Quantum Prediction of Transport Dynamics in Discretized State Spaces

Felix Govaers

arxiv: 2604.24161 · v1 · submitted 2026-04-27 · 🪐 quant-ph · cs.IT· math.IT· stat.CO

Quantum Prediction of Transport Dynamics in Discretized State Spaces

Felix Govaers This is my paper

Pith reviewed 2026-05-08 04:24 UTC · model grok-4.3

classification 🪐 quant-ph cs.ITmath.ITstat.CO

keywords quantum algorithmFokker-Planck equationBayesian state estimationquantum Fourier transformWick rotationunitary propagationdiscretized state spaceprobability density

0 comments

The pith

A quantum algorithm encodes probability densities in qubit amplitudes and propagates the Fokker-Planck equation unitarily by handling drift exactly and approximating diffusion via a Wick-rotation surrogate.

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

The paper develops a gate-based quantum method for the prediction step of Bayesian state estimation on a discretized position-velocity grid. Probability densities are stored directly in quantum amplitudes, so the representable space grows exponentially with qubit count. Drift transport is realized exactly through quantum Fourier transforms and phase gates, while the nonlinear diffusion term is replaced by a linear unitary phase evolution derived from a Wick rotation. Numerical tests on example scenarios show the resulting distributions stay close to the exact classical Fokker-Planck solution. The approach therefore offers a route to high-dimensional filtering problems that would otherwise demand heavy classical tensor methods.

Core claim

The central claim is that the Fokker-Planck evolution on a finite grid can be realized as a fully unitary quantum circuit: the linear drift operator is applied exactly in amplitude space after a quantum Fourier transform, and the nonlinear diffusion is replaced by a dispersive phase evolution obtained through Wick rotation, producing a complete unitary propagator whose output matches the classical solution closely in tested cases.

What carries the argument

The unitary surrogate for the diffusion term, obtained by Wick rotation, which converts the nonlinear diffusion operator into a linear phase evolution in the Fourier domain while preserving unitarity for gate-based implementation.

If this is right

The number of representable states grows exponentially with the number of qubits, allowing compact storage of high-dimensional densities that would otherwise require complex classical tensor decompositions.
The drift component reproduces classical transport dynamics exactly.
The full propagation remains unitary and can be implemented efficiently with standard quantum gates.
Numerical agreement with the exact Fokker-Planck solution holds for the tested one- and two-dimensional scenarios.

Where Pith is reading between the lines

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

If the surrogate remains stable in higher dimensions, the same circuit structure could be applied to other linear or mildly nonlinear transport PDEs on quantum hardware.
The exponential scaling suggests the method could be combined with quantum amplitude estimation or measurement routines to produce full quantum Bayesian filters.
Hardware implementations would need to verify that circuit depth stays within coherence limits for the target grid sizes.

Load-bearing premise

The Wick-rotation surrogate for the nonlinear diffusion term stays accurate over multiple time steps and in higher-dimensional grids without accumulating errors that would invalidate the Bayesian predictions.

What would settle it

A side-by-side numerical run in three or more spatial dimensions over many time steps that compares the final probability distribution produced by the quantum circuit against the exact solution of the Fokker-Planck equation and measures whether the total variation distance exceeds a chosen threshold.

Figures

Figures reproduced from arXiv: 2604.24161 by Felix Govaers.

**Figure 1.** Figure 1: Prediction circuit for nx = 4 qubits for the position and nv = 4 qubits for the velocity register, respectively. B. Combined drift–diffusion propagation Combining the drift operator derived in Section IV with the unitary diffusion operator yields the full propagation scheme |ψ ′ ⟩ = Ux Uv |ψ⟩, (69) where Ux denotes the drift operator acting in the position register and Uv denotes the dispersive diffusion… view at source ↗

**Figure 2.** Figure 2: Marginal densities px(x) and pv(v) at the initialization (top), after the exact classical solution using the matrix exponential (middle) and the resulting marginal densities from the quantum circuit (bottom) for Scenario 1 (left) and 2 (right). 3) a forward QFT on the velocity register. The QFT on nv qubits requires O(n 2 v ) gates. The diagonal phase operator acts only on the velocity register and requir… view at source ↗

read the original abstract

We propose a gate-based quantum algorithm for the prediction step of Bayesian state estimation based on the Fokker-Planck equation on a discretized position-velocity state space. The probability density is encoded in the amplitudes of a quantum state, enabling a compact representation of high-dimensional distributions. Exploiting the circulant structure of finite-difference operators, the evolution is realized in the spectral domain using quantum Fourier transforms and phase rotations. A key result is that the drift component can be implemented exactly in amplitude space, leading to an accurate reproduction of the classical transport dynamics. In contrast, the diffusion term does not admit a linear representation in amplitude space due to the nonlinear relation between probability density and wave function. To enable a quantum implementation, we introduce a unitary surrogate based on a Wick rotation, transforming diffusion into a dispersive phase evolution. This yields a fully unitary propagation that can be implemented efficiently on a gate-based quantum computer. The proposed method is evaluated numerically for different scenarios and shows strong agreement with the exact solution of the Fokker-Planck equation. The approach demonstrates the potential of quantum computing for Bayesian state estimation, as the representable state space grows exponentially with the number of qubits. This allows the efficient representation and propagation of probability densities that would otherwise require complex tensor decompositions on classical hardware, making the method a promising candidate for high-dimensional filtering problems.

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 gives a quantum algorithm for Fokker-Planck prediction in Bayesian filtering that handles drift exactly via QFT on circulant operators but relies on an unanalyzed Wick-rotation surrogate for the nonlinear diffusion term.

read the letter

This paper puts forward a gate-based quantum algorithm for the prediction step in high-dimensional Bayesian state estimation. It encodes the probability density directly in quantum amplitudes on a discretized position-velocity grid and uses the quantum Fourier transform to implement the drift term exactly because the finite-difference operators are circulant. That exactness for transport is the clean part of the construction and avoids the usual discretization artifacts in that component alone. The exponential scaling of the representable state space with qubit count is the standard quantum advantage argument, and the abstract reports numerical checks that line up with the classical Fokker-Planck solution on the tested cases. Those are the two things worth noting up front. The diffusion term cannot be represented linearly in amplitude space, so the authors introduce a unitary surrogate obtained by Wick rotation that converts the Laplacian on the density into a dispersive phase on the wave function. This keeps the whole propagation unitary and therefore implementable on current hardware. The approach is new in its specific combination of amplitude encoding, circulant QFT for drift, and this particular surrogate for diffusion. The soft spot is exactly the one the stress-test flags: the surrogate is an enabling approximation rather than an exact rewriting, and the paper provides no quantitative breakdown of local error growth, accumulation over repeated timesteps, or behavior on larger grids. The abstract claims strong agreement, but without those metrics it is difficult to judge whether the method remains reliable for the multi-step filtering loops it targets. This is aimed at researchers working on quantum algorithms for stochastic PDEs or quantum-enhanced filtering in robotics and control. A reader already following quantum PDE solvers might pick up the circulant-QFT detail, but anyone needing production-level accuracy in long trajectories should wait for tighter error bounds. I would send it to peer review. The proposal is concrete enough that referees can check the surrogate's practical limits and ask for the missing analysis.

Referee Report

2 major / 2 minor

Summary. The paper proposes a gate-based quantum algorithm for the prediction step of Bayesian state estimation via the Fokker-Planck equation on a discretized position-velocity state space. Probability densities are encoded in quantum amplitudes; the drift term is realized exactly using quantum Fourier transforms on circulant finite-difference operators, while the nonlinear diffusion term is replaced by a unitary surrogate obtained via Wick rotation that converts it to dispersive phase evolution. The method is claimed to be fully unitary and efficient, with numerical evaluations showing strong agreement with the exact classical Fokker-Planck solution and exponential scaling advantages for high-dimensional problems.

Significance. If the Wick-rotation surrogate proves sufficiently accurate without prohibitive error accumulation, the work would offer a concrete route to quantum-assisted high-dimensional filtering by exploiting exponential state-space growth and exact drift implementation. The exact amplitude-space drift via QFT and circulant structure is a clear technical strength that avoids fitting parameters.

major comments (2)

[Abstract] Abstract: the assertion of 'strong agreement with the exact solution of the Fokker-Planck equation' is unsupported by any reported error metrics, discretization parameters, number of timesteps, or surrogate-error accumulation analysis, which directly undermines assessment of whether the central claim holds for repeated prediction steps.
[Method (diffusion surrogate)] Method description of diffusion term: the Wick-rotation unitary surrogate for the nonlinear term D∇²p is introduced explicitly as an approximation that 'enables a quantum implementation' without derivation of local truncation error, stability bounds, or multi-step accumulation analysis; this is load-bearing for the claim that the full propagation reproduces Fokker-Planck dynamics accurately enough for Bayesian use.

minor comments (2)

[Abstract] Abstract: the phrase 'different scenarios' is used without specifying the test cases, dimensions, or pointing to the corresponding figures or tables.
[Abstract] The abstract states that the representable state space 'grows exponentially with the number of qubits' but does not discuss the qubit overhead required for the velocity component or the encoding of the joint position-velocity density.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive comments on our manuscript. We appreciate the positive assessment of the exact drift implementation and the potential for high-dimensional filtering. We address each major comment below and describe the revisions we will make.

read point-by-point responses

Referee: [Abstract] Abstract: the assertion of 'strong agreement with the exact solution of the Fokker-Planck equation' is unsupported by any reported error metrics, discretization parameters, number of timesteps, or surrogate-error accumulation analysis, which directly undermines assessment of whether the central claim holds for repeated prediction steps.

Authors: We agree that the abstract claim would be more robust with explicit quantitative support. The numerical evaluations in the results section show visual agreement between the quantum propagation and the classical Fokker-Planck solution, but the abstract itself does not cite error metrics or parameters. In the revised manuscript we will update the abstract to reference specific quantitative measures (e.g., relative L2 and maximum absolute errors), the discretization parameters (grid sizes and qubit counts), the number of timesteps used in the reported simulations, and a brief statement on observed surrogate-error accumulation. These additions will allow readers to evaluate suitability for repeated prediction steps. revision: yes
Referee: [Method (diffusion surrogate)] Method description of diffusion term: the Wick-rotation unitary surrogate for the nonlinear term D∇²p is introduced explicitly as an approximation that 'enables a quantum implementation' without derivation of local truncation error, stability bounds, or multi-step accumulation analysis; this is load-bearing for the claim that the full propagation reproduces Fokker-Planck dynamics accurately enough for Bayesian use.

Authors: We concur that a more detailed error analysis of the diffusion surrogate is necessary. The manuscript presents the Wick-rotation construction as a unitary surrogate but does not derive its truncation properties or accumulation behavior. In the revised version we will expand the method section to include a derivation of the local truncation error, stability considerations for the resulting unitary operator, and numerical results quantifying error growth over multiple timesteps. This will clarify the regime in which the surrogate remains sufficiently accurate for Bayesian filtering applications. revision: yes

Circularity Check

0 steps flagged

No significant circularity; Wick surrogate introduced explicitly as approximation

full rationale

The paper presents a new algorithmic proposal for quantum implementation of the Fokker-Planck prediction step. Drift is realized exactly via circulant operators and QFT in amplitude space. Diffusion is handled by an explicitly introduced unitary surrogate via Wick rotation, described as a transformation to enable unitarity rather than an exact rewriting or derivation from the target nonlinear dynamics. Numerical agreement with the exact FP solution is reported as validation, not as a self-referential fit. No load-bearing self-citations, no fitted parameters renamed as predictions, and no ansatz smuggled via prior work appear in the derivation chain. The construction remains self-contained as an enabling approximation with external benchmarking.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the assumption that finite-difference operators are circulant (standard for uniform grids) and that a Wick-rotated unitary can serve as a usable surrogate for nonlinear diffusion. No free parameters or new physical entities are introduced.

axioms (2)

domain assumption Finite-difference operators on the discretized position-velocity grid are circulant.
Invoked to justify the use of quantum Fourier transforms for spectral implementation.
ad hoc to paper A unitary operator obtained by Wick rotation provides an acceptable approximation to the nonlinear diffusion term.
Introduced explicitly to obtain a fully unitary quantum circuit.

pith-pipeline@v0.9.0 · 5533 in / 1405 out tokens · 56147 ms · 2026-05-08T04:24:31.731579+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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Koch,Tracking and Sensor Data Fusion: Methodological Frame- work and Selected Applications

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B. Ristic, S. Arulampalam, and N. Gordon,Beyond the Kalman Filter: Particle Filters for Tracking Applications. Artech House, 2004

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F. Daum, L. Dai, J. Huang, and A. Noushin, “Bulletproofing Bayesian particle flow against stiffness,” inSignal Processing, Sensor/Information Fusion, and T arget Recognition XXXIII, I. Kadar, E. P . Blasch, and L. L. Grewe, Eds., vol. 13057, International Society for Optics and Photonics. SPIE, 2024, p. 1305708. [Online]. Available: https://doi.org/10.111...

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Tensor Decomposition based Multi Target Tracking in Cluttered Environments,

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Risken and H

H. Risken and H. Haken,The Fokker-Planck Equation: Methods of Solution and Applications Second Edition. Springer, 1989

work page 1989

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On a fast cpd-tensor operator for target tracking,

J. Gehlen and F. Govaers, “On a fast cpd-tensor operator for target tracking,” in2025 28th International Conference on Information Fusion (FUSION), 2025, pp. 1–8

work page 2025

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On a quantum realization of the bayesian filtering using the log-homotopy flow,

F. Govaers, “On a quantum realization of the bayesian filtering using the log-homotopy flow,” in2024 IEEE International Confer- ence on Multisensor Fusion and Integration for Intelligent Systems (MFI), 2024, pp. 1–6

work page 2024

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On anti-symmetry in multiple target tracking,

W. Koch, “On anti-symmetry in multiple target tracking,” in2018 21st International Conference on Information Fusion (FUSION), 2018, pp. 957–964

work page 2018

[13] [13]

A Quantum Algorithm for the Prediction Step of a Bayesian Recursion,

F. Govaers, “A Quantum Algorithm for the Prediction Step of a Bayesian Recursion,” inProceedings of the 24th International Confernece on Information Fusion, Venice, Italy, July 2024

work page 2024

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Lloyd, Universal quantum simulators

S. Lloyd, “Universal quantum simulators,”Science, vol. 273, no. 5278, pp. 1073–1078, 1996. [Online]. Available: https://www.science.org/doi/abs/10.1126/science.273.5278.1073

work page doi:10.1126/science.273.5278.1073 1996

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A quantum algorithm for the diffusion step of grid-based filter,

Y. Choe, C. G. Park, J. Dun ´ık, J. Krej ˇc´ı, J. Matou ˇsek, and M. Brandner, “A quantum algorithm for the diffusion step of grid-based filter,” 2026. [Online]. Available: https://arxiv.org/abs/2603.00741

work page arXiv 2026

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