The Study of Entangled States in Quantum Computation and Quantum Information Science

TL;DR

This thesis investigates the properties and applications of entangled states in quantum computation and information science, including graph state classification, quantum error correction, and efficient entangling measurement circuits.

Contribution

It provides new characterizations of graph state equivalence classes, demonstrates limitations of transversal gates in quantum error correction, and constructs scalable circuits for entangling measurements.

Findings

Graph states with certain graph structures have simple equivalence classifications.

Transversal gates are insufficient for universal quantum computation on most QECCs.

Efficient polynomial-scale circuits for entangling measurements are constructed.

Abstract

This thesis explores the use of entangled states in quantum computation and quantum information science. Entanglement, a quantum phenomenon with no classical counterpart, has been identified as an important and quantifiable resource in many areas of theoretical quantum information science, including quantum error correction, quantum cryptography, and quantum algorithms. We first investigate the equivalence classes of a particular class of entangled states (known as graph states due to their association with mathematical graphs) under local operations. We prove that for graph states corresponding to graphs with neither cycles of length 3 nor 4, the equivalence classes can be characterized in a very simple way. We also present software for analyzing and manipulating graph states. We then study quantum error-correcting codes whose codewords are highly entangled states. An important area of investigation concerning QECCs is to determine which resources are necessary in order to carry out any computation on the code to an arbitrary degree of accuracy, while simultaneously maintaining a high degree of resistance to noise. We prove that transversal gates, which are designed to prevent the propagation of errors through a system, are insufficient to achieve universal computation on almost all QECCs. Finally, we study the problem of creating efficient quantum circuits for creating entangling measurements. Entangling measurements can be used to harness the apparent extra computing power of quantum systems by allowing us to extract information about the global, collective properties of a quantum state using local measurements. We construct explicit quantum circuits that create entangling measurements, and show that these circuits scale polynomially in the input parameters.