Norman Yao

We prove that recognizing the phase of matter of an unknown quantum state is quantum computationally hard. More specifically, we show that the worst-case runtime of any phase recognition algorithm must grow exponentially in the correlation length $\xi$ of the state. This exponential growth renders the problem practically infeasible even for moderate constant values of the correlation length $\xi$, and leads to super-polynomial quantum computational time in the system size $n$ whenever $\xi = \omega(\log n)$. 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 to quantum systems with symmetries. We prove that symmetric pseudorandom unitaries exist under standard cryptographic conjectures, and can be constructed in extremely low circuit depths for any discrete on-site group. We also provide extensions of our results to systems with translation invariance and purely classical phases of matter. A key technical limitation is that the locality of the parent Hamiltonian of the states we consider is linear in $\xi$; removing this constraint remains an important open question.