Quantum complexity of time evolution with chaotic Hamiltonians
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Springer
Received: October 10, 2019 Accepted: January 2, 2020 Published: January 21, 2020
Quantum complexity of time evolution with chaotic Hamiltonians
a
David Rittenhouse Laboratory, University of Pennsylvania, 209 S.33rd Street, Philadelphia, PA 19104, U.S.A. b Theoretische Natuurkunde, Vrije Universiteit Brussel (VUB), and International Solvay Institutes, Pleinlaan 2, Brussels B-1050, Belgium
E-mail: [email protected], [email protected], [email protected], [email protected] 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 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. Keywords: AdS-CFT Correspondence, Field Theories in Lower Dimensions ArXiv ePrint: 1905.05765
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP01(2020)134
JHEP01(2020)134
Vijay Balasubramanian,a,b Matthew DeCross,a Arjun Kara and Onkar Parrikara
Contents 1 Introduction
1 5 8 11 14
3 Conjugate points and the eigenstate complexity hypothesis 3.1 Solving the Jacobi equation 3.2 Conjugate points as zero modes 3.3 Conjugate points in the bi-invariant case 3.4 Turning on cost factors
16 18 19 20 21
4 Discussion 4.1 Late-time saturation 4.2 Quantum computation 4.3 Quantum chaos
28 29 31 32
A Majorana Fermion basis for su(2N/2 )
33
B Conjugate points in perturbation theory
36
C Some more details on ECH
38
1
Introduction
In recent years the late time dynamics of general relativity have been examined through various lenses. Two of the most prominent directions in th
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