On the complexity of sampling from shallow Brownian circuits

• URL: http://arxiv.org/abs/2411.04169v1
• Date: Wed, 06 Nov 2024 19:00:00 GMT
• Title: On the complexity of sampling from shallow Brownian circuits
• Authors: Gregory Bentsen, Bill Fefferman, Soumik Ghosh, Michael J. Gullans, Yinchen Liu,
• Abstract summary: We study a constant-time Brownian circuit model which shares many similarities with constant-depth random quantum circuits. We show that the output distributions of Brownian circuits at shallow depths follow a Porter-Thomas distribution, just like in the case of deep 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.
• Score: 0.0
• License:
• Abstract: 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.

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