On the Entanglement Entropy Distribution of a Hybrid Quantum Circuit

Abstract

We investigate the distribution of entanglement entropy in hybrid quantum circuits consisting of random unitary gates and local measurements applied at a finite rate. We demonstrate that higher moments of the entanglement entropy distribution, such as a ratio between the variance and the mean and skewness, capture nontrivial features of the measurement-induced dynamics that are invisible to the mean entropy alone. We demonstrate that these quantities exhibit distinct and robust behaviors across the volume-law and area-law phases, and can serve as effective diagnostics of measurement-induced entanglement transitions. We propose a phenomenological model describing the effect of measurements in the area-law regime, which, when combined with the directed polymer in a random environment description of the volume-law phase, well matches numerical simulations across the entire phase diagram.

Figures (13)

• Figure 1: Schematic diagram of the entanglement entropy spectrum of hybrid circuits as the measurement rate increases. The changing shape can be quantified by the skewness. Each value of the index of dispersion (IoD) corresponds to each of the distributions shown.
• Figure 2: (Left) Histograms of entanglement entropy distributions of the hybrid circuits at measurement rates $p = 0.023, 0.16$, and $0.22$, which lie in the volume-law phase, at criticality, and in the area-law phase, respectively. (Middle) IoD curve and (Right) skewness curve with increasing rate of measurement, $p$, with yellow markers indicating values of $p$ that correspond to the histograms.
• Figure 3: A hybrid quantum circuit consisting of random unitary gates sampled from uniformly distributed two qubit Clifford gates arranged in a brickwork fashion and projective measurements acting on each qubit at a rate $p$.
• Figure 4: Collapse of the entanglement entropy of different system sizes according to the scaling form in Eq. \ref{['eq:Scaling']}. We reaffirm the known critical point, $p_c = 0.16$ and $\nu = 1.3$.
• Figure 5: Variance curves of both Haar random and Clifford random hybrid quantum circuits. The critical measurement rate for the Haar (Clifford) random circuit is indicated by a solid (dashed) vertical line. $p_c = 0.26$ for Haar random circuit and $p_c = 0.16$ for Clifford random circuit.