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arXiv · 2206.14205

Linear Growth of Circuit Complexity from Brownian Dynamics

Shao-Kai Jian·Gregory Bentsen·Brian Swingle
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Abstract

We calculate the frame potential for Brownian clusters of $N$ spins or fermions with time-dependent all-to-all interactions. In both cases the problem can be mapped to an effective statistical mechanics problem which we study using a path integral approach. We argue that the $k$th frame potential comes within $\epsilon$ of the Haar value after a time of order $t \sim k N + k \log k + \log \epsilon^{-1}$. Using a bound on the diamond norm, this implies that such circuits are capable of coming very close to a unitary $k$-design after a time of order $t \sim k N$. We also consider the same question for systems with a time-independent Hamiltonian and argue that a small amount of time-dependent randomness is sufficient to generate a $k$-design in linear time provided the underlying Hamiltonian is quantum chaotic. These models provide explicit examples of linear complexity growth that are also analytically tractable.

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Shao-Kai Jian, Gregory Bentsen, Brian Swingle. 2022-06-28. Linear Growth of Circuit Complexity from Brownian Dynamics. https://arxiv.org/abs/2206.14205

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