  Quantum dynamics that explores 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 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.
