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Open Systems & Information Dynamics

Rigorous Estimation of Computational Complexity for OMV SAT Algorithm

Satoshi Iriyama, Masanori Ohya

Published Jun 1, 2008• 12 citations
Publisher

Abstract

For SAT problem, which is known to be NP-complete, Ohya and Masuda found a quantum algorithm calculating a given boolean function for all truth assignments. They showed that we can decide whether this boolean function is satisfiable or not in polynomial time if a certain superposed state can be detected physically, which turns out to be related to an amplification process. Then Ohya and Volovich realized this amplification by means of chaos dynamics [4, 5]. In this paper, we study the complexity of the SAT algorithm by rigorously counting the steps of OMV algorithm discussed previously in [1, 2, 4, 5].

DOI
10.1142/s1230161208000158

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For SAT problem, which is known to be NP-complete, Ohya and Masuda found a quantum algorithm calculating a given boolean function for all truth assignments. They showed that we can decide whether this boolean function is satisfiable or not in polynomial time if a certain superposed state can be detected physically, which turns out to be related to an amplification process. Then Ohya and Volovich realized this amplification by means of chaos dynamics [4, 5]. In this paper, we study the complexity of the SAT algorithm by rigorously counting the steps of OMV algorithm discussed previously in [1, 2, 4, 5].
Ohya and Volovich found the quantum algorithm with a chaos dynamics, called the OV SAT algorithm which enabled to solve NP-Complete problem in polynomial time. It is proved that the unitary operator of the quantum algorithm can be constructed by a product of so called fundamental gates. To discuss the computational complexity rigorously, we intro- duced a generalized quantum Turing machine(GQTM) where the computational process is given by quantum channels including a dispative dynamics and the configuration is represented by density operators. In this paper, we explain the GQTM and the OV SAT algorithm, then we discuss the computational complexity of OV SAT algorithm. Finally, we introduce some resent results and topics by use of GQTM.
Ohya and Volovich found the quantum algorithm with a chaos dynamics, called the OV SAT algorithm which enabled to solve NP-Complete problem in polynomial time. It is proved that the unitary operator of the quantum algorithm can be constructed by a product of so called fundamental gates. To discuss the computational complexity rigorously, we introduced a generalized quantum Turing machine(GQTM) where the computational process is given by quantum channels including a dispative dynamics and the configuration is represented by density operators. In this paper, we explain the GQTM and the OV SAT algorithm, then we discuss the computational complexity of OV SAT algorithm. Finally, we introduce some resent results and topics by use of GQTM.
We analyze the Ohya-Masuda quantum algorithm that solves the so-called “satisfiability” problem, which is an NP-complete problem of the complexity theory. We distinguish three steps in the algorithm, and analyze the second step, in which a coherent superposition of states (a “pure” state) transforms into an “incoherent” mixture presented by a density matrix. We show that, if “nonideal” (in analogy with “nonideal” quantum measurement), this transformation can make the algorithm to fail in some cases. On this basis we give some general notions on the physical implementation of the Ohya-Masuda algorithm.