Hardness of recognizing phases of matter

TL;DR

This work proves that recognizing the phase of matter for an unknown quantum state is quantum computationally hard, with time needed growing exponentially in the correlation range $\xi$ and super-polynomially in system size $n$ when $\xi=\omega(\log n)$. The authors generalize pseudorandom unitaries to systems with symmetries, constructing symmetric PRUs in poly$(n)$ depth and even polylogarithmic depth under suitable assumptions, and show translation-invariant variants. These PRUs render fixed-point states of both trivial and nontrivial phases indistinguishable to any efficient quantum observer, establishing worst-case hardness for phase recognition across quantum, mixed, and classical phases. The results imply the optimality of existing phase-recognition algorithms that scale with $\xi$ and raise open questions about the role of locality, continuous symmetries, and physically typical states in enabling practical phase identification. Overall, the paper links cryptographic constructions to fundamental questions in quantum many-body physics, providing a rigorous hardness baseline for phase recognition.

Abstract

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.

Figures (4)

• Figure 1: Illustration of our main results. (a) We consider the question: Given experimental access to many copies of a quantum state $\ket{\psi}$, can one recognize what phase of matter $\ket{\psi}$ is in? Examples of phases of matter include the trivial phase, symmetry-breaking phases, symmetry-protected topological (SPT) phases, and topological order. Our main result is that the complexity of recognizing phases of matter grows exponentially in the correlation range $\xi$ of the state, becoming super-polynomial in the system size $n$ as soon as $\xi = \mathop{\mathrm{poly}}\nolimits(\log n)$. (b) We achieve this result by extending the study of pseudorandom unitaries (PRUs) to quantum systems with symmetries. We show that any fixed point state of any phase of matter can become indistinguishable from a symmetric Haar-random state after a low-depth symmetric circuit is applied.
• Figure 2: Schematic of our main results on symmetric pseudorandom unitaries (PRUs). (a) Any symmetric unitary is block-diagonal in the irreducible representations (irreps) $\lambda_1,\ldots,\lambda_r$ of the symmetry group. Our $\mathop{\mathrm{poly}}\nolimits n$-depth construction of symmetric PRUs applies independent random unitaries in each block, by performing a sequence of controlled PRUs conditioned on the irrep of the entire system (Theorem \ref{['thm:poly']}). (b) To construct symmetric PRUs in $\mathop{\mathrm{poly}}\nolimits(\log n)$ depth, we first prove that two small symmetric PRUs can "glue" into a symmetric PRU on a larger system (Lemma \ref{['lemma:gluing']}). This is used to prove that the two-layer circuit in which each unitary is drawn independently from a symmetric PRU ensemble is a low-depth symmetric PRU (Theorem \ref{['thm:polylog']}). At a technical level, the symmetric gluing lemma follows by identifying the terms $R_{\boldsymbol g} \sigma$ and $R_{\boldsymbol h} \tau$ arising from the first and second unitaries' twirls, where $\boldsymbol g, \boldsymbol h \in G^{\otimes k}$ are symmetry group elements and $\sigma, \tau \in S_k$ are permutation operators. (c) A translation-invariant modification of the two-layer circuit, which underlies our construction of $\mathop{\mathrm{poly}}\nolimits(\log n)$-depth symmetric PRUs with translation invariance (Theorem \ref{['thm:TI']}).
• Figure 3: Three phases of matter and their order parameters after application of a low-depth symmetric PRU. (a) A symmetry-breaking phase (example: a GHZ state) features long-range correlations and a quantized non-zero mutual information. (b) A symmetry-protected topological (SPT) phase (example: a 1D cluster state) features string order parameters and a quantized non-zero conditional mutual information. (c) Topological orders (example: a toric code state) feature loop order parameters and a quantized non-zero topological entanglement entropy. In all three cases, the order parameters and quantized correlations are retained after the low-depth PRU. However, they cannot be detected by any polynomial-time observer without knowledge of the unitary that was applied.
• Figure 4: Extension of our results to classical phases of matter. (a) We consider a reversible one-layer brickwork circuit composed of symmetric pseudorandom permutations acting on patches of $\xi = \omega(\log n)$ bits each. The input to the circuit is any fixed point probability distribution, $p_0(x)$, tensored with $n$ mixed ancilla bits (black-white squares). We prove that for any input $p_0(x)$, the output of the circuit is indistinguishable from the maximally mixed distribution, i.e. the trivial phase. (b) Samples from the $\mathbbm{Z}_2$-symmetry-breaking fixed point distribution before and after the circuit. The circuit transforms the order parameter from a simple single-body observable into a pseudorandom function on $2\xi = \mathop{\mathrm{poly}}\nolimits(\log n)$ bits. This hides the phase of matter from any $\mathop{\mathrm{poly}}\nolimits n$-time observer.

Theorems & Definitions (49)

• Theorem 1: Hardness of recognizing quantum phases of matter
• Theorem 2: Symmetric pseudorandom unitaries
• Lemma 1: Gluing symmetric random unitaries
• Theorem 3: Symmetric random unitaries in extremely low depth
• Theorem 4: Translation-invariant symmetric pseudorandom unitaries
• Theorem 5: Hardness of recognizing mixed state phases of matter
• Theorem 6: Hardness of recognizing classical phases of matter
• Proposition 1: Exact symmetric Haar twirl
• proof
• Lemma 2: Approximate symmetric Haar twirl