Critical properties of the measurement-induced transition in random quantum circuits

Abstract

We numerically study the measurement-driven quantum phase transition of Haar-random quantum circuits in 1 + 1 dimensions. By analyzing the tripartite mutual information we are able to make a precise estimate of the critical measurement rate pc = 0.17(1). We extract estimates for the associated bulk critical exponents that are consistent with the values for percolation, as well as those for stabilizer circuits, but differ from previous estimates for the Haar-random case. Our estimates of the surface order parameter exponent appear different from that for stabilizer circuits or percolation, but we are unable to definitively rule out the scenario where all exponents in the three cases match. Moreover, in the Haar case the prefactor for the entanglement entropies Sn depends strongly on the Rényi index n; for stabilizer circuits and percolation this dependence is absent. Results on stabilizer circuits are used to guide our study and identify measures with weak finite-size effects. We discuss how our numerical estimates constrain theories of the transition.Nonequilibrium quantum systems can undergo various phase transitions in their dynamics; characterizing such transitions is a key open question in modern quantum statistical physics. So far, these nonequilibrium phase transitions have been studied primarily for isolated quantum systems [1, 2] and for steady states of dissipative systems [3,4]. One much-studied case is the many-body localization transition [2], which can be seen either (i) as a dynamical transition at which thermalization slows down and stops as a parameter (e.g., the disorder strength in a spin chain) is tuned or (ii) as an entanglement transition at which the many-body eigenstates of the system change from volume-law to area-law entangled. Recently, a different type of entanglement transition was discovered [5][6][7] in the steady-state entanglement of the states produced by individual quantum trajectories [8-11] of a repeatedly-measured quantum many-body system. As the system is measured at an increasing rate, this single-trajectory entanglement goes from volume-law to area-law (see Fig. 1(a)), as has been demonstrated both numerically, and analytically in certain tractable limits [6, 7,[12][13][14][15][16][17]. This measurementdriven non-equilibrium quantum phase transition can also be interpreted as a purification transition [18] that can collapse a mixed state to a pure state through a sufficiently large rate of local projective measurements.A measurement driven transition is expected to occur for any form of quantum chaotic dynamics, e.g. in both random circuit [6, 7] and Hamiltonian [19] dynamics. Current studies have mainly focused on quantum circuits, acting on an array of qudits (of local Hilbert space dimension q); these are believed to be generic models of chaotic quantum dynamics [20][21][22][23][24][25][26]. Various choices of gates have been explored numerically [6, 7, 15]. In specific limiting cases, analytic results (or large-scale simu-lations) exist. Specifically, the transition in...

Measurement-induced entanglement transitions in many-body localized systems

“…1(a). With measurements in the X basis, the transition from volume to area-law entanglement occurs at a nonzero measurement probability p X c > 0, similar to previously studied chaotic systems [9][10][11][12][13][14][15][16]18,19,[21][22][23][24][25]32,33]. On the other hand, with measurements in the Z basis, the volume law is destroyed for any nonzero p, similar to previously studied integrable systems [12,17,20].…”

“…As discussed in Refs. [18,33], the advantage of using the tripartite information here is that it avoids any log N divergences in the entanglement entropy at criticality, which have been observed in hybrid Haar-random circuit models [18]. Instead, I 3 is expected to scale as I 3 ∝ −N in the volume-law phase, reach an O(1) constant at criticality, and then vanish in the area-law phase.…”

Efficient classical simulation of noisy random quantum circuits in one dimension

“…Recently, there have been numerous studies on 1D random quantum circuits subject to local projective (or weak) measurements [68][69][70][71][72][73][74][75][76][77][78][79][80][81]. In this model, local projective (or weak) measurements (which happens with a probability p m at each qubit site in a given time step) reduce non-trivial quantum entanglement between disjoint subsystems.…”

“…Then, the remaining MPO parameters Γ α n−1 can be updated in the same way as in the bulk case, following the procedure described in Eqs. (68)- (76).…”

Entanglement Membrane in Chaotic Many-Body Systems

“…The effective spin model may also be extended to dynamics with measurement, or other types of interaction with an environment, for example, to clarify the relationship between different universality classes of measurementinduced criticality [88,89,[93][94][95][96][97][98][99][100][101][102][103].…”