Non-Hermitian (NH) quantum systems demonstrate striking differences from their Hermitian counterparts, leading to claims of NH advantage in areas ranging from metrology to entanglement generation. We show that in the context of quantum computation, any such NH advantage is unlikely to be scalable as an efficient computational resource: if coherent normalized non-unitary evolution could be realized with only polynomial overhead, then the resulting model could implement postselection, implying implausibly strong complexity-theoretic power under standard assumptions. We define NHBQP(U) as the computational power of poly-size quantum circuits that, in addition to a standard universal unitary gate set, may apply a fixed gate U on $O(1)$ qubits that is not proportional to a unitary, with the state renormalized after each use of U. We prove this model is powerful enough to decide PostBQP. In the standard uniform circuit-family model this characterization is tight: for any fixed such U, NHBQP(U)=PostBQP=PP. PostBQP is believed intractable, so this suggests that any scalable NH computational advantage must come with a cost limiting its efficiency. Additionally, we study locality-preserving purifications of restricted classes of non-unitary systems. Using this framework, we show that unitary gates with postselection can simulate not only evolution under NH Hamiltonians but arbitrary quantum trajectories. Any NH model whose purification lies in a strongly simulable unitary family (e.g., Clifford, matchgate, or low-bond-dimension tensor-network circuits) remains efficiently classically simulable, provided the relevant postselected events occur with probability $Ω(2^{-\text{poly}(n)})$. Thus adding non-Hermiticity to a universal unitary system makes it infeasibly computationally powerful, while adding it to a strongly simulable system adds no computational power in this setting.

翻译：非厄米（NH）量子系统展现出与其厄米对应物显著不同的特性，这使得在从计量学到纠缠生成等多个领域中有人宣称存在NH优势。我们表明，在量子计算的背景下，任何此类NH优势都不太可能作为高效的计算资源实现扩展：假如归一化的相干非幺正演化能以仅多项式开销实现，那么得到的模型将能够实现后选择，这意味着在标准假设下它拥有过于强大到难以置信的计算复杂性能力。我们定义NHBQP(U)为多尺寸量子电路的计算能力，这些电路除了标准通用幺正门集外，还可对$O(1)$个量子比特应用一个固定但非正比于幺正的门U，且每次使用U后对状态进行重归一化。我们证明该模型足以强到判定PostBQP。在标准均匀电路族模型中，这一刻画是紧的：对于任何此类固定的U，有NHBQP(U)=PostBQP=PP。PostBQP被认为难以处理，因此这表明任何可扩展的NH计算优势都必须以限制其效率的成本为代价。此外，我们研究了受限非幺正系统的局域性保持纯化方案。利用这一框架，我们证明带有后选择的幺正门不仅能模拟NH哈密顿量下的演化，还能模拟任意量子轨迹。任何纯化方案属于可强模拟幺正族（例如Clifford门、匹配门或低键维张量网络电路）的NH模型，只要相关后选择事件以概率$Ω(2^{-\text{poly}(n)})$发生，就仍然保持经典高效可模拟性。因此，向通用幺正系统添加非厄米性会使其计算能力变得不可行地强大，而向强可模拟系统添加非厄米性则在此设定下不增加任何计算能力。