Chaos and Measurement-Induced Criticality on Stabiliser Circuits

Abstract

Repeated local projective measurements are known to induce an entanglement transition in interacting many-body systems with unitary dynamics. We investigate this transition on stabiliser circuits evolving under Clifford gates. Following a brief review of results in the random circuit setting, we introduce an extension that models ‘deferred measurements’ by storing different quantum trajectories in ancillae. Our results demonstrate that the entanglement transition is lost when measurements are delayed; moreover, entanglement growth becomes localised in space, such that mutual information is nonzero only within a correlation length that diverges as the measurement probability approaches zero. Motivated by the extensive use of stabiliser states to study this class of problems, we turn our attention to the thermalising properties of Clifford circuits as diagnosed by their spectral statistics. We present an efficient algorithm for computing the spectral form factor for a Clifford Floquet and find that this form factor has an exponential ramp with immediate onset, sub-linear ramp time, and late-time mean greater than that of the Circular Unitary Ensemble. Our results show that, though Cliffords are known to successfully reproduce some aspects of chaos in Haar unitary circuits, they lack conventional signatures of thermalisation.