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：我们通过双变量量子信号处理（QSP）扩展，利用非对易信号算子，实现了对非厄米哈密顿量 $H_{\mathrm{eff}} = H_R + iH_I$（其中 $H_R$ 为厄米算子，$H_I \succeq 0$）的查询最优量子模拟。该算法将相互作用绘景戴森级数编码为双环面上的多项式，并通过结构化多变量QSP（M-QSP）电路实现。一个常数比率条件确保了任意非对易信号算子下M-QSP电路的标量角度求解。一种保持次数的平方和谱分解方法允许构造双变量中的标量互补多项式。角度在经典预计算步骤中确定性求解，运行时间为 $\mathcal{O}(d_R \cdot d_I)$ 次经典操作。算子范数 $α_R$ 和 $β_I$ 以加法形式贡献查询复杂度 $\mathcal{O}((α_R + β_I)T + \log(1/\varepsilon)/\log\log(1/\varepsilon))$，与独立预言机模型的信息论下界匹配，其中 $H_R$ 和 $H_I$ 通过独立块编码访问。后选择成功概率为 $e^{-2β_I T}\|e^{-iH_{\mathrm{eff}}T}|ψ_0\rangle\|^2\cdot (1 - \mathcal{O}(\varepsilon))$，分解为来自固有势垒的状态相关因子 $\|e^{-iH_{\mathrm{eff}}T}|ψ_0\rangle\|^2$ 和来自多项式块编码的 $e^{-2β_I T}$ 附加开销。