88,265 research outputs found
Optimisation of Quantum Evolution Algorithms
Given a quantum Hamiltonian and its evolution time, the corresponding unitary
evolution operator can be constructed in many different ways, corresponding to
different trajectories between the desired end-points. A choice among these
trajectories can then be made to obtain the best computational complexity and
control over errors. As an explicit example, Grover's quantum search algorithm
is described as a Hamiltonian evolution problem. It is shown that the
computational complexity has a power-law dependence on error when a
straightforward Lie-Trotter discretisation formula is used, and it becomes
logarithmic in error when reflection operators are used. The exponential change
in error control is striking, and can be used to improve many importance
sampling methods. The key concept is to make the evolution steps as large as
possible while obeying the constraints of the problem. In particular, we can
understand why overrelaxation algorithms are superior to small step size
algorithms.Comment: 7 pages. Talk presented at the 32nd International Symposium on
Lattice Field Theory, 23-28 June 2014, Columbia University, New York, US
Quantum circuit implementation of the Hamiltonian versions of Grover's algorithm
We analyze three different quantum search algorithms, the traditional
Grover's algorithm, its continuous-time analogue by Hamiltonian evolution, and
finally the quantum search by local adiabatic evolution. We show that they are
closely related algorithms in the sense that they all perform a rotation, at a
constant angular velocity, from a uniform superposition of all states to the
solution state. This make it possible to implement the last two algorithms by
Hamiltonian evolution on a conventional quantum circuit, while keeping the
quadratic speedup of Grover's original algorithm.Comment: 5 pages, 3 figure
Natural evolution strategies and variational Monte Carlo
A notion of quantum natural evolution strategies is introduced, which
provides a geometric synthesis of a number of known quantum/classical
algorithms for performing classical black-box optimization. Recent work of
Gomes et al. [2019] on heuristic combinatorial optimization using neural
quantum states is pedagogically reviewed in this context, emphasizing the
connection with natural evolution strategies. The algorithmic framework is
illustrated for approximate combinatorial optimization problems, and a
systematic strategy is found for improving the approximation ratios. In
particular it is found that natural evolution strategies can achieve
approximation ratios competitive with widely used heuristic algorithms for
Max-Cut, at the expense of increased computation time
The Majorization Arrow in Quantum Algorithm Design
We apply majorization theory to study the quantum algorithms known so far and
find that there is a majorization principle underlying the way they operate.
Grover's algorithm is a neat instance of this principle where majorization
works step by step until the optimal target state is found. Extensions of this
situation are also found in algorithms based in quantum adiabatic evolution and
the family of quantum phase-estimation algorithms, including Shor's algorithm.
We state that in quantum algorithms the time arrow is a majorization arrow.Comment: REVTEX4.b4 file, 4 color figures (typos corrected.
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
Noise resistance of adiabatic quantum computation using random matrix theory
Besides the traditional circuit-based model of quantum computation, several
quantum algorithms based on a continuous-time Hamiltonian evolution have
recently been introduced, including for instance continuous-time quantum walk
algorithms as well as adiabatic quantum algorithms. Unfortunately, very little
is known today on the behavior of these Hamiltonian algorithms in the presence
of noise. Here, we perform a fully analytical study of the resistance to noise
of these algorithms using perturbation theory combined with a theoretical noise
model based on random matrices drawn from the Gaussian Orthogonal Ensemble,
whose elements vary in time and form a stationary random process.Comment: 9 pages, 3 figure
Adiabatic quantum search algorithm for structured problems
The study of quantum computation has been motivated by the hope of finding
efficient quantum algorithms for solving classically hard problems. In this
context, quantum algorithms by local adiabatic evolution have been shown to
solve an unstructured search problem with a quadratic speed-up over a classical
search, just as Grover's algorithm. In this paper, we study how the structure
of the search problem may be exploited to further improve the efficiency of
these quantum adiabatic algorithms. We show that by nesting a partial search
over a reduced set of variables into a global search, it is possible to devise
quantum adiabatic algorithms with a complexity that, although still
exponential, grows with a reduced order in the problem size.Comment: 7 pages, 0 figur
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