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Quantum complexity of time evolution with chaotic Hamiltonians

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  • Published: 21 January 2020
  • Volume 2020, article number 134, (2020)
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Quantum complexity of time evolution with chaotic Hamiltonians
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  • Vijay Balasubramanian1,2,
  • Matthew DeCross1,
  • Arjun Kar1 &
  • …
  • Onkar Parrikar1 

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A preprint version of the article is available at arXiv.

Abstract

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 off- diagonal 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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Authors and Affiliations

  1. David Rittenhouse Laboratory, University of Pennsylvania, 209 S.33rd Street, Philadelphia, PA, 19104, USA

    Vijay Balasubramanian, Matthew DeCross, Arjun Kar & Onkar Parrikar

  2. Theoretische Natuurkunde, Vrije Universiteit Brussel (VUB), and International Solvay Institutes, Pleinlaan 2, B-1050, Brussels, Belgium

    Vijay Balasubramanian

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  1. Vijay Balasubramanian
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Correspondence to Onkar Parrikar.

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ArXiv ePrint: 1905.05765

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Balasubramanian, V., DeCross, M., Kar, A. et al. Quantum complexity of time evolution with chaotic Hamiltonians. J. High Energ. Phys. 2020, 134 (2020). https://doi.org/10.1007/JHEP01(2020)134

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  • Received: 10 October 2019

  • Accepted: 02 January 2020

  • Published: 21 January 2020

  • DOI: https://doi.org/10.1007/JHEP01(2020)134

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Keywords

  • AdS-CFT Correspondence
  • Field Theories in Lower Dimensions
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