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International Joint Conference on Neural Networks (IJCNN)

Study on the performance of neuromorphic adiabatic quantum computation algorithms

Aiko Ono, Shigeo Sato, Mitsunaga Kinjo, Koji Nakajima

Published Jun 1, 2008• 1 citation
Publisher

Abstract

Quantum computation algorithms indicate possibility that non-deterministic polynomial time (NP-time) problems can be solved much faster than by classical methods. Farhi et al. [2], [3] have proposed an adiabatic quantum computation (AQC) for solving the three-satisfiability problem (3-SAT). We have proposed a neuromorphic quantum computation algorithm based on AQC, in which an analogy to an artificial neural network (ANN) is considered in order to design a Hamiltonian. However, in the neuromorphic AQC, the relation between its computation time and the probability of correct answers is not clear yet. In this paper, we study both of residual energy and the probability of finding solution as a function of computation time. The results show that the performance of the neuromorphic AQC depends on the characteristic of Hamiltonians.

DOI
10.1109/ijcnn.2008.4634148

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Quantum computation algorithms indicate possibility that non-deterministic polynomial time (NP-time) problems can be solved much faster than by classical methods. Farhi et al. [2], [3] have proposed an adiabatic quantum computation (AQC) for solving the three-satisfiability problem (3-SAT). We have proposed a neuromorphic quantum computation algorithm based on AQC, in which an analogy to an artificial neural network (ANN) is considered in order to design a Hamiltonian. However, in the neuromorphic AQC, the relation between its computation time and the probability of correct answers is not clear yet. In this paper, we study both of residual energy and the probability of finding solution as a function of computation time. The results show that the performance of the neuromorphic AQC depends on the characteristic of Hamiltonians.
Quantum computation algorithms indicate possibility that non-deterministic polynomial time problems can be solved much faster than classical methods. We have proposed a neuromorphic quantum computation algorithm based on adiabatic quantum computation, in which an analogy to an artificial neural network is considered in order to design a Hamiltonian. However, in the neuromorphic AQC, the relation between its computation time and success rate has not been clear. In this paper, we study residual energy and the probability of correct answers as a function of computation time. The residual energy behaves as expected from the adiabatic theorem. On the other hand, the success rate strongly depends on energy level crossings of excited states during Hamiltonian evolution. The results indicate that computation time must be adjusted according to a target problem.
Farhi et al. have proposed an adiabatic quantum computation (AQC), which can be applied to NP-problems if one can know an appropriate Hamiltonian for a target problem. We have proposed a neuromorphic adiabatic quantum computation (NAQC) as the AQC with energy dissipation and an efficient method for designing a final Hamiltonian in consideration of the analogy with a neural network. The NAQC can be applied to optimization problems if its cost function can be expressed in a quadratic form. And successful operations have been confirmed by numerical simulations. In addition, local adiabatic evolution for quantum search have proposed by Roland et al. in order to speed-up the calculation time. In this paper, we show preliminary results for NAQC with local adiabatic evolution by numerical simulations.
Real parallel computing with a quantum computer attracts vast interest due to its extreme high potential. We propose a neuromorphic quantum computation algorithm based on an adiabatic Hamiltonian evolution with energy dissipation. This algorithm can be applied to problems if a cost function can be expressed in a quadratic form. This requirement results from the fact that our Hamiltonian is designed by following a method similar to an artificial neural network (ANN). The state of an ANN is often trapped at local minima, and the network outputs an error. Since the state of a quantum system with the proposed algorithm is always in the ground state according to the adiabatic theorem, it is not necessary to be concerned that the quantum state is trapped at local minima. However, there is no guarantee that a quantum algorithm based on an adiabatic Hamiltonian evolution with degeneration or level crossing is successfully executed. We show successful numerical simulation results with the proposed algorithm by introducing energy dissipation to keep the quantum state staying in the ground state, and then we show an application to the nnn-queen problem, which is one of the combinatorial optimization problems.
The advantage of quantum mechanical dynamics in information processing has attracted much interest, and dedicated studies on quantum computation algorithms indicate that a quantum computer has remarkable computational power in certain tasks. Quantum properties such as quantum superposition and quantum tunneling are worth studying because they may overcome the weakness of gradient descent method in classical neural networks. Also, the technique established for neural networks can be useful for developing a quantum algorithm. In this chapter, first the authors show the effectiveness of incorporating quantum dynamics and then propose neuromorphic adiabatic quantum computation algorithm based on the adiabatic change of Hamiltonian. The proposed method can be viewed as one of complex-valued neural networks because a qubit operates like a neuron. Next, the performance of the proposed algorithm is studied by applying it to a combinatorial optimization problem. Finally, they discuss the learning ability and hardware implementation.
A quantum computer employing a single quantum as a qubit executes real parallel computation. Several algorithms have been proposed for quantum computation. However, these algorithms are applicable only to a limited number of applications. Therefore, a general purpose algorithm should be studied and developed for practical use in the near future. We focus on the adiabatic evolution algorithm in order to incorporate an artificial neural network(ANN)-like method and discuss how to use this algorithm for solving an optimization problem.
Quantum computation by adiabatic evolution, as described in quant-ph/0001106, will solve satisfiability problems if the running time is long enough. In certain special cases (that are classically easy) we know that the quantum algorithm requires a running time that grows as a polynomial in the number of bits. In this paper we present numerical results on randomly generated instances of an NP-complete problem and of a problem that can be solved classically in polynomial time. We simulate a quantum computer (of up to 16 qubits) by integrating the Schrodinger equation on a conventional computer. For both problems considered, for the set of instances studied, the required running time appears to grow slowly as a function of the number of bits.