Factoring Discrete Quantum Walks on Distance Regular Graphs into Continuous Quantum Walks

• URL: http://arxiv.org/abs/2008.01224v2
• Date: Sun, 16 Oct 2022 22:15:08 GMT
• Title: Factoring Discrete Quantum Walks on Distance Regular Graphs into Continuous Quantum Walks
• Authors: Hanmeng Zhan
• Abstract summary: We consider a discrete-time quantum walk, called the Grover walk, on a distance regular graph $X$. We show that the square of the transition matrix of the Grover walk on $X$ is a product of at most $d$ commuting transition matrices of continuous-time quantum walks.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We consider a discrete-time quantum walk, called the Grover walk, on a distance regular graph $X$. Given that $X$ has diameter $d$ and invertible adjacency matrix, we show that the square of the transition matrix of the Grover walk on $X$ is a product of at most $d$ commuting transition matrices of continuous-time quantum walks, each on some distance digraph of the line digraph of $X$. We also obtain a similar factorization for any graph $X$ in a Bose Mesner algebra.

Related papers

• Polynomial-depth quantum algorithm for computing matrix determinant [46.13392585104221]
  We propose an algorithm for calculating the determinant of a square matrix, and construct a quantum circuit realizing it. Each row of the matrix is encoded as a pure state of some quantum system. The admitted matrix is therefore arbitrary up to the normalization of quantum states of those systems.
  arXiv  Detail & Related papers  (2024-01-29T23:23:27Z)
• Discrete-time quantum walks on Cayley graphs of Dihedral groups using generalized Grover coins [0.0]
  We study discrete-time quantum walks on Cayley graphs corresponding to Dihedral groups. We show that the walks are periodic only for coins that are permutation or negative of a permutation matrix.
  arXiv  Detail & Related papers  (2023-09-26T18:53:35Z)
• Re-Analyze Gauss: Bounds for Private Matrix Approximation via Dyson Brownian Motion [28.431572772564518]
  Given a symmetric matrix $M$ and a vector $lambda$, we present new bounds on the Frobenius-distance utility of the Gaussian mechanism for approximating $M$ by a matrix. Our bounds depend on both $lambda$ and the gaps in the eigenvalues of $M$, and hold whenever the top $k+1$ eigenvalues of $M$ have sufficiently large gaps.
  arXiv  Detail & Related papers  (2022-11-11T18:54:01Z)
• Periodicity of bipartite walk on biregular graphs with conditional spectra [0.0]
  We study a class of discrete quantum walks, known as bipartite walks. Any discrete quantum walk is given by the powers of a unitary matrix $U$ indexed by arcs or edges of the underlying graph. We apply periodicity results of bipartite walks to get a characterization of periodicity of Grover's walk on regular graphs.
  arXiv  Detail & Related papers  (2022-11-04T21:02:30Z)
• Quantum teleportation in the commuting operator framework [63.69764116066747]
  We present unbiased teleportation schemes for relative commutants $N'cap M$ of a large class of finite-index inclusions $Nsubseteq M$ of tracial von Neumann algebras. We show that any tight teleportation scheme for $N$ necessarily arises from an orthonormal unitary Pimsner-Popa basis of $M_n(mathbbC)$ over $N'$.
  arXiv  Detail & Related papers  (2022-08-02T00:20:46Z)
• On quantum algorithms for the Schr\"odinger equation in the semi-classical regime [27.175719898694073]
  We consider Schr"odinger's equation in the semi-classical regime. Such a Schr"odinger equation finds many applications, including in Born-Oppenheimer molecular dynamics and Ehrenfest dynamics.
  arXiv  Detail & Related papers  (2021-12-25T20:01:54Z)
• Quantum simulation of perfect state transfer on weighted cubelike graphs [0.0]
  A continuous-time quantum walk on a graph evolves according to the unitary operator $e-iAt$. Perfect state transfer (PST) in a quantum walk is the transfer of a quantum state from one node to another node with $100%$ fidelity.
  arXiv  Detail & Related papers  (2021-10-30T10:42:54Z)
• Quantum walks driven by quantum coins with two multiple eigenvalues [0.0]
  We consider a spectral analysis on the quantum walks on graph $G=(V,E)$ with the local coin operators $C_u_uin V$ and the flip flop shift. We show that this quantum walk can be into a cellular automaton on $ell2(V;mathbbCp)$ whose time evolution is described by a self adjoint operator $T$ and its remainder.
  arXiv  Detail & Related papers  (2021-10-02T03:23:51Z)
• A Matrix Chernoff Bound for Markov Chains and Its Application to Co-occurrence Matrices [30.49132891333353]
  We prove a Chernoff-type bound for sums of matrix-valued random variables sampled via a regular (aperiodic and irreducible) finite Markov chain. Our proof is based on the matrix expander (regular undirected graph) Chernoff bound [Garg et al. STOC '18] and scalar Chernoff-Hoeffding bounds for Markov chains. Our matrix Chernoff bound for Markov chains can be applied to analyze the behavior of co-occurrence statistics for sequential data.
  arXiv  Detail & Related papers  (2020-08-06T05:44:54Z)
• A crossover between open quantum random walks to quantum walks [0.0]
  The walk connects an open quantum random walk and a quantum walk with parameters $Min mathbbN$ controlling a decoherence effect. We analytically show that a typical behavior of quantum walks appears even in a small gap of the parameter from the open quantum random walk.
  arXiv  Detail & Related papers  (2020-07-02T07:42:24Z)
• Continuous-time quantum walks in the presence of a quadratic perturbation [55.41644538483948]
  We address the properties of continuous-time quantum walks with Hamiltonians of the form $mathcalH= L + lambda L2$. We consider cycle, complete, and star graphs because paradigmatic models with low/high connectivity and/or symmetry.
  arXiv  Detail & Related papers  (2020-05-13T14:53:36Z)

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.