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arXiv:2605.12450

Simulation of Non-Hermitian Hamiltonians with Bivariate Quantum Signal Processing

Joshua M. Courtney

quant-phcs.CCcs.DSSubmitted May 12, 2026• Announced May 13, 2026• 1 citation
v1 · May 12, 2026v2 · Jun 15, 2026v3 · Jul 21, 2026
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Abstract

We achieve query-optimal quantum simulations of non-Hermitian Hamiltonians $H_{\mathrm{eff}} = H_R + iH_I$, where $H_R$ is Hermitian and $H_I \succeq 0$, 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 $\mathcal{O}(d_R \cdot d_I)$ classical operations. Operator norms $α_R\,,β_I$ contribute additively with query complexity $\mathcal{O}((α_R + β_I)T + \log(1/\varepsilon)/\log\log(1/\varepsilon))$ matching an information-theoretic lower bound in the separate-oracle model, where $H_R$ and $H_I$ are accessed through independent block encodings. The postselection success probability is $e^{-2β_I T}\|e^{-iH_{\mathrm{eff}}T}|ψ_0\rangle\|^2\cdot (1 - \mathcal{O}(\varepsilon))$, decomposing into a state-dependent factor $\|e^{-iH_{\mathrm{eff}}T}|ψ_0\rangle\|^2$ from the intrinsic barrier and an $e^{-2β_I T}$ overhead from polynomial block-encoding.

Comments
64 pages, 3 figures, 4 tables

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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.
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.
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.
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.
Quantum signal processing is a powerful framework in quantum algorithms, playing a central role in Hamiltonian simulation and related applications. The sequence of polynomials implemented at each step of this protocol provides a polynomial basis for block-encoding any polynomial of a unitary. We characterize the achievable polynomial bases in terms of their orthogonality or biorthogonality with respect to a linear functional admitting an integral representation. Explicit expressions for the quantum signal processing angles are derived for families of polynomial sequences, including Hermite, Jacobi, and Rogers-Szegő polynomials. We show that 2n+22n+22n+2 rotation angles are required to encode a sequence of polynomials in these classes up to degree nnn. We use this result to show that an εεε-approximation of a smooth function fff can be block-encoded using O(log⁡(1/ε))O(\log(1/ε))O(log(1/ε)) gates via its Hermite series expansion. The connections established with the theory of orthogonal and biorthogonal polynomials lead to a new method for solving the quantum signal processing angle-finding problem, yielding explicit expressions for the angles. They also provide a complete characterization of the polynomials achievable by SU(1,1)\mathrm{SU}(1,1)SU(1,1)-QSP in terms of their roots. Biorthogonality properties are shown to hold in the bivariate QSP setting, yielding a set of necessary conditions for achievable polynomials.
We give a quantum algorithm for Lindbladian simulation given a block encoding of the Hamiltonian HHH and a projected unitary encoding of the stacked jump operator B=∑k=1m∣k⟩⊗LkB=\sum_{k=1}^m \lvert k\rangle\otimes L_kB=∑k=1m​∣k⟩⊗Lk​, with normalization factors αHα_HαH​ and αBα_BαB​, respectively. For evolution time ttt, set τ=(αH+αB2)tτ=(α_H+α_B^2)tτ=(αH​+αB2​)t. The algorithm approximates the evolution channel to diamond-norm error ε\varepsilonε using O ⁣(τ+log⁡(1/ε)log⁡ ⁣(e+log⁡(1/ε)/τ))O\!\left(τ+\frac{\log(1/\varepsilon)}{\log\!\left(e+\log(1/\varepsilon)/τ\right)}\right)O(τ+log(e+log(1/ε)/τ)log(1/ε)​) oracle queries, matching the query lower bound for Hamiltonian simulation. The number of additional one- and two-qubit gates is linear in the query complexity up to polylogarithmic factors. The query- and gate-complexity bounds extend to Lipschitz-continuous time-dependent Lindbladians under coherent time-indexed oracle access. Our construction uses a one-query transducer that implements a product of rational approximations to short-time evolution when supplied with a catalyst. We bound the error from omitting the catalyst by exploiting orthogonality between different sequences of Kraus labels. The gate implementation combines a compressed Kraus-label representation, which stores only the positions and values of the nonzero labels, with the rotation factorization of Chen et al.
Quantum signal processing (QSP) and quantum singular value transformation (QSVT) currently stand as the most efficient techniques for implementing functions of block-encoded matrices, a central task that lies at the heart of most prominent quantum algorithms. However, current QSP approaches face several challenges, such as the restrictions imposed on the family of achievable polynomials and the difficulty of calculating the required phase angles for specific transformations. In this paper, we present a generalized quantum signal processing (GQSP) approach, employing general SU(2) rotations as our signal-processing operators, rather than relying solely on rotations in a single basis. Our approach lifts all practical restrictions on the family of achievable transformations, with the sole remaining condition being that |P|≤1 , a restriction necessary due to the unitary nature of quantum computation. Furthermore, GQSP provides a straightforward recursive formula for determining the rotation angles needed to construct the polynomials in cases where P and Q are known. In cases where only P is known, we provide an efficient optimization algorithm capable of identifying in under a minute of GPU time, a corresponding Q for polynomials of degree on the order of 107 . We further illustrate GQSP simplifies QSP-based strategies for Hamiltonian simulation, offer an optimal solution to the ϵ -approximate fractional query problem that requires O((1/δ)+log(1/ϵ)) queries to perform where O(1/δ) is a proved lower bound, and introduces novel approaches for implementing bosonic operators. Moreover, we propose a novel framework for the implementation of normal matrices, demonstrating its applicability through synthesis of diagonal matrices, as well as the development of a new algorithm for convolution through synthesis of circulant matrices using only O(dlogN+log2N) 1 and 2-qubit gates for a filter of lengths d . Published by the American Physical Society 2024