Quantum walks driven by quantum coins with two multiple eigenvalues

• URL: http://arxiv.org/abs/2110.00716v1
• Date: Sat, 2 Oct 2021 03:23:51 GMT
• Title: Quantum walks driven by quantum coins with two multiple eigenvalues
• Authors: Norio Konno, Iwao Sato, Etsuo Segawa, Yutaka Shikano
• Abstract summary: 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.
• Score: 0.0
• License: http://creativecommons.org/publicdomain/zero/1.0/
• Abstract: We consider a spectral analysis on the quantum walks on graph $G=(V,E)$ with the local coin operators $\{C_u\}_{u\in V}$ and the flip flop shift. The quantum coin operators have commonly two distinct eigenvalues $\kappa,\kappa'$ and $p=\dim(\ker(\kappa-C_u))$ for any $u\in V$ with $1\leq p\leq \delta(G)$, where $\delta(G)$ is the minimum degrees of $G$. We show that this quantum walk can be decomposed into a cellular automaton on $\ell^2(V;\mathbb{C}^p)$ whose time evolution is described by a self adjoint operator $T$ and its remainder. We obtain how the eigenvalues and its eigenspace of $T$ are lifted up to as those of the original quantum walk. As an application, we express the eigenpolynomial of the Grover walk on $\mathbb{Z}^d$ with the moving shift in the Fourier space.

Related papers

• Study And Implementation of Unitary Gates in Quantum Computation Using Schrodinger Dynamics [0.0]
  This thesis explores the concept of realizing quantum gates using physical systems like atoms and oscillators perturbed by electric and magnetic fields. In all design procedures, the gates that appear are infinite-dimensional, with an interaction between the atom and the electromagnetic field modulated by a controllable function of time.
  arXiv  Detail & Related papers  (2024-08-30T06:08:35Z)
• Quantum connection, charges and virtual particles [65.268245109828]
  A quantum bundle $L_hbar$ is endowed with a connection $A_hbar$ and its sections are standard wave functions $psi$ obeying the Schr"odinger equation. We will lift the bundles $L_Cpm$ and connection $A_hbar$ on them to the relativistic phase space $T*R3,1$ and couple them to the Dirac spinor bundle describing both particles and antiparticles.
  arXiv  Detail & Related papers  (2023-10-10T10:27:09Z)
• Quantum Speedups for Zero-Sum Games via Improved Dynamic Gibbs Sampling [30.53587208999909]
  We give a quantum algorithm for computing an $epsilon$-approximate Nash equilibrium of a zero-sum game in a $m times n$ payoff matrix with bounded entries. Given a standard quantum oracle for accessing the payoff matrix our algorithm runs in time $widetildeO(sqrtm + ncdot epsilon-2.5 + epsilon-3)$ and outputs a classical representation of the $epsilon$-approximate Nash equilibrium.
  arXiv  Detail & Related papers  (2023-01-10T02:56:49Z)
• Algebraic Aspects of Boundaries in the Kitaev Quantum Double Model [77.34726150561087]
  We provide a systematic treatment of boundaries based on subgroups $Ksubseteq G$ with the Kitaev quantum double $D(G)$ model in the bulk. The boundary sites are representations of a $*$-subalgebra $Xisubseteq D(G)$ and we explicate its structure as a strong $*$-quasi-Hopf algebra. As an application of our treatment, we study patches with boundaries based on $K=G$ horizontally and $K=e$ vertically and show how these could be used in a quantum computer
  arXiv  Detail & Related papers  (2022-08-12T15:05:07Z)
• Factorized Hilbert-space metrics and non-commutative quasi-Hermitian observables [0.0]
  It is well known that an (in general, non-commutative) set of non-Hermitian operators $Lambda_j$ with real eigenvalues need not necessarily represent observables. We describe a specific class of quantum models in which these operators plus the underlying physical Hilbert-space metric $Theta$ are all represented.
  arXiv  Detail & Related papers  (2022-06-27T18:33:03Z)
• Beyond the Berry Phase: Extrinsic Geometry of Quantum States [77.34726150561087]
  We show how all properties of a quantum manifold of states are fully described by a gauge-invariant Bargmann. We show how our results have immediate applications to the modern theory of polarization.
  arXiv  Detail & Related papers  (2022-05-30T18:01:34Z)
• Complementarity in quantum walks [0.08896991256227595]
  We study discrete-time quantum walks on $d$-cycles with a position and coin-dependent phase-shift. For prime $d$ there exists a strong complementarity property between the eigenvectors of two quantum walk evolution operators. We show that the complementarity is still present in the continuous version of this model, which corresponds to a one-dimensional Dirac particle.
  arXiv  Detail & Related papers  (2022-05-11T12:47:59Z)
• 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)
• Quantum double aspects of surface code models [77.34726150561087]
  We revisit the Kitaev model for fault tolerant quantum computing on a square lattice with underlying quantum double $D(G)$ symmetry. We show how our constructions generalise to $D(H)$ models based on a finite-dimensional Hopf algebra $H$.
  arXiv  Detail & Related papers  (2021-06-25T17:03:38Z)
• A new type of spectral mapping theorem for quantum walks with a moving shift on graphs [0.2578242050187029]
  The spectral mapping theorem for quantum walks can only be applied for walks employing a shift operator whose square is the identity. We acquire a new spectral mapping theorem for the Grover walk with a shift operator whose cube is the identity on finite graphs.
  arXiv  Detail & Related papers  (2021-03-09T05:45:25Z)
• Wavefunction structure in quantum many-fermion systems with $k$-body interactions: conditional $q$-normal form of strength functions [0.40357531559576965]
  For finite quantum many-particle systems modeled with say $m$ fermions in $N$ single particle states, the wavefunction structure is studied using random matrix theory. It is shown that the first four moments of $F_kappa(E)$ are essentially same as that of the conditional $q$-normal distribution.
  arXiv  Detail & Related papers  (2020-11-09T11:53:39Z)

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.