Nicolas Macris

Nielsen proposed an upper bound on the complexity, defined as the total number of 1 and 2-qubit gates, of simulating a unitary evolution through a quantum circuit. In this work, we study a bound on this complexity. We provide evidence that the fluctuations of this quantity distinguish between integrable and chaotic systems. For random Hamiltonians with GOE, GUE, or Poisson level statistics, the expected height and variance of the plateau reached by the bound for sufficiently large times are universal regardless of the spectral density and level statistics. However, there is an intermediate time scale (depending on the number of qubits) such that the amplitude of fluctuations, and hence the variance, shows a dependence on the chaotic versus integrable level statistics. Our results are analytical and rely on expressions for the spectral form factor of two-point correlation functions. They corroborate previous numerical observations and heuristic arguments of Craps, De ClercK, Evnin, Hacker, and Pavlov. We also analyze various solvable fermion systems and find similar phenomena.