Quantum Fourier Analysis

• URL: http://arxiv.org/abs/2002.03477v1
• Date: Mon, 10 Feb 2020 00:25:53 GMT
• Title: Quantum Fourier Analysis
• Authors: Arthur Jaffe, Chunlan Jiang, Zhengwei Liu, Yunxiang Ren, and Jinsong Wu
• Abstract summary: Quantum Fourier analysis is a new subject that combines an algebra with analytic estimates. This provides interesting tools to investigate phenomena such as quantum symmetry. We cite several applications of the quantum Fourier analysis in subfactor theory, in category theory, and in quantum information.
• Score: 1.776439648597615
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: {\em Quantum Fourier analysis} is a new subject that combines an algebraic Fourier transform (pictorial in the case of subfactor theory) with analytic estimates. This provides interesting tools to investigate phenomena such as quantum symmetry. We establish bounds on the quantum Fourier transform $\FS$, as a map between suitably defined $L^{p}$ spaces, leading to a new uncertainty principle for relative entropy. We cite several applications of the quantum Fourier analysis in subfactor theory, in category theory, and in quantum information. We suggest a new topological inequality, and we outline several open problems.

Related papers

• Highly-efficient quantum Fourier transformations for some nonabelian groups [0.0]
  We present fast quantum Fourier transformations for a number of nonabelian groups of interest for high energy physics. For each group, we derive explicit quantum circuits and estimate resource scaling for fault-tolerant implementations.
  arXiv  Detail & Related papers  (2024-07-31T18:00:04Z)
• Quantum Hamilton-Jacobi Theory, Spectral Path Integrals and Exact-WKB [0.0]
  Hamilton-Jacobi theory is a powerful formalism, but its utility is not explored in quantum theory beyond the correspondence principle. We propose a new way to perform path integrals in quantum mechanics by using a quantum version of Hamilton-Jacobi theory.
  arXiv  Detail & Related papers  (2024-06-12T02:50:43Z)
• Fast Fourier transforms and fast Wigner and Weyl functions in large quantum systems [0.0]
  Two methods for fast Fourier transforms are used in a quantum context. The first method is for systems with dimension of the Hilbert space $D=dn$ with $d$ an odd integer. The second method is also used for the fast calculation of Wigner and Weyl functions, in quantum systems with large finite dimension of the Hilbert space.
  arXiv  Detail & Related papers  (2024-05-08T15:54:35Z)
• Nonlocal Quantum Field Theory and Quantum Entanglement [0.0]
  We discuss the nonlocal nature of quantum mechanics and the link with relativistic quantum mechanics such as quantum field theory. We use here a nonlocal quantum field theory (NLQFT) which is finite, satisfies Poincar'e invariance, unitarity and microscopic causality.
  arXiv  Detail & Related papers  (2023-07-21T20:42:07Z)
• Geometric relative entropies and barycentric Rényi divergences [16.385815610837167]
  monotone quantum relative entropies define monotone R'enyi quantities whenever $P$ is a probability measure. We show that monotone quantum relative entropies define monotone R'enyi quantities whenever $P$ is a probability measure.
  arXiv  Detail & Related papers  (2022-07-28T17:58:59Z)
• Speeding up Learning Quantum States through Group Equivariant Convolutional Quantum Ans\"atze [13.651587339535961]
  We develop a framework for convolutional quantum circuits with SU$(d)$symmetry. We prove Harrow's statement on equivalence between $nameSU(d)$ and $S_n$ irrep bases.
  arXiv  Detail & Related papers  (2021-12-14T18:03:43Z)
• $\PT$ Symmetry and Renormalisation in Quantum Field Theory [62.997667081978825]
  Quantum systems governed by non-Hermitian Hamiltonians with $PT$ symmetry are special in having real energy eigenvalues bounded below and unitary time evolution. We show how $PT$ symmetry may allow interpretations that evade ghosts and instabilities present in an interpretation of the theory within a Hermitian framework.
  arXiv  Detail & Related papers  (2021-03-27T09:46:36Z)
• Learning Set Functions that are Sparse in Non-Orthogonal Fourier Bases [73.53227696624306]
  We present a new family of algorithms for learning Fourier-sparse set functions. In contrast to other work that focused on the Walsh-Hadamard transform, our novel algorithms operate with recently introduced non-orthogonal Fourier transforms. We demonstrate effectiveness on several real-world applications.
  arXiv  Detail & Related papers  (2020-10-01T14:31:59Z)
• Quantum information theory and Fourier multipliers on quantum groups [0.0]
  We compute the exact values of the minimum output entropy and the completely bounded minimal entropy of quantum channels acting on matrix algebras. Our results use a new and precise description of bounded Fourier multipliers from $mathrmL1(mathbbG)$ into $mathrmLp(mathbbG)$ for $1 p leq infty$ where $mathbbG$ is a co-amenable locally compact quantum group.
  arXiv  Detail & Related papers  (2020-08-27T09:47:10Z)
• Sample-efficient learning of quantum many-body systems [17.396274240172122]
  We study the problem of learning the Hamiltonian of a quantum many-body system given samples from its Gibbs state. We give the first sample-efficient algorithm for the quantum Hamiltonian learning problem.
  arXiv  Detail & Related papers  (2020-04-15T18:01:59Z)
• Quantum Statistical Complexity Measure as a Signalling of Correlation Transitions [55.41644538483948]
  We introduce a quantum version for the statistical complexity measure, in the context of quantum information theory, and use it as a signalling function of quantum order-disorder transitions. We apply our measure to two exactly solvable Hamiltonian models, namely: the $1D$-Quantum Ising Model and the Heisenberg XXZ spin-$1/2$ chain. We also compute this measure for one-qubit and two-qubit reduced states for the considered models, and analyse its behaviour across its quantum phase transitions for finite system sizes as well as in the thermodynamic limit by using Bethe ansatz.
  arXiv  Detail & Related papers  (2020-02-05T00:45:21Z)

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.