Quantum walks driven by many coins

Brun, Todd A.; Carteret, Hilary A.; Ambainis, Andris

doi:10.1103/physreva.67.052317

Cited by 175 publications

(193 citation statements)

References 29 publications

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“…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.…”

Section: Coined Quantum Walk On An Infinite Linementioning

confidence: 99%

“…This has the advantage of the dynamics remaining purely unitary, rendering the calculations simpler. The first such study, Brun et al (2003c), considered multiple coins used in sequence, with the sequence repeating after all the coins had been used once. This produces a quantum walk that still spreads linearly with the number of steps, but with the rate of spreading reduced inversely by the number of coins.…”

Section: Multiple Coins In the Walk On The Linementioning

confidence: 99%

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Decoherence in quantum walks – a review

Kendon¹

2007

Math. Struct. in Comp. Science

327

385

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The development of quantum walks in the context of quantum computation, as generalisations of random walk techniques, has led rapidly to several new quantum algorithms. These all follow a unitary quantum evolution, apart from the final measurement. Since logical qubits in a quantum computer must be protected from decoherence by error correction, there is no need to consider decoherence at the level of algorithms. Nonetheless, enlarging the range of quantum dynamics to include non-unitary evolution provides a wider range of possibilities for tuning the properties of quantum walks. For example, small amounts of decoherence in a quantum walk on the line can produce more uniform spreading (a top-hat distribution), without losing the quantum speed up. This paper reviews the work on decoherence, and more generally on non-unitary evolution, in quantum walks and suggests what future questions might prove interesting to pursue in this area.

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Section: Coined Quantum Walk On An Infinite Linementioning

confidence: 99%

Section: Multiple Coins In the Walk On The Linementioning

confidence: 99%

Decoherence in quantum walks – a review

Kendon¹

2007

Math. Struct. in Comp. Science

327

385

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“…We know little about the spiky probability distribution in the later case. The existence of the spikes are already reported in the two-dimensional Grover walk [7,8], one-dimensional quantum walk as many coins [9], but, no probability distributions are calculated explicitly. In this paper we consider quantum walk with four internal states, which is the same as the quantum walk with many coins essentially, and compute analyti-cally the probability distribution.…”

Section: Introductionmentioning

confidence: 99%

Localization of multi-state quantum walk in one dimension

Inui

Konno

2005

Physica A: Statistical Mechanics and its Applications

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Particle trapping in multi-state quantum walk on a circle is studied. The timeaveraged probability distribution of a particle which moves four different lattice sites according to four internal states is calculated exactly. In contrast with "Hadamard walk" with only two internal states, the particle remains at the initial position with high probability. The time-averaged probability of finding the particle decreases exponentially as distance from a center of a spike. This implies that the particle is trapped in a narrow region. This striking difference is minutely explained from difference between degeneracy of eigenvalues of the time-evolution matrices. The dependence of the particle distribution on initial conditions is also considered.

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“…The dynamics of quantum walks of both types has been studied in detail for walks on an infinite linefor the continuous-time case in Refs. [7,11,12] and for the discrete-time case in [13,14,15,16,17]. There has also been considerable work on other regular graphs.…”

Section: Introductionmentioning

confidence: 99%

Quantum walks on quotient graphs

2007

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A discrete-time quantum walk on a graph Γ is the repeated application of a unitary evolution operator to a Hilbert space corresponding to the graph. If this unitary evolution operator has an associated group of symmetries, then for certain initial states the walk will be confined to a subspace of the original Hilbert space. Symmetries of the original graph, given by its automorphism group, can be inherited by the evolution operator. We show that a quantum walk confined to the subspace corresponding to this symmetry group can be seen as a different quantum walk on a smaller quotient graph. We give an explicit construction of the quotient graph for any subgroup H of the automorphism group and illustrate it with examples. The automorphisms of the quotient graph which are inherited from the original graph are the original automorphism group modulo the subgroup H used to construct it. The quotient graph is constructed by removing the symmetries of the subgroup H from the original graph. We then analyze the behavior of hitting times on quotient graphs. Hitting time is the average time it takes a walk to reach a given final vertex from a given initial vertex. It has been shown in earlier work [Phys. Rev. A 74, 042334 (2006)] that the hitting time for certain initial states of a quantum walks can be infinite, in contrast to classical random walks. We give a condition which determines whether the quotient graph has infinite hitting times given that they exist in the original graph. We apply this condition for the examples discussed and determine which quotient graphs have infinite hitting times. All known examples of quantum walks with hitting times which are short compared to classical random walks correspond to systems with quotient graphs much smaller than the original graph; we conjecture that the existence of a small quotient graph with finite hitting times is necessary for a walk to exhibit a quantum speed-up.

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Quantum walks driven by many coins

Cited by 175 publications

References 29 publications

Decoherence in quantum walks – a review

Decoherence in quantum walks – a review

Localization of multi-state quantum walk in one dimension

Quantum walks on quotient graphs

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