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.

翻译：我们实现了对非厄米哈密顿量 $H_{\mathrm{eff}} = H_R + iH_I$ 的查询最优量子模拟，其中 $H_R$ 为厄米算符且 $H_I \succeq 0$，方法采用非对易信号算子的双变量扩展量子信号处理（QSP）。该算法将交互图像戴森级数编码为双环面上的多项式，通过结构化多变量QSP（M-QSP）电路实现。常量比条件保证了具有任意非对易信号算子的M-QSP电路的标量角度求解。保度平方和谱分解实现了双变量标量互补多项式。角度在经典预计算步骤中确定性计算，运行时间为 $\mathcal{O}(d_R \cdot d_I)$ 次经典操作。算符范数 $α_R\,,β_I$ 以查询复杂度 $\mathcal{O}((α_R + β_I)T + \log(1/\varepsilon)/\log\log(1/\varepsilon))$ 呈加性贡献，与分离预言机模型中的信息论下界匹配——该模型下 $H_R$ 和 $H_I$ 通过独立块编码访问。后选择成功概率为 $e^{-2β_I T}\|e^{-iH_{\mathrm{eff}}T}|ψ_0\rangle\|^2\cdot (1 - \mathcal{O}(\varepsilon))$，分解为本征势垒引起的状态依赖因子 $\|e^{-iH_{\mathrm{eff}}T}|ψ_0\rangle\|^2$ 与多项式块编码带来的 $e^{-2β_I T}$ 额外开销。