Linear growth of circuit complexity from Brownian dynamics

Journal Article · 27 August 2023 · Journal of High Energy Physics (Online)

DOI: https://doi.org/10.1007/jhep08(2023)190 · OSTI ID:2421204

Jian, Shao-Kai ^[1]; Bentsen, Gregory ^[2]; Swingle, Brian ^[2]

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1. Tulane Univ., New Orleans, LA (United States); Univ. of California, Oakland, CA (United States)
2. Brandeis Univ., Waltham, MA (United States)

How rapidly can a many-body quantum system generate randomness? Using path integral methods, we demonstrate that Brownian quantum systems have circuit complexity that grows linearly with time. In particular, we study Brownian clusters of N spins or fermions with time-dependent all-to-all interactions, and calculate the Frame Potential to characterize complexity growth in these models. In both cases the problem can be mapped to an effective statistical mechanics problem which we study using path integral methods. Within this framework it is straightforward to show that the kth Frame Potential comes within ϵ of the Haar value after a time of order t ~ kN + k log k + log ϵ^–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 ~ kN. 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 analytically tractable and are directly applicable to practical applications calling for unitary k-designs.

• Research Organization: Univ. of California, Oakland, CA (United States)
• Sponsoring Organization: USDOE Office of Science (SC)
• Grant/Contract Number: SC0019380
• OSTI ID: 2421204
• Journal Information: Journal of High Energy Physics (Online), Journal Name: Journal of High Energy Physics (Online) Journal Issue: 8 Vol. 2023; ISSN 1029-8479
• Publisher: Springer NatureCopyright Statement
• Country of Publication: United States
• Language: English