We study quantum circuits consisting of unitary gates, projective measurements, and control operations that steer the system toward a pure absorbing state. Two types of phase transition occur as the rate of these control operations is increased: a measurement-induced entanglement transition, and a directed percolation transition into the absorbing state (taken here to be a product state). In this work, we show analytically that these transitions are generically distinct, with the quantum trajectories becoming disentangled before the absorbing state transition is reached, and we analyze their critical properties. We introduce a simple class of models where the measurements in each quantum trajectory define an effective tensor network (ETN)---a subgraph of the initial spacetime graph where nontrivial time evolution takes place. By analyzing the entanglement properties of the ETN, we show that the entanglement and absorbing-state transitions coincide only in the limit of the infinite local Hilbert-space dimension. Focusing on a Clifford model which allows numerical simulations for large system sizes, we verify our predictions and study the finite-size crossover between the two transitions at large local Hilbert space dimension. We give evidence that the entanglement transition is governed by the same fixed point as in hybrid circuits without feedback.
We study quantum circuits consisting of unitary gates, projective measurements, and control operations that steer the system toward a pure absorbing state. Two types of phase transition occur as the rate of these control operations is increased: a measurement-induced entanglement transition, and a directed percolation transition into the absorbing state (taken here to be a product state). In this work, we show analytically that these transitions are generically distinct, with the quantum trajectories becoming disentangled before the absorbing state transition is reached, and we analyze their critical properties. We introduce a simple class of models where the measurements in each quantum trajectory define an effective tensor network (ETN)---a subgraph of the initial spacetime graph where nontrivial time evolution takes place. By analyzing the entanglement properties of the ETN, we show that the entanglement and absorbing-state transitions coincide only in the limit of the infinite local Hilbert-space dimension. Focusing on a Clifford model which allows numerical simulations for large system sizes, we verify our predictions and study the finite-size crossover between the two transitions at large local Hilbert space dimension. We give evidence that the entanglement transition is governed by the same fixed point as in hybrid circuits without feedback.
Monitored quantum circuits in which entangling unitary dynamics compete with projective local measurements can host measurement-induced phase transitions witnessed by entanglement measures at late times. Adding feedback conditioned on the measurement outcomes gives rise to another type of phase transition witnessed by local order parameters and correlation functions. These transitions, known as control or absorbing-state transitions, generically occur within the area-law entanglement phase and are thought to be governed by classical physics in that their critical exponents match those of the classical limit of the model. In this work, we examine quantum features of these transitions, focusing on a Bernoulli circuit model with a well-defined classical limit, in the presence of unitary dynamics, measurements, and feedback. First we demonstrate that, in the local basis defined by the absorbing state, the steady-state quantum coherence undergoes a phase transition at the control transition, where its logarithm changes discontinuously from volume- to area-law scaling. Second, we analyze the control transition from the perspective of fluctuations in observables, which carry two contributions: classical fluctuations over circuit realizations (present in the classical limit), and quantum fluctuations over trajectories and states (both absent in the classical limit). Both contributions can be estimated in experiments without post-selection. The circuit-to-circuit fluctuations, the dominant contribution, carry the critical behavior of the classical limit. However, the subleading quantum fluctuations that represent fluctuations between different quantum ``worlds'' also go critical at the control transition. These critical quantum fluctuations at the control transition also occur in other models, and we discuss how they can be measured experimentally without post-selection.
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.
Changes in the entanglement structure and critical phenomena are hallmarks of quantum phase transitions. Here, we discuss how they appear in transitions between classes of states with distinct entanglement patterns beyond the paradigm of stable equilibrium phases of matter. Using a mapping between stochastic automata and isometric Tensor Network States (isoTNS), we construct a two-dimensional quantum state from the Domany-Kinzel automaton, which is a (1+1)D process with an absorbing phase transition in the directed percolation class. At the critical point of the automaton, the corresponding isoTNS hosts algebraic correlations in all spatial directions. The continuous parent Hamiltonian of this state has a degenerate ground state manifold. It consists of a product state (the absorbing state) and a second state that undergoes a transition from pairwise entanglement between distant regions, similarly to the W state, to a state with trivial entanglement. Our results demonstrate how the correspondence between isoTNS and classical stochastic evolution can be used to probe the Hilbert space structure beyond stable ground state manifolds.
Nascent quantum computers motivate the exploration of quantum many-body systems in nontraditional scenarios. For example, it has become natural to explore the dynamics of systems evolving under both unitary evolution and measurement. Such systems can undergo dynamical phase transitions in the entanglement properties of quantum trajectories conditional on the measurement outcomes. Here, we explore dynamics in which one attempts to (locally) use those measurement outcomes to steer the system toward a target state, and we study the resulting phase diagram as a function of the measurement and feedback rates. Steering succeeds when the measurement and feedback rates exceed a threshold, yielding an absorbing-state transition in the trajectory-averaged density matrix. We argue that the absorbing-state transition generally occurs at different critical parameters from the entanglement transition in individual trajectories and has distinct critical properties. The efficacy of steering depends on the nature of the target state: in particular, for local dynamics targeting long-range correlated states, steering is necessarily slow and the entanglement and steering transitions are well separated in parameter space.
Adaptive quantum circuits—where a quantum many-body state is controlled using measurements and conditional unitary operations—are a powerful paradigm for state preparation and quantum error-correction tasks. They can support two types of nonequilibrium quantum phase transitions: measurement-induced transitions between volume- and area-law-entangled steady states and control-induced transitions where the system falls into an absorbing state or, more generally, an orbit visiting several absorbing states. Within this context, nonlocal conditional operations can alter the critical properties of the two transitions and the topology of the phase diagram. Here, we consider the scenario where the measurements are , in order to engineer efficient control onto dynamical trajectories. Motivated by Rydberg-atom arrays, we consider a locally constrained model with global sublattice magnetization measurements and local correction operations to steer the system’s dynamics onto a many-body orbit with finite recurrence time. The model has a well-defined classical limit, which we leverage to aid our analysis of the control transition. As a function of the density of local correction operations, we find control and entanglement transitions with continuously varying critical exponents. For sufficiently high densities of local correction operations, we find that both transitions acquire a dynamical critical exponent z < 1 , reminiscent of criticality in long-range power-law interacting systems. At low correction densities, we find that the criticality reverts to a short-range nature with z ≳ 1 . In the long-range regime, the control and entanglement transitions are indistinguishable to within the resolution of our finite-size numerics, while in the short-range regime we find evidence that the transitions become distinct. We conjecture that the effective long-range criticality mediated by collective measurements is essential in driving the two transitions together.
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.