4,523 research outputs found
New Developments in Quantum Algorithms
In this survey, we describe two recent developments in quantum algorithms.
The first new development is a quantum algorithm for evaluating a Boolean
formula consisting of AND and OR gates of size N in time O(\sqrt{N}). This
provides quantum speedups for any problem that can be expressed via Boolean
formulas. This result can be also extended to span problems, a generalization
of Boolean formulas. This provides an optimal quantum algorithm for any Boolean
function in the black-box query model.
The second new development is a quantum algorithm for solving systems of
linear equations. In contrast with traditional algorithms that run in time
O(N^{2.37...}) where N is the size of the system, the quantum algorithm runs in
time O(\log^c N). It outputs a quantum state describing the solution of the
system.Comment: 11 pages, 1 figure, to appear as an invited survey talk at MFCS'201
Quantum Walks on the Line with Phase Parameters
In this paper, a study on discrete-time coined quantum walks on the line is
presented. Clear mathematical foundations are still lacking for this quantum
walk model. As a step towards this objective, the following question is being
addressed: {\it Given a graph, what is the probability that a quantum walk
arrives at a given vertex after some number of steps?} This is a very natural
question, and for random walks it can be answered by several different
combinatorial arguments. For quantum walks this is a highly non-trivial task.
Furthermore, this was only achieved before for one specific coin operator
(Hadamard operator) for walks on the line. Even considering only walks on
lines, generalizing these computations to a general SU(2) coin operator is a
complex task. The main contribution is a closed-form formula for the amplitudes
of the state of the walk (which includes the question above) for a general
symmetric SU(2) operator for walks on the line. To this end, a coin operator
with parameters that alters the phase of the state of the walk is defined.
Then, closed-form solutions are computed by means of Fourier analysis and
asymptotic approximation methods. We also present some basic properties of the
walk which can be deducted using weak convergence theorems for quantum walks.
In particular, the support of the induced probability distribution of the walk
is calculated. Then, it is shown how changing the parameters in the coin
operator affects the resulting probability distribution.Comment: In v2 a small typo was fixed. The exponent in the definition of N_j
in Theorem 3 was changed from -1/2 to 1. 20 pages, 3 figures. Presented at
10th Asian Conference on Quantum Information Science (AQIS'10). Tokyo, Japan.
August 27-31, 201
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
Quantum central limit theorem for continuous-time quantum walks on odd graphs in quantum probability theory
The method of the quantum probability theory only requires simple structural
data of graph and allows us to avoid a heavy combinational argument often
necessary to obtain full description of spectrum of the adjacency matrix. In
the present paper, by using the idea of calculation of the probability
amplitudes for continuous-time quantum walk in terms of the quantum probability
theory, we investigate quantum central limit theorem for continuous-time
quantum walks on odd graphs.Comment: 19 page, 1 figure
Adiabatic Quantum State Generation and Statistical Zero Knowledge
The design of new quantum algorithms has proven to be an extremely difficult
task. This paper considers a different approach to the problem, by studying the
problem of 'quantum state generation'. This approach provides intriguing links
between many different areas: quantum computation, adiabatic evolution,
analysis of spectral gaps and groundstates of Hamiltonians, rapidly mixing
Markov chains, the complexity class statistical zero knowledge, quantum random
walks, and more.
We first show that many natural candidates for quantum algorithms can be cast
as a state generation problem. We define a paradigm for state generation,
called 'adiabatic state generation' and develop tools for adiabatic state
generation which include methods for implementing very general Hamiltonians and
ways to guarantee non negligible spectral gaps. We use our tools to prove that
adiabatic state generation is equivalent to state generation in the standard
quantum computing model, and finally we show how to apply our techniques to
generate interesting superpositions related to Markov chains.Comment: 35 pages, two figure
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