Simulation of Non-Hermitian Hamiltonians with Bivariate Quantum Signal Processing

• URL: http://arxiv.org/abs/2605.12450v1
• Date: Tue, 12 May 2026 17:40:56 GMT
• Title: Simulation of Non-Hermitian Hamiltonians with Bivariate Quantum Signal Processing
• Abstract summary: We achieve query-optimal quantum simulations of non-Hermitian Hamiltonians $H_mathrmeff = H_R + iH_I$, where $H_R$ is Hermitian and $H_I succeq 0$.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

Related papers

• Optimal T-Count for Block Encodings of Fermionic and Spin Hamiltonians [4.914113120082008]
  We study the cost of constructing block encodings of structured fermionic and spin Hamiltonians in a unitary Clifford$+T$ model.<n>Our main technical tool is an ancilla-compression theorem: any block encoding of an $n$-qubit operator with $a$ clean ancillas and at most $s$ $T$ gates can be compressed to use at most $mina,n+2s$ ancillas.
  arXiv  Detail & Related papers  (2026-09-10T07:01:08Z)
• Two-Tower Quantum Matrix Chain Multiplication: Trading Qubits for Depth [73.08853228981701]
  Matrix chain multiplication is a problem in scientific computing, machine learning, and graph analysis.<n>We present emphTwo-Tower Matrix, a quantum subroutine that encodes the product $mathcalW$ of the $K$ matrices into a quantum state in circuit depth.
  arXiv  Detail & Related papers  (2026-07-14T18:42:40Z)
• Faster quantum linear system solver beyond the condition number [50.84794327094274]
  We present two quantum algorithms that produce the normalized solution $|xrangle$ of linear system $Ax=| b rangle$ to accuracy $$ with complexity independent of the condition number $=lVert A-1rVert$.<n>Our filtering-based solver is extremely simple with a favorable runtime prefactor.
  arXiv  Detail & Related papers  (2026-07-08T17:49:40Z)
• Optimal Bounds, Barriers, and Extensions for Non-Hermitian Bivariate Quantum Signal Processing [0.0]
  We find the anti-Hermitian query complexity $d_I = (betaI T + log/varepsilon)/loglog (1/varepsilon)$ to be tight, established via Chebyshev coefficient, modified Bessel functions, and Lambert$W$ inversion.<n>A constant-access-operator construction achieves intrinsic barrier $e-2T$ on a restricted domain, though extension to the full bitorus remains open.
  arXiv  Detail & Related papers  (2026-05-12T19:03:36Z)
• A Factorization Identity for Twisted Multinomial Coefficients with Application to Pilot States in Hamiltonian Decoded Quantum Interferometry [0.0]
  We introduce the twisted multinomial coefficient, in which each inversion between letters $i$ and $j$ carries a pair-dependent weight.<n>We show that the factorization yields an exact matrix product state (MPS) of bond dimension $k+1$ for the expansion coefficients of $hk$ in a twisted algebra.
  arXiv  Detail & Related papers  (2026-04-01T15:27:23Z)
• Optimal Scalar Quantization for Matrix Multiplication: Closed-Form Density and Phase Transition [50.36362492608702]
  We study entrywise scalar quantization of two matrices prior to multiplication.<n>We obtain a closed-form optimal point density [ star(u) propto exp!left(-fracu26right)bigl( (1-2)+2u22bigr), qquad u=fracx_X,, and prove a correlation-driven phase transition.
  arXiv  Detail & Related papers  (2026-03-20T01:53:44Z)
• Spectral Gaps with Quantum Counting Queries and Oblivious State Preparation [47.600794349481966]
  In this work, we present a quantum algorithm which approximates values up to additive error $epsilonDelta_k$ using a logarithmic number of qubits.<n>A key technical step in the analysis is the preparation of a suitable random initial state, which ultimately allows us to efficiently count the number of eigenvalues that are smaller than a threshold.
  arXiv  Detail & Related papers  (2025-08-28T17:04:18Z)
• Quantum oracles for the finite element method [45.200826131319815]
  This study examines the quantum routines required for the implementation of oracles used in the block-encoding of the $N times N stiffness and mass matrices.<n>We show how to construct the necessary oracles, which require the calculation of element geometry, square root and the implementation of conditional operations.
  arXiv  Detail & Related papers  (2025-04-28T14:28:31Z)
• Low-degree approximation of QAC$^0$ circuits [0.0]
  We show that the parity function cannot be computed in QAC$0$. We also show that any QAC circuit of depth $d$ that approximately computes parity on $n$ bits requires $2widetildeOmega(n1/d)$.
  arXiv  Detail & Related papers  (2024-11-01T19:04:13Z)
• Unconditionally separating noisy $\mathsf{QNC}^0$ from bounded polynomial threshold circuits of constant depth [6.8680041558282054]
  We show that parallel quantum computation can exhibit greater computational power than previously recognized.<n>We bridge the theory of non-local games in higher dimensions with computational advantage on emerging quantum computers.
  arXiv  Detail & Related papers  (2024-08-29T09:40:55Z)
• Learning a Single Neuron with Adversarial Label Noise via Gradient Descent [50.659479930171585]
  We study a function of the form $mathbfxmapstosigma(mathbfwcdotmathbfx)$ for monotone activations. The goal of the learner is to output a hypothesis vector $mathbfw$ that $F(mathbbw)=C, epsilon$ with high probability.
  arXiv  Detail & Related papers  (2022-06-17T17:55:43Z)
• Low-degree learning and the metric entropy of polynomials [44.99833362998488]
  We prove that any (deterministic or randomized) algorithm which learns $mathscrF_nd$ with $L$-accuracy $varepsilon$ requires at least $Omega(sqrtvarepsilon)2dlog n leq log mathsfM(mathscrF_n,d,|cdot|_L,varepsilon) satisfies the two-sided estimate $$c (1-varepsilon)2dlog
  arXiv  Detail & Related papers  (2022-03-17T23:52:08Z)
• Random matrices in service of ML footprint: ternary random features with no performance loss [55.30329197651178]
  We show that the eigenspectrum of $bf K$ is independent of the distribution of the i.i.d. entries of $bf w$. We propose a novel random technique, called Ternary Random Feature (TRF) The computation of the proposed random features requires no multiplication and a factor of $b$ less bits for storage compared to classical random features.
  arXiv  Detail & Related papers  (2021-10-05T09:33:49Z)
• Continuous Submodular Maximization: Beyond DR-Submodularity [48.04323002262095]
  We first prove a simple variant of the vanilla coordinate ascent, called Coordinate-Ascent+. We then propose Coordinate-Ascent++, that achieves tight $(1-1/e-varepsilon)$-approximation guarantee while performing the same number of iterations. The computation of each round of Coordinate-Ascent++ can be easily parallelized so that the computational cost per machine scales as $O(n/sqrtvarepsilon+nlog n)$.
  arXiv  Detail & Related papers  (2020-06-21T06:57:59Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.