Fig. 1: Overview.

a A quantum computer simulates dynamics either from a chaotic ensemble or a pseudochaotic ensemble. Any poly-time quantum algorithm \({{{\mathcal{A}}}}\) on a polynomial number of copies with poly-time classical post-processing fails to pin down whether the dynamics is chaotic or not by measuring OTOC. An example of pseudochaotic dynamics is obtained by conjugating the dynamics in a subspace by a random permutation. To make this dynamics pseudochaotic, the dimension of the subspace 2^k should be given by \(k=\omega (\log n)\) with the number of qubits n, which is much smaller than the entire space dimension 2ⁿ. A circuit implementation of this dynamics requires \({{{\rm{polylog}}}}(n)\) depth with all-to-all connectivity. b We can schematically classify how the late-time 1/O_VW scales with n into three different regimes. In a chaotic system (orange color), this scaling is exponential in n. For a system with local scrambling (blue color), the scaling is at most polynomial in n. Pseudochaotic dynamics (peach color) exhibits the scaling which falls between these two. c The 2^k-dimensional subspace is mapped to the entire space by isometries {O_a}. \({{{{\mathcal{V}}}}}_{{{{\rm{sub}}}}}\) is the space spanned by an ensemble of unitary operators {u} in the subspace, which could be non-chaotic. Through the action of O_a, \({{{{\mathcal{V}}}}}_{{{{\rm{sub}}}}}\) is mapped to \({{{\mathcal{V}}}}\) within the entire Hilbert space, preserving its dimension. Remarkably, even if the subspace dimension is negligibly smaller than the entire space dimension, its ensemble average over random isometries cannot be distinguished from chaotic dynamics by any poly-time quantum algorithms with access to polynomially many copies of evolved states.