Simulation of Non-Hermitian Hamiltonians with Bivariate Quantum Signal Processing
2026-05-12 • Computational Complexity
Computational ComplexityData Structures and Algorithms
AI summaryⓘ
The authors develop a way to efficiently simulate a special type of quantum system described by non-Hermitian Hamiltonians, which are used when energy can be lost or gained. They extend a technique called quantum signal processing to handle more complex, two-variable cases involving non-commuting operators. Their method achieves the best possible query efficiency based on fundamental limits and uses a classical precomputation step to find necessary parameters. The approach carefully accounts for error and success probabilities related to the system's natural decay.
Quantum simulationNon-Hermitian HamiltoniansQuantum signal processingDyson seriesBlock encodingOperator normPostselectionSpectral factorizationQuery complexity
Authors
Joshua M. Courtney
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.