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Construction of Unitary Matrices and Bounding Minimal Quantum Gate Fidelity Using Genetic Algorithms

Connor Gregor

Published Sep 1, 2018• 2 citations
OpenAlex

Abstract

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 $2 \\times 2$ 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.

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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.
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.
This paper presents a comparative study regarding the evolutionary design of quantum operators in the form of unitary matrices. Three different techniques allowing the generation of unitary matrices are investigated whose parameters are tuned by means of evolutionary algorithms. Such problem may be considered as a numerical optimisation task because the unitary matrices are represented by complex numbers with real-valued coefficients. Specifically, Genetic Algorithm and Evolution Strategy are applied, each in four different setups, and evaluated on three case studies: the 2-qubit Controlled-NOT gate, 3-qubit entanglement operator and 4-qubit detector of an element with the maximum amplitude. The evolutionary algorithms in combination with the given representation techniques are evaluated in order to determine the abilities of various experimental setups to solve the given tasks. The main contribution is the utilisation of QR decomposition, a technique for generating unitary matrices from theoretically any sequence of complex numbers, that has been applied for the first time in this paper for the evolution of quantum operators. It will be demonstrated that the QR decomposition exhibits a good performance especially in solving more complex quantum operators in comparison with other representations.
Due to the non- intuitive nature of Quantum algorithms, it becomes difficult for a classically trained person to efficiently construct new ones. So rather than designing new algorithms manually, lately, Genetic algorithms (GA) are being implemented for this purpose. GA is a technique to automatically solve a problem using principles of Darwinian evolution. This has been implemented to explore the possibility of evolving an n-qubit circuit when the circuit matrix has been provided using a set of single, two and three qubit gates. Using a variable length population and universal stochastic selection procedure, a number of possible solution circuits, with different number of gates can be obtained for the same input matrix during different runs of GA. The given algorithm has also been successfully implemented to obtain two and three qubit Boolean circuits using Quantum gates. The results demonstrate the effectiveness of the GA procedure even when the search spaces are large.
Methods of optimal control are applied to a model system of interacting two-level particles (e.g., spin-half atomic nuclei or electrons or two-level atoms) to produce high-fidelity quantum gates while simultaneously negating the detrimental effect of decoherence. One set of particles functions as the quantum information processor, whose evolution is controlled by a time-dependent external field. The other particles are not directly controlled and serve as an effective environment, coupling to which is the source of decoherence. The control objective is to generate target one- and two-qubit unitary gates in the presence of strong environmentally-induced decoherence and under physically motivated restrictions on the control field. The quantum-gate fidelity, expressed in terms of a novel state-independent distance measure, is maximized with respect to the control field using combined genetic and gradient algorithms. The resulting high-fidelity gates demonstrate the feasibility of precisely guiding the quantum evolution via optimal control, even when the system complexity is exacerbated by environmental coupling. It is found that the gate duration has an important effect on the control mechanism and resulting fidelity. An analysis of the sensitivity of the gate performance to random variations in the system parameters reveals a significant degree of robustness attained by the optimal control solutions.
The goal of this work is to understand the application of the evolutionary programming approach to the problem of quantum circuit design. This problem is motivated by the following observations: In order to keep up with the seemingly insatiable demand for computing power our computing devices will continue to shrink, all the way down to the atomic scale, at which point they become quantum mechanical systems. In fact, this event, known as Moore?s Horizon, is likely to occur in less than 25 years. The recent discovery of several quantum algorithms which can solve some interesting problems more efficiently than any known classical algorithm. While we are not yet certain that quantum computers will ever be practical to build, there do now exist the first few astonishing experimental devices capable of briefly manipulating small quantities of quantum information. The programming of these devices is already a nontrivial problem, and as these devices and their algorithms become more complicated this problem will quickly become a significant challenge. The Evolutionary Programming (EP) approach to problem solving seeks to mimic the processes of evolutionary biology which have resulted in the awesome complexity of living systems, almost all of which are well beyond our current analysis and engineering capabilities. This approach is motivated by the highly successful application of Koza?s Genetic Programming (GP) approach to a variety of circuit design problems, and specifically the preliminary reports byWilliams and Gray and also Rubinstein who applied GP to quantum circuit design. Accompanying this work is software for evolutionary quantum circuit design which incorporates several advances over previous approaches, including: A formal language for describing parallel quantum circuits out of an arbitary elementary gate set, including gates with one or more parameters. A fitness assessment procedure that measures both average case fidelity with a respect for global phase equivalences, and implementation cost. A Memetic Programming (MP) based reproductive strategy that uses a combination of global genetic and local memetic searches to effectively search through diverse circuit topologies and optimize the parameterized gates they contain. Several benchmark experiments are performed on small problems which support the conclusion that Evolutionary Programming is a viable approach to quantum circuit design and that further experiments utilizing more computational resources and more problem insight can be expected to yield many new and interesting quantum circuits.