Hermitian Matrix Function Synthesis without Block-Encoding

Anuradha Mahasinghe; Frederic Cadet; Jingbo Wang; Kaushika De Silva; Peter Chin; Xavier Cadet

arxiv: 2512.18249 · v2 · submitted 2025-12-20 · 🪐 quant-ph

Hermitian Matrix Function Synthesis without Block-Encoding

Anuradha Mahasinghe , Kaushika De Silva , Xavier Cadet , Peter Chin , Frederic Cadet , Jingbo Wang This is my paper

Pith reviewed 2026-05-16 20:20 UTC · model grok-4.3

classification 🪐 quant-ph

keywords Hermitian matricespolynomial functionsquantum signal processingblock-encodingquantum algorithmsGQSPHamiltonian simulation

0 comments

The pith

A method using generalized quantum signal processing implements arbitrary polynomials of Hermitian matrices without block-encoding.

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

The paper shows how to realize polynomial functions on Hermitian matrices by applying the generalized quantum signal processing framework directly. This avoids the need to block-encode the matrix and the associated ancillary qubit overheads and post-selection costs that plague linear combination of unitaries approaches. The resulting success probability remains stable regardless of the polynomial degree. A sympathetic reader would care because this could simplify quantum algorithms for Hamiltonian simulation, linear systems, and state preparation by reducing circuit complexity and resource demands. The approach derives closed-form expressions for symmetric polynomial expansions and uses linear combinations of GQSP circuits.

Core claim

By leveraging the Generalized Quantum Signal Processing (GQSP) framework, arbitrary polynomials of a Hermitian matrix can be implemented through linear combinations of circuits without block-encoding, yielding a stable, degree-independent success probability and closed-form expressions for symmetric expansions.

What carries the argument

Generalized Quantum Signal Processing (GQSP) applied directly to Hermitian matrices, combined with linear combinations of circuits to realize the polynomial transformation.

If this is right

Reduces resource overhead by eliminating block-encoding preparation and ancillary qubits.
Achieves stable success probability independent of polynomial degree.
Opens new pathways for quantum algorithm design in settings where Hermitian operators arise from symmetric combinations of unitaries.
Supports applications in Hamiltonian simulation, quantum linear system solving, and high-fidelity state preparation.
Derives closed-form expressions for symmetric polynomial expansions.

Where Pith is reading between the lines

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

May extend to other matrix functions beyond polynomials if GQSP generalizations exist.
Could simplify implementations in quantum machine learning kernels that rely on such functions.
Testable by comparing circuit depths and success rates against QSVT-based methods on small Hermitian matrices.
Implies potential for lower error rates in noisy intermediate-scale quantum devices due to reduced overhead.

Load-bearing premise

The generalized quantum signal processing framework can be applied directly to Hermitian matrices without introducing new instabilities or additional overheads.

What would settle it

An explicit calculation or simulation showing that the success probability decreases with increasing polynomial degree or that block-encoding is still required for stability.

Figures

Figures reproduced from arXiv: 2512.18249 by Anuradha Mahasinghe, Frederic Cadet, Jingbo Wang, Kaushika De Silva, Peter Chin, Xavier Cadet.

read the original abstract

Implementing polynomial functions of Hermitian matrices on quantum hardware is a foundational task in quantum computing, critical for accurate Hamiltonian simulation, quantum linear system solving, high-fidelity state preparation, machine learning kernels, and other advanced quantum algorithms. Existing state-of-the-art techniques, including Qubitization, Quantum Singular Value Transformation (QSVT), and Quantum Signal Processing (QSP), rely heavily on block-encoding the Hermitian matrix. These methods are often constrained by the complexity of preparing the block-encoded state, the overhead associated with the required ancillary qubits, or the challenging problem of angle synthesis for the polynomial's phase factors, which limits the achievable circuit depth and overall efficiency. In this work, we propose a novel and resource-efficient approach to implement arbitrary polynomials of a Hermitian matrix by leveraging the Generalized Quantum Signal Processing (GQSP) framework. Our method circumvents the need for block-encoding and avoids the compounding post-selection overheads characteristic of LCU-based constructions, achieving a stable, degree-independent success probability. We derive closed-form expressions for symmetric polynomial expansions and demonstrate how linear combinations of GQSP circuits can realize the desired transformation. This approach reduces resource overhead and opens new pathways for quantum algorithm design for functions of Hermitian matrices, particularly in settings where the Hermitian operator arises naturally from symmetric combinations of unitaries.

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 claims a GQSP route to Hermitian matrix polynomials that skips block-encoding and keeps success probability stable, but the linear-combination step for general cases looks likely to reintroduce degree-dependent post-selection.

read the letter

The main takeaway is that the authors use the Generalized Quantum Signal Processing framework to implement polynomial functions of Hermitian matrices without first block-encoding the operator. They give closed-form expressions for the symmetric-polynomial case and argue that linear combinations of GQSP circuits handle the general case while avoiding the usual post-selection blow-up from LCU constructions. That would matter for Hamiltonian simulation and quantum linear systems if the overhead really stays constant with degree. The symmetric closed forms are the clearest concrete piece; they look like a useful incremental step for problems where the target polynomial has that symmetry. The rest of the argument rests on the claim that the linear combination can be realized without new instabilities or degree-dependent success probability. The abstract does not show the explicit circuit for that combination step. If the combination is done by superposing over the individual GQSP circuits, the post-selection probability is bounded by the squared norm of the coefficient vector, which for a degree-d polynomial will generally change with d. Nothing in the provided text derives a normalization or embedding that keeps the probability flat. No derivations, proofs, or numerical checks appear, so it is impossible to tell whether they have a workaround or simply overlooked the scaling. The citation pattern correctly distinguishes the approach from standard Qubitization, QSVT, and QSP, but the novelty claim hinges on the missing construction details. This is worth a serious referee for people building quantum algorithms around matrix functions, especially those already using GQSP variants. The topic is relevant and the block-encoding avoidance is a reasonable target, but the central claim needs the combination step spelled out before it can be evaluated. I would send it to review rather than desk-reject.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes implementing arbitrary polynomials of Hermitian matrices via the Generalized Quantum Signal Processing (GQSP) framework without block-encoding. It derives closed-form expressions for symmetric polynomials and realizes general polynomials through linear combinations of GQSP circuits, claiming a stable success probability independent of polynomial degree and reduced overhead relative to LCU or standard QSP/QSVT methods.

Significance. If the constructions and probability claims hold, the approach could lower resource costs for Hamiltonian simulation, quantum linear systems, and state preparation by removing block-encoding preparation and ancillary overheads, enabling more efficient circuits in settings where Hermitian operators arise from symmetric unitary combinations.

major comments (2)

[Abstract and the section deriving the linear-combination step] The central claim of degree-independent success probability for linear combinations of GQSP circuits (Abstract) lacks an explicit construction or normalization argument. Preparing a superposition over the individual GQSP circuits to realize the linear combination would bound the post-selection probability by the squared norm of the coefficient vector; for a degree-d polynomial this norm generally varies with d, contradicting the degree-independent claim unless a specific embedding or rescaling is derived.
[Section presenting the closed-form expressions] The closed-form expressions for symmetric polynomial expansions (Abstract) are stated to exist but no explicit formulas, derivation, or verification that they map to the target polynomial without introducing new instabilities are supplied in the provided text; this is load-bearing for the claim that GQSP can be applied directly to Hermitian matrices.

minor comments (2)

[Methods or Results] Add a small-scale numerical example or circuit diagram illustrating the GQSP circuit for a low-degree symmetric polynomial to clarify the construction.
[Introduction] Clarify the precise relationship between GQSP and standard QSP phase-factor synthesis to highlight where the block-encoding avoidance originates.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments, which help strengthen the presentation of our results. We address each major comment below and confirm that the requested clarifications and explicit derivations will be incorporated in the revised manuscript.

read point-by-point responses

Referee: [Abstract and the section deriving the linear-combination step] The central claim of degree-independent success probability for linear combinations of GQSP circuits (Abstract) lacks an explicit construction or normalization argument. Preparing a superposition over the individual GQSP circuits to realize the linear combination would bound the post-selection probability by the squared norm of the coefficient vector; for a degree-d polynomial this norm generally varies with d, contradicting the degree-independent claim unless a specific embedding or rescaling is derived.

Authors: We thank the referee for this observation. The linear combination is realized via a rescaled embedding of the coefficient vector that exploits the bounded norm of the symmetric polynomial expansions derived from the GQSP framework; this embedding ensures the squared norm remains bounded by a constant independent of degree d. The post-selection probability is therefore stable. We will add an explicit construction of the embedding together with the normalization argument in the revised manuscript. revision: yes
Referee: [Section presenting the closed-form expressions] The closed-form expressions for symmetric polynomial expansions (Abstract) are stated to exist but no explicit formulas, derivation, or verification that they map to the target polynomial without introducing new instabilities are supplied in the provided text; this is load-bearing for the claim that GQSP can be applied directly to Hermitian matrices.

Authors: We acknowledge that the initial submission did not include the full explicit formulas and derivation. The closed-form expressions are obtained by expanding the target polynomial in the basis of symmetric Chebyshev polynomials of the first kind and then mapping each term to a GQSP sequence; the resulting circuit preserves Hermiticity and introduces no additional instabilities beyond those already present in standard QSP. We will insert the complete formulas, the step-by-step derivation, and a short verification that the composition reproduces the target polynomial in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation extends GQSP without self-referential reduction

full rationale

The paper claims to implement arbitrary polynomials of Hermitian matrices via GQSP without block-encoding, deriving closed-form symmetric expansions and using linear combinations of GQSP circuits. No quoted step reduces a prediction or uniqueness claim to a fitted parameter, self-citation chain, or redefinition (e.g., no parameter fitted to data then renamed as prediction, no ansatz smuggled via overlapping-author citation, no success probability forced by construction). The approach is presented as building directly on the established GQSP framework with explicit derivations for the Hermitian case, keeping the central claims independent of the paper's own outputs. This is the common honest non-finding for papers that extend prior frameworks without internal redefinition.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only. No free parameters, invented entities, or non-standard axioms are mentioned. The approach assumes standard quantum computing primitives and the prior existence of the GQSP framework.

axioms (1)

domain assumption Standard assumptions of quantum circuit model and existence of GQSP framework
Method relies on GQSP being available as a black-box primitive.

pith-pipeline@v0.9.0 · 5538 in / 1135 out tokens · 21746 ms · 2026-05-16T20:20:30.583148+00:00 · methodology

discussion (0)

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 echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

A=1/2(U+U†) … An=Rn(U)+Rn(U†) where Rn(x) is the explicit binomial sum … P(A)=∑cn(Rn(U)+Rn(U†))
IndisputableMonolith/Foundation/BranchSelection.lean branch_selection refines

?

refines
Relation between the paper passage and the cited Recognition theorem.

closed-form expressions for symmetric polynomial expansions … linear combinations of GQSP circuits … stable, degree-independent success probability

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.

Reference graph

Works this paper leans on

26 extracted references · 26 canonical work pages

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work page 2017
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Ground-state preparation and energy estima- tion of quantum many-body systems

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Hamiltonian Simulation with Optimal Sam- pleComplexity

Guang Hao Low and Isaac L. Chuang. “Hamiltonian Simulation with Optimal Sam- pleComplexity”.In:Quantum Information & Computation16.11-12(2016),pp.819– 862

work page 2016

[2] [2]

Hamiltonian simulation by qubitization

Guang Hao Low and Isaac L. Chuang. “Hamiltonian simulation by qubitization”. In:Quantum3 (2019), p. 163

work page 2019

[3] [3]

Manca, S

Dominic W. Berry et al. “Simulating Hamiltonian dynamics with a truncated Tay- lor series”. In:Physical Review Letters114.9 (2015), p. 090502.doi: 10.1103/Phys- RevLett.114.090502

work page doi:10.1103/phys- 2015

[4] [4]

Quantum algorithm for systems of linear equations with exponentially improved dependence on precision

Andrew M. Childs, Robin Kothari, and Rolando D. Somma. “Quantum algorithm for systems of linear equations with exponentially improved dependence on precision”. In:SIAM Journal on Computing46.6 (2017), pp. 1920–1950

work page 2017

[5] [5]

Optimal Hamiltonian simulation by quantum signal processing

Guang Hao Low and Isaac L. Chuang. “Optimal Hamiltonian simulation by quantum signal processing”. In:Physical Review Letters118 (2017), p. 010501

work page 2017

[6] [6]

Ground-state preparation and energy estima- tion of quantum many-body systems

Yulong Dong, Lin Lin, and Yu Tong. “Ground-state preparation and energy estima- tion of quantum many-body systems”. In:NPJ Quantum Information7.1 (2021), p. 24.doi: 10.1038/s41534-021-00356-4

work page doi:10.1038/s41534-021-00356-4 2021

[7] [7]

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing , pages =

András Gilyén et al. “Quantum singular value transformation and beyond: Expo- nential improvements for quantum algorithms”. In:Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing. ACM, 2019, pp. 193–204.doi: 10.1145/3313276.3316366

work page doi:10.1145/3313276.3316366 2019

[8] [8]

Nonlinear transformation of com- plex amplitudes via quantum singular value transformation

Naixu Guo, Kosuke Mitarai, and Keisuke Fujii. “Nonlinear transformation of com- plex amplitudes via quantum singular value transformation”. In:arXiv:2107.10764 (2024)

work page arXiv 2024

[9] [9]

Generalized quantum signal processing

Davoud Motlagh and Nathan Wiebe. “Generalized quantum signal processing”. In: PRX Quantum5 (2024), p. 020368

work page 2024

[10] [10]

Complementarypolynomialsinquan- tum signal processing

BenjaminBerntsonandChristianSünderhauf.“Complementarypolynomialsinquan- tum signal processing”. In:Communications in Mathematical Physics406 (2025), p. 62

work page 2025

[11] [11]

Robust black-box quantum-state preparation via quantum signal processing

Lorenzo Laneve. “Robust black-box quantum-state preparation via quantum signal processing”. In:arXiv:2305.04705(2023)

work page arXiv 2023

[12] [12]

Efficient phase-factor evaluation in quantum signal processing

Yulong Dong et al. “Efficient phase-factor evaluation in quantum signal processing”. In:Physical Review A103.4 (2021), p. 042419

work page 2021

[13] [13]

Quantum algorithm for systems of linear equations with exponentially improved dependence on precision

Andrew M. Childs, Robin Kothari, and Rolando D. Somma. “Quantum algorithm for systems of linear equations with exponentially improved dependence on precision”. In:SIAM Journal on Computing46 (2017), pp. 1920–1950

work page 2017

[14] [14]

Robust angle finding for generalized quantum signal processing

Shuntaro Yamamoto and Nobuyuki Yoshioka. “Robust angle finding for generalized quantum signal processing”. In:arXiv:2402.03016(2024)

work page arXiv 2024

[15] [15]

Efficient quantum circuits for Toeplitz and Hankel matrices

Anuradha Mahasinghe and Jingbo B. Wang. “Efficient quantum circuits for Toeplitz and Hankel matrices”. In:Journal of Physics A: Mathematical and Theoretical49 (2016), p. 275301

work page 2016

[16] [16]

Efficient quantum circuits for dense circulant and circulant-like operators

Shi Shi Zhou and Jingbo B. Wang. “Efficient quantum circuits for dense circulant and circulant-like operators”. In:Royal Society Open Science4 (2017), p. 160906. 13

work page 2017

[17] [17]

Vibration analysis of cyclic symmetrical systems by quan- tum algorithms

Anuradha Mahasinghe. “Vibration analysis of cyclic symmetrical systems by quan- tum algorithms”. In:Mathematical Problems in Engineering2019 (2019), p. 4969351. doi: 10.1155/2019/4969351

work page doi:10.1155/2019/4969351 2019

[18] [18]

Linear combination of unitaries and block encodings: A tutorial exposition

Juan Miguel Arrazola et al. “Linear combination of unitaries and block encodings: A tutorial exposition”. In:Communications in Mathematical Physics(2023)

work page 2023

[19] [19]

Explicit quantum circuits for block encodings of certain sparse matrices

Daan Camps et al. “Explicit quantum circuits for block encodings of certain sparse matrices”.In:SIAM Journal on Matrix Analysis and Applications45(2024),pp.801– 827

work page 2024

[20] [20]

Block-encoding structured matrices for data input in quantum computing

Christian Sünderhauf, Earl Campbell, and Joan Camps. “Block-encoding structured matrices for data input in quantum computing”. In:Quantum8 (2024), p. 1234

work page 2024

[21] [21]

Quantum computing for option portfolio anal- ysis

Y. Wu, Jingbo B. Wang, and Y. Li. “Quantum computing for option portfolio anal- ysis”. In:arXiv:2406.00486(2024)

work page arXiv 2024

[22] [22]

arXiv preprint arXiv:2501.04567 (2025)

JanRullkötteretal.“Resource-efficientvariationalblock-encoding”.In:arXiv:2501.04567 (2025)

work page arXiv 2025

[23] [23]

Block encoding bosons by signal processing

Charles F. Kane et al. “Block encoding bosons by signal processing”. In:Quantum 9 (2025), p. 1747

work page 2025

[24] [24]

arXiv:2312.00723 [quant-ph] doi:10

Christian Sünderhauf. “Generalized quantum singular value transformation”. In: arXiv:2312.00723(2023)

work page arXiv 2023

[25] [25]

In:Discussions with Yusen Wu(2025)

work page 2025

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Higher-orderquantumoperations

PhilipTarantoetal.“Higher-orderquantumoperations”.In:arXiv preprint arXiv:2503.09063 (2025). 14

work page arXiv 2025