量子张量网络态的非稳定子性在二维情况下是难解的
Nonstabilizerness of quantum tensor network states is intractable in two dimensions
- Scuola Superiore Meridionale（南方高等学院）
- Istituto Nazionale di Fisica Nucleare, Sezione di Napoli（意大利国家核物理研究所那不勒斯分部）
- Max Planck Institute of Quantum Optics（马克斯·普朗克量子光学研究所）
- Munich Center for Quantum Science and Technology（慕尼黑量子科学与工程中心）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明在二维情况下，即使键维数恒定，计算张量网络态的稳定子熵是#P困难的，判定稳定子成员资格是C_=P完全的，表明不存在多项式资源的经典或量子算法。
AI中文摘要：
非稳定子性是量子系统超越经典可模拟区域所必需的一种资源。随着稳定子α-雷尼熵的出现，非稳定子性也成为一种多体诊断工具，与纠缠互补。虽然判定任意量子态是否具有非稳定子性已被证明是困难的，但这样的态已经需要指数级（相对于量子比特数）的描述，因此超出了多体物理的研究范围。在此，我们转而考虑二维张量网络（TN）态，这些态紧凑地描述了满足纠缠面积定律的态，并询问它们是否允许针对同一任务的更简单算法。与一维情况相反，我们证明，即使在小的、恒定的键维数下，计算二维张量网络态的稳定子熵对于任何整数α≥2都是#P困难的，而判定稳定子成员资格是C_=P完全的。因此，在标准复杂性假设下，没有任何经典或量子算法能在最坏情况下用多项式资源解决这两个问题中的任何一个。即使将问题表述为达到常数加性精度，这两个困难结果仍然成立。
英文摘要：
Nonstabilizerness is a necessary resource for quantum systems to lie beyond the classically simulable regime. With the advent of stabilizer $α$-Rényi entropies, nonstabilizerness has also become a many-body diagnostic, complementary to entanglement. While deciding whether an arbitrary quantum state has nonstabilizerness is provably hard, such states already require a description exponentially large in the number of qubits and are thus out of reach for many-body physics. Here we instead consider 2D tensor network (TN) states, which compactly capture states obeying an entanglement area law, and ask whether they admit a simpler algorithm for the same task. In contrast to the 1D case, we prove that, even at small, constant bond dimension, computing the stabilizer entropy of 2D TN states is $\#\mathrm{P}$-hard for any integer $α\ge 2$, and deciding stabilizer membership is $\mathrm{C_=P}$-complete. The corresponding constant-accuracy problems remain $\mathrm{C_= P}$-hard. We further prove that deciding whether a PEPS can be transformed into a stabilizer state by local unitaries over a specified bipartition is $\mathrm{C_= P}$-hard. Under standard complexity assumptions, these results rule out classical or quantum algorithms with polynomial resources for all three tasks in the worst case.
