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Evolving Quantum Algorithms with Genetic Programming

Duncan MacKinnon

Published Dec 20, 2017• 3 citations
OpenAlex

Abstract

In this thesis we present a genetic programming scheme for evolving quantum programs in the form of arbitrary unitary operators. Expanding on the work done by Hutsell and Greenwood, and later by Krawec in developing unitary operator evolution schemes, we present an alternative representation for unitary operators which has a better search topology than the representations used in these previous implementations. This alternative representation is tested on a general unitary evolution problem, and displays better convergence than previously implemented representations.

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In this thesis we present a genetic programming scheme for evolving quantum programs in the form of arbitrary unitary operators. Expanding on the work done by Hutsell and Greenwood, and later by Krawec in developing unitary operator evolution schemes, we present an alternative representation for unitary operators which has a better search topology than the representations used in these previous implementations. This alternative representation is tested on a general unitary evolution problem, and displays better convergence than previously implemented representations.
The paper presents a new representation and corresponding set of genetic operators for a scheme to evolve quantum circuits with various properties. The scheme is a variant on the techniques of genetic programming and genetic algorithms, having components borrowed from each. By recognising the foundation of a quantum circuit as being a collection of gates, each operating on various categories of qubits and each taking parameters, the scheme can successfully search for most circuits. The algorithm is applied to the problem of entanglement production.
Automatic quantum circuit design is motivated by the difficulties in manual design, because quantum algorithms are highly non-intuitive and practical quantum computer hardware is not yet available. Thus, quantum computers have to be simulated on classical hardware which naturally entails an exponential growth of computational costs and allows only to simulate small quantum systems, i. e., with only few qubits. Huge search spaces render evolutionary approaches nearly unable to achieve breakthrough solutions in the development of new quantum algorithms. Consequently, at present we must be content to evolve essentially already existing (black-box) quantum algorithms. This thesis presents empirical results on the evolution of quantum circuits using genetic programming. For that purpose, a linear and a linear-tree GP system (allowing intermediate measurements) with integrated quantum computer simulator were implemented. Their practicality in evolving quantum circuits is shown in different experiments for 1-SAT (solutions act like Hogg's algorithm) and the Deutsch-Jozsa problem. These experiments confirm that the evolution of quantum circuits is practically feasible only for sufficiently small problem instances. In this context, scalability and the detection of scalability becomes very important. It is shown that scalable quantum circuits are evolvable to a certain degree: a general quantum circuit can be inferred manually from the evolved solutions for small instances of the given problem. Besides, further experiments indicate that 're-evolution' is effective for the evolution of scalable quantum circuits. With this method the start population of a problem instance is inoculated with evolved solutions for a smaller problem instance. Furthermore, investigations of fitness landscapes and selection strategies are made, with the aim of improving the efficiency of evolutionary search. A notable result is that using the crossover operator damages rather than benefits evolution of quantum circuits.
In this thesis a novel representation of a unitary matrix intended to act as a quantum program generated by evolutionary programming shall be presented. The representation consists of a number of 2times22 \\times 22times2 matrices being evolved and combined such that a unitary matrix capable of carrying out a predetermined function on a pure quantum state is generated. In this thesis the algorithm will demonstrate proficiency at performing tasks such as cloning a quantum state or creating order amidst a quantum state in superposition. Furthermore, in the field of quantum information, quantum channels are implemented in order to either perform quantum operations to a quantum state or for one to transmit a quantum state from a sender to a recipient. When this happens, the physical implementation of a quantum channel will often differ from the intended quantum channel that was to be implemented. In cases such as these, the quantum gate fidelity is calculated in order to determine how decoherent the resulting state is from what it should have been. In this thesis a procedure that uses a genetic algorithm to place an upper bound on minimal quantum gate fidelity will be presented.
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.