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arxiv: 2508.21390 · v2 · submitted 2025-08-29 · 🧮 math.NA · cs.NA· quant-ph

Generalized quantum singular value transformation with application in quantum conjugate gradient least squares algorithm

Yu-Qiu Liu , Hefeng Wang , Hua Xiang This is my paper

Pith reviewed 2026-05-18 20:59 UTC · model grok-4.3

classification 🧮 math.NA cs.NAquant-ph
keywords generalized quantum singular value transformationquantum signal processingblock encodingconjugate gradient least squareshybrid quantum algorithmmatrix functionsquantum linear algebra
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The pith

Generalized quantum singular value transformation relaxes parity restrictions on polynomials to handle functions of general matrices and enables a hybrid quantum CGLS algorithm.

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

The paper extends generalized quantum signal processing from unitary matrices to general matrices, producing a method called generalized quantum singular value transformation. This approach removes the parity limitations that apply to standard quantum singular value transformation when constructing block encodings of matrix functions. The authors then combine this extension with the link between generalized and standard matrix functions to build a classical-quantum hybrid version of the conjugate gradient least squares algorithm. A sympathetic reader would care because the method widens the set of matrix operations that quantum circuits can implement directly, opening routes to quantum-assisted solvers for linear least squares problems.

Core claim

The authors extend generalized quantum signal processing to general matrices and name the result generalized quantum singular value transformation. GQSVT implements block encodings of matrix functions while relaxing the parity requirements that constrain quantum singular value transformation. By using the relationship between generalized matrix functions and standard matrix functions, the paper constructs a classical-quantum hybrid quantum conjugate gradient least squares algorithm that employs this GQSVT construction.

What carries the argument

Generalized quantum singular value transformation (GQSVT), which carries the argument by generalizing signal processing techniques to produce block encodings of matrix functions on arbitrary matrices without parity constraints on the target polynomials.

If this is right

  • Quantum circuits can now implement a larger family of matrix functions without separate parity adjustments.
  • Hybrid quantum-classical solvers become feasible for least squares problems by interleaving classical conjugate gradient steps with GQSVT blocks.
  • Block-encoding constructions gain flexibility for non-unitary operators that arise in linear algebra applications.
  • The same GQSVT technique can be reused for other iterative methods that rely on matrix function evaluations.

Where Pith is reading between the lines

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

  • Similar hybrid constructions could be explored for other Krylov subspace methods beyond CGLS.
  • Numerical benchmarks on small matrices would reveal the practical overhead of the quantum blocks relative to fully classical implementations.
  • Extending the method to noisy intermediate-scale quantum devices would test whether the relaxed parity rules reduce circuit depth enough to matter in practice.

Load-bearing premise

The relationship between generalized matrix functions and standard matrix functions can be used directly to build a working hybrid CGLS algorithm from a GQSVT block encoding.

What would settle it

A concrete counterexample in which the hybrid CGLS procedure based on GQSVT produces incorrect solutions or fails to converge on a small, well-conditioned least squares problem that classical CGLS solves accurately.

Figures

Figures reproduced from arXiv: 2508.21390 by Hefeng Wang, Hua Xiang, Yu-Qiu Liu.

Figure 1
Figure 1. Figure 1: Quantum circuit for GQSP, where the angles [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The GQSVT circuit for a polynomial of degree [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: (a) the detailed circuit of operator M, (b) an example for CI−Π˜ NOT when a = 3. Denote the circuit in dashed box in [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Circuit of the swap test for block encodings [ [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
read the original abstract

Quantum signal processing (QSP) and generalized quantum signal processing (GQSP) are essential tools for implementing the block encoding of matrix functions. The achievable polynomials of QSP have restrictions on parity, while GQSP eliminates these restrictions. But GQSP only constructs functions of unitary matrices. In this paper, we further investigate GQSP and extend it to general matrices. Compared with the quantum singular value transformation (QSVT), our proposed method relaxes the requirements on the parity of polynomials. We refer to this extension as generalized quantum singular value transformation (GQSVT). Subsequently, by utilizing the relationship between generalized matrix functions and standard matrix functions, we propose a classical-quantum hybrid quantum conjugate gradient least squares (CGLS) algorithm using GQSVT.

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

3 major / 1 minor

Summary. The paper extends generalized quantum signal processing (GQSP) to a generalized quantum singular value transformation (GQSVT) applicable to general matrices via block encodings, relaxing the parity restrictions on polynomials that apply to standard QSVT. It then invokes a relationship between generalized and standard matrix functions to construct a classical-quantum hybrid conjugate gradient least squares (CGLS) algorithm.

Significance. If the GQSVT construction and its error-controlled integration into iterative solvers can be made rigorous, the work would provide a useful relaxation of polynomial constraints in quantum linear algebra primitives, potentially improving flexibility for hybrid algorithms targeting least-squares problems. The hybrid CGLS proposal could bridge classical iteration with quantum block encodings, but only if query complexity and convergence under approximation error are established.

major comments (3)
  1. [Section 3] Section 3 (GQSVT definition and construction): the extension from GQSP on unitaries to singular-value transformation on general matrices is asserted via block encodings, but no explicit theorem or derivation shows how the parity relaxation is achieved while preserving the singular-value mapping; the relationship to standard matrix functions is invoked without a supporting lemma or error analysis.
  2. [Section 4] Section 4 (hybrid CGLS algorithm): the substitution of the GQSVT block encoding into the classical CGLS iteration (repeated applications of A and A^T) is proposed without an accumulated-error bound; it is not shown that the approximation error from each quantum call remains controlled across conjugate-gradient steps or that the classical updates remain stable under the block-encoding noise model.
  3. [Section 4] Section 4, final paragraph: the claim that the generalized-to-standard matrix-function relationship directly yields a working hybrid solver lacks a complexity analysis or convergence guarantee; no query-complexity or iteration-count bound is derived that accounts for the parity relaxation.
minor comments (1)
  1. [Throughout] Notation for block encodings and polynomial degrees is introduced without a consolidated table or consistent symbol list, making cross-references between the GQSVT construction and the CGLS application difficult to follow.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed comments, which help clarify the presentation of our results. We address each major comment point by point below, indicating where revisions will be incorporated to strengthen the rigor of the GQSVT construction and the hybrid CGLS analysis.

read point-by-point responses

  1. Referee: [Section 3] Section 3 (GQSVT definition and construction): the extension from GQSP on unitaries to singular-value transformation on general matrices is asserted via block encodings, but no explicit theorem or derivation shows how the parity relaxation is achieved while preserving the singular-value mapping; the relationship to standard matrix functions is invoked without a supporting lemma or error analysis.

    Authors: The construction in Section 3 applies GQSP directly to a block encoding of the target matrix, inheriting the parity relaxation from the GQSP framework while the singular-value mapping is preserved by the properties of the block encoding and the implicit SVD. The relationship to standard matrix functions is used to connect the generalized and conventional cases. We agree that the current exposition would benefit from greater formality. In the revised manuscript we will insert an explicit theorem stating the GQSVT mapping together with a supporting lemma that quantifies the approximation error under the block-encoding model. revision: yes

  2. Referee: [Section 4] Section 4 (hybrid CGLS algorithm): the substitution of the GQSVT block encoding into the classical CGLS iteration (repeated applications of A and A^T) is proposed without an accumulated-error bound; it is not shown that the approximation error from each quantum call remains controlled across conjugate-gradient steps or that the classical updates remain stable under the block-encoding noise model.

    Authors: The manuscript introduces the hybrid iteration but does not yet supply a full accumulated-error analysis. We acknowledge this limitation. In the revision we will derive an error-propagation bound that tracks the approximation error introduced by each GQSVT call through the conjugate-gradient steps and demonstrate stability of the classical updates under the standard block-encoding noise model. revision: yes

  3. Referee: [Section 4] Section 4, final paragraph: the claim that the generalized-to-standard matrix-function relationship directly yields a working hybrid solver lacks a complexity analysis or convergence guarantee; no query-complexity or iteration-count bound is derived that accounts for the parity relaxation.

    Authors: The final paragraph invokes the generalized-to-standard relationship to justify applicability to CGLS. While the parity relaxation is expected to permit lower-degree polynomials and therefore fewer queries, the manuscript does not contain explicit query-complexity or convergence bounds that incorporate this relaxation. We will add a dedicated complexity subsection that derives query and iteration bounds accounting for the relaxed parity condition and the hybrid error model. revision: yes

Circularity Check

0 steps flagged

No circularity: GQSVT extension and hybrid CGLS are constructive applications of prior frameworks

full rationale

The paper starts from established QSP/GQSP/QSVT literature, relaxes the parity restriction on polynomials to handle non-unitary matrices via block encodings, and names the result GQSVT. It then invokes an external relationship between generalized and standard matrix functions to assemble a hybrid classical-quantum CGLS routine. No step reduces a claimed output to a fitted parameter, self-defined quantity, or self-citation chain that is itself unverified; the derivation remains forward and self-contained against the cited external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so no concrete free parameters, ad-hoc axioms, or invented entities can be extracted. The work relies on standard quantum signal processing assumptions and the unstated relationship between generalized and standard matrix functions.

axioms (1)
  • domain assumption Block-encoding and quantum signal processing primitives from prior literature are assumed valid and composable.
    The paper builds directly on QSP and GQSP without re-deriving them.

pith-pipeline@v0.9.0 · 5659 in / 1163 out tokens · 57502 ms · 2026-05-18T20:59:51.984178+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    We refer to this extension as generalized quantum singular value transformation (GQSVT). Subsequently, by utilizing the relationship between generalized matrix functions and standard matrix functions, we propose a classical-quantum hybrid quantum conjugate gradient least squares (CGLS) algorithm using GQSVT.

  • IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    We further generalize GQSP by employing arbitrary SU(2) rotations and four classes of controlled unitary, and combined with qubitization and singular value decomposition, we realize the block encoding of indefinite parity polynomials of general operations.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Constrained Optimal Polynomials for Quantum Linear System Solvers

    math.NA 2026-04 unverdicted novelty 7.0

    Constrained Uniform Polynomial (CUP) and Constrained Adaptive Polynomial (CAP) solvers achieve lower error than standard QSVT and Chebyshev methods in noise-limited regimes by optimizing accuracy versus block-encoding...

Reference graph

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26 extracted references · 26 canonical work pages · cited by 1 Pith paper · 1 internal anchor

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