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Entropy
Volume 18
Issue 6
10.3390/e18060228
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Article

Discrete Time Dirac Quantum Walk in 3+1 Dimensions

by
Giacomo Mauro D’Ariano
1,*,
Nicola Mosco
2,
Paolo Perinotti
2 and
Alessandro Tosini
2
1
QUIT group, Dipartimento di Fisica, via Bassi 6, Pavia 27100, Italy
2
INFN Gruppo IV, Sezione di Pavia, via Bassi 6, Pavia 27100, Italy
*
Author to whom correspondence should be addressed.
Entropy 2016, 18(6), 228; https://doi.org/10.3390/e18060228
Submission received: 4 March 2016 / Revised: 28 May 2016 / Accepted: 30 May 2016 / Published: 14 June 2016
(This article belongs to the Special Issue Quantum Information and Communication: From Foundations to Applications)
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Abstract

In this paper we consider quantum walks whose evolution converges to the Dirac equation in the limit of small wave-vectors. We show exact Fast Fourier implementation of the Dirac quantum walks in one, two, and three space dimensions. The behaviour of particle states—defined as states smoothly peaked in some wave-vector eigenstate of the walk—is described by an approximated dispersive differential equation that for small wave-vectors gives the usual Dirac particle and antiparticle kinematics. The accuracy of the approximation is provided in terms of a lower bound on the fidelity between the exactly evolved state and the approximated one. The jittering of the position operator expectation value for states having both a particle and an antiparticle component is analytically derived and observed in the numerical implementations.
Keywords: quantum walk; Dirac equation quantum walk; Dirac equation
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MDPI and ACS Style

D’Ariano, G.M.; Mosco, N.; Perinotti, P.; Tosini, A. Discrete Time Dirac Quantum Walk in 3+1 Dimensions. Entropy 2016, 18, 228. https://doi.org/10.3390/e18060228

AMA Style

D’Ariano GM, Mosco N, Perinotti P, Tosini A. Discrete Time Dirac Quantum Walk in 3+1 Dimensions. Entropy. 2016; 18(6):228. https://doi.org/10.3390/e18060228

Chicago/Turabian Style

D’Ariano, Giacomo Mauro, Nicola Mosco, Paolo Perinotti, and Alessandro Tosini. 2016. "Discrete Time Dirac Quantum Walk in 3+1 Dimensions" Entropy 18, no. 6: 228. https://doi.org/10.3390/e18060228

APA Style

D’Ariano, G. M., Mosco, N., Perinotti, P., & Tosini, A. (2016). Discrete Time Dirac Quantum Walk in 3+1 Dimensions. Entropy, 18(6), 228. https://doi.org/10.3390/e18060228

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