Quantum complexity of time evolution with chaotic Hamiltonians

被引:92
作者
Balasubramanian, Vijay [1 ,2 ,3 ]
DeCross, Matthew [1 ]
Kar, Arjun [1 ]
Parrikar, Onkar [1 ]
机构
[1] Univ Penn, David Rittenhouse Lab, 209 S 33rd St, Philadelphia, PA 19104 USA
[2] Vrije Univ Brussel VUB, Theoret Nat Kunde, Pleinlaan 2, B-1050 Brussels, Belgium
[3] Int Solvay Inst, Pleinlaan 2, B-1050 Brussels, Belgium
基金
美国国家科学基金会;
关键词
AdS-CFT Correspondence; Field Theories in Lower Dimensions;
D O I
10.1007/JHEP01(2020)134
中图分类号
O412 [相对论、场论]; O572.2 [粒子物理学];
学科分类号
摘要
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 not subset of BQP/poly and PSPACE not subset of BQSUBEXP/subexp, and the "fast-forwarding" of quantum Hamiltonians.
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页数:44
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