scholarly journals Maze Solving by a Quantum Walk with Sinks and Self-Loops: Numerical Analysis

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2263
Author(s):  
Leo Matsuoka ◽  
Kenta Yuki ◽  
Hynek Lavička ◽  
Etsuo Segawa
Keyword(s):  
Numerical Analysis ◽  
Discrete Time ◽  
Shortest Paths ◽  
Quantum Walk ◽  
Amplitude Distribution ◽  
Quantum Systems ◽  
Natural Phenomena ◽  
Convergence Process ◽  
Time Quantum ◽  
Autonomous Optimization

Maze-solving by natural phenomena is a symbolic result of the autonomous optimization induced by a natural system. We present a method for finding the shortest path on a maze consisting of a bipartite graph using a discrete-time quantum walk, which is a toy model of many kinds of quantum systems. By evolving the amplitude distribution according to the quantum walk on a kind of network with sinks, which is the exit of the amplitude, the amplitude distribution remains eternally on the paths between two self-loops indicating the start and the goal of the maze. We performed a numerical analysis of some simple cases and found that the shortest paths were detected by the chain of the maximum trapped densities in most cases of bipartite graphs. The counterintuitive dependence of the convergence steps on the size of the structure of the network was observed in some cases, implying that the asymmetry of the network accelerates or decelerates the convergence process. The relation between the amplitude remaining and distance of the path is also discussed briefly.

scholarly journals General methods and properties to evaluate continuum limits of the 1D discrete time quantum walk

2020 ◽  
Vol 19 (10) ◽  
Author(s):  
Michael Manighalam ◽  
Mark Kon
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Quantum Walk ◽  
Quantum Systems ◽  
Quantum Walks ◽  
Time Limit ◽  
Continuous Limit ◽  
Continuous Space ◽  
Time Quantum ◽  
Space Limit

Abstract Models of quantum walks which admit continuous time and continuous spacetime limits have recently led to quantum simulation schemes for simulating fermions in relativistic and nonrelativistic regimes (Molfetta GD, Arrighi P. A quantum walk with both a continuous-time and a continuous-spacetime limit, 2019). This work continues the study of relationships between discrete time quantum walks (DTQW) and their ostensive continuum counterparts by developing a more general framework than was done in Molfetta and Arrighi (A quantum walk with both a continuous-time and a continuous-spacetime limit, 2019) to evaluate the continuous time limit of these discrete quantum systems. Under this framework, we prove two constructive theorems concerning which internal discrete transitions (“coins”) admit nontrivial continuum limits. We additionally prove that the continuous space limit of the continuous time limit of the DTQW can only yield massless states which obey the Dirac equation. Finally, we demonstrate that for general coins the continuous time limit of the DTQW can be identified with the canonical continuous time quantum walk when the coin is allowed to transition through the continuous limit process.

scholarly journals Fractional recurrence in discrete-time quantum walk

Open Physics ◽  
2010 ◽  
Vol 8 (6) ◽  
Author(s):  
C. Chandrashekar
Keyword(s):  
Wave Packet ◽  
Discrete Time ◽  
Quantum Walk ◽  
Quantum Systems ◽  
Discrete Energy ◽  
Energy Eigenvalues ◽  
Continuous Energy ◽  
Time Quantum ◽  
Recurrence Theorem ◽  
Complete Quantum

AbstractQuantum recurrence theorem holds for quantum systems with discrete energy eigenvalues and fails to hold in general for systems with continuous energy. We show that during quantum walk process dominated by interference of amplitude corresponding to different paths fail to satisfy the complete quantum recurrence theorem. Due to the revival of the fractional wave packet, a fractional recurrence characterized using quantum Pólya number can be seen.

scholarly journals A Discontinuity of the Energy of Quantum Walk in Impurities

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1134
Author(s):  
Kenta Higuchi ◽  
Takashi Komatsu ◽  
Norio Konno ◽  
Hisashi Morioka ◽  
Etsuo Segawa
Keyword(s):  
Unit Circle ◽  
Stationary State ◽  
Discrete Time ◽  
Quantum Walk ◽  
Time Limit ◽  
The Other ◽  
Unitary Matrix ◽  
Initial State ◽  
Time Quantum ◽  
Long Time

We consider the discrete-time quantum walk whose local dynamics is denoted by a common unitary matrix C at the perturbed region {0,1,⋯,M−1} and free at the other positions. We obtain the stationary state with a bounded initial state. The initial state is set so that the perturbed region receives the inflow ωn at time n(|ω|=1). From this expression, we compute the scattering on the surface of −1 and M and also compute the quantity how quantum walker accumulates in the perturbed region; namely, the energy of the quantum walk, in the long time limit. The frequency of the initial state of the influence to the energy is symmetric on the unit circle in the complex plain. We find a discontinuity of the energy with respect to the frequency of the inflow.

Limitations of discrete-time quantum walk on a one-dimensional infinite chain

2018 ◽  
Vol 382 (13) ◽  
pp. 899-903
Author(s):  
Jia-Yi Lin ◽  
Xuanmin Zhu ◽  
Shengjun Wu
Keyword(s):  
Discrete Time ◽  
Quantum Walk ◽  
Infinite Chain ◽  
One Dimensional ◽  
Time Quantum

scholarly journals A limit theorem for a splitting distribution of a quantum walk

2018 ◽  
Vol 16 (03) ◽  
pp. 1850023
Author(s):  
Takuya Machida
Keyword(s):  
Limit Theorem ◽  
Probability Distribution ◽  
Random Walks ◽  
Discrete Time ◽  
Limit Distribution ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Unitary Evolution ◽  
Time Quantum ◽  
Long Time

Discrete-time quantum walks are considered a counterpart of random walks and their study has been getting attention since around 2000. In this paper, we focus on a quantum walk which generates a probability distribution splitting to two parts. The quantum walker with two coin states spreads at points, represented by integers, and we analyze the chance of finding the walker at each position after it carries out a unitary evolution a lot of times. The result is reported as a long-time limit distribution from which one can see an approximation to the finding probability.

DISCRETE TIME QUANTUM WALK ON A LINE WITH TWO PARTICLES

2006 ◽  
Vol 04 (03) ◽  
pp. 573-583 ◽  
Author(s):  
L. SHERIDAN ◽  
N. PAUNKOVIĆ ◽  
Y. OMAR ◽  
S. BOSE
Keyword(s):  
Discrete Time ◽  
Relative Phase ◽  
Initial Conditions ◽  
Quantum Walk ◽  
Degree Of Freedom ◽  
Initial State ◽  
Time Quantum

We introduce the idea of a quantum walk with two particles and study it for the case of a discrete time walk on a line. We consider both separable and maximally entangled initial conditions, and show how the entanglement and the relative phase between the states describing the coin degree of freedom of each particle will influence the evolution of the quantum walk. In particular, these factors will have consequences for the distance between the particles and the probability of finding them at a given point, yielding results that cannot be obtained from a separable initial state, be it pure or mixed. Finally, we review briefly proposals for implementations.

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