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AIP conference proceedings

Complementarity in Quantum Walks

Viv M. Kendon

Published Jan 1, 2004• 0 citations
Publisher

Abstract

By generalizing a quantum walk to include measurement of the path taken by the quantum walker, we show that quantum walks interpolate between a fully coherent quantum dynamics and a classical 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.

DOI
10.1063/1.1834399

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By generalizing a quantum walk to include measurement of the path taken by the quantum walker, we show that quantum walks interpolate between a fully coherent quantum dynamics and a classical 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.
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.
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.
We explore a discrete-time, coined quantum walk on a quantum network where the coherent superposition of walker moves originates from the unitary interaction of the walker coin with the qubit degrees of freedom in the quantum network. The walk dynamics leads to a growth of entanglement between the walker and the network on one hand and, on the other, between the network qubits among themselves. The initial entanglement among the network qubits plays a crucial role in determining the asymptotic values of these entanglement measures and the quantum walk statistics. Specifically, the entanglement entropy of the walker-network state and the negativity of the quantum network-qubit state saturate to values increasing with the initial network entanglement. The asymptotic time-averaged walker-position probability distribution shows increasing localization around the initial walker position with higher initial network entanglement. A potential application of these results as a characterization tool for quantum network properties is suggested.
We consider two independent quantum walks on separate lines augmented by partial or full swapping of coins after each step. For classical random walks, swapping or not swapping coins makes little difference to the random-walk characteristics, but we show that quantum walks with partial swapping of coins have complicated yet elegant interwalker correlations. Specifically we study the joint position distribution of the reduced two-walker state after tracing out the coins and analyze total, classical, and quantum correlations in terms of mutual information, quantum mutual information, and measurement-induced disturbance. Our analysis shows intriguing quantum features without classical analogs.