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arxiv: 2510.06837 · v2 · submitted 2025-10-08 · 🪐 quant-ph · physics.comp-ph· physics.flu-dyn

A Quantum Linear Systems Pathway for Solving Differential Equations

Abhishek Setty This is my paper

Pith reviewed 2026-05-18 09:42 UTC · model grok-4.3

classification 🪐 quant-ph physics.comp-phphysics.flu-dyn
keywords quantum linear systemsblock encodingQSVTdifferential equationsheat equationBurgers equationquantum algorithmscomputational fluid dynamics
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The pith

Differential equations are solved by turning them into linear systems handled with block encoding plus QSVT on quantum hardware.

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

The paper develops a pathway to solve differential equations on quantum computers by first discretizing them into linear systems. These systems are then addressed using block encoding to make the matrices quantum-accessible, followed by Quantum Singular Value Transformation to extract the solutions. The method is shown on a tridiagonal linear system and then extended to the heat equation with mixed boundary conditions and to a Carleman-linearized form of the nonlinear Burgers equation. Scaling analysis identifies regimes where quantum approaches could compete with classical solvers, while circuit depth and post-selection probability estimates are given for implementation on IBM superconducting processors.

Core claim

A systematic pathway for solving differential equations within the quantum linear systems framework by combining block encoding with Quantum Singular Value Transformation (QSVT), demonstrated on a complex tridiagonal linear system and extended to the heat equation with mixed boundary conditions and Carleman-linearized nonlinear Burgers' equation, together with scaling analysis and hardware resource estimates.

What carries the argument

Block encoding combined with Quantum Singular Value Transformation (QSVT) applied to linear systems obtained from discretized differential equations.

Load-bearing premise

Differential equations can be converted into linear systems whose block encodings and QSVT circuits have depths and post-selection probabilities practical for near-term or future quantum hardware without prohibitive overhead from discretization or linearization.

What would settle it

Implementing the block encoding and QSVT circuits on IBM hardware for the presented heat or Burgers problems and finding two-qubit gate depths or post-selection probabilities that render the computation infeasible.

Figures

Figures reproduced from arXiv: 2510.06837 by Abhishek Setty.

Figure 1
Figure 1. Figure 1: Quantum linear systems pathway for solving differential equations using QSVT. Here [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Quantum circuit for solving the linear system [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Solution of the complex tridiagonal linear system Section 4.1. a Circuit for block encoding [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Solution of the heat equation Section 4.2. a Circuit for block encoding [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: a Matrix representation of L† for solving Burgers’ equation. Colored sub-blocks highlight distinct matrix components, while the uncolored block corresponds to A2 1 = 0 in Eq. (F.5). Data items are labeled [ai] 13 i=0. Continuous colored lines indicate repeating ai values along diagonals with fixed offsets, and black square dots mark single, non-repeating entries. b Initial condition at t = 0, and compariso… view at source ↗
read the original abstract

We present a systematic pathway for solving differential equations within the quantum linear systems framework by combining block encoding with Quantum Singular Value Transformation (QSVT). The approach is demonstrated on a complex tridiagonal linear system and extended to problems in computational fluid dynamics: the heat equation with mixed boundary conditions and Carleman-linearized nonlinear Burgers' equation. Our scaling analysis of the heat equation identifies regimes where classical computation remains feasible and estimates circuit depths required to achieve potential quantum advantage. We further evaluate post-selection success probabilities for the presented examples and provide hardware resource estimates for block encoding and QSVT circuits in terms of two-qubit gate depth, evaluated on IBM superconducting processors with heavy-hex and square lattice topologies. These results highlight both the practical limitations of current hardware and key directions for depth reduction and scalable quantum linear solvers.

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 / 2 minor

Summary. The manuscript presents a systematic pathway for solving differential equations within the quantum linear systems framework by combining block encoding with Quantum Singular Value Transformation (QSVT). It demonstrates the approach on a complex tridiagonal linear system and extends it to computational fluid dynamics problems: the heat equation with mixed boundary conditions and the Carleman-linearized nonlinear Burgers' equation. Scaling analysis for the heat equation identifies classical feasibility regimes and estimates circuit depths for potential quantum advantage; post-selection success probabilities and hardware resource estimates (two-qubit gate depths on IBM heavy-hex and square lattices) are also provided.

Significance. If the central claims hold, the work offers a concrete bridge from quantum linear solvers to CFD applications, with explicit hardware estimates that clarify near-term limitations and depth-reduction needs. The extension to nonlinear problems via Carleman linearization and the reproducible resource calculations for specific topologies are notable strengths that could guide future implementations.

major comments (2)
  1. [Carleman linearization for Burgers' equation] Carleman linearization section for Burgers' equation: the scaling analysis and resource estimates focus on circuit depth and post-selection probabilities but provide no explicit error bounds on the truncation order N as a function of Reynolds number or grid size. Without these, the approximation error from linearization remains uncontrolled and is load-bearing for the claim that the truncated system faithfully represents the original nonlinear dynamics without erasing potential quantum advantage.
  2. [Scaling analysis of the heat equation] Heat equation scaling analysis: the identification of regimes where classical computation remains feasible and the circuit-depth estimates for quantum advantage depend on unstated modeling choices for discretization and mixed boundary conditions; the manuscript should derive or cite how these affect block-encoding efficiency and overall system size.
minor comments (2)
  1. Notation for QSVT parameters and block-encoding operators could be introduced more explicitly in the methods section to improve readability for readers unfamiliar with the prior quantum linear systems literature.
  2. Figure captions for circuit diagrams and resource tables should include explicit two-qubit gate counts and post-selection probabilities to allow direct comparison with the text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive summary and constructive major comments. We address each point below and will revise the manuscript accordingly to strengthen the rigor and clarity of the presentation.

read point-by-point responses

  1. Referee: Carleman linearization section for Burgers' equation: the scaling analysis and resource estimates focus on circuit depth and post-selection probabilities but provide no explicit error bounds on the truncation order N as a function of Reynolds number or grid size. Without these, the approximation error from linearization remains uncontrolled and is load-bearing for the claim that the truncated system faithfully represents the original nonlinear dynamics without erasing potential quantum advantage.

    Authors: We agree that explicit error bounds on the truncation order N would improve the manuscript. Our current demonstrations use a fixed truncation order chosen to keep the approximation error small for the presented Reynolds numbers and grid sizes. In the revised version we will add a dedicated subsection deriving or citing bounds on the Carleman truncation error (drawing from existing analyses in the literature) as a function of Reynolds number and grid resolution, thereby controlling the approximation error and clarifying the regimes in which the quantum advantage claims remain valid. revision: yes

  2. Referee: Heat equation scaling analysis: the identification of regimes where classical computation remains feasible and the circuit-depth estimates for quantum advantage depend on unstated modeling choices for discretization and mixed boundary conditions; the manuscript should derive or cite how these affect block-encoding efficiency and overall system size.

    Authors: We acknowledge that the scaling analysis would benefit from greater explicitness. The manuscript already specifies a second-order central finite-difference discretization on a uniform grid with mixed (Dirichlet-Neumann) boundary conditions, which produces a tridiagonal system whose sparsity pattern directly determines the block-encoding cost. In the revision we will expand this section to derive the resulting matrix dimension and block-encoding query complexity explicitly from the discretization parameters and boundary conditions, and we will cite standard references on finite-difference schemes for the heat equation to make these modeling choices transparent. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation applies standard QSVT and block encoding to DEs

full rationale

The paper constructs a pathway by combining block encoding with QSVT to solve linear systems arising from differential equations, including demonstrations on tridiagonal matrices, the heat equation, and Carleman-linearized Burgers' equation. Scaling analysis and resource estimates (circuit depth, post-selection probabilities) are derived from the problem discretization and hardware models rather than being tautological to any fitted inputs or self-citations. No load-bearing step reduces by construction to a prior result from the same authors, a renamed empirical pattern, or a parameter fit presented as a prediction. The approach remains self-contained against external benchmarks in quantum linear systems literature, with the central claims resting on explicit circuit constructions and evaluations rather than definitional equivalence.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The approach relies on standard assumptions from quantum computing and numerical methods for linear systems; no new free parameters or invented entities are introduced in the abstract description.

axioms (2)
  • domain assumption Block encoding of the linear system matrices can be performed with polynomial overhead in system size.
    Implicit in the use of block encoding for quantum linear systems solvers applied to discretized DEs.
  • standard math QSVT can be applied to implement the required matrix functions with controlled error.
    Relies on established properties of Quantum Singular Value Transformation from prior literature.

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    RX (θ) RZ(−2ϕ ′ d) H (a) 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 x 1/x True QSP 1/κ (b) Figure A.6: a Circuit for QSP in theW X convention Eq. (A.3), withθ=−2 cos −1 x. The central dots indicate continuation of the pattern. b Approximation of the inverse function using QSP overx∈[0,1]. For demonstration, de- sired inverse values are truncated. The best approximat...

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