Abstract
For a continuous-time quantum walk on a line the variance of the position observable grows quadratically in time, whereas, for its classical counterpart on the same graph, it exhibits a linear, diffusive, behaviour. A quantum walk, thus, propagates at a rate which is linear in time, as compared to the square root rate for a classical random walk. Indeed, it has been suggested that there are graphs that can be traversed by a quantum walker exponentially faster than by the classical random analogue. In this note we adopt the approach of exploring the conditions to impose on a Markov process in order to emulate its quantum counterpart: the central issue that emerges is the problem of taking into account, in the numerical generation of each sample path, the causative effect of the ensemble of trajectories to which it belongs. How to deal numerically with this problem is shown in a paradigmatic example.
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References
Childs, A., Farhi, E., Gutmann, S.: An example of the difference between quantum and classical random walks. Quantum Information Processing 1, 35–43 (2002)
Perets, H., Lahini, Y., Pozzi, F., Sorel, M., Morandotti, R., Silberberg, Y.: Realization of quantum walks with negligible decoherence in waveguide lattices, arXiv:quant-ph/0707.0741v2 (2007)
Guerra, F., Morato, L.: Quantization of dynamical systems and stochastic control theory. Phys. Rev. D 27, 1774–1786 (1983)
Guerra, F., Marra, R.: Discrete stochastic variational principles and quantum mechanics. Phys. Rev. D 29(8), 1647–1655 (1984)
Smolin, L.: Could quantum mechanics be an approximation to another theory?, arXiv-quant-ph/0609109 (2006)
Farhi, E., Gutmann, S.: Quantum computation and decision trees. Phys. Rev. A 58, 915–928 (1998)
Childs, A., Cleve, R., Deotto, E., Farhi, E., Gutmann, S., Spielman, D.: Exponential algorithmic speed up by quantum walk. In: STOC 2003. Proc. 35th ACM symp., pp. 59–68 (2003)
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de Falco, D., Tamascelli, D. (2008). Quantum Walks: A Markovian Perspective. In: Geffert, V., Karhumäki, J., Bertoni, A., Preneel, B., Návrat, P., Bieliková, M. (eds) SOFSEM 2008: Theory and Practice of Computer Science. SOFSEM 2008. Lecture Notes in Computer Science, vol 4910. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77566-9_45
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DOI: https://doi.org/10.1007/978-3-540-77566-9_45
Publisher Name: Springer, Berlin, Heidelberg
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