Quantum random walks with decoherent coins

Abstract

The quantum random walk has been much studied recently, largely due to its highly nonclassical behavior. In this paper, we study one possible route to classical behavior for the discrete quantum walk on the line: the presence of decoherence in the quantum "coin" which drives the walk. We find exact analytical expressions for the time dependence of the first two moments of position, and show that in the long-time limit the variance grows linearly with time, unlike the unitary walk. We compare this to the results of direct numerical simulation, and see how the form of the position distribution changes from the unitary to the usual classical result as we increase the strength of the decoherence.

Decoherence in quantum walks – a review

“…Only if a new coin is used for every step of the walk does it become equivalent to the classical random walk. This is in contrast with the behaviour obtained by decohering the coin (Brun et al 2003a), which always results in classical limiting behaviour, as pointed out in Brun et al (2003b). Classical behaviour is thus associated with an environment so large that one never comes close to the Poincaré recurrence time over the timescales considered.…”

“…In much the same way as we now know almost everything about the properties and possible states of two qubits -though quantum computers will clearly need far more than two qubits to be useful -the simple quantum walk on a line has now been thoroughly studied (see, for example, Ambainis et al (2001), Bach et al (2004), Yamasaki et al (2002), Kendon and Tregenna (2003), Brun et al (2003a;2003c), Konno et al (2004), Konno (2002) and Carteret et al (2003)), though there is no suggestion that it will lead to useful quantum walk algorithms by itself.…”

“…Lopéz and Paz present an elegant analysis of decoherence in the quantum walk using a discrete Wigner function to bring out the pattern of correlations and their decay when the coin is subject to decoherence. Their decoherence model follows much the same approach as that in Brun et al (2003a), which will be described in Section 5.2. In their study, Lopéz and Paz restrict the dynamics to cycles.…”

“…This will also provide an example of how to use Fourier transforms to simplify the dynamics and obtain the solution. We follow the method of Brun et al (2003aBrun et al ( 2003b to solve equation (48) for the coined quantum walk on a line with dephasing applied to the coin. We thus have S given by equation (6),…”

Investigation of continuous-time quantum walk via spectral distribution associated with adjacency matrix

“…The walk on infinite line, | q k (t) | 2 has the form of a binomial distribution, with a width which spreads like t 1/2 (i.e, ν = 1/2), therefore the variance grows linearly with time. But the variance in the quantum walk on infinite line, by contrast, grows quadratically with time, and the distribution | q k (t) | 2 has a complicated, oscillatory form [28]. Now, by using the probability amplitudes of Eq.…”