Quantum complexity of time evolution with chaotic Hamiltonians

Balasubramanian, Vijay; DeCross, Matthew; Kar, Arjun; Parrikar, Onkar

doi:10.1007/jhep01(2020)134

Title: Quantum complexity of time evolution with chaotic Hamiltonians

Journal Article · 31 December 2019 · Journal of High Energy Physics (Online)

DOI: https://doi.org/10.1007/jhep01(2020)134 · OSTI ID:1800271

Balasubramanian, Vijay ^[1]; DeCross, Matthew ^[2]; Kar, Arjun ^[2]; Parrikar, Onkar ^[2]

Univ. of Pennsylvania, Philadelphia, PA (United States). David Rittenhouse Lab.; Vrije Univ., Brussel (Belgium). Theoretische Natuurkunde. International Solvay Inst.; OSTI
Univ. of Pennsylvania, Philadelphia, PA (United States). David Rittenhouse Lab.

We study the quantum complexity of time evolution in large-N chaotic systems, with the SYK model as our main example. This complexity is expected to increase linearly for exponential time prior to saturating at its maximum value, and is related to the length of minimal geodesics on the manifold of unitary operators that act on Hilbert space. Using the Euler-Arnold formalism, we demonstrate that there is always a geodesic between the identity and the time evolution operator e -iHt whose length grows linearly with time. This geodesic is minimal until there is an obstruction to its minimality, after which it can fail to be a minimum either locally or globally. We identify a criterion — the Eigenstate Complexity Hypothesis (ECH) — which bounds the overlap between offdiagonal energy eigenstate projectors and the k-local operators of the theory, and use it to argue that the linear geodesic will at least be a local minimum for exponential time. We show numerically that the large-N SYK model (which is chaotic) satisfies ECH and thus has no local obstructions to linear growth of complexity for exponential time, as expected from holographic duality. In contrast, we also study the case with N = 2 fermions (which is integrable) and find short-time linear complexity growth followed by oscillations. Our analysis relates complexity to familiar properties of physical theories like their spectra and the structure of energy eigenstates and has implications for the hypothesized computational complexity class separations PSPACE * BQP/poly and PSPACE * BQSUBEXP/subexp, and the “fast-forwarding” of quantum Hamiltonians.

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Research Organization:: Duke Univ., Durham, NC (United States); Univ. of Pennsylvania, Philadelphia, PA (United States)

Sponsoring Organization:: USDOE Office of Science (SC)

Grant/Contract Number:: FG02-05ER41367; SC0020360

OSTI ID:: 1800271

Journal Information:: Journal of High Energy Physics (Online), Journal Name: Journal of High Energy Physics (Online) Journal Issue: 1 Vol. 2020; ISSN 1029-8479

Publisher:: Springer BerlinCopyright Statement

Country of Publication:: United States

Language:: English

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What kind of “complexity” is dual to holographic complexity? Yang, Run-Qiu; An, Yu-Sen; Niu, Chao The European Physical Journal C, Vol. 82, Issue 3 https://doi.org/10.1140/epjc/s10052-022-10151-0	journal	March 2022
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