Simulation of Non-Hermitian Hamiltonians with Bivariate Quantum Signal Processing

We achieve query-optimal quantum simulations of non-Hermitian Hamiltonians $H_{\mathrm{e}\mathrm{f}\mathrm{f}}=H_R+i{H}_I$, where $H_R$ is Hermitian and $H_I⪰0$, 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 ${\alpha }_R,{\beta }_I$ contribute additively with query complexity $\mathcal{O}(({\alpha }_R+{\beta }_I)T+\log (1\mathrm{/}\epsilon )\mathrm{/}\log \log (1\mathrm{/}\epsilon ))$ 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 $e^{-2{\beta }_{I}T}\mathrm{\parallel }{e}^{-i{H}_{\mathrm{e}\mathrm{f}\mathrm{f}}T}\mathrm{\mid }{\psi }_0⟩{\mathrm{\parallel }}^2\cdot (1-\mathcal{O}(\epsilon ))$, decomposing into a state-dependent factor $\mathrm{\parallel }{e}^{-i{H}_{\mathrm{e}\mathrm{f}\mathrm{f}}T}\mathrm{\mid }{\psi }_0⟩{\mathrm{\parallel }}^2$ from the intrinsic barrier and an $e^{-2{\beta }_{I}T}$ overhead from polynomial block-encoding.