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arxiv: 2606.00368 · v1 · pith:GZLNMYCVnew · submitted 2026-05-29 · 🪐 quant-ph

Quantum Signal Processing for Linear PDEs: Circuit Design and Experimental Validation

Hyeongjin Kim , Revathi Jambunathan , Jan Balewski , Daan Camps This is my paper

Pith reviewed 2026-06-28 21:44 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum signal processinglinear PDEsquantum Fourier transformadvection equationwave equationPoisson equationquantum circuitshardware experiments
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The pith

Quantum signal processing augments frequency-domain circuits to solve advection, wave, and Poisson equations accurately on noisy IBMQ hardware.

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

The paper constructs end-to-end quantum circuits that solve linear PDEs by propagating dynamics in frequency space with the quantum Fourier transform. It contrasts compact first-order approximations, which carry uncontrollable error, against deeper circuits that use quantum signal processing polynomials to achieve tunable algorithmic error. Numerical simulations and hardware runs on IBMQ devices are used to benchmark the advection, wave, and Poisson equations, showing that the QSP version yields accurate results under realistic noise. The method is further extended to non-homogeneous Dirichlet boundaries and tested numerically on a Poisson problem whose source comes from plasma simulations.

Core claim

End-to-end quantum circuits compiled to machine instructions can propagate linear PDE dynamics in frequency space via the quantum Fourier transform; when the required operators are approximated by quantum signal processing polynomials rather than first-order methods, the resulting solutions remain accurate on current hardware despite noise, as demonstrated for the advection, wave, and Poisson equations and extended to non-homogeneous boundary conditions.

What carries the argument

Quantum signal processing (QSP) polynomials that approximate the frequency-domain evolution operators, used together with the quantum Fourier transform to implement the PDE dynamics.

If this is right

  • Tunable algorithmic error is available for the advection, wave, and Poisson equations at the expense of deeper circuits.
  • Accurate solutions are obtained on real hardware for these three equations under the tested discretizations.
  • The same circuit framework extends to non-homogeneous Dirichlet conditions.
  • The approach accepts source terms taken directly from high-fidelity classical plasma simulations.

Where Pith is reading between the lines

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

  • Similar QSP-augmented frequency-domain circuits could be applied to other linear evolution equations whose spatial operators become multiplication after a Fourier transform.
  • If hardware noise continues to decrease, the deeper QSP circuits may become preferable to first-order approximations for problems requiring high accuracy.
  • Hybrid quantum-classical workflows could use the quantum circuit only for the linear solve step while handling nonlinear terms or boundary updates classically.

Load-bearing premise

The first-order and QSP approximations, when paired with the quantum Fourier transform, keep total error controllable for the chosen spatial discretizations and boundary conditions of the advection, wave, and Poisson equations.

What would settle it

Executing the compiled QSP circuits on IBMQ hardware for the Poisson equation with the plasma-derived source and finding that the measured solution deviates from the reference by more than the sum of algorithmic and hardware-error bounds would falsify the claim of controllable accuracy.

read the original abstract

Quantum algorithms offer new avenues for solving partial differential equations (PDEs). While the potential for end-to-end quantum advantage is at present not well understood, recent literature presents explicit circuit constructions for solving certain classes of linear PDEs in the frequency domain and thus offers concrete examples to study. In this work, we develop end-to-end implementations of these quantum circuits compiled to machine-level instructions and benchmark them in both numerical simulations and IBMQ hardware experiments. We focus on the advection, wave, and Poisson equations and study quantum circuits that propagate the dynamics in frequency space via the quantum Fourier transform using approximate methods based on a first-order approximation which offer compact representations with uncontrollable approximation error, and polynomial approximation methods based on quantum signal processing (QSP) leading to deeper circuits with tunable algorithmic error. In addition, we experimentally demonstrate that the QSP-augmented algorithm can provide accurate solutions under realistic hardware constraints. Finally, we extend our method to address non-homogeneous Dirichlet boundary conditions and verify it numerically for a Poisson equation with source term obtained from high-fidelity physics simulations of a capacitively coupled plasma.

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 / 2 minor

Summary. The manuscript develops end-to-end quantum circuit implementations for solving the advection, wave, and Poisson equations in the frequency domain via the quantum Fourier transform. It contrasts compact first-order approximations (uncontrollable error) with deeper QSP-based polynomial approximations (tunable error), benchmarks both via numerical simulations and IBMQ hardware runs, claims that the QSP version yields accurate solutions under realistic hardware noise, and numerically extends the approach to non-homogeneous Dirichlet boundary conditions using a source term drawn from plasma simulations.

Significance. If the hardware results are robust, the work supplies concrete circuit-level evidence that QSP can keep total error controllable on current superconducting hardware for these linear PDEs, moving beyond purely theoretical constructions and offering practical guidance on depth-versus-accuracy trade-offs for quantum scientific computing.

major comments (1)
  1. [Hardware experiments] Hardware validation section: the central experimental claim that QSP-augmented circuits produce accurate solutions under realistic IBMQ constraints is load-bearing for the paper's contribution, yet the abstract provides no quantitative scaling of observed error versus QSP degree, no error-bar details, and no statement of data-exclusion rules or mitigation techniques; without these, it is impossible to confirm that algorithmic error remains dominant over hardware noise as circuit depth grows.
minor comments (2)
  1. [Abstract] The abstract states that first-order methods have 'uncontrollable approximation error' while QSP methods have 'tunable algorithmic error'; a brief comparison table or plot of error versus degree for both families would clarify the distinction for readers.
  2. [Boundary conditions extension] The extension to non-homogeneous Dirichlet conditions is verified only numerically; if space permits, a short remark on why hardware validation was not attempted for this case would help scope the claims.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive feedback. We address the single major comment below and will revise the manuscript to incorporate the requested details.

read point-by-point responses

  1. Referee: [Hardware experiments] Hardware validation section: the central experimental claim that QSP-augmented circuits produce accurate solutions under realistic IBMQ constraints is load-bearing for the paper's contribution, yet the abstract provides no quantitative scaling of observed error versus QSP degree, no error-bar details, and no statement of data-exclusion rules or mitigation techniques; without these, it is impossible to confirm that algorithmic error remains dominant over hardware noise as circuit depth grows.

    Authors: We agree that the abstract and hardware section would benefit from these quantitative details to make the experimental claims fully verifiable. In the revised manuscript we will (i) update the abstract to include a concise statement of the observed error scaling versus QSP degree on hardware, (ii) add error bars to all hardware-result figures together with a description of how they were computed, (iii) explicitly state the data-exclusion rules (e.g., discarding shots whose readout fidelity fell below a calibrated threshold), and (iv) document the mitigation techniques applied (readout-error mitigation via calibration matrices and, where used, dynamical decoupling). These additions will allow readers to confirm that algorithmic error remains dominant over hardware noise for the depths reported. revision: yes

Circularity Check

0 steps flagged

No circularity; experimental validation is independent of derivation inputs

full rationale

The paper constructs circuits for advection, wave, and Poisson equations via QFT combined with either first-order approximation or QSP polynomial methods, then benchmarks them via numerical simulation and IBMQ hardware runs. The load-bearing claim is the experimental demonstration that QSP-augmented circuits yield controllable error under realistic noise, which is established by direct execution on external hardware rather than by any equation that reduces a reported accuracy metric to a parameter fitted inside the same derivation. No self-definitional steps, fitted-input predictions, or load-bearing self-citations appear; the cited QSP technique is standard external literature. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on standard quantum-circuit compilation assumptions and IBMQ noise characteristics; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Quantum Fourier transform and QSP phase functions can be compiled to native gates with controllable total error on current hardware
    Invoked when claiming that the QSP-augmented circuits remain accurate under realistic constraints.

pith-pipeline@v0.9.1-grok · 5727 in / 1279 out tokens · 29922 ms · 2026-06-28T21:44:56.398342+00:00 · methodology

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Reference graph

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