Symmetry enriched phases of quantum circuits

TL;DR

This paper explores the phases of quantum circuits with symmetry and measurements, revealing novel phases with no equilibrium counterparts and classifying their properties through numerical and theoretical analysis.

Contribution

It introduces the concept of dynamical symmetries enlarging physical symmetries, predicting new quantum phases in measurement-driven circuits that are not possible in equilibrium.

Findings

Existence of long-range order in volume-law phases

Prediction of topological area-law phases protected by dynamical symmetries

Identification of a $U(1)$ critical phase and Kosterlitz-Thouless transition in fermionic circuits

Abstract

Quantum circuits consisting of random unitary gates and subject to local measurements have been shown to undergo a phase transition, tuned by the rate of measurement, from a state with volume-law entanglement to an area-law state. From a broader perspective, these circuits generate a novel ensemble of quantum many-body states at their output. In this paper, we characterize this ensemble and classify the phases that can be established as steady states. Symmetry plays a nonstandard role in that the physical symmetry imposed on the circuit elements does not on its own dictate the possible phases. Instead, it is extended by dynamical symmetries associated with this ensemble to form an enlarged symmetry. Thus, we predict phases that have no equilibrium counterpart and could not have been supported by the physical circuit symmetry alone. We give the following examples. First, we classify the phases of a circuit operating on qubit chains with $\mathbb{Z}_2$ symmetry. One striking prediction, corroborated with numerical simulation, is the existence of distinct volume-law phases in one dimension, which nonetheless support true long-range order. We furthermore argue that owing to the enlarged symmetry, this system can in principle support a topological area-law phase, protected by the combination of the circuit symmetry and a dynamical permutation symmetry. Second, we consider a Gaussian fermionic circuit that only conserves fermion parity. Here the enlarged symmetry gives rise to a $U(1)$ critical phase at moderate measurement rates and a Kosterlitz-Thouless transition to area-law phases. We comment on the interpretation of the different phases in terms of the capacity to encode quantum information. We discuss close analogies to the theory of spin glasses pioneered by Edwards and Anderson as well as crucial differences that stem from the quantum nature of the circuit ensemble.