Measure and Forget Dynamics in Random Circuits

Abstract

"Forgetful" measurements-physically similar to dephasing-are of interest both for applications to fault-tolerant quantum computing and fundamentally, in studying how entanglement and entropy spread. This paper investigates measurement-induced phase transitions (MIPT) in random Clifford circuits when measurement outcomes are partially forgotten. Our findings reveal a local thermalization rate that remains constant regardless of system size. We also numerically calculate the decay behavior at the turning points in the entropy diagram. We observe a counterintuitive phenomenon where the entropy reaches a threshold and stops evolving, even as the system size increases. This challenges an intuition, drawn from previous studies of noisy random circuits, that noise will cause the thermalization of the whole system. Additionally, we identify the disappearance of the purification transition and discuss the implications of these entanglement dynamics for quantum error-correction codes.

Figures (9)

• Figure 1: (a) A configuration of local two-qubit unitaries is uniformly drawn from the Clifford group and arranged in a 1D brickwork pattern, with periodic boundary conditions. Between each layer of gates, "$Z$ forgetting" happens with probability $p_f$ at each site. The initial state is pure. (b) Entropy changes as a function of forgetting rate for a fixed circuit depth 8. The system shows a local thermalization rate that is independent of system size $N$. (c) Entropy changes as a function of forgetting rate for a fixed system size 8. The turning point where the system first reaches the upper bound shifts toward the left dramatically with increasing circuit depth. (d) Plotting $S(\rho)/N$ as a function of circuit depth (time) for a fixed forgetting rate, the curves for different system sizes are essentially the same. (e) The forgetting rate corresponding to the turning point shown in (c), as a function of circuit depth. The values of each data point are shown in the middle table. One can observe that, when the system depth is very large (i.e., approaches the thermal limit), even a small forgetting rate will totally thermalize the system. The decay shown in this plot obeys a power law $\alpha d^{v}$, where $d$ stand for the depth. For $N=64$, $\alpha\approx4,\ v\approx-5/4$.
• Figure 2: The circuit employed to explore the interplay between the measurement and forget processes. Between each "brickwork" layer of random Clifford gates, we introduce supplementary strata of random "$Z$ Measurements" and "$Z$ Forgets." The boundary conditions are periodic.
• Figure 3: The entropy diagram under the competition of the measurement and forget processes, where the initial state is pure. The vertical axis is the measurement rate $p_m$ in the circuit. The horizontal axis is the forgetting rate $p_f$. The color bar shows the entropy per site $S/N$, with a bluer shade signifying greater entropy. Across (a)--(d), for a constant depth of 8, the structure of the diagrams remains the same even as the system size $N$ grows. Increasing $N$ imparts a smoother character to the diagram, but its fundamental structure persists.
• Figure 4: The entropy diagram under the competition of the measurement and forget processes, with a pure initial state. The vertical axis represents the measurement rate $p_m$ within the circuit. The horizontal axis portrays the forgetting rate $p_f$ in the circuit. The color bar visualizes the entropy per site $S/N$, with a bluer shade signifying greater entropy. From (a)--(d) we conclude that, for a fixed system size $N=64$, the diagram remains essentially unchanged when the depth surpasses $N=64$. The observations from (e) and (f) indicate that this dynamics is also independent of the overall system size.
• Figure 5: The basic setup of the distillation protocol. The qubits are input at the bottom, with a number of layers of monitored circuits added on that is proportional to the time. At a certain point, one (or more) of the qubits is pulled out, and replaced with one half of an EPR pair (or pairs). To distill entanglement between the removed qubit(s) and the other half of the of the EPR pair(s), one applies the recent layers of the monitored circuit in reverse and compares the new measurement results to the previous ones. By summing the two sets of measurement outcomes $s=m\cdot \Bar{m}$, one can find a recovery operator to distill the entanglement YoshidaCode. Distillability corresponds to recoverability of classical information in a dual error-correcting code.