2024/11/06 by Gregory Bentsen, Bentsen, Gregory, Bill Fefferman +7 · 1 citation
Computer Science · #FOS: Physical sciences #Machine Learning and Algorithms #Neural Networks and Applications #Quantum Physics (quant-ph) #Statistical Mechanics (cond-mat.stat-mech) #Stochastic Gradient Optimization Techniques
paper · pdf · doi:10.48550/arxiv.2411.04169
openalex publication_date 2024/11/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
While many statistical properties of deep random quantum circuits can be deduced, often rigorously and other times heuristically, by an approximation to global Haar-random unitaries, the statistics of constant-depth random quantum circuits are generally less well-understood due to a lack of amenable tools and techniques. We circumvent this barrier by considering a related constant-time Brownian circuit model which shares many similarities with constant-depth random quantum circuits but crucially allows for direct calculations of higher order moments of its output distribution. Using mean-field (large-n) techniques, we fully characterize the output distributions of Brownian circuits at shallow depths and show that they follow a Porter-Thomas distribution, just like in the case of deep circuits, but with a truncated Hilbert space. The access to higher order moments allows for studying the expected and typical Linear Cross-entropy (XEB) benchmark scores achieved by an ideal quantum computer versus the state-of-the-art classical spoofers for shallow Brownian circuits. We discover that for these circuits, while the quantum computer typically scores within a constant factor of the expected value, the classical spoofer suffers from an exponentially larger variance. Numerical evidence suggests that the same phenomenon also occurs in constant-depth discrete random quantum circuits, like those defined over the all-to-all architecture. We conjecture that the same phenomenon is also true for random brickwork circuits in high enough spatial dimension.