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arxiv: 2508.13510 · v2 · submitted 2025-08-19 · 🪐 quant-ph

Schr\"odingerization for quantum linear systems problems with near-optimal dependence on matrix queries

Yin Yang , Yue Yu , Long Zhang This is my paper

Pith reviewed 2026-05-18 23:11 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum linear systemsSchrödingerizationHamiltonian simulationlinear combination of Hamiltonian simulationsblock preconditioningquantum algorithmscondition number
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The pith

The solution to Ax = b can be recovered as a linear combination of Hamiltonian simulations applied to Schrödingerized problems in one higher dimension.

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

The paper establishes a quantum algorithm for linear systems by showing that the solution vector can be expressed through the linear combination of Hamiltonian simulations of Schrödingerization-form problems. Schrödingerization converts the original non-unitary dynamics into unitary Schrödinger-type evolution by lifting the system into one extra dimension via a warped phase transformation. For positive definite matrices the solution appears as the steady state of an associated ODE system; for general Hermitian matrices a similar LCHS representation holds without applying the mapping directly to the least-squares formulation. Block preconditioning is then used to keep the effective condition number from growing too fast, producing an algorithm whose query complexity scales nearly linearly with the condition number.

Core claim

We demonstrate that in both the positive definite and general Hermitian cases, the solution x can be expressed as the LCHS of Schrödingerization-form problems. When A is positive definite the solution is the steady-state solution to a system of linear ODEs that is solved using the LCHS method. When A is general Hermitian the inverse is still represented in LCHS form, but Schrödingerization is applied to a carefully chosen formulation rather than the least-squares system itself so that the condition number does not increase prohibitively.

What carries the argument

The Schrödingerization mapping, which lifts a linear ODE or PDE with non-unitary dynamics to a Schrödinger-type system in one higher dimension via warped phase transformation, allowing the solution to be recovered as a linear combination of Hamiltonian simulations.

If this is right

  • For positive definite A the steady-state solution of the associated ODE system is obtained directly via LCHS of the Schrödingerized problem.
  • For general Hermitian A the inverse is recovered from an LCHS form that uses a Fourier kernel but avoids direct Schrödingerization of the least-squares system.
  • Block preconditioning reduces the dependence on the condition number to nearly linear scaling.
  • The resulting query complexity is near-optimal in the number of matrix queries.

Where Pith is reading between the lines

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

  • The same lifting technique might be tested on mildly non-Hermitian systems if a suitable block preconditioner can be found.
  • Classical numerical analysts could implement the underlying continuous ODE formulation to check whether the error bounds hold outside the quantum setting.
  • The one-dimension-higher representation may suggest new classical preconditioners that exploit the extra variable introduced by the warped phase transformation.

Load-bearing premise

The LCHS representation of the inverse or pseudoinverse stays valid after the Schrödingerization mapping without a prohibitive growth in effective condition number.

What would settle it

Apply the algorithm to a small, explicitly solvable positive definite matrix and verify that the output vector matches the classical solution to within the error bound predicted by the analysis.

Figures

Figures reproduced from arXiv: 2508.13510 by Long Zhang, Yin Yang, Yue Yu.

Figure 1
Figure 1. Figure 1: Finite difference discretization for the Poisson equation in one dimension [PITH_FULL_IMAGE:figures/full_fig_p018_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Exact and numerical solutions for P1-Lagrange elements 2 3 4 5 6 7 8 log(1/h) 10-2 10-1 100 |u-uh |1 O (h0.94) ||u-uh || O (h1.9) (a) 2 4 6 8 10 12 14 16 log(1/h) 10-2 10-1 100 |u-uh |1 O (h0.56) ||u-uh || O (h0.29) (b) 2 4 6 8 10 12 14 16 log(1/h) 10-3 10-2 10-1 100 |u-uh |1 O (h0.96) ||u-uh || O (h1.9) (c) [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Convergence rates for P1-Lagrange elements in the L 2 and H1 norms We consider the discretization using the Virtual Element Method, which generalizes the stan￾dard finite element method to accommodate arbitrary polytopal meshes. For the detailed con￾struction of the numerical method, we refer the reader to [24]. The exact and numerical solutions for the linear virtual element method with 32 polygons are sh… view at source ↗
Figure 4
Figure 4. Figure 4: Exact and numerical solutions for linear virtual element method [PITH_FULL_IMAGE:figures/full_fig_p021_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Exact and numerical solutions for the radiative transfer equation [PITH_FULL_IMAGE:figures/full_fig_p022_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Convergence rates for P1-Lagrange elements for the preconditioned system For the P2-Lagrange elements, the multilevel methods still use linear finite elements as the coarse spaces. This results in the following nested finite element spaces: V0 ⊂ V1 ⊂ · · · ⊂ VJ ⊂ Vh, where V0, · · · , VJ correspond to P1-Lagrange elements, and Vh represents the space for higher-order elements. It is important to note that … view at source ↗
Figure 7
Figure 7. Figure 7: Exact and numerical solutions for P2-Lagrange elements 2 2.5 3 3.5 4 log(1/h) 10-3 10-2 10-1 |u-uh |1 O (h1.4) ||u-uh || O (h1.9) [PITH_FULL_IMAGE:figures/full_fig_p024_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Convergence rates for P2-Lagrange elements -20 -15 -10 -5 0 5 10 15 0 5 10 15 20 25 30 (a) Np = 211 -20 -15 -10 -5 0 5 10 15 0 5 10 15 20 25 30 (b) Np = 213 -20 -15 -10 -5 0 5 10 15 0 5 10 15 20 25 30 (c) Np = 215 [PITH_FULL_IMAGE:figures/full_fig_p024_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Errors of ∥u − e |p|w∥ for different values of p (see [19]) Fig. 9a for the first mesh. It is evident that the solution can be recovered theoretically for p ≥ 1/2. However, when p is relatively large, the accuracy may degrade, as shown in [PITH_FULL_IMAGE:figures/full_fig_p024_9.png] view at source ↗
read the original abstract

We develop a quantum algorithm for linear algebraic equations $ A\bb{x} = \bb{b} $ from the perspective of Schr\"odingerization-form problems, which are characterized by a system of linear convection equations in one higher dimension. When $ A $ is positive definite, the solution $ \bb{x} $ can be interpreted as the steady-state solution to a system of linear ordinary differential equations (ODEs). This ODE system can be solved by using the linear combination of Hamiltonian simulation (LCHS) method in \cite{ACL2023LCH2}, which serves as the continuous implementation of the Fourier transform in the Schr\"odingerization method from \cite{JLY22SchrShort, JLY22SchrLong}. Schr\"odingerization transforms linear partial differential equations (PDEs) and ODEs with non-unitary dynamics into Schr\"odinger-type systems via the so-called warped phase transformation that maps the equation into one higher dimension. When $ A $ is a general Hermitian matrix, the inverse matrix can still be represented in the LCHS form in \cite{ACL2023LCH2}, but with a kernel function based on the Fourier approach in \cite{Childs2017QLSA}. Although this LCHS form provides the steady-state solution to a system of linear ODEs associated with the least-squares equation, applying Schr\"odingerization to this least-squares system is not appropriate, as it results in a much larger condition number. We demonstrate that in both cases, the solution $ \bb{x} $ can be expressed as the LCHS of Schr\"odingerization-form problems. We provide a detailed implementation and error analysis. Furthermore, we incorporate a block preconditioning technique to achieve nearly linear scaling in the condition number, thereby attaining near-optimal query complexity.

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 develops a quantum algorithm for solving linear systems Ax=b by recasting the problem as Schrödingerization-form linear convection equations in one higher dimension. For positive-definite A the solution is obtained as the steady-state of an ODE system solved via LCHS (ACL2023LCH2); for general Hermitian A the inverse is represented via the Fourier-kernel LCHS of Childs2017QLSA, with Schrödingerization applied to the resulting ODE rather than directly to the least-squares system. Block preconditioning is introduced to obtain near-optimal query complexity with respect to the condition number κ.

Significance. If the error analysis and condition-number bounds hold, the work supplies a new route to quantum linear-system solvers that achieves near-optimal dependence on matrix queries while unifying Schrödingerization with existing LCHS techniques. The explicit use of warped-phase mappings and block preconditioning to control the effective Hamiltonian norm is a concrete technical contribution that could be extended to other non-unitary linear-algebra primitives.

major comments (2)
  1. [§4] §4 (general Hermitian case): the argument that Schrödingerization of the LCHS-form ODE avoids a prohibitive increase in effective condition number is stated but the explicit bound relating the warped-phase parameter, the Fourier-kernel width, and the original κ is not derived; without this bound it is unclear whether the total query complexity remains O(κ polylog(κ/ε)) after accounting for the block-encoding cost of the higher-dimensional Hamiltonian.
  2. [§5.2] §5.2 (block preconditioning): the claim that the preconditioner reduces the effective condition number to near-linear scaling is supported only by a high-level cost analysis; a concrete lemma bounding the norm of the preconditioned Schrödingerized operator (analogous to Eq. (12) for the positive-definite case) is missing, which is load-bearing for the near-optimal query-complexity statement.
minor comments (2)
  1. [§3] The abstract and §3 refer to “detailed implementation and error analysis” yet the main text contains only schematic diagrams; explicit pseudocode or circuit diagrams for the combined LCHS+Schrödingerization block would improve readability.
  2. Notation for the warped-phase variable and the Fourier kernel is introduced inconsistently between the positive-definite and Hermitian sections; a single table of symbols would eliminate ambiguity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comments point by point below, indicating the revisions we will make to strengthen the presentation and rigor of the results.

read point-by-point responses

  1. Referee: [§4] §4 (general Hermitian case): the argument that Schrödingerization of the LCHS-form ODE avoids a prohibitive increase in effective condition number is stated but the explicit bound relating the warped-phase parameter, the Fourier-kernel width, and the original κ is not derived; without this bound it is unclear whether the total query complexity remains O(κ polylog(κ/ε)) after accounting for the block-encoding cost of the higher-dimensional Hamiltonian.

    Authors: We thank the referee for this observation. The manuscript in §4 explains that Schrödingerization is applied to the LCHS-form ODE (rather than directly to the least-squares formulation) precisely to control the effective condition number, leveraging the warped-phase transformation and the Fourier-kernel representation from Childs et al. (2017). While this qualitative argument is provided along with the overall error analysis, we acknowledge that an explicit quantitative bound relating the warped-phase parameter, the Fourier-kernel width, and the original κ is not derived in detail. In the revised manuscript we will add this derivation as a new lemma (or subsection) within §4. The bound will be obtained by combining the norm estimates for the Schrödingerized operator with the properties of the LCHS kernel, confirming that the block-encoding cost of the higher-dimensional Hamiltonian does not inflate the complexity beyond O(κ polylog(κ/ε)). revision: yes

  2. Referee: [§5.2] §5.2 (block preconditioning): the claim that the preconditioner reduces the effective condition number to near-linear scaling is supported only by a high-level cost analysis; a concrete lemma bounding the norm of the preconditioned Schrödingerized operator (analogous to Eq. (12) for the positive-definite case) is missing, which is load-bearing for the near-optimal query-complexity statement.

    Authors: We appreciate the referee highlighting this gap. Section 5.2 introduces block preconditioning to achieve near-linear scaling in the condition number and supports the claim with a high-level cost analysis that combines the preconditioner with the Schrödingerized LCHS implementation. We agree that a concrete lemma explicitly bounding the norm of the preconditioned Schrödingerized operator (in direct analogy to Eq. (12) for the positive-definite case) is currently absent and is necessary to make the near-optimal query-complexity statement fully rigorous. In the revised version we will insert this lemma in §5.2, deriving the norm bound from the block-encoding properties of the preconditioner and the warped-phase mapping, thereby providing the missing load-bearing step. revision: yes

Circularity Check

0 steps flagged

No significant circularity; central claims build on independent external citations

full rationale

The paper's derivation expresses the solution x as the LCHS output of Schrödingerization-form problems for both positive-definite and general Hermitian cases, then adds block preconditioning for near-optimal query complexity. This rests on the LCHS representation from the cited ACL2023LCH2, the Schrödingerization mapping from JLY22SchrShort/JLY22SchrLong, and the Fourier kernel from Childs2017QLSA. These are external prior results with no indication that the new combination reduces by construction to a fit, self-definition, or self-citation chain within the present work. The abstract and text explicitly distinguish the approach (e.g., avoiding direct Schrödingerization of the least-squares system due to condition-number issues) and supply separate implementation and error analysis. No load-bearing step equates to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are explicitly introduced in the abstract; the work builds on existing LCHS and Schrödingerization frameworks.

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