Adiabatic quantum search algorithm for structured problems
Jérémie Roland,Nicolas J. Cerf +1 more
TL;DR: It is shown 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.
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Abstract: 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 speedup 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.
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References
A Quantum Adiabatic Evolution Algorithm Applied to Random Instances of an NP-Complete Problem
TL;DR: For the small examples that the authors could simulate, the quantum adiabatic algorithm worked well, providing evidence that quantum computers (if large ones can be built) may be able to outperform ordinary computers on hard sets of instances of NP-complete problems.
Strengths and Weaknesses of Quantum Computing
TL;DR: It is proved that relative to an oracle chosen uniformly at random with probability 1 the class $\NP$ cannot be solved on a quantum Turing machine (QTM) in time $o(2^{n/2})$.
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Quantum search by local adiabatic evolution
TL;DR: In this article, the authors apply the time-dependent Hamiltonian approach to Grover's problem and find that by adjusting the evolution rate of the Hamiltonian so as to keep the evolution adiabatic on each infinitesimal time interval, the total running time is of order N, where N is the number of items in the database.
Nested quantum search and structured problems
TL;DR: A quantum algorithm is known that solves an unstructured search problem in a number of iterations of order square root of d, where d is the dimension of the search space as mentioned in this paper.
Quantum-circuit model of Hamiltonian search algorithms
Jérémie Roland,Nicolas J. Cerf +1 more
TL;DR: These algorithms are closely related 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, which makes it possible to implement the two Hamiltonian-evolution algorithms on a conventional quantum circuit, while keeping the quadratic speedup of Grover's original algorithm.