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arXiv:2610.01125

Multivariate Quantum Signal Processing

Guang Hao Low

quant-phSubmitted Oct 1, 2026• Announced Oct 2, 2026• 0 citations
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Abstract

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 $H=\sum_j H_j$ with weighted quasinorm cost $\approx t\langle\vecλ,\vec C\rangle_{1/2}+\mathcal{O}(\|\vec C\|_1\log(1/ε))$ for block-encoding normalizations $\vecλ$ and costs $\vec C$; (2) sparse simulation with optimal time--norm $Θ(\sqrt{d}\|H\|_{\rm 1\to2}t)$ 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 $t\langle\vecλ_{\rm eff},\vec C_{\rm eff}\rangle_{1/2}$ 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.

Comments
305 pages, 34 figues, 14 tables

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Polynomial transformations are basic primitives in quantum algorithms: quantum signal processing and singular value transformation compile univariate polynomials into circuits with query complexity essentially set by degree. Multivariable transformations of noncommuting matrices, however, lack a comparable synthesis theory. We develop a complete constructive theory under joint block access for all polynomials contractive on the prescribed matrix domain. Given a compact finite-state description of a degree-$D$ polynomial $P$ and $0<τ<1$, we synthesize $P$ with $(1+τ)$-optimal normalization using $O(D/\sqrtτ)$ queries, and exactly $D$ under row-block access, matching a degree lower bound. The complete circuit realization is classically computable in polynomial time with polylogarithmic dependence on accuracy. The key to our construction is a finite algorithmic Schur--Agler theorem: the coefficient recurrence defines a polynomial-dimensional continuation space supporting a complete semidefinite certificate for contractivity on the prescribed domain. Factoring this certificate yields the contractive realization underlying the query algorithm. Beyond matrix transformations, our framework lifts multivariable polynomial synthesis to quantum channel transformations. Given coherent Kraus access $K=\{K_a\}_a$, we synthesize any finite jointly contractive family of noncommutative polynomial maps $K\mapsto\{F_b(K)\}_b$ as completely positive operations, allowing coherent interference among Kraus histories. Furthermore, channel-level transformations specified by causal Choi data can be synthesized explicitly as fixed-order quantum combs. Collectively, these results point to a broader program: multivariable approximation as a language for multi-operator quantum algorithms and higher-order quantum information processing.

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Multivariate quantum signal processing (M-QSP) has recently been shown to be applicable for non-Hermitian Hamiltonian simulation, opening several problems regarding the optimization landscape, angle-finding, and constant-factor analysis. We resolve several of these problems here. We find the anti-Hermitian query complexity $d_I = Θ(\betaI T + \log(1/\varepsilon)/\log\log(1/\varepsilon))$ to be tight, established via Chebyshev coefficient bounds, modified Bessel function asymptotics, and Lambert~WWW inversion. Fast-forwarding to $d_I = \mathcal{O}(\sqrt{\betaI T})$ is impossible in the bivariate polynomial model, though a linear state-dependent improvement to dI=OβeffT+log⁡(1/ε)/log⁡log⁡(1/ε))d_I = \mathcal{O} β_{\mathrm{eff}} T + \log(1/\varepsilon)/\log\log(1/\varepsilon))dI​=Oβeff​T+log(1/ε)/loglog(1/ε)) is achievable. The optimization landscape of M-QSP admits spurious local minima, but a warm-start basin guarantee ensures the two-stage algorithm converges. CRC-exploiting block peeling reduces angle-finding from O(d3)\mathcal{O}(d^3)O(d3) to O(d2)\mathcal{O}(d^2)O(d2) classical operations, and optimized error allocation yields a leading constant of approximately~222 relative to the information-theoretic lower bound. A constant-ratio condition extends to non-identical signal operators, enabling time-dependent non-Hermitian simulation with query complexity $\mathcal{O}(\int_0^T(\alphaR(s) + \betaI(s))\,ds + \log(1/\varepsilon)/\log\log(1/\varepsilon))$. Block-encoding overhead $e^{-2\betaI T}$ holds across all function classes within the walk-operator oracle model, and dilational methods (Schrödingerization) achieve the walk-operator barrier. A precisely characterized direct-access construction achieves the intrinsic barrier e−2ωTe^{-2ωT}e−2ωT (with $ω< \betaI$ for non-commuting Hamiltonians) on a restricted domain, though extension to the full bitorus remains open.
We achieve query-optimal quantum simulations of non-Hermitian Hamiltonians Heff=HR+iHIH_{\mathrm{eff}} = H_R + iH_IHeff​=HR​+iHI​, where HRH_RHR​ is Hermitian and HI⪰0H_I \succeq 0HI​⪰0, using a bivariate extension of quantum signal processing (QSP) with non-commuting signal operators. The algorithm encodes the interaction-picture Dyson series as a polynomial on the bitorus, implemented through a structured multivariable QSP (M-QSP) circuit. A constant-ratio condition guarantees scalar angle-finding for M-QSP circuits with arbitrary non-commuting signal operators. A degree-preserving sum-of-squares spectral factorization permits scalar complementary polynomials in two variables. Angles are deterministically calculated in a classical precomputation step, running in O(dR⋅dI)\mathcal{O}(d_R \cdot d_I)O(dR​⋅dI​) classical operations. Operator norms αR ,βIα_R\,,β_IαR​,βI​ contribute additively with query complexity O((αR+βI)T+log⁡(1/ε)/log⁡log⁡(1/ε))\mathcal{O}((α_R + β_I)T + \log(1/\varepsilon)/\log\log(1/\varepsilon))O((αR​+βI​)T+log(1/ε)/loglog(1/ε)) matching an information-theoretic lower bound in the separate-oracle model, where HRH_RHR​ and HIH_IHI​ are accessed through independent block encodings. The postselection success probability is e−2βIT∥e−iHeffT∣ψ0⟩∥2⋅(1−O(ε))e^{-2β_I T}\|e^{-iH_{\mathrm{eff}}T}|ψ_0\rangle\|^2\cdot (1 - \mathcal{O}(\varepsilon))e−2βI​T∥e−iHeff​T∣ψ0​⟩∥2⋅(1−O(ε)), decomposing into a state-dependent factor ∥e−iHeffT∣ψ0⟩∥2\|e^{-iH_{\mathrm{eff}}T}|ψ_0\rangle\|^2∥e−iHeff​T∣ψ0​⟩∥2 from the intrinsic barrier and an e−2βITe^{-2β_I T}e−2βI​T overhead from polynomial block-encoding.
Many problems in linear algebra-such as those arising from non-Hermitian physics, transcorrelated quantum chemistry and differential equations-can be solved on a quan-tum computer by processing eigenvalues of the non-normal input matrices. However, the existing Quantum Singular Value Transformation (QSVT) framework is ill-suited to this task, as eigenvalues and singular values are different in general. We present a Quantum EigenValue Transformation (QEVT) framework for applying arbitrary polynomial transformations on eigenvalues of block-encoded non-normal operators, and a related Quantum EigenValue Estimation (QEVE) algorithm for operators with real spectra. QEVT has query complexity to the block encoding nearly recovering that of the QSVT for a Hermitian input, and QEVE achieves the Heisenberg-limited scaling for diagonalizable input matrices. As applications, we develop a linear differential equation solver with strictly linear time query complexity for average-case diagonalizable operators, as well as a ground state preparation algorithm that upgrades previous nearly optimal results for Hermitian Hamiltonians to diagonalizable matrices with real spectra. Underpinning our algorithms is an efficient method to prepare a quantum super-position of Faber polynomials, which generalize the nearly-best uniform approximation properties of Chebyshev polynomials to the complex plane. Of independent interest, we also develop techniques to generatennnFourier coefficients with 0 (poly log (nnn)) gates compared to prior approaches with linear cost.
Polynomial transformations of data encoded in quantum operations are a building block of quantum algorithms, and their query complexity measures how often those operations are used. For multiple variables, existing methods either realize restricted polynomial families or implement monomials separately at a query cost exponential in the number of variables. We present a quantum circuit that implements every multivariate trigonometric polynomial up to an explicit normalization. The circuit processes one variable coherently over the frequency components of the remaining variables, allowing all components to share the same signal queries. Each variable is queried exactly as many times as its degree in the polynomial, and we prove that this query complexity is optimal. We further extend the construction to commuting unitaries. The resulting circuit applies the multivariate polynomial simultaneously across their shared eigenbasis, using the minimum numbers of forward and inverse queries to each unitary. We also study trainable rotations with fixed preparation and a local readout of the last input. At query depth at least two, independent uniform initialization gives every active readout gradient a variance lower bound set by its squared branch weight and the inverse square root of the depth, uniformly over data states and dimensions. For supervised squared loss, retaining sample correlations gives an initialization gradient-variance bound in terms of targets and input spectra. This also bounds the expected loss decrease after the first exact gradient step with a specified step size. These results provide a scalable and practical solution for developing multivariate quantum algorithms and quantum learning models.
Abstract. Many problems in linear algebra—such as those arising from non-Hermitian physics, transcorrelated quantum chemistry, and differential equations—can be solved on a quantum computer by processing eigenvalues of the non-normal input matrices. However, the existing Quantum Singular Value Transformation (QSVT) framework is ill-suited for this task because eigenvalues and singular values are different in general. We present a Quantum EigenValue Transformation (QEVT) framework for applying arbitrary polynomial transformations on eigenvalues of block-encoded non-normal operators and a related Quantum EigenValue Estimation (QEVE) algorithm for operators with real spectra. QEVT has query complexity to the block encoding nearly recovering that of the QSVT for a Hermitian input, and QEVE achieves the Heisenberg-limited scaling for diagonalizable input matrices. As applications, we develop a linear differential equation solver with strictly linear time query complexity for average-case diagonalizable operators, as well as a ground state preparation algorithm that upgrades previous nearly optimal results for Hermitian Hamiltonians to diagonalizable matrices with real spectra. Underpinning our algorithms is an efficient method to prepare a quantum superposition of Faber polynomials, which generalize the nearly-best uniform approximation properties of Chebyshev polynomials to the complex plane. Of independent interest, we also develop techniques to generate [Formula: see text] Fourier coefficients with [Formula: see text] gates compared to prior approaches with linear cost.
Polynomial transformations are basic primitives in quantum algorithms: quantum signal processing and singular value transformation compile univariate polynomials into circuits with query complexity essentially set by degree. Multivariable transformations of noncommuting matrices, however, lack a comparable synthesis theory. We develop a complete constructive theory under joint block access for all polynomials contractive on the prescribed matrix domain. Given a compact finite-state description of a degree-DDD polynomial PPP and 0<τ<10<τ<10<τ<1, we synthesize PPP with (1+τ)(1+τ)(1+τ)-optimal normalization using O(D/τ)O(D/\sqrtτ)O(D/τ​) queries, and exactly DDD under row-block access, matching a degree lower bound. The complete circuit realization is classically computable in polynomial time with polylogarithmic dependence on accuracy. The key to our construction is a finite algorithmic Schur--Agler theorem: the coefficient recurrence defines a polynomial-dimensional continuation space supporting a complete semidefinite certificate for contractivity on the prescribed domain. Factoring this certificate yields the contractive realization underlying the query algorithm. Beyond matrix transformations, our framework lifts multivariable polynomial synthesis to quantum channel transformations. Given coherent Kraus access K={Ka}aK=\{K_a\}_aK={Ka​}a​, we synthesize any finite jointly contractive family of noncommutative polynomial maps K↦{Fb(K)}bK\mapsto\{F_b(K)\}_bK↦{Fb​(K)}b​ as completely positive operations, allowing coherent interference among Kraus histories. Furthermore, channel-level transformations specified by causal Choi data can be synthesized explicitly as fixed-order quantum combs. Collectively, these results point to a broader program: multivariable approximation as a language for multi-operator quantum algorithms and higher-order quantum information processing.