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arXiv:2609.32262

Contour-integral and Fourier transform based multivariable quantum eigenvalue transformation for commuting matrices

Shan Jiang, Dong An

quant-phmath.NASubmitted Sep 26, 2026• Announced Sep 29, 2026• 0 citations• Revised Oct 3, 2026 (v2)
v1 · Sep 26, 2026v2 · Oct 3, 2026
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Abstract

We study the problem of implementing multivariable matrix-valued functions on a quantum computer and propose two quantum algorithms for multivariable matrix eigenvalue transformations acting on tuples of pairwise commuting matrices. The first algorithm is based on multivariable contour integrals and applies to arbitrary holomorphic functions on $\mathbb{C}^n$. The second algorithm is based on high-dimensional Fourier transforms and designed for smooth functions of commuting Hermitian matrices. For both algorithms, we discuss and analyze the complexity of their quantum implementation leveraging quantum singular value transformation, compression gadgets, and linear combination of unitaries. We additionally study their hybrid quantum--classical variants that reduce the number of required ancilla qubits to logarithmic dependence on the number of variables, at the cost of increased query complexity. As an application, we show that multivariable matrix polynomials can be implemented with no additional explicit degree dependence, depending only on global properties of the polynomials.

Comments
36 pages including appendix

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We study the problem of implementing multivariable matrix-valued functions on a quantum computer and propose two quantum algorithms for multivariable matrix eigenvalue transformations acting on tuples of pairwise commuting matrices. The first algorithm is based on multivariable contour integrals and applies to arbitrary holomorphic functions on Cn\mathbb{C}^nCn. The second algorithm is based on high-dimensional Fourier transforms and designed for smooth functions of commuting Hermitian matrices. For both algorithms, we discuss and analyze the complexity of their quantum implementation leveraging quantum singular value transformation, compression gadgets, and linear combination of unitaries. We additionally study their hybrid quantum--classical variants that reduce the number of required ancilla qubits to logarithmic dependence on the number of variables, at the cost of increased query complexity. As an application, we show that multivariable matrix polynomials can be implemented with no additional explicit degree dependence, depending only on global properties of the polynomials.
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Quantum signal processing allows for quantum eigenvalue transformation with Hermitian matrices, in which each eigenspace component of an input vector gets transformed according to its eigenvalue. In this work, we introduce the multivariate quantum eigenvalue transformation for functions of commuting Hermitian matrices. We then present a framework for working with polynomial matrix functions in which we may solve MQET, and give the application of computing functions of normal matrices using a quantum computer.
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We develop Multivariate Quantum Signal Processing for designing matrix-valued transformations of multiple noncommuting nonnormal block-encoded matrices. Our framework equips quantum transducers with the analytic structure of multiport Infinite-Impulse-Response (IIR) filters and establishes a transfer-function methodology for designing quantum query algorithms. Efficiently computable analytic bounds that hold uniformly across the oracle promise yield efficient quantum circuits with logarithmic error dependence and near-unity success probability, paralleling the approximation-theoretic guarantees of univariate QSP/QSVT. We provide a complete characterization of achievable transfer functions and a principled library of analytic IIR components. Our modular designs greatly improve or resolve the per-oracle query complexity of open multi-oracle problems: (1) Hamiltonian simulation under H=∑jHjH=\sum_j H_jH=∑j​Hj​ with weighted quasinorm cost ≈t⟨λ⃗,C⃗⟩1/2+O(∥C⃗∥1log⁡(1/ε))\approx t\langle\vecλ,\vec C\rangle_{1/2}+\mathcal{O}(\|\vec C\|_1\log(1/ε))≈t⟨λ,C⟩1/2​+O(∥C∥1​log(1/ε)) for block-encoding normalizations λ⃗\vecλλ and costs C⃗\vec CC; (2) sparse simulation with optimal time--norm Θ(d∥H∥1→2t)Θ(\sqrt{d}\|H\|_{\rm 1\to2}t)Θ(d​∥H∥1→2​t) and additive error dependence; (3) solving linear systems and (4) differential equations under minimal matrix assumptions with optimal weighted state-preparation and multi-term cost; (5) ground-state preparation and energy estimation; (6) generalized eigenvalue problems. We apply these to exactly simulate downfolded Hamiltonians with leading cost t⟨λ⃗eff,C⃗eff⟩1/2t\langle\vecλ_{\rm eff},\vec C_{\rm eff}\rangle_{1/2}t⟨λeff​,Ceff​⟩1/2​ further reduced by fractional dynamic correlation energy contributions from coarse-grained Hamiltonians, and show similar classical-data-input-reduction results for multiscale graph classification via Kron reduction. We also show the online generalization with anisotropic Kalman filtering of streaming quantum data.