Quantum computers herald the arrival of a new era in which previously intractable computational problems will be solved efficiently. However, quantum technology is held down by decoherence, a phenomenon that is omnipresent in the quantum paradigm and that renders quantum information useless when left unchecked. The science of quantum error correction, a discipline that seeks to combine and protect quantum information from the effects of decoherence using structures known as codes, has arisen to meet this challenge. Stabilizer codes, a particular subclass of quantum codes, have enabled fast progress in the field of quantum error correction by allowing parallels to be drawn with the widely studied field of classical error correction. This has resulted in the construction of the quantum counterparts of well-known capacity-approaching classical codes like sparse codes and quantum turbo codes. However, quantum codes obtained in this manner do not entirely evoke the stupendous error correcting abilities of their classical counterparts. This occurs because classical strategies ignore important differences between the quantum and classical paradigms, an issue that needs to be addressed if quantum error correction is to succeed in its battle with decoherence. In this dissertation we study a phenomenon exclusive to the quantum paradigm, known as degeneracy, and its effects on the performance of sparse quantum codes. Furthermore, we also analyze and present methods to improve the performance of a specific family of sparse quantum codes in various different scenarios.
Quantum computers herald the arrival of a new era in which previously intractable computational problems will be solved efficiently. However, quantum technology is held down by decoherence, a phenomenon that is omnipresent in the quantum paradigm and that renders quantum information useless when left unchecked. The science of quantum error correction, a discipline that seeks to combine and protect quantum information from the effects of decoherence using structures known as codes, has arisen to meet this challenge. Stabilizer codes, a particular subclass of quantum codes, have enabled fast progress in the field of quantum error correction by allowing parallels to be drawn with the widely studied field of classical error correction. This has resulted in the construction of the quantum counterparts of well-known capacity-approaching classical codes like sparse codes and quantum turbo codes. However, quantum codes obtained in this manner do not entirely evoke the stupendous error correcting abilities of their classical counterparts. This occurs because classical strategies ignore important differences between the quantum and classical paradigms, an issue that needs to be addressed if quantum error correction is to succeed in its battle with decoherence. In this dissertation we study a phenomenon exclusive to the quantum paradigm, known as degeneracy, and its effects on the performance of sparse quantum codes. Furthermore, we also analyze and present methods to improve the performance of a specific family of sparse quantum codes in various different scenarios.
The well-documented capacity-approaching performance of sparse codes in the realm of classical communications has inspired the search for their quantum counterparts. Sparse quantum codes are generally built as the amalgamation of two robust classical codes and are decoded via classical decoding algorithms. However, the quantum paradigm presents phenomena that act in a deleterious manner on sparse quantum codes when they are decoded based on classical methodologies. One such phenomenon is known as degeneracy, and it is a major contributor to why sparse quantum codes do not entirely evoke the stupendous error correcting abilities of their classical counterparts. In this paper, we adopt a group theoretical approach to discuss the issue of degeneracy as it relates to sparse quantum codes. Furthermore, we compare the decoding process of sparse quantum codes with that of sparse classical codes, illustrating the challenges that appear in the quantum domain. Finally, we provide a detailed example to illustrate the effects of degeneracy on sparse quantum codes and the challenges of designing an optimum decoder for these schemes.
The field of quantum computing holds immense promise for solving complex problems that are currently intractable for classical computers. However, the practical realization of quantum computers faces a critical challenge: the inherent fragility of qubits, the fundamental units of quantum information. Quantum systems are susceptible to various types of errors and decoherence due to their interaction with the surrounding environment, which can lead to a rapid degradation of quantum information. This chapter delves into the pivotal aspect of quantum error correction techniques as a means to mitigate the adverse effects of errors and decoherence in quantum computing systems. Begins by elucidating the theoretical foundations of quantum error correction, exploring the principles that underlie the encoding of quantum information into larger, redundant codes that are resilient to errors. A comprehensive overview of prominent quantum error correction codes, such as the stabilizer codes and topological codes, is provided, along with a discussion of their respective strengths and limitations.
Quantum technologies have shown immeasurable potential to effectively solve several information processing tasks such as prime number factorization, unstructured database search or complex macromolecule simulation. As a result of such capability to solve certain problems that are not classically tractable, quantum machines have the potential revolutionize the modern world via applications such as drug design, process optimization, unbreakable communications or machine learning. However, quantum information is prone to suffer from errors caused by the so-called decoherence, which describes the loss in coherence of quantum states associated to their interactions with the surrounding environment. This decoherence phenomenon is present in every quantum information task, be it transmission, processing or even storage of quantum information. Consequently, the protection of quantum information via quantum error correction codes (QECC) is of paramount importance to construct fully operational quantum computers. Understanding environmental decoherence processes and the way they are modeled is fundamental in order to construct effective error correction methods capable of protecting quantum information. In this thesis, the nature of decoherence is studied and mathematically modelled; and QECCs are designed and optimized so that they exhibit better error correction capabilities.
Quantum computers would prove a ground-breaking effect in several re- search fields to the advantage of our society due to their proved capacity for solving some problems deemed as too complex for classical comput- ers. Consequently, there is a generalized academic effort for constructing a quantum computer. Nonetheless, a real quantum computer does not suffice for implementing quantum algorithms reliably, it must be fault-tolerant. Quantum computers undergo noise due to a phenomenon generally named quantum decoherence, a fault-tolerant quantum computer has the capabil- ity to suppress the effects of decoherence to an extent. For that to happen, a quantum computer should consider quantum error correction, a process in which decoherence is studied and attempted to be corrected. Within the context of quantum error correction, the information from quantum proces- sors is to be stored in a larger, more redundant system named a quantum error correcting code. Afterwards, one can obtain a vector named syndrome which provides partial information on the effect decoherence has had on a code. The process of recovering the decoherence or error that the code has undergone is named decoding. This thesis studies decoders for quan- tum error correcting codes, their performance, complexity and adaption to different types of quantum decoherence.
Quantum information is prone to suffer from errors caused by the so-called decoherence, which describes the loss in coherence of quantum states associated to their interactions with the surrounding environment. This decoherence phenomenon is present in every quantum information task, be it transmission, processing or even storage of quantum information. Consequently, the protection of quantum information via quantum error correction codes (QECC) is of paramount importance to construct fully operational quantum computers. Understanding environmental decoherence processes and the way they are modeled is fundamental in order to construct effective error correction methods capable of protecting quantum information. Moreover, quantum channel models that are efficiently implementable and manageable on classical computers are required in order to design and simulate such error correction schemes. In this article, we present a survey of decoherence models, reviewing the manner in which these models can be approximated into quantum Pauli channel models, which can be efficiently implemented on classical computers. We also explain how certain families of quantum error correction codes can be entirely simulated in the classical domain, without the explicit need of a quantum computer. A quantum error correction code for the approximated channel is also a correctable code for the original channel, and its performance can be obtained by Monte Carlo simulations on a classical computer.
Quantum algorithms have the potential to provide exponential speedups over some of the best known classical algorithms. These speedups may enable quantum devices to solve currently intractable problems such as those in the fields of optimization, material science, chemistry, and biology. Thus, the realization of large-scale, reliable quantum-computers will likely have a significant impact on the world. For this reason, the focus of this dissertation is on the development of quantum-computing applications and robust, scalable quantum-architectures. I begin by presenting an overview of the language of quantum computation. I then, in joint work with Ojas Parekh, analyze the performance of the quantum approximate optimization algorithm (QAOA) on a graph problem called Max Cut. Next, I present a new stabilizer simulation algorithm that gives improved runtime performance for topological stabilizer codes. After that, in joint work with Andrew Landahl, I present a new set of procedures for performing logical operations called "color-code lattice-surgery." Finally, I describe a software package I developed for studying, developing, and evaluating quantum error-correcting codes under realistic noise.