2,155 research outputs found
On the hitting times of quantum versus random walks
In this paper we define new Monte Carlo type classical and quantum hitting
times, and we prove several relationships among these and the already existing
Las Vegas type definitions. In particular, we show that for some marked state
the two types of hitting time are of the same order in both the classical and
the quantum case.
Further, we prove that for any reversible ergodic Markov chain , the
quantum hitting time of the quantum analogue of has the same order as the
square root of the classical hitting time of . We also investigate the
(im)possibility of achieving a gap greater than quadratic using an alternative
quantum walk.
Finally, we present new quantum algorithms for the detection and finding
problems. The complexities of both algorithms are related to the new,
potentially smaller, quantum hitting times. The detection algorithm is based on
phase estimation and is particularly simple. The finding algorithm combines a
similar phase estimation based procedure with ideas of Tulsi from his recent
theorem for the 2D grid. Extending his result, we show that for any
state-transitive Markov chain with unique marked state, the quantum hitting
time is of the same order for both the detection and finding problems
Decoherence in Quantum Walks on the Hypercube
We study a natural notion of decoherence on quantum random walks over the
hypercube. We prove that in this model there is a decoherence threshold beneath
which the essential properties of the hypercubic quantum walk, such as linear
mixing times, are preserved. Beyond the threshold, we prove that the walks
behave like their classical counterparts.Comment: 7 pages, 3 figures; v2:corrected typos in references; v3:clarified
section 2.1; v4:added references, expanded introduction; v5: final journal
versio
Hitting Time of Quantum Walks with Perturbation
The hitting time is the required minimum time for a Markov chain-based walk
(classical or quantum) to reach a target state in the state space. We
investigate the effect of the perturbation on the hitting time of a quantum
walk. We obtain an upper bound for the perturbed quantum walk hitting time by
applying Szegedy's work and the perturbation bounds with Weyl's perturbation
theorem on classical matrix. Based on the definition of quantum hitting time
given in MNRS algorithm, we further compute the delayed perturbed hitting time
(DPHT) and delayed perturbed quantum hitting time (DPQHT). We show that the
upper bound for DPQHT is actually greater than the difference between the
square root of the upper bound for a perturbed random walk and the square root
of the lower bound for a random walk.Comment: 9 page
Mixing Times in Quantum Walks on Two-Dimensional Grids
Mixing properties of discrete-time quantum walks on two-dimensional grids
with torus-like boundary conditions are analyzed, focusing on their connection
to the complexity of the corresponding abstract search algorithm. In
particular, an exact expression for the stationary distribution of the coherent
walk over odd-sided lattices is obtained after solving the eigenproblem for the
evolution operator for this particular graph. The limiting distribution and
mixing time of a quantum walk with a coin operator modified as in the abstract
search algorithm are obtained numerically. On the basis of these results, the
relation between the mixing time of the modified walk and the running time of
the corresponding abstract search algorithm is discussed.Comment: 11 page
Quantum walks: a comprehensive review
Quantum walks, the quantum mechanical counterpart of classical random walks,
is an advanced tool for building quantum algorithms that has been recently
shown to constitute a universal model of quantum computation. Quantum walks is
now a solid field of research of quantum computation full of exciting open
problems for physicists, computer scientists, mathematicians and engineers.
In this paper we review theoretical advances on the foundations of both
discrete- and continuous-time quantum walks, together with the role that
randomness plays in quantum walks, the connections between the mathematical
models of coined discrete quantum walks and continuous quantum walks, the
quantumness of quantum walks, a summary of papers published on discrete quantum
walks and entanglement as well as a succinct review of experimental proposals
and realizations of discrete-time quantum walks. Furthermore, we have reviewed
several algorithms based on both discrete- and continuous-time quantum walks as
well as a most important result: the computational universality of both
continuous- and discrete- time quantum walks.Comment: Paper accepted for publication in Quantum Information Processing
Journa
Search via Quantum Walk
We propose a new method for designing quantum search algorithms for finding a
"marked" element in the state space of a classical Markov chain. The algorithm
is based on a quantum walk \'a la Szegedy (2004) that is defined in terms of
the Markov chain. The main new idea is to apply quantum phase estimation to the
quantum walk in order to implement an approximate reflection operator. This
operator is then used in an amplitude amplification scheme. As a result we
considerably expand the scope of the previous approaches of Ambainis (2004) and
Szegedy (2004). Our algorithm combines the benefits of these approaches in
terms of being able to find marked elements, incurring the smaller cost of the
two, and being applicable to a larger class of Markov chains. In addition, it
is conceptually simple and avoids some technical difficulties in the previous
analyses of several algorithms based on quantum walk.Comment: 21 pages. Various modifications and improvements, especially in
Section
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