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arxiv: 2604.19688 · v1 · submitted 2026-04-21 · 🪐 quant-ph

Quantum Eigenvalue Transformations for Arbitrary Matrices

Xabier Guti\'errez , Lorenzo Laneve , Mikel Sanz This is my paper

Pith reviewed 2026-05-10 03:02 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum signal processingblock encodingeigenvalue transformationJordan normal formquantum algorithmssingular value transformationmatrix polynomials
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The pith

Special block encodings let quantum signal processing apply polynomials to eigenvalues of any matrix.

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

The paper aims to remove the restriction that quantum signal processing works only on unitaries and quantum singular value transformation works only on Hermitian matrices, so that polynomials can be applied to the eigenvalues of arbitrary square matrices. It introduces n-regular block encodings whose powers match those of the encoded matrix up to a chosen order n. The authors prove that quantum signal processing applied to any such encoding produces the polynomial transformation on the matrix itself, and that this holds even when the matrix is not diagonalizable because the action follows the Jordan normal form. They also supply an explicit construction that converts any ordinary block encoding into an n-regular one at the cost of O(log n) extra qubits. A reader would care because this supplies a uniform route to matrix functions on general linear operators inside quantum algorithms.

Core claim

The central claim is that an n-regular block encoding of an arbitrary square matrix A turns the application of quantum signal processing into the direct application of a polynomial of degree at most n to A, with the transformation acting on the eigenvalues associated with the Jordan normal form of A and independent of any further details of A's structure.

What carries the argument

The n-regular block encoding: a unitary whose successive powers up to order n reproduce the corresponding powers of the block-encoded matrix.

If this is right

  • Any polynomial of degree at most n can be applied to the eigenvalues of an arbitrary square matrix through quantum signal processing.
  • The transformation works without requiring the matrix to be diagonalizable or any explicit knowledge of its Jordan structure.
  • Standard block encodings can be upgraded to n-regular form using only O(log n) ancillary qubits and gates.
  • The resulting operator equals the polynomial applied to the original matrix regardless of the matrix's internal details.

Where Pith is reading between the lines

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

  • This approach could be combined with existing quantum linear-system solvers to handle non-Hermitian operators directly.
  • The same regularity idea might extend to rectangular matrices or to functions beyond polynomials.
  • Small quantum simulations on two-by-two Jordan blocks would provide an immediate experimental test of the eigenvalue mapping.

Load-bearing premise

That an n-regular block encoding can be built from a standard one without adding errors that would stop the polynomial from correctly transforming the eigenvalues of non-diagonalizable matrices.

What would settle it

A direct computation or small-scale circuit run on a non-diagonalizable Jordan block matrix showing that the output state after the quantum signal processing step fails to match the eigenvalues obtained by applying the target polynomial to the Jordan form would falsify the claim.

Figures

Figures reproduced from arXiv: 2604.19688 by Lorenzo Laneve, Mikel Sanz, Xabier Guti\'errez.

Figure 1
Figure 1. Figure 1: FIG. 1: Circuit for quantum eigenvalue transformation of general matrices based on (G)QSP. If [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
read the original abstract

Quantum Signal Processing (QSP) and Quantum Singular Value Transformation (QSVT) provide an efficient framework for implementing polynomials of block-encoded matrices, and thus offer a systematic approach to quantum algorithm design. However, despite a number of recent advances, important limitations remain. In particular, QSP can only transform unitary matrices, by applying a polynomial to their eigenvalues, while QSVT is a singular-value transformation and thus one can only obtain the polynomial of Hermitian matrices. As a consequence, these techniques do not directly apply to an arbitrary non-Hermitian matrix that is not diagonalizable. In this work, we propose a simple yet powerful method to extend these ideas to arbitrary square matrices by acting on their eigenvalues. To this end, we introduce the notion of an $n$-regular block encoding, namely, a block encoding whose $k$-th power reproduces the $k$-th power of the encoded matrix for every $0 < k < n$. We show that applying QSP to any unitary with this property is equivalent to applying a polynomial of degree at most $n$ to the block-encoded matrix, independently of its internal structure. Moreover, we provide a simple construction that transforms any block encoding into an $n$-regular one using only $O(\log n)$ ancillary qubits and operations. Finally, we show that this construction induces the desired transformation on the eigenvalues associated with the Jordan normal form of the matrix.

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 introduces n-regular block encodings and shows that QSP applied to any unitary with this property implements a polynomial of degree at most n on the block-encoded matrix A, independently of its internal structure. It provides an explicit construction converting any block encoding into an n-regular one with O(log n) ancillary qubits and operations, and claims this induces the desired eigenvalue transformation on the Jordan normal form of arbitrary (including defective) matrices.

Significance. If the central construction and equivalence hold exactly, the result would meaningfully extend QSP/QSVT beyond unitary and Hermitian cases, enabling systematic polynomial transformations on the eigenvalues of general square matrices via Jordan calculus. The O(log n) overhead and structure-independent claim are efficient and broadly applicable strengths; the work supplies a concrete algorithmic primitive that could support new quantum linear-algebra routines.

major comments (2)
  1. [§3] §3 (Construction of the n-regular block encoding): the argument that the O(log n)-ancilla unitary U satisfies [U^k]_{top-left} = A^k exactly for all k ≤ n must be verified for defective matrices. For a Jordan block J = D + N with nilpotent N ≠ 0, any ancillary leakage into the generalized eigenspace would alter the action of higher powers on N and therefore change p(A) away from the intended Jordan-functional-calculus result. An explicit invariance lemma or direct computation on the generalized eigenvectors is required.
  2. [Theorem 4.1] Theorem 4.1 (Equivalence of QSP on n-regular unitary to polynomial action on A): the proof that the top-left block of the QSP circuit equals p(A) relies on the n-regular property holding without error. If the construction in §3 fails to preserve exact powers for non-diagonalizable A, the claimed independence from internal structure does not follow. The manuscript should supply a short error-bound or commutator calculation showing that the ancillary registers do not mix the Jordan chains.
minor comments (2)
  1. [Definition 2.3] The definition of n-regular block encoding (Eq. (3) or equivalent) would be clearer if written as an explicit block-matrix condition on U^k rather than a verbal statement.
  2. [Figure 2] Figure 2 (circuit for the n-regular construction) would benefit from explicit labeling of the O(log n) ancilla registers and a brief caption explaining how the controlled operations enforce the power-reproduction property.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for providing constructive feedback. We are pleased that the referee recognizes the potential significance of extending QSP to arbitrary matrices. We address each of the major comments below and outline the revisions we plan to make.

read point-by-point responses

  1. Referee: [§3] §3 (Construction of the n-regular block encoding): the argument that the O(log n)-ancilla unitary U satisfies [U^k]_{top-left} = A^k exactly for all k ≤ n must be verified for defective matrices. For a Jordan block J = D + N with nilpotent N ≠ 0, any ancillary leakage into the generalized eigenspace would alter the action of higher powers on N and therefore change p(A) away from the intended Jordan-functional-calculus result. An explicit invariance lemma or direct computation on the generalized eigenvectors is required.

    Authors: We appreciate the referee's emphasis on rigorously verifying the construction for defective matrices. Our construction of the n-regular block encoding is designed such that the ancillary qubits are entangled in a controlled manner that ensures the top-left block exactly reproduces A^k for k ≤ n, without introducing errors from generalized eigenspaces. This holds because the additional operations are applied in a way that commutes with the Jordan structure in the relevant subspace. To make this explicit, we will add a new lemma in §3 that provides a direct computation on generalized eigenvectors, demonstrating invariance of the Jordan chains under the powered unitary. This will confirm that no ancillary leakage alters the nilpotent part. revision: yes

  2. Referee: [Theorem 4.1] Theorem 4.1 (Equivalence of QSP on n-regular unitary to polynomial action on A): the proof that the top-left block of the QSP circuit equals p(A) relies on the n-regular property holding without error. If the construction in §3 fails to preserve exact powers for non-diagonalizable A, the claimed independence from internal structure does not follow. The manuscript should supply a short error-bound or commutator calculation showing that the ancillary registers do not mix the Jordan chains.

    Authors: We agree that strengthening the proof of Theorem 4.1 with an explicit calculation would enhance clarity. The proof proceeds by showing that each step in the QSP sequence preserves the n-regular property, leading to the top-left block being p(A) by the definition of polynomial application via the Jordan functional calculus. We will incorporate a short commutator calculation in the revised manuscript to explicitly show that the ancillary registers do not mix the Jordan chains, thereby confirming the structure-independent nature of the transformation. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation relies on explicit new definition and construction

full rationale

The paper defines n-regular block encoding directly via the power-reproduction property on blocks, supplies an O(log n)-ancilla construction to realize it from any standard block encoding, and derives the QSP-polynomial equivalence from that definition plus the algebraic action of QSP on the unitary. The Jordan-functional-calculus claim follows from the preserved block powers rather than from any self-citation chain, fitted parameter, or renamed prior result. No load-bearing step reduces to its own input by construction; the central claims are supported by the provided construction and are therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the existence and efficient constructibility of n-regular block encodings and on the preservation of Jordan-form eigenvalue action under the QSP polynomial.

axioms (1)
  • domain assumption Standard block-encoding properties and QSP polynomial implementation hold for the unitary part of the n-regular encoding.
    Invoked when stating equivalence between QSP on the unitary and polynomial action on the matrix.
invented entities (1)
  • n-regular block encoding no independent evidence
    purpose: Enables polynomial eigenvalue transformations on arbitrary matrices via QSP.
    New definition introduced to overcome limitations of standard block encodings for non-unitary, non-Hermitian cases.

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Reference graph

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