Quantum detection of recurrent dynamics

Quantum dynamics that explore an unexpectedly small fraction of Hilbert space is inherently interesting. Integrable systems, quantum scars, MBL, hidden tensor structures, and systems with gauge symmetries are examples. Beyond dimension and volume, spectral features such as an [Formula: see text]-density of periodic eigenvalues, or other spectral features, can also imply observable recurrence. Low volume dynamics will recur near its initial state [Formula: see text] more rapidly, i.e. [Formula: see text], is more likely to occur for modest values of [Formula: see text], when the (forward) orbit [Formula: see text] is of relatively low dimension [Formula: see text] and relatively small [Formula: see text]-volume. We describe simple quantum algorithms to detect such approximate recurrence. Applications include detection of certain cases of hidden tensor factorizations [Formula: see text]. “Hidden” refers to an unknown conjugation, e.g. [Formula: see text], which will obscure the low-volume nature of the dynamics. Hidden tensor structures have been observed to emerge both in a high energy context of operator-level spontaneous symmetry breaking [M. Freedman and M. Shokrian-Zini, J. High Energy Phys. 2021 (2021) 140; M. Freedman and M. Shokrian-Zini, J. High Energy Phys. 2021 (2021) 102; M. Freedman and M. Shokrian-Zini, arXiv:2112.08613; M. Shokrian-Zini, A. R. Brown and M. Freedman, J. High Energy Phys. 2023 (2023) 82], and at the opposite end of the intellectual world in linguistics [P. Smolensky, Artif. Intell. 46(1–2) (1990) 159; R. T. McCoy, T. Linzen, E. Dunbar and P. Smolensky, RNNs implicitly implement tensor product representations, in The Seventh Int. Conf. Learning Representations (ICLR, 2019), pp. 1–22]. We collect some observations on the computational difficulty of locating these structures and detecting related spectral information. A technical result, Appendix A, is that the language describing unitary circuits with no spectral gap (NUSG) around 1 is QMA-complete. Appendix B connects the Kolmogorov–Arnold representation theorem to hidden tensor structures.