12,996 research outputs found
Period Finding with Adiabatic Quantum Computation
We outline an efficient quantum-adiabatic algorithm that solves Simon's
problem, in which one has to determine the `period', or xor-mask, of a given
black-box function. We show that the proposed algorithm is exponentially faster
than its classical counterpart and has the same complexity as the corresponding
circuit-based algorithm. Together with other related studies, this result
supports a conjecture that the complexity of adiabatic quantum computation is
equivalent to the circuit-based computational model in a stronger sense than
the well-known, proven polynomial equivalence between the two paradigms. We
also discuss the importance of the algorithm and its theoretical and
experimental implications for the existence of an adiabatic version of Shor's
integer factorization algorithm that would have the same complexity as the
original algorithm.Comment: 6 page
Scaling of running time of quantum adiabatic algorithm for propositional satisfiability
We numerically study quantum adiabatic algorithm for the propositional
satisfiability. A new class of previously unknown hard instances is identified
among random problems. We numerically find that the running time for such
instances grows exponentially with their size. Worst case complexity of quantum
adiabatic algorithm therefore seems to be exponential.Comment: 7 page
Experimental implementation of local adiabatic evolution algorithms by an NMR quantum information processor
Quantum adiabatic algorithm is a method of solving computational problems by
evolving the ground state of a slowly varying Hamiltonian. The technique uses
evolution of the ground state of a slowly varying Hamiltonian to reach the
required output state. In some cases, such as the adiabatic versions of
Grover's search algorithm and Deutsch-Jozsa algorithm, applying the global
adiabatic evolution yields a complexity similar to their classical algorithms.
However, using the local adiabatic evolution, the algorithms given by J. Roland
and N. J. Cerf for Grover's search [ Phys. Rev. A. {\bf 65} 042308(2002)] and
by Saurya Das, Randy Kobes and Gabor Kunstatter for the Deutsch-Jozsa algorithm
[Phys. Rev. A. {\bf 65}, 062301 (2002)], yield a complexity of order
(where N=2 and n is the number of qubits). In this paper we report
the experimental implementation of these local adiabatic evolution algorithms
on a two qubit quantum information processor, by Nuclear Magnetic Resonance.Comment: Title changed, Adiabatic Grover's search algorithm added, error
analysis modifie
Universality of Entanglement and Quantum Computation Complexity
We study the universality of scaling of entanglement in Shor's factoring
algorithm and in adiabatic quantum algorithms across a quantum phase transition
for both the NP-complete Exact Cover problem as well as the Grover's problem.
The analytic result for Shor's algorithm shows a linear scaling of the entropy
in terms of the number of qubits, therefore difficulting the possibility of an
efficient classical simulation protocol. A similar result is obtained
numerically for the quantum adiabatic evolution Exact Cover algorithm, which
also shows universality of the quantum phase transition the system evolves
nearby. On the other hand, entanglement in Grover's adiabatic algorithm remains
a bounded quantity even at the critical point. A classification of scaling of
entanglement appears as a natural grading of the computational complexity of
simulating quantum phase transitions.Comment: 30 pages, 17 figures, accepted for publication in PR
A universal adiabatic quantum query algorithm
Quantum query complexity is known to be characterized by the so-called
quantum adversary bound. While this result has been proved in the standard
discrete-time model of quantum computation, it also holds for continuous-time
(or Hamiltonian-based) quantum computation, due to a known equivalence between
these two query complexity models. In this work, we revisit this result by
providing a direct proof in the continuous-time model. One originality of our
proof is that it draws new connections between the adversary bound, a modern
technique of theoretical computer science, and early theorems of quantum
mechanics. Indeed, the proof of the lower bound is based on Ehrenfest's
theorem, while the upper bound relies on the adiabatic theorem, as it goes by
constructing a universal adiabatic quantum query algorithm. Another originality
is that we use for the first time in the context of quantum computation a
version of the adiabatic theorem that does not require a spectral gap.Comment: 22 pages, compared to v1, includes a rigorous proof of the
correctness of the algorithm based on a version of the adiabatic theorem that
does not require a spectral ga
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