Quantum Detection of Recurrent Dynamics

TL;DR

This paper introduces quantum algorithms for detecting low-dimensional recurrent dynamics and hidden tensor structures in quantum systems, with applications in physics and linguistics, and discusses the computational complexity of related problems.

Contribution

It presents new quantum algorithms for approximate recurrence detection and hidden tensor structure identification, and proves the QMA-completeness of certain spectral gap problems.

Findings

Quantum algorithms can detect approximate recurrence in low-volume quantum dynamics.

Hidden tensor structures can be identified even when obscured by unknown conjugations.

Detecting spectral features related to recurrence is computationally hard, being QMA-complete.

Abstract

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 $O(1)$-density of periodic eigenvalues, or other spectral features, can also imply observable recurrence. Low volume dynamics will recur near its initial state $| \psi_0\rangle$ more rapidly, i.e. $\lVert\mathrm{U}^k | \psi_0\rangle - | \psi_0\rangle \rVert < \epsilon$, is more likely to occur for modest values of $k$, when the (forward) orbit $\operatorname{closure}(\{\mathrm{U}^k\}_{k=1,2,\dots})$ is of relatively low dimension $d$ and relatively small $d$-volume. We describe simple quantum algorithms to detect such approximate recurrence. Applications include detection of certain cases of hidden tensor factorizations $\mathrm{U} \cong V^\dagger(\mathrm{U}_1\otimes \cdots \otimes \mathrm{U}_n)V$. "Hidden" refers to an unknown conjugation, e.g. $\mathrm{U}_1 \otimes \cdots \otimes \mathrm{U}_v \rightarrow V^\dagger(\mathrm{U}_1 \otimes \cdots \otimes \mathrm{U}_n)V$, 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 [FSZ21a, FSZ21b, FSZ21c, SZBF23], and at the opposite end of the intellectual world in linguistics [Smo09, MLDS19]. 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.