Chaos and complexity by design

Roberts, Daniel A.; Yoshida, Beni

doi:10.1007/JHEP04(2017)121

Chaos and complexity by design

Oct 16, 2016

77 pages

Published in:

JHEP 04 (2017) 121

Published: Apr 20, 2017

e-Print:

1610.04903 [quant-ph]

DOI:

10.1007/JHEP04(2017)121

View in:

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Abstract: (Springer)

We study the relationship between quantum chaos and pseudorandomness by developing probes of unitary design. A natural probe of randomness is the “frame poten-tial,” which is minimized by unitary k-designs and measures the 2-norm distance between the Haar random unitary ensemble and another ensemble. A natural probe of quantum chaos is out-of-time-order (OTO) four-point correlation functions. We show that the norm squared of a generalization of out-of-time-order 2k-point correlators is proportional to the kth frame potential, providing a quantitative connection between chaos and pseudorandomness. Additionally, we prove that these 2k-point correlators for Pauli operators completely determine the k-fold channel of an ensemble of unitary operators. Finally, we use a counting argument to obtain a lower bound on the quantum circuit complexity in terms of the frame potential. This provides a direct link between chaos, complexity, and randomness.

Note:

46+many pages, and all the figures too. v2: the director's cut -- more jokes, less typos

AdS-CFT Correspondence
Gauge-gravity correspondence
Random Systems
Holography and condensed matter physics (AdS/CMT)
unitarity: operator
chaos
correlation function

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