Quantum Computation with Topological Codes

Christopher Anthony Cesare

This dissertation is concerned with quantum computation using many-body quantum systems encoded in topological codes. The interest in these topological systems has increased in recent years as devices in the lab begin to reach the fidelities required for performing arbitrarily long quantum algorithms. The most well-studied system, Kitaev's toric code, provides both a physical substrate for performing universal fault-tolerant quantum computations and a useful pedagogical tool for explaining the way other topological codes work. In this dissertation, I first review the necessary formalism for quantum information and quantum stabilizer codes, and then I introduce two families of topological codes: Kitaev's toric code and Bombin's color codes. I then present three chapters of original work. First, I explore the distinctness of encoding schemes in the color codes. Second, I introduce a model of quantum computation based on the toric code that uses adiabatic interpolations between static Hamiltonians with gaps constant in the system size. Lastly, I describe novel state distillation protocols that are naturally suited for topological architectures and show that they provide resource savings in terms of the number of required ancilla states when compared to more traditional approaches to quantum gate approximation.

Clarice Dias de Albuquerque, Eduardo Brandani da Silva, Waldir Silva Soares

Jiannis K. Pachos

Aleksander Kubica

This thesis is devoted to studying a class of quantum error-correcting codes — topological quantum codes. We explore the question of how one can achieve fault- tolerant quantum computation with topological codes. We treat quantum error-correcting codes not only as a compelling ingredient needed to build a quantum computer, but also as a useful theoretical tool in other areas of physics. In particular, we explore what insights topological codes can provide into challenging questions, such as the classification of quantum phases of matter. In this thesis, we focus on a family of topological codes — color codes, which are particularly intriguing due to the rich physics they display and their computational power. We start by introducing color codes and explaining their basic properties. Then, we show how to perform fault-tolerant universal quantum computation with three-dimensional color codes by transverse gates and code switching. We later compare the resource overhead of the code-switching approach with that of a state distillation scheme. We discuss how to perform error correction with the toric and color codes, as well as introduce local decoders for those two families of codes. By exploiting a connection between error correction and statistical mechanics we estimate the storage threshold error rates for bit-flip and phase-flip noise in the three-dimensional color code. We finish by showing that the color and toric code families in d dimensions are equivalent in a sense of local unitary transformations and explore implications of this equivalence.

Tapio Simula

Krishna Kumar

Quantum computing stands at the precipice of revolutionizing numerous fields, yet its widespread practical application is obstructed by high error rates, far exceeding those encountered in classical computing. For quantum computers to become viable, these error rates must be significantly reduced, which is a critical challenge in the field. This paper explores the evolving landscape of quantum error correction, focusing on the topological code as a leading strategy for improving the fidelity of quantum computations. topological codes utilize a two-dimensional array of qubits and stabilizer operations to detect and correct quantum errors, offering robust scalability for large-scale quantum systems. The theoretical foundations of topological codes are examined along with their implementation challenges and recent advancements in decoding algorithms. Additionally, stabilizer codes are analyzed and their synergistic role in enhancing quantum error correction schemes. By addressing both theoretical concepts and practical considerations, this study provides insights into overcoming quantum error correction obstacles, moving closer to reliable, fault-tolerant quantum computing systems. The paper also highlights emerging techniques that may shape the future of quantum computing reliability.