylogarithmic overhead in 1/ε whose constants stay independent of dimension d and grid size N_x.

What would settle it

A concrete calculation or experiment on a sequence of increasing d and N_x that shows the total number of queries or gates grows exponentially rather than linearly with d N_x would falsify the claimed complexity.

Figures

Figures reproduced from arXiv: 2606.28752 by Nana Liu, Shi Jin, Yue Yu.

Figure 1
Figure 1. Figure 1: Quantum circuit for the Schr¨odingerization approach in [ [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Numerical results for the regularized Wasserstein proximal problem in 1D. [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Numerical results for the regularized Wasserstein proximal problem in 2D. [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
read the original abstract

We develop a quantum algorithm for the regularized Wasserstein proximal operator, which is a fundamental tool in optimal transport and mean-field games. The regularization introduces a small diffusive term into the continuity equation of the Benamou-Brenier formulation, which results in a forward-backward PDE system consisting of a Fokker-Planck equation and a viscous Hamilton-Jacobi equation with a quadratic Hamiltonian. Through the Cole-Hopf transformation, both equations are converted to forward heat equations, whose coupling requires a Hadamard division to prepare the initial data for the second heat equation and a Hadamard product to recover the terminal density. We solve these heat equations via the Schr\"odingerization method and implement the Hadamard division and product operations using simple matrix-vector multiplication representations. The complete quantum algorithm prepares an $\varepsilon$-approximation of the terminal density state with $\mathcal{O}(d N_x T \log^2(1/\varepsilon))$ query complexity, up to constants depending on the potential and initial density, where $d$ is the spatial dimension, $N_x$ is the number of grid points per spatial dimension and $T$ is the evolution time. The complexity depends only {\it linearly} on $d N_x$, yielding an {\it exponential} speedup over classical methods, whose cost scales as $N_x^d$ per time step. Numerical experiments validate the effectiveness of the proposed algorithm.

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

1 major / 0 minor

Summary. The paper develops a quantum algorithm for the regularized Wasserstein proximal operator arising in optimal transport and mean-field games. Starting from the Benamou-Brenier formulation with diffusive regularization, it obtains a forward-backward PDE system (Fokker-Planck and viscous Hamilton-Jacobi), applies the Cole-Hopf transformation to convert both equations to forward heat equations, handles their coupling via Hadamard division (to prepare initial data) and Hadamard product (to recover the terminal density), and solves the heat equations with the Schrödingerization method. The central claim is that the complete algorithm prepares an ε-approximation of the terminal density state with query complexity O(d N_x T log²(1/ε)), up to constants depending on the potential and initial density, yielding an exponential speedup over classical methods whose cost scales as N_x^d per time step.

Significance. If the stated complexity bound holds with the claimed polylog(1/ε) overhead and d-, N_x-independent constants, the result would constitute a meaningful contribution to quantum algorithms for high-dimensional optimal transport, directly addressing the curse of dimensionality that limits classical grid-based methods. The explicit use of Schrödingerization to obtain linear scaling in d N_x, together with the concrete matrix-vector realizations of the Hadamard operations, is a technical strength worth crediting.

major comments (1)
  1. [Abstract] Abstract (complexity paragraph): The O(d N_x T log²(1/ε)) query complexity is load-bearing for the exponential-speedup claim, yet the manuscript provides no explicit matrix constructions or norm/condition-number bounds for the Hadamard-division and Hadamard-product operators. Without these bounds it is impossible to verify that their quantum simulation cost is absorbed into the stated polylog(1/ε) factor and remains independent of d and N_x, as required by the weakest assumption underlying the linear scaling.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need for additional detail on the Hadamard operations to support the complexity claim. We address the comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract (complexity paragraph): The O(d N_x T log²(1/ε)) query complexity is load-bearing for the exponential-speedup claim, yet the manuscript provides no explicit matrix constructions or norm/condition-number bounds for the Hadamard-division and Hadamard-product operators. Without these bounds it is impossible to verify that their quantum simulation cost is absorbed into the stated polylog(1/ε) factor and remains independent of d and N_x, as required by the weakest assumption underlying the linear scaling.

    Authors: We agree that explicit matrix constructions together with norm and condition-number bounds are required to rigorously confirm that the Hadamard operations contribute only a polylog(1/ε) factor independent of d and N_x. The manuscript states that these operations are realized via simple matrix-vector multiplications, but does not supply the concrete matrices or the accompanying bounds. In the revised version we will add a dedicated subsection that (i) gives the explicit matrix representations for both the Hadamard division (used to prepare initial data) and the Hadamard product (used to recover the terminal density), and (ii) proves that the operator norms and condition numbers of these matrices are bounded by quantities depending only on the potential and the initial density, hence independent of d and N_x. With these additions the simulation cost of the Hadamard steps is absorbed into the stated O(log²(1/ε)) overhead, preserving the overall O(d N_x T log²(1/ε)) query complexity. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation rests on external Schrödingerization and standard transformations

full rationale

The paper converts the regularized proximal operator PDEs via the known Cole-Hopf transformation to coupled heat equations, solves them with the Schrödingerization method (treated as an external technique), and realizes Hadamard division/product via matrix-vector multiplications. The O(d N_x T log²(1/ε)) query complexity is stated to follow from these steps with polylog(1/ε) overhead and constants independent of d and N_x in the regime of interest. No self-definitional loop, fitted parameter renamed as prediction, or load-bearing self-citation chain appears in the abstract or described chain; the central claim retains independent content from the cited external method and does not reduce to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the Cole-Hopf transform converting the Fokker-Planck/viscous HJ system into coupled heat equations and on Schrödingerization delivering the stated query complexity for those equations; both are treated as established background.

axioms (2)
  • domain assumption Cole-Hopf transformation converts the regularized continuity and Hamilton-Jacobi system into forward heat equations whose coupling is handled by Hadamard division and product.
    Invoked in the abstract to reduce the problem to heat equations.
  • domain assumption Schrödingerization solves the resulting heat equations with O(N_x T log²(1/ε)) queries per dimension.
    Basis for the overall complexity claim.

pith-pipeline@v0.9.1-grok · 5784 in / 1395 out tokens · 44974 ms · 2026-06-30T09:12:27.883237+00:00 · methodology

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Reference graph

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