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arXiv:2302.04140 · SciPost Physics Core

A simple coin for a 2d entangled walk

Ahmadullah Zahed, Kallol Sen

Published Jul 10, 2023• 3 citations
arXivHTMLPDF

Abstract

We analyze the effect of a simple coin operator, built out of Bell pairs, in a 2d Discrete Quantum Random Walk (DQRW) problem. The specific form of the coin enables us to find analytical and closed form solutions to the recursion relations of the DQRW. The coin induces entanglement between the spin and position degrees of freedom, which oscillates with time and reaches a constant value asymptotically. We probe the entangling properties of the coin operator further, by two different measures. First, by integrating over the space of initial tensor product states, we determine the Entangling Power of the coin operator. Secondly, we compute the Generalized Relative Rényi Entropy between the corresponding density matrices for the entangled state and the initial pure unentangled state. Both the Entangling Power and Generalized Relative Rényi Entropy behaves similar to the entanglement with time. Finally, in the continuum limit, the specific coin operator reduces the 2d DQRW into two 1d massive fermions coupled to synthetic gauge fields, where both the mass term and the gauge fields are built out of the coin parameters.

DOI
10.21468/scipostphyscore.6.3.048

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We analyze the effect of a simple coin operator, built out of Bell pairs, in a 2d Discrete Quantum Random Walk (DQRW) problem. The specific form of the coin enables us to find analytical and closed form solutions to the recursion relations of the DQRW. The coin induces entanglement between the spin and position degrees of freedom, which oscillates with time and reaches a constant value asymptotically. We probe the entangling properties of the coin operator further, by two different measures. First, by integrating over the space of initial tensor product states, we determine the Entangling Power of the coin operator. Secondly, we compute the Generalized Relative Rényi Entropy between the corresponding density matrices for the entangled state and the initial pure unentangled state. Both the Entangling Power and Generalized Relative Rényi Entropy behaves similar to the entanglement with time. Finally, in the continuum limit, the specific coin operator reduces the 2d DQRW into two 1d massive fermions coupled to synthetic gauge fields, where both the mass term and the gauge fields are built out of the coin parameters.
The evolution operator of a discrete-time quantum walk involves a conditional shift in position space which entangles the 'coin' and position degrees of freedom of the walker. After several steps, the coin-position entanglement (CPE) converges to a well-defined value which depends on the initial state. In this work we provide an analytical method which allows for the exact calculation of the asymptotic reduced density operator and the corresponding CPE for a discrete-time quantum walk on a two-dimensional lattice. We use the von Neumann entropy of the reduced density operator as an entanglement measure. The method is applied to the case of a Hadamard walk for which the dependence of the resulting CPE on initial conditions is obtained. Initial states leading to the maximum or minimum CPE are identified and the relation between the coin or position entanglement present in the initial state of the walker and the final level of CPE is discussed. The CPE obtained from separable initial states satisfies an additivity property in terms of CPE of the corresponding one-dimensional cases. Non-local initial conditions are also considered and we find that the extreme case of an initial uniform position distribution leads to the largest CPE variation.
Discrete-time quantum walk evolve by a unitary operator which involves two operators a conditional shift in position space and a coin operator. This operator entangles the coin and position degrees of freedom of the walker. In this paper, we investigate the asymptotic behavior of the coin position entanglement (CPE) for an inhomogeneous quantum walk which determined by two orthogonal matrices in one-dimensional lattice. Free parameters of coin operator together provide many conditions under which a measurement perform on the coin state yield the value of entanglement on the resulting position quantum state. We study the problem analytically for all values that two free parameters of coin operator can take and the conditions under which entanglement becomes maximal are sought.
Abstract In this article we show that for any quantum walker with m -dimensional coin subspace, we have m 2 × m 2 specific constant matrix C where it completely determines the asymptotic reduced density matrix of the walker. We show that for any initial state with P 0 projector, the reduced density matrix can be obtained by T r 1 P 0 ⊗ I C or equivalently T r 2 I ⊗ P 0 C . It is worth mentioning that the characteristic matrix C is independent of the initial state and just depends on the coin operator, so by finding this matrix for a specific type of quantum walks (QW), the long-time behavior of it, such as local state of the coin after a long-time walk and asymptotic entanglement between the coin and position will be completely known for any initial state. We have found the characteristic matrix C for the general coin operator, U 2 , as well as the exact form of this matrix for the local initial state.
We report a systematic study of asymptotic behavior of entanglement between the position and coin degrees of freedom for a one-dimensional discrete quantum walk. Effects of coin bias, initial coin asymmetry and phase difference and non-locality of initial position state on the fluctuations and asymptotic value of entanglement are investigated by using Fourier approach. Fluctuations in entropy are found to die out as power law in time as t-α with α = 1/2,3/2 and 5/2 depending on the initial state parameters of the system.
Quantum walks, both discrete (coined) and continuous time, form the basis of several recent quantum algorithms. Here we use numerical simulations to study the properties of discrete, coined quantum walks. We investigate the variation in the entanglement between the coin and the position of the particle by calculating the entropy of the reduced density matrix of the coin. We consider both dynamical evolution and asymptotic limits for coins of dimensions from two to eight on regular graphs. For low coin dimensions, quantum walks which spread faster (as measured by the mean square deviation of their distribution from uniform) also exhibit faster convergence towards the asymptotic value of the entanglement between the coin and particle's position. For high-dimensional coins, the DFT coin operator is more efficient at spreading than the Grover coin. We study the entanglement of the coin on regular finite graphs such as cycles, and also show that on complete bipartite graphs, a quantum walk with a Grover coin is always periodic with period four. We generalize the 'glued trees' graph used by Childs et al (2003 Proc. STOC , pp 59–68) to higher branching rate (fan out) and verify that the scaling with branching rate and with tree depth is polynomial.
A discrete-time quantum walk (DTQW) is a lattice model of a single particle whose two-component internal state is repeatedly mixed by a unitary "coin" and shifted in space, chirality by chirality. In this work, we present a framework for coupling such a walk to a background electromagnetic (U(1)) gauge field. Instead of introducing traditional link variables common to the existing literature, we embed the gauge phase directly within the coin operator already required by the free walk, thereby eliminating the need for extra local degrees of freedom. We demonstrate this construction in two stages: first, using only coin phases to generate a purely spatial potential (A_1) while structurally forcing the temporal potential (A_0) to vanish (the temporal, or Weyl, gauge); and second, incorporating a single post-shift phase to recover a fully general potential (A_mu = (A_0, A_1). In both regimes, we show that the gauge transformation law for the coin phases is rigorously forced by the consistency requirement between the left- and right-moving sectors rather than assumed. We verify the construction non-perturbatively through the exact lattice dispersion relation, the walk’s convergence rate to the continuum Dirac equation, and its intrinsic avoidance of the fermion doubling problem.