Time evolution of complexity: a critique of three methods

Ali, Tibra (0000 0000 8658 0851, Perimeter Institute, 31 Caroline Street North, Waterloo, Ontario, N2L 2Y5, Canada) ; Bhattacharyya, Arpan (0000 0004 0372 2033, Center for Gravitational Physics, Yukawa Institute for Theoretical Physics (YITP), Kyoto University, Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto, 606-8502, Japan) ; Haque, S. (0000 0004 1936 9596, Department of Physics, University of Windsor, 401 Sunset Avenue, Windsor, Ontario, N9B 3P4, Canada) ; Kim, Eugene (0000 0004 1936 9596, Department of Physics, University of Windsor, 401 Sunset Avenue, Windsor, Ontario, N9B 3P4, Canada) ; Moynihan, Nathan (0000 0004 1937 1151, The Laboratory for Quantum Gravity & Strings, Department of Mathematics & Applied Mathematics, University of Cape Town, Private Bag, Rondebosch, 7701, South Africa)

12 April 2019

Abstract: In this work, we propose a testing procedure to distinguish between the different approaches for computing complexity. Our test does not require a direct comparison between the approaches and thus avoids the issue of choice of gates, basis, etc. The proposed testing procedure employs the information-theoretic measures Loschmidt echo and Fidelity; the idea is to investigate the sensitivity of the complexity (derived from the different approaches) to the evolution of states. We discover that only circuit complexity obtained directly from the wave function is sensitive to time evolution, leaving us to claim that it surpasses the other approaches. We also demonstrate that circuit complexity displays a universal behaviour — the complexity is proportional to the number of distinct Hamiltonian evolutions that act on a reference state. Due to this fact, for a given number of Hamiltonians, we can always find the combination of states that provides the maximum complexity; consequently, other combinations involving a smaller number of evolutions will have less than maximum complexity and, hence, will have resources. Finally, we explore the evolution of complexity in non-local theories; we demonstrate the growth of complexity is sustained over a longer period of time as compared to a local theory.

Published in: JHEP 1904 (2019) 087
Published by: Springer/SISSA
DOI: 10.1007/JHEP04(2019)087
arXiv: 1810.02734
License: CC-BY-4.0

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