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arXiv:2110.14403 · SciPost Physics Core

Measurement-induced criticality in extended and long-range unitary circuits

Shraddha Sharma, Xhek Turkeshi, Rosario Fazio, Marcello Dalmonte

Published Apr 21, 2022• 102 citations
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Abstract

We explore the dynamical phases of unitary Clifford circuits with variable-range interactions, coupled to a monitoring environment. We investigate two classes of models, distinguished by the action of the unitary gates, which either are organized in clusters of finite-range two-body gates, or are pair-wise interactions randomly distributed throughout the system with a power-law distribution. We find the range of the interactions plays a key role in characterizing both phases and their measurement-induced transitions. For the cluster unitary gates we find a transition between a phase with volume-law scaling of the entanglement entropy and a phase with area-law entanglement entropy. Our results indicate that the universality class of the phase transition is compatible to that of short range hybrid Clifford circuits. Oppositely, in the case of power-law distributed gates, we find the universality class of the phase transition changes continuously with the parameter controlling the range of interactions. In particular, for intermediate values of the control parameter, we find a non-conformal critical line which separates a phase with volume-law scaling of the entanglement entropy from one with sub-extensive scaling. Within this region, we find the entanglement entropy and the logarithmic negativity present a cross-over from a phase with algebraic growth of entanglement with system size, and an area-law phase.

DOI
10.21468/scipostphyscore.5.2.023

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We explore the dynamical phases of unitary Clifford circuits with variable-range interactions, coupled to a monitoring environment. We investigate two classes of models, distinguished by the action of the unitary gates, which either are organized in clusters of finite-range two-body gates, or are pair-wise interactions randomly distributed throughout the system with a power-law distribution. We find the range of the interactions plays a key role in characterizing both phases and their measurement-induced transitions. For the cluster unitary gates we find a transition between a phase with volume-law scaling of the entanglement entropy and a phase with area-law entanglement entropy. Our results indicate that the universality class of the phase transition is compatible to that of short range hybrid Clifford circuits. Oppositely, in the case of power-law distributed gates, we find the universality class of the phase transition changes continuously with the parameter controlling the range of interactions. In particular, for intermediate values of the control parameter, we find a non-conformal critical line which separates a phase with volume-law scaling of the entanglement entropy from one with sub-extensive scaling. Within this region, we find the entanglement entropy and the logarithmic negativity present a cross-over from a phase with algebraic growth of entanglement with system size, and an area-law phase.
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.
Measurement-induced entanglement transitions arise in quantum circuits of unitary gates and measurements, referred to as hybrid circuits. On average, unitary dynamics generate entanglement, and measurements suppress entanglement. In one-dimensional hybrid circuits, this competition leads to a transition between an area-law phase when the probability of measurements is higher, and a volume-law entangled phase when the probability is lower. In this thesis, the phase transition is studied using two different approaches. First, state vector simulations are performed for hybrid circuits composed of Haar-random unitary gates and random Clifford gates. This allows for direct access to the state vectors and entanglement entropy, but is only feasible for smaller size systems (up to 12 qubits). Second, a traffic flow model is implemented as an approximation to stabiliser dynamics of Clifford circuits, which is studied for larger systems (up to 2048 qubits). Scaling analyses are carried out for both approaches to investigate the dependence of the steady-state entropy on system size and measurement probability. Data collapses of simulation data for different circuit sizes are obtained using both a logarithmic form S−αlnN and by also including an additional correction S−αlnN+βN^(−ω) where ω = 1, leading to a more accurate collapse for smaller sizes. Both the state vector model and traffic flow model lead to a phase transition in entanglement entropy, but with different critical parameters. This indicates that the traffic flow model and the quantum circuit models do not belong to the same universality class. The results clarify the efficiency but also limitations of classical models for describing measurement-induced entanglement transitions.
Measurement-only circuits provide a minimal setting in which repeated local projections can either generate or suppress many-body entanglement, giving rise to measurement-induced phase transitions and dynamical regimes, that might have no unitary counterpart. Here we investigate entanglement and information transitions in one-dimensional measurement-only Clifford circuits with long-range two-qubit parity checks. By tuning both the measurement range and density per layer, we uncover a broad set of phases whose classification requires probes beyond entanglement entropy, such as mutual information, tripartite mutual information, purification from an ancilla, and Bell-cluster statistics. We map phase diagrams using large-scale Clifford simulations for two protocols: a random-basis design in which each measurement is randomly chosen from {XX,YY,ZZ}\lbrace XX,YY,ZZ \rbrace{XX,YY,ZZ}, and a single-basis design in which the basis is fixed within each layer but varies between layers, hence introducing more structure to the circuit. We map the trajectory-averaged entanglement entropy to a two-dimensional statistical mechanics model by extending a replica-based method to random-basis measurement-only circuits, and show that a continuous-time limit yields an effective long-range XX hamiltonian in the steady state. This connection links the observed volume-law to sub-volume-law entanglement transition to the boundary between a continuous symmetry broken phase and a critical XY phase. Strikingly, in structured (single-basis) circuits we find a phase in which volume-law and long-range entanglement coexists with rapid, size-independent purification of an ancilla qubit, and the absence of scrambling, highlighting measurement-only circuits as a promising route to efficiently preparing highly entangled and technologically useful quantum states.
Many-body unitary dynamics interspersed with repeated measurements display a rich phenomenology hallmarked by measurement-induced phase transitions. Employing feedback-control operations that steer the dynamics toward an absorbing state, we study the entanglement entropy behavior at the absorbing state phase transition. For short-range control operations, we observe a transition between phases with distinct subextensive scalings of entanglement entropy. In contrast, the system undergoes a transition between volume-law and area-law phases for long-range feedback operations. The fluctuations of entanglement entropy and of the order parameter of the absorbing state transition are fully coupled for sufficiently strongly entangling feedback operations. In that case, entanglement entropy inherits the universal dynamics of the absorbing state transition. This is, however, not the case for arbitrary control operations, and the two transitions are generally distinct. We quantitatively support our results by introducing a framework based on stabilizer circuits with classical flag labels. Our results shed new light on the problem of observability of measurement-induced phase transitions.
An entanglement transition can occur without a corresponding transition in purification. We demonstrate this separation in spatially local monitored Clifford circuits with a fixed number of layers per Floquet period, where a spatially random pattern of Clifford gates and Pauli measurements is repeated exactly in time. Rare spatial structures persist under this repetition and break the correspondence between entanglement and purification found in conventional spacetime-random circuits. In one spatial dimension, the late-time entanglement produced with an initial product state remains area-law, while the system remains extensively mixed in the thermodynamic and long-time limits if the initial state is maximally mixed. In two or more dimensions, we find a transition between volume-law and area-law entanglement while the system remains mixed on both sides of this transition. An extensive amount of quantum information survives indefinitely in both phases and evolves unitarily despite the repeated measurements. We introduce the logical-support radius to characterize its spatial spreading and identify the volume-law and area-law phases with delocalized and localized logical dynamics, respectively. We further introduce the logical Krylov dimension, which counts the independent logical operators dynamically generated from an initially local operator. Its scaling resolves the transition and provides evidence for a new universality class of measurement-induced phase transitions.
We propose that the quantum conditional mutual information (QCMI), computed with a suitably chosen partition of the system, serves as a powerful probe for detecting measurement-induced entanglement phase transitions in monitored quantum circuits. To demonstrate this, we investigate monitored variable-range Clifford circuits and identify the phase boundary between volume-law and area-law entanglement phases by performing finite-size scaling analyses of the QCMI. Assuming that the entanglement entropy exhibits a logarithmic dependence on system size at criticality in short-range interacting cases, we further show that the QCMI allows for the simultaneous determination of both the critical point and the universal coefficient of the logarithmic term in the entanglement entropy via a crossing-point analysis. For the shortest-range interacting case studied, we obtain the thermodynamic-limit value of the coefficient as c~=1.519(3)\tilde{c}=1.519(3)c~=1.519(3), which is significantly smaller than values reported in previous studies.