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arXiv:2210.11957

Measurement-induced phase transitions in $(d+1)$-dimensional stabilizer circuits

Piotr Sierant, Marco Schirò, Maciej Lewenstein, Xhek Turkeshi

Published Oct 21, 2022• 2 citations
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Abstract

The interplay between unitary dynamics and local quantum measurements results in unconventional non-unitary dynamical phases and transitions. In this paper we investigate the dynamics of $(d+1)$-dimensional hybrid stabilizer circuits, for $d=1,2,3$. We characterize the measurement-induced phases and their transitions using large-scale numerical simulations focusing on entanglement measures, purification dynamics, and wave-function structure. Our findings demonstrate the measurement-induced transition in $(d+1)$ spatiotemporal dimensions is conformal and close to the percolation transition in $(d+1)$ spatial dimensions.

DOI
10.48550/arxiv.2210.11957

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Measurement-induced phase transitions in (d+1)(d+1)(d+1)-dimensional stabilizer circuits
The interplay between unitary dynamics and local quantum measurements results in unconventional non-unitary dynamical phases and transitions. In this paper we investigate the dynamics of (d+1)(d+1)(d+1)-dimensional hybrid stabilizer circuits, for d=1,2,3d=1,2,3d=1,2,3. We characterize the measurement-induced phases and their transitions using large-scale numerical simulations focusing on entanglement measures, purification dynamics, and wave-function structure. Our findings demonstrate the measurement-induced transition in (d+1)(d+1)(d+1) spatiotemporal dimensions is conformal and close to the percolation transition in (d+1)(d+1)(d+1) spatial dimensions.
The interplay between unitary dynamics and local quantum measurements results in unconventional nonunitary dynamical phases and transitions. In this paper we investigate the dynamics of (d+1)(d+1)(d+1)-dimensional hybrid stabilizer circuits, for d=1,2,3d=1,2,3d=1,2,3. We characterize the measurement-induced phases and their transitions using large-scale numerical simulations focusing on entanglement measures, purification dynamics, and wave-function structure. Our findings demonstrate the measurement-induced transition in (d+1)(d+1)(d+1) spatiotemporal dimensions is conformal and has critical properties close to the percolation transition in (d+1)(d+1)(d+1) spatial dimensions.
Entanglement transitions in quantum dynamics present a novel class of phase transitions in non-equilibrium systems. When a many-body quantum system undergoes hybrid quantum dynamics, consisting of unitary evolution interspersed with monitored random measurements, the steady-state can exhibit a phase transition between volume- and area-law entanglement. The role of dimension in the nature of these transitions is an open problem. There is a dimensional correspondence between measurement-induced transitions in non-unitary quantum circuits in ddd spatial dimensions and classical statistical mechanical models in d+1d+1d+1 dimensions, where the time dimension in the quantum problem is mapped to a spatial dimension in the classical model. In this work we show that the role of dimension is considerably richer by unveiling a form of `dimensional hybridity': critical properties of the steady-state entanglement are governed by a combination of exponents consistent with ddd-dimensional percolation and (d+1)(d+1)(d+1)-dimensional percolation. We uncover this dimensional hybridity in 1+1D and 2+1D circuits using a graph-state based simulation algorithm where the entanglement structure is encoded in an underlying graph, providing access to the geometric structure of entanglement. We locate the critical point using the tripartite information, revealing area-law entanglement scaling at criticality, and showing that the entanglement transition coincides with the purification transition. The emergence of this `dimensional hybridity' in these non-unitary quantum circuits sheds new light on the universality of measurement-induced transitions, and opens the way for analyzing the quantum error correcting properties of random unitary circuits in higher dimensions.
Random quantum circuits provide a framework for probing universal dynamical phenomena in chaotic quantum many-body systems, by circumventing the microscopic details of Hamiltonian evolution. Bipartite entanglement measures can be used to characterize the dynamical phases that emerge in these systems. Typically, under random unitary evolution, bipartite entanglement entropies exhibit a linear growth, and attain steady-state values that scale extensively with the system volume. However, when local projective measurements are introduced, a suppression of steady-state bipartite entropies occurs, leading to a sub-extensive scaling with the system volume. This class of entanglement scaling transitions, from an extensive to a sub-extensive scaling regime, has sparked interest in a broader class of measurement-induced phase transitions, with a potential to further our understanding of fault-tolerance in quantum computing, and in realizing synthetic quantum phases on digital quantum simulators. In this work, we characterize entanglement scaling transitions in one-dimensional random quantum circuits composed of q-level systems, with the local Hilbert space dimension q ≥ 2. This predominantly numerical study employs generalizations of the stabilizer formalism and the Clifford group to higher-dimensional Hilbert spaces to simulate the measurement-induced transitions. We find that the nature of the entanglement scaling transition remains invariant under changes in the local Hilbert space dimension. However, the critical measurement rate inducing the transition increases with q, asymptotically approaching 1/2 in the limit q → ∞. To further explore this classical limit, we develop a minimal dynamical model that captures entanglement structures under unitary evolution. We analyze the entanglement structures in thermal stabilizer states using this model, and propose an approach for incorporating measurements within this framework.
We investigate the dynamics of two-dimensional quantum spin systems under the combined effect of random unitary gates and local projective measurements. When considering steady states, a measurement-induced transition occurs between two distinct dynamical phases, one characterized by a volume-law scaling of entanglement entropy, the other by an area law. Employing stabilizer states and Clifford random unitary gates, we numerically investigate square lattices of linear dimension up to L=48L=48L=48 for two distinct measurement protocols. For both protocols, we observe a transition point where the dominant contribution in the entanglement entropy displays multiplicative logarithmic violations to the area law. We obtain estimates of the correlation length critical exponent at the percent level; these estimates suggest universal behavior and are incompatible with the universality class of 3D percolation.
We study the influence of feedback operations on the dynamics of (d+1)-dimensional monitored random quantum circuit. Competition between unitary dynamics and measurements leads to an entanglement phase transition, while feedback steers the dynamics towards an absorbing state, yielding an absorbing state phase transition. Based on previous results in one spatial dimension (Phys. Rev. Lett. 130, 120402 (2023)), we discuss the interplay between the two types of transitions for d ≥ 2 in the presence of (i) short-range feedback operations or (ii) additional global control operations. In both cases, the absorbing state transition belongs to the d-dimensional directed percolation universality class. In contrast, the entanglement transition depends on the feedback operation type and reveals dynamics' inequivalent features. The entanglement and absorbing state phase transition remain separated for short-range feedback operations. When global control operations are applied, we find the two critical points coinciding; nevertheless, the universality class may still differ, depending on the choice of control operation.
Competition between unitary dynamics that scrambles quantum information nonlocally and local measurements that probe and collapse the quantum state can result in a measurement-induced entanglement phase transition. Here we study this phenomenon in an analytically tractable all-to-all Brownian hybrid circuit model composed of qubits. The system is initially entangled with an equal sized reference, and the subsequent hybrid system dynamics either partially preserves or totally destroys this entanglement depending on the measurement rate. Our approach can access a variety of entropic observables which are distinguished by the averaging procedure, and for concreteness we focus on a particular purity quantity for which the averaging is particularly simple. We represent the purity as a path integral coupling four replicas with twisted boundary conditions. Saddle-point analysis reveals a second-order phase transition corresponding to replica permutation symmetry breaking below a critical measurement rate. The transition is mean-field-like and we characterize the critical properties near the transition in terms of a simple Ising field theory in 0+10+10+1 dimensions. In addition to studying the purity of the entire system, we study subsystem purities and relate these results to manifestations of quantum error correction in the model. We also comment on the experimental feasibility for simulating this averaged purity, and corroborate our results with exact diagonalization for modest system sizes.