We prove that recognizing the phase of matter of an unknown quantum state is quantum computationally hard. More specifically, we show that the quantum computational time of any phase recognition algorithm must grow exponentially in the range of correlations $ξ$ of the unknown state. This exponential growth renders the problem practically infeasible for even moderate correlation ranges, and leads to super-polynomial quantum computational time in the system size $n$ whenever $ξ= ω(\log n)$. Our results apply to a substantial portion of all known phases of matter, including symmetry-breaking phases and symmetry-protected topological phases for any discrete on-site symmetry group in any spatial dimension. To establish this hardness, we extend the study of pseudorandom unitaries (PRUs) to quantum systems with symmetries. We prove that symmetric PRUs exist under standard cryptographic conjectures, and can be constructed in extremely low circuit depths. We also establish hardness for systems with translation invariance and purely classical phases of matter. A key technical limitation is that the locality of the parent Hamiltonians of the states we consider is linear in $ξ$; the complexity of phase recognition for Hamiltonians with constant locality remains an important open question.

翻译：我们证明了识别未知量子态的物质相在量子计算意义下是困难的。具体而言，我们证明了任何相识别算法的量子计算时间必须随未知态关联长度$ξ$呈指数增长。这种指数增长使得该问题即使对于中等关联长度也实际不可行，并且当$ξ= ω(\log n)$时，会导致算法在系统尺寸$n$上具有超多项式量子计算时间。我们的结果适用于所有已知物质相的相当大部分，包括任意空间维度中任意离散局域对称群的对称破缺相和对称保护拓扑相。为确立这一困难性，我们将伪随机幺正算符（PRU）的研究推广至具有对称性的量子系统。我们证明了在标准密码学猜想下对称PRU是存在的，并且可以在极低的电路深度中构造。我们还确立了具有平移不变性及纯经典物质相系统的困难性。一个关键的技术限制在于我们所考虑态的父母哈密顿量的局域性与$ξ$呈线性关系；对于具有常数局域性的哈密顿量，其相识别问题的复杂性仍然是一个重要的开放性问题。