508 research outputs found
Quantum Simulations of Classical Random Walks and Undirected Graph Connectivity
AbstractWhile it is straightforward to simulate a very general class of random processes space-efficiently by non-unitary quantum computations (e.g., quantum computations that allow intermediate measurements to occur), it is not currently known to what extent restricting quantum computations to be unitary affects the space required for such simulations. This paper presents a method by which a limited class of random processes—random walks on undirected graphs—can be simulated by unitary quantum computations in a space-efficient (and time-efficient) manner. By means of such simulations, it is demonstrated that the undirected graph connectivity problem for regular graphs can be solved by one-sided error quantum Turing machines that run in logspace and require a single measurement at the end of their computations. It follows that symmetric logspace is contained in a quantum analogue of randomized logspace that disallows intermediate measurements
Decoherence in Discrete Quantum Walks
We present an introduction to coined quantum walks on regular graphs, which
have been developed in the past few years as an alternative to quantum Fourier
transforms for underpinning algorithms for quantum computation. We then
describe our results on the effects of decoherence on these quantum walks on a
line, cycle and hypercube. We find high sensitivity to decoherence, increasing
with the number of steps in the walk, as the particle is becoming more
delocalised with each step. However, the effect of a small amount of
decoherence can be to enhance the properties of the quantum walk that are
desirable for the development of quantum algorithms, such as fast mixing times
to uniform distributions.Comment: 15 pages, Springer LNP latex style, submitted to Proceedings of DICE
200
Decoherence in quantum walks - a review
The development of quantum walks in the context of quantum computation, as
generalisations of random walk techniques, led rapidly to several new quantum
algorithms. These all follow 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.Comment: 52 pages, invited review, v2 & v3 updates to include significant work
since first posted and corrections from comments received; some non-trivial
typos fixed. Comments now limited to changes that can be applied at proof
stag
Exponential algorithmic speedup by quantum walk
We construct an oracular (i.e., black box) problem that can be solved
exponentially faster on a quantum computer than on a classical computer. The
quantum algorithm is based on a continuous time quantum walk, and thus employs
a different technique from previous quantum algorithms based on quantum Fourier
transforms. We show how to implement the quantum walk efficiently in our
oracular setting. We then show how this quantum walk can be used to solve our
problem by rapidly traversing a graph. Finally, we prove that no classical
algorithm can solve this problem with high probability in subexponential time.Comment: 24 pages, 7 figures; minor corrections and clarification
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