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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 Simon Problem

Shigeru Nakayama, Peng Gang, Ichiro Iimura

Published Dec 1, 2013• 0 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 Simon problem more efficiently by a method with higher observation probability.

DOI
10.1109/pdcat.2013.20

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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 Simon 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 Bernstein-Vazirani problem more efficiently by a method with higher observation probability.
Having the advantages of inherent robustness and fault tolerance, adiabatic quantum algorithms have attracted much attention in recent years. In this paper, we report the NMR experimental realization of adiabatic algorithms for the modified Simon's problem in a NMR quantum information processor. We first realized an adiabatic quantum algorithm by Rao [Phys. Rev. A 67, 052306 (2003)] and then proposed a quantum algorithm by using the method of enhanced symmetry of the Hamiltonian given in Schaller and Schtzhold [Quantum Inf. Comput. 10, 109 (2010)]. We demonstrated this symmetry-enhanced algorithm in experiment. We studied the effects of the number of Trotter evolution steps on the validity of adiabatic evolution and decoherence. In practical applications, a trade-off between pulse imperfection and adiabatic requirement must be made in the Trotter evolution step number. An optimal evolution step number and run time were indicated by our results. The experimental demonstration also showed that the symmetry-enhanced algorithm gave a better performance.
Adiabatic quantum computation (AQC) is a universal model for quantum computation which seeks to transform the initial ground state of a quantum system into a final ground state encoding the answer to a computational problem. AQC initial Hamiltonians conventionally have a uniform superposition as ground state. We diverge from this practice by introducing a simple form of heuristics: the ability to start the quantum evolution with a state which is a guess to the solution of the problem. With this goal in mind, we explain the viability of this approach and the needed modifications to the conventional AQC (CAQC) algorithm. By performing a numerical study on hard-to-satisfy 6 and 7 bit random instances of the satisfiability problem (3-SAT), we show how this heuristic approach is possible and we identify that the performance of the particular algorithm proposed is largely determined by the Hamming distance of the chosen initial guess state with respect to the solution. Besides the possibility of introducing educated guesses as initial states, the new strategy allows for the possibility of restarting a failed adiabatic process from the measured excited state as opposed to restarting from the full superposition of states as in CAQC. The outcome of the measurement can be used as a more refined guess state to restart the adiabatic evolution. This concatenated restart process is another heuristic that the CAQC strategy cannot capture.
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 Schrodinger 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.