Quantum Detection of Recurrent Dynamics

• URL: http://arxiv.org/abs/2407.16055v1
• Date: Mon, 22 Jul 2024 21:13:45 GMT
• Title: Quantum Detection of Recurrent Dynamics
• Authors: Michael H. Freedman,
• Abstract summary: We describe simple quantum algorithms to detect such approximate recurrence. Hidden tensor structures have been observed to emerge both in a high energy context of operator-level spontaneous symmetry breaking. We collect some observations on the computational difficulty of locating these structures and detecting related spectral information.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

Related papers

• Near-Optimal and Tractable Estimation under Shift-Invariance [0.21756081703275998]
  Class of all such signals is but extremely rich: it contains all exponential oscillations over $mathbbCn$ with total degree $s$. We show that the statistical complexity of this class, as measured by the radius squared minimax frequencies of the $(delta)$-confidence $ell$-ball, is nearly the same as for the class of $s$-sparse signals, namely $Oleft(slog(en) + log(delta-1)right) cdot log(en/s)
  arXiv  Detail & Related papers  (2024-11-05T18:11:23Z)
• Tensor network approximation of Koopman operators [0.0]
  We propose a framework for approximating the evolution of observables of measure-preserving ergodic systems. Our approach is based on a spectrally-convergent approximation of the skew-adjoint Koopman generator. A key feature of this quantum-inspired approximation is that it captures information from a tensor product space of dimension $(2d+1)n$.
  arXiv  Detail & Related papers  (2024-07-09T21:40:14Z)
• Geometry of degenerate quantum states, configurations of $m$-planes and invariants on complex Grassmannians [55.2480439325792]
  We show how to reduce the geometry of degenerate states to the non-abelian connection $A$. We find independent invariants associated with each triple of subspaces. Some of them generalize the Berry-Pancharatnam phase, and some do not have analogues for 1-dimensional subspaces.
  arXiv  Detail & Related papers  (2024-04-04T06:39:28Z)
• A Unified Framework for Uniform Signal Recovery in Nonlinear Generative Compressed Sensing [68.80803866919123]
  Under nonlinear measurements, most prior results are non-uniform, i.e., they hold with high probability for a fixed $mathbfx*$ rather than for all $mathbfx*$ simultaneously. Our framework accommodates GCS with 1-bit/uniformly quantized observations and single index models as canonical examples. We also develop a concentration inequality that produces tighter bounds for product processes whose index sets have low metric entropy.
  arXiv  Detail & Related papers  (2023-09-25T17:54:19Z)
• Detection-Recovery Gap for Planted Dense Cycles [72.4451045270967]
  We consider a model where a dense cycle with expected bandwidth $n tau$ and edge density $p$ is planted in an ErdHos-R'enyi graph $G(n,q)$. We characterize the computational thresholds for the associated detection and recovery problems for the class of low-degree algorithms.
  arXiv  Detail & Related papers  (2023-02-13T22:51:07Z)
• Quantum and classical low-degree learning via a dimension-free Remez inequality [52.12931955662553]
  We show a new way to relate functions on the hypergrid to their harmonic extensions over the polytorus. We show the supremum of a function $f$ over products of the cyclic group $exp(2pi i k/K)_k=1K$. We extend to new spaces a recent line of work citeEI22, CHP, VZ22 that gave similarly efficient methods for learning low-degrees on hypercubes and observables on qubits.
  arXiv  Detail & Related papers  (2023-01-04T04:15:40Z)
• Markovian Repeated Interaction Quantum Systems [0.0]
  We study a class of dynamical semigroups $(mathbbLn)_ninmathbbN$ that emerge, by a Feynman--Kac type formalism, from a random quantum dynamical system. As a physical application, we consider the case where the $mathcalL_omega$'s are the reduced dynamical maps describing the repeated interactions of a system with thermal probes.
  arXiv  Detail & Related papers  (2022-02-10T20:52:40Z)
• Uncertainties in Quantum Measurements: A Quantum Tomography [52.77024349608834]
  The observables associated with a quantum system $S$ form a non-commutative algebra $mathcal A_S$. It is assumed that a density matrix $rho$ can be determined from the expectation values of observables. Abelian algebras do not have inner automorphisms, so the measurement apparatus can determine mean values of observables.
  arXiv  Detail & Related papers  (2021-12-14T16:29:53Z)
• Random matrices in service of ML footprint: ternary random features with no performance loss [55.30329197651178]
  We show that the eigenspectrum of $bf K$ is independent of the distribution of the i.i.d. entries of $bf w$. We propose a novel random technique, called Ternary Random Feature (TRF) The computation of the proposed random features requires no multiplication and a factor of $b$ less bits for storage compared to classical random features.
  arXiv  Detail & Related papers  (2021-10-05T09:33:49Z)
• Spectral statistics in constrained many-body quantum chaotic systems [0.0]
  We study the spectral statistics of spatially-extended many-body quantum systems with on-site Abelian symmetries or local constraints. In particular, we analytically argue that in a system of length $L$ that conserves the $mth$ multipole moment, $t_mathrmTh$ scales subdiffusively as $L2(m+1)$.
  arXiv  Detail & Related papers  (2020-09-24T17:59:57Z)
• Emergent universality in critical quantum spin chains: entanglement Virasoro algebra [1.9336815376402714]
  Entanglement entropy and entanglement spectrum have been widely used to characterize quantum entanglement in extended many-body systems. We show that the Schmidt vectors $|v_alpharangle$ display an emergent universal structure, corresponding to a realization of the Virasoro algebra of a boundary CFT.
  arXiv  Detail & Related papers  (2020-09-23T21:22:51Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.