Measurement-induced entanglement transitions in quantum dynamics represent a new class of nonequilibrium transitions, akin to thresholds in quantum error correction. In this work, the authors explore the universal properties of entanglement transitions in one- and two-dimensional monitored Clifford circuits, using a graph-state-based algorithm to unravel geometric properties of entanglement. A study of entanglement clusters in the steady state reveals that, despite similarities in the bulk, the surface critical exponents show strong deviations from classical percolation.
Measurement-induced entanglement transitions in quantum dynamics represent a new class of nonequilibrium transitions, akin to thresholds in quantum error correction. In this work, the authors explore the universal properties of entanglement transitions in one- and two-dimensional monitored Clifford circuits, using a graph-state-based algorithm to unravel geometric properties of entanglement. A study of entanglement clusters in the steady state reveals that, despite similarities in the bulk, the surface critical exponents show strong deviations from classical percolation.
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 d spatial dimensions and classical statistical mechanical models in d+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 d-dimensional percolation and (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.
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=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 structure of multipartite entanglement in monitored quantum circuits exhibiting measurement-induced phase transitions (MIPTs). Using a one-dimensional Clifford circuit subject to local measurements with a probability p, we show numerically that the entanglement depth, corresponding to the size of the largest cluster of entangled qubits scales as a power law with system size on both sides of the transition. The power law exponent is 1 in the entangling phase and continuously decreases to 0 as p→1 in the disentangling phase. In addition, we find that the spatial support of the largest cluster exhibits an approximate fractal geometry with a tunable fractal dimension controlled by the measurement rate. We argue that this structure arises from a competition between unitary-driven coagulation of entangled clusters and measurement-induced fragmentation, giving rise to a fractal steady state reminiscent of classical coagulation-fragmentation models. Away from the MIPT critical point, the fractal dimension matches the entanglement depth power law exponent. These results show that multipartite entanglement structure provides a fresh perspective on the emergent quantum correlations in monitored quantum circuits and noisy quantum dynamics.
The role of quantum fluctuations in modifying the critical behavior of nonequilibrium phase transitions is a fundamental but unsolved question. In this study, we examine the absorbing state phase transition of a 1D chain of qubits undergoing a contact process that involves both coherent and classical dynamics. We adopt a discrete-time quantum model with states that can be described in the stabilizer formalism, and therefore allows for an efficient simulation of large system sizes. The extracted critical exponents indicate that the absorbing state phase transition of this Clifford circuit model belongs to the directed percolation universality class. This suggests that the inclusion of quantum fluctuations does not necessarily alter the critical behavior of nonequilibrium phase transitions of purely classical systems. Finally, we extend our analysis to a non-Clifford circuit model, where a tentative scaling analysis in small systems reveals critical exponents that are also consistent with the directed percolation universality class. Published by the American Physical Society 2024
A new class of quantum entanglement transitions separating phases with different entanglement entropy scaling has been observed in recent numerical studies. Despite the numerical efforts, an analytical understanding of such transitions has remained elusive. Here, the authors propose a theory for the area-law to volume-law entanglement transition in many-body systems that undergo both random unitary evolutions and projective measurements. Using the replica method, the authors map analytically this entanglement transition to an ordering transition in a classical statistical mechanics model. They derive the general entanglement scaling properties at the transition and show a solvable limit where this transition can be mapped onto two-dimensional percolation.
In many-body physics, an explicit wavefunction often provides the most thorough understanding, yet for monitored random Clifford circuit it has remained missing. Here we develop a framework that grants direct access to the typical output states. As every stabilizer state is local-Clifford-equivalent to a graph state, the graph adjacency matrix, a classical bit matrix, provides a complete description of the quantum state. In the large-N limit, we show that these graphs converge to the Erdős--Rényi random graph G(N,1/2), which allows us to resolve the open problem of Greenberger--Horne--Zeilinger (GHZ) entanglement generated by deep random Clifford circuits. We obtain analytically the mean GHZ content ⟨g3⟩=1.204 for even N and 1.325 for odd N. For monitored Clifford circuits with a 1D layout, we uncover an emergent Erdős--Rényi subgraph G(Nsub,1/2) in the output states of the volume-law phase, where Nsub/N≈1−p/pc with p the measurement rate and pc the critical point of the measurement-induced phase transition (MIPT). The output state is thus equivalent to the output of an unmonitored random Clifford circuit on Nsub qubits, weakly perturbed by the remaining N−Nsub qubits carrying little entanglement. This result directly accounts for the quantum error-correcting capability of the volume-law phase, and implies the same GHZ statistics for the whole volume-law phase. We further identify a clustering effect for qubits in the dense subgraph, which we reproduce with an infection-recovery toy model that exhibits a measurement-induced absorbing-state phase transition. Finally, a mean-field argument on the graph locates the MIPT critical point at pc=0.1608, in excellent agreement with the numerical value pc≈0.16.