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International Conference on Parallel and Distributed Computing, Applications and Technologies (PDCAT)

Study on Quantum Parallel Processing by Adiabatic Quantum Computation in Bernstein-Vazirani Problem

Shigeru Nakayama, Peng Gang, Ichiro Iimura

Published Dec 1, 2012• 3 citations
Publisher

Abstract

Adiabatic quantum computation has been proposed as quantum parallel processing with adiabatic evolution by using a superposition state to solve combinatorial optimization problem, then it has been applied to many problems like satisfiability problem. Among them, Deutsch and Deutsch-Jozsa problems have been tried to be solved by using adiabatic quantum computation. In this paper, we modify the adiabatic quantum computation and propose to solve Bernstein-Vazirani problem more efficiently by a method with higher observation probability.

DOI
10.1109/pdcat.2012.140

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Adiabatic quantum computation has been proposed as quantum parallel processing with adiabatic evolution by using a superposition state to solve combinatorial optimization problem, then it has been applied to many problems like satisfiability problem. Among them, Deutsch and Deutsch-Jozsa problems have been tried to be solved by using adiabatic quantum computation. In this paper, we modify the adiabatic quantum computation and propose to solve Bernstein-Vazirani problem more efficiently by a method with higher observation probability.
Adiabatic quantum computation has been proposed as quantum parallel processing with adiabatic evolution by using a superposition state to solve combinatorial optimization problem, then it has been applied to many problems like satisfiability problem. Among them, Deutsch and Deutsch-Jozsa problems have been tried to be solved by using adiabatic quantum computation. In this paper, we modify the adiabatic quantum computation and propose to solve Simon problem more efficiently by a method with higher observation probability.
We present two efficient quantum adiabatic algorithms for Bernstein–Vazirani problem and Simon’s problem. We show that the time complexities of the algorithms for Bernstein–Vazirani problem and Simon’s problem are O(1) and O(n), respectively, which are the same complexities as the corresponding algorithms in quantum circuit model. In these two algorithms, the adiabatic Hamiltonians are realized by unitary interpolation instead of standard linear interpolation. Comparing with the adiabatic algorithms using linear interpolation, the energy gaps of our algorithms keep constant. Therefore, the complexities are much easier to analyze using this method.
Quantum computation is a part of quantum information theory. By using quantum mechanical phenomena it is possible to speed up the solving of some problems with quantum algorithms over classical probabilistic algorithms. In some cases quantum algorithms can utilise quantum parallelism in order to speed up problem solving. In this thesis experiments on IBM quantum computers are compared with ideal error free simulations. In the first part quantum teleportation is shown in practice, similarly the performance of Deutsch–Jozsa, Bernstein–Vazirani, Simon’s and Grover’s search algorithms are shown in action on IBM quantum computers and each is compared against an ideal simulation. The last experiment shows how the binary success probability of an unsharp measurement using Lüders instrument can be improved by adding sequential measurements by using an ideal simulation and a simulation using simple noise model.
The model of adiabatic quantum computation is a new paradigm for designing quantum algorithms. This model is based on the quantum adiabatic theorem, where a quantum computer evolves the quantum system slowly to switch gradually from an initial Hamiltonian with a ground state which is easy to construct, to a final Hamiltonian whose ground state encodes the solution of the problem to be solved. This dissertation presents an experimental study on the discrete quantum adiabatic computation in combinatorial search problems, we take k-Satisfiability problem as target problem to be examined. For k ? 3, k-SAT is NP-complete, i.e., among the most difficult NP problems in the worst case. Here we proposes a new monotonic variation method for the phase shift and mixing functions in the adiabatic quantum algorithm called the Quadric variation method, in order to speed up the algorithm search and decrease the overall resulting search cost. In addition, we present a better parameter configuration for the algorithm to be used with quadric variation as well as the previously proposed methods. The experiment are carried out in solving random instances of 3-SAT problems. The results indicate that the proposed method of using monotonic quadric variation in the phase functions of the adiabatic algorithm greatly improve the search behavior of the adiabatic algorithm and reduces the resulting search cost, also the results indicate that the better configuration of the algorithm parameters could greatly enhance the search behavior for the proposed method as well as the previously proposed methods.
The dynamics of nonlinear systems qualitatively change depending on their parameters, which is called bifurcation. A quantum-mechanical nonlinear oscillator can yield a quantum superposition of two oscillation states, known as a Schrödinger cat state, via quantum adiabatic evolution through its bifurcation point. Here we propose a quantum computer comprising such quantum nonlinear oscillators, instead of quantum bits, to solve hard combinatorial optimization problems. The nonlinear oscillator network finds optimal solutions via quantum adiabatic evolution, where nonlinear terms are increased slowly, in contrast to conventional adiabatic quantum computation or quantum annealing, where quantum fluctuation terms are decreased slowly. As a result of numerical simulations, it is concluded that quantum superposition and quantum fluctuation work effectively to find optimal solutions. It is also notable that the present computer is analogous to neural computers, which are also networks of nonlinear components. Thus, the present scheme will open new possibilities for quantum computation, nonlinear science, and artificial intelligence.