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arXiv:quant-ph/0404043 · Physical Review A

Complementarity and quantum walks

Viv M. Kendon, Barry C. Sanders

Published Feb 14, 2005• 66 citations
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Abstract

We show that quantum walks interpolate between a coherent ``wave walk'' and a random walk depending on how strongly the walker's coin state is measured; i.e., the quantum walk exhibits the quintessentially quantum property of complementarity, which is manifested as a tradeoff between knowledge of which path the walker takes vs the sharpness of the interference pattern. A physical implementation of a quantum walk (the quantum quincunx) should thus have an identifiable walker and the capacity to demonstrate the interpolation between wave walk and random walk depending on the strength of measurement.

DOI
10.1103/physreva.71.022307

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We show that quantum walks interpolate between a coherent ``wave walk'' and a random walk depending on how strongly the walker's coin state is measured; i.e., the quantum walk exhibits the quintessentially quantum property of complementarity, which is manifested as a tradeoff between knowledge of which path the walker takes vs the sharpness of the interference pattern. A physical implementation of a quantum walk (the quantum quincunx) should thus have an identifiable walker and the capacity to demonstrate the interpolation between wave walk and random walk depending on the strength of measurement.
A quantum walk is a mathematical model and a quantum version of a random walk. For the past two decades, it has been studied from mathematical and physical viewpoints. It has many desirable properties. This paper reevaluates quantum walk from a quantum mechanical perspective, as Feynman essentially proposed it as a toy model in his path integral method. Then we demonstrate that the quantum walk model exhibits both particle-like and wave-like properties by showing reflection and partial reflection as a particle and interference as a wave. In particular, for an initial state supported at two sites ±ℓ\pm\ell±ℓ, we explicitly derive the weak limit distribution of the Hadamard walk and show that the Konno distribution is modulated by the interference factor 2cos⁡2(ℓk(x))2\cos^2(\ell k(x))2cos2(ℓk(x)). The resulting interference profile is reminiscent of far-field (Fraunhofer-type) diffraction.
The development of quantum algorithms based on quantum versions of random walks is placed in the context of the emerging field of quantum computing. Constructing a suitable quantum version of a random walk is not trivial; pure quantum dynamics is deterministic, so randomness only enters during the measurement phase, i.e. when converting the quantum information into classical information. The outcome of a quantum random walk is very different from the corresponding classical random walk owing to the interference between the different possible paths. The upshot is that quantum walkers find themselves further from their starting point than a classical walker on average, and this forms the basis of a quantum speed up, which can be exploited to solve problems faster. Surprisingly, the effect of making the walk slightly less than perfectly quantum can optimize the properties of the quantum walk for algorithmic applications. Looking to the future, even with a small quantum computer available, the development of quantum walk algorithms might proceed more rapidly than it has, especially for solving real problems.
Quantum walk (QW), which is considered as the quantum counterpart of the classical random walk (CRW), is actually the quantum extension of CRW from the single-coin interpretation. The sequential unitary evolution engenders correlation between different steps in QW and leads to a non-binomial position distribution. In this paper, we propose an alternative quantum extension of CRW from the ensemble interpretation, named quantum random walk (QRW), where the walker has many unrelated coins, modeled as two-level systems, initially prepared in the same state. We calculate the walker’s position distribution in QRW for different initial coin states with the coin operator chosen as Hadamard matrix. In one-dimensional case, the walker’s position is the asymmetric binomial distribution. We further demonstrate that in QRW, coherence leads the walker to perform directional movement. For an initially decoherenced coin state, the walker’s position distribution is exactly the same as that of CRW. Moreover, we study QRW in 2D lattice, where the coherence plays a more diversified role in the walker’s position distribution.