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arXiv:1403.2827 · Journal of the Korean Physical Society

A genetic-algorithm-based method to find unitary transformations for any desired quantum computation and application to a one-bit oracle decision problem

Jeongho Bang, Seokwon Yoo

Published Dec 1, 2014• 13 citations
arXivHTMLPDF

Abstract

We propose a genetic-algorithm-based method to find the unitary transformations for any desired quantum computation. We formulate a simple genetic algorithm by introducing the “genetic parameter vector” of the unitary transformations to be found. In the genetic algorithm process, all components of the genetic parameter vectors are supposed to evolve to the solution parameters of the unitary transformations. We apply our method to find the optimal unitary transformations and to generalize the corresponding quantum algorithms for a realistic problem, the one-bit oracle decision problem, or the often-called Deutsch problem. By numerical simulations, we can faithfully find the appropriate unitary transformations to solve the problem by using our method. We analyze the quantum algorithms identified by the found unitary transformations and generalize the variant models of the original Deutsch’s algorithm.

DOI
10.3938/jkps.65.2001

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We propose a genetic-algorithm-based method to find the unitary transformations for any desired quantum computation. We formulate a simple genetic algorithm by introducing the “genetic parameter vector” of the unitary transformations to be found. In the genetic algorithm process, all components of the genetic parameter vectors are supposed to evolve to the solution parameters of the unitary transformations. We apply our method to find the optimal unitary transformations and to generalize the corresponding quantum algorithms for a realistic problem, the one-bit oracle decision problem, or the often-called Deutsch problem. By numerical simulations, we can faithfully find the appropriate unitary transformations to solve the problem by using our method. We analyze the quantum algorithms identified by the found unitary transformations and generalize the variant models of the original Deutsch’s algorithm.
A new approach based on Evolution Strategies (ES) is proposed to evolve quantum unitary operators which represent the computational algorithm a quantum computer would perform to solve an arbitrary problem. This approach has superior attributes over previous endeavors by using real valued vectors and not limited to a predefined set of quantum gates. We show this method can be used to solve a variety of key problems from evolving simple quantum gates, quantum oracles, to generalized problem instances. This method is highly extensible and useful for propagating future research of quantum unitary operator development.
Quantum Genetic Algorithms (QGA) integrate genetic programming and quantum computing to address search and optimization problems. The standard strategy of the hybrid QGA approach is to add quantum resources to classical genetic algorithms (GA), thus improving their efficacy (i.e., quantum optimization of a classical algorithm). However, the extent of such improvements is still unclear. Conversely, Reduced Quantum Genetic Algorithm (RQGA) is a fully quantum algorithm that reduces the GA search for the best fitness in a population of potential solutions to running Grover's algorithm. Unfortunately, RQGA finds the best fitness value and its corresponding chromosome (i.e., the solution or one of the solutions of the problem) in exponential runtime, [[EQUATION]], where [[EQUATION]] is the number of qubits in the individuals' quantum register. This paper introduces a novel QGA optimization strategy, namely a classical optimization of a fully quantum algorithm, to address the Reduced Quantum Genetic Algorithm (RQGA) complexity problem. Accordingly, we control the complexity of theReduced Quantum Genetic Algorithm (RQGA) algorithm by selecting a limited number of qubits in the individuals' register and fixing the remaining ones as classical values of '0' and '1' with a genetic algorithm. We also improve the performance of RQGA by discarding unfit solutions and bounding the search only in the area of valid individuals. As a result, our Hybrid Quantum Algorithm with Genetic Optimization (HQAGO) solves search problems in [[EQUATION]] oracle queries, where [[EQUATION]] is the number of fixed classical bits in the individuals' register.
Genetic quantum algorithms (GQA) are population based evolutionary algorithms that imitate quantum physics by introducing quantum bits for a basic probabilistic genotypic representation and hence better population diversity, and quantum gates for evolving the population of solutions. While quantum inspired gates play an important role in the evolutionary process, there is no specific method for their design, i.e. they are mostly developed through ad hoc procedures. Here, we propose a multi-objective meta-level GQA in order to determine parameters of QG which is applicable for a wide variety of optimization problems. Specifically, a two-layer GQA is constructed, in which the lower layer's objective is to optimize the four junction types: Dejong, Peak, Easoms, and Griewank. And the higher layer's objective is to determine optimal parameters for QG that helps the proposed algorithm find best solutions. GQA optimization performance, with optimized parameter of QG is compared with GA on several benchmark problems and the superiority of the proposed method is statistically shown.
By violating the extended Church-Turing thesis, the quantum computing (QC) paradigm has emerged as a serious contender to classical computing. We now know that certain classical exponential problems (NP (nondeterministic polynomial time)-complete) can be solved in polynomial time by QC. Shor’s integer factorization and Grover’s method for unstructured search are perhaps the most well-known quantum algorithms. QC has been successfully applied to a variety of domains including optimization problems. Optimization can be solved by using deterministic/numerical as well as stochastic methods. Genetic algorithms (GA) belong to the latter category. Incorporating quantum mechanical properties such as superposition, entanglement, interference into the structure of GA has resulted in a more efficient evolutionary computing method called the quantum-inspired genetic algorithm. This method has been successfully applied to image processing, signal processing, economic load dispatch, coalition formation, task scheduling, etc. This chapter presents an overview of the current trends and developments in the domain of quantum-inspired genetic algorithms with the aim of investigating areas of further research in this sphere.