We extend standard fault tolerance theory and Kitaev’s model for quantum computation so as to enable quantitative determination of design parameters for quantum computers that ensure that the overall computation yields a correct final result with some prescribed probability, as opposed to merely ensuring that the desired final quantum state is obtained. Our extension allows us to explicitly calculate the number of levels of error correction concatenation needed to achieve a correct final result with some prescribed success probability and demonstrate a clear connection between error correction and fault tolerance.
We extend standard fault tolerance theory and Kitaev’s model for quantum computation so as to enable quantitative determination of design parameters for quantum computers that ensure that the overall computation yields a correct final result with some prescribed probability, as opposed to merely ensuring that the desired final quantum state is obtained. Our extension allows us to explicitly calculate the number of levels of error correction concatenation needed to achieve a correct final result with some prescribed success probability and demonstrate a clear connection between error correction and fault tolerance.
In this paper we extend both standard fault tolerance theory and Kitaev's model for quantum computation, combining them so as to yield quantitative results that reveal the interplay between the two. Our analysis establishes a methodology that allows us to quantitatively determine design parameters for a quantum computer, the values of which ensure that an overall computation of interest yields a correct *final result* with some prescribed probability of success, as opposed to merely ensuring that the desired *final quantum state* is obtained. As a specific example of the practical application of our approach, we explicitly calculate the number of levels of error correction concatenation needed to achieve a correct final result for the overall computation with some prescribed success probability. Since our methodology allows one to determine parameters required in order to achieve the correct final result for the overall quantum computation, as opposed to merely ensuring that the desired final quantum state is produced, our method enables the determination of complete quantum computational resource requirements associated to the actual solution of practical problems.
In this paper we introduce a universal operator theoretic framework for quantum fault tolerance. This incorporates a top-down approach that implements a system-level criterion based on specification of the full system dynamics, applied at every level of error correction concatenation. This leads to more accurate determinations of error thresholds than could previously be obtained. This is demonstrated both formally and with an explicit numerical example. The basis for our approach is the Quantum Computer Condition (QCC), an inequality governing the evolution of a quantum computer. We show that all known coding schemes are actually special cases of the QCC. We demonstrate this by introducing a new, operator theoretic form of entanglement assisted quantum error correction, which incorporates as special cases all known error correcting protocols, and is itself a special case of the QCC.
This thesis is concerned with certain theoretical problems that arise naturally in the context of fault-tolerant quantum computation. Fault-tolerance can be defined as the art of building reliable devices from unreliable components, and is of particular importance for quantum computers that aim to precisely control the dynamics of extremely sensitive quantum systems. A model of quantum computation is a specification of the basic building blocks by which a quantum computation is implemented. The best known model is the quantum circuit model, where computations are implemented by means of unitary quantum gates that are applied to two-level quantum systems known as qubits. In a physical implementation of a quantum circuit, the gates and qubits will inevitably be affected by noise. Fault-tolerant quantum circuits are designed to be resilient against the effects of this noise, provided that it is not too strong. Fault-tolerance in the quantum circuit model is well developed thanks to the theory of quantum error-correcting codes. These codes allow for the correction of small numbers of errors introduced by a variety of noise processes. In a fault-tolerant quantum circuit, qubits are replaced with encoded qubits, and quantum gates with encoded gates that are immediately followed by special quantum circuits for error correction. Provided the rate at which errors occur is below a constant threshold value, the accumulation of errors can be checked so that the correct output of the computation correctly determined. The threshold acts as both a measure of how good the design of a quantum circuit is, and also as a target for experimenters aiming to implement quantum circuits. Much current research in quantum computation is aimed at designing quantum circuits that increase the noise threshold, hopefully to the point where it becomes within the reach of experimenters. A fault-tolerant encoded quantum gate must limit the propagation of errors so that the code’s corrective capabilities are not overwhelmed. It is not so easy to design encoded gates that satisfy this property, and to date only a finite handful of such gates are known. In a faulttolerant quantum circuit, all quantum gates must be decomposed or compiled in terms of those that may be implemented fault-tolerantly. In the first part of this thesis, we present two results that may be applied to this problem of gate compilation. The first is a generic method based on the Solovay-Kitaev theorem that may be applied to all quantum gates, but is most effective for those that act on single qubits. We present the Solovay-Kitaev theorem in its simples known form as an algorithm, together with novel constructions that can be used to implement it. Following this we give a two specialized methods for two-qubit gate decomposition, based on the Cartan decomposition of the Lie group SU(4). The cluster-state model of computation is an alternative to the quantum circuit model, and makes use of quantum measurements and highly entangled cluster-states to implement a quantum computation. The fault-tolerant techniques developed for quantum circuits are not immediately applicable in this model, so in order for it to be a realistic candidate for performing computations we prove that such techniques are possible. In the second part of the thesis we prove that constant fault-tolerance thresholds may be achieved in the cluster-state model, and in particular in an adaption to an optical implementation. Following this we design a complete error correction scheme for optical cluster-state computation, and numerically determine the threshold of this model in the face of the dominant noise models likely to affect such an implementation.
The model of quantum computation developed by Kitaev (1 ,∮4.1) shows that a perfect, error-free, quantum computer can lead to reliable classical outputs, despite the need to apply a necessarily probabilistic measurement. In this paper we extend the analysis to account for necessarily imperfect quantum computation. The analysis presented here is required to establish the utility of practical quantum computation even given the assumption that fault-tolerance techniques are successfully applied. This is due to the fact that the application of currently known fault-tolerance techniques does not permanently and completely remove errors. To this end we have introduced a mathematical relation that compares the accuracy of a necessarily imperfect quantum computation to a prescribed performance bound. Finally, we discuss several mathematical aspects of this bound and its usefulness in analyzing quantum computing protocols.
Quantum computers promise to solve computational problems that are inaccessible to classical machines. The reliable execution of large-scale quantum algorithms requires robust operations in the presence of noise. Quantum error correction addresses this challenge by encoding quantum information redundantly across several physical qubits, such that errors can be detected and corrected during a computation. However, no single error-correcting code intrinsically supports an inherently robust, fault-tolerant implementation of all gates needed for universal quantum computing. Moreover, most modern approaches to both quantum error correction and the implementation of universal logical operations on encoded qubits rely on mid-circuit measurements and feed-forward control. These types of operations are technologically demanding, often slow, and prone to high error rates, and they limit the overall success probability of quantum algorithms. In this thesis, we develop fault-tolerant protocols and circuits for quantum error correction and universal logical computation, with an emphasis on implementations compatible with current hardware systems. We first focus on fault-tolerant quantum error correction on small codes. By comparing different protocols, we identify trade-offs between compact circuit constructions and their robustness to noise. We then assess their practical performance by comparing numerical simulations of noisy circuits with experimental data. Beyond the preservation of encoded information, we study how universal logic can be implemented in a fault-tolerant manner on encoded qubits. A central approach considered in this thesis is code switching, in which logical information is transferred between two distinct error-correcting codes that support complementary sets of inherently fault-tolerant logical gates. In combination, these codes give access to a complete universal fault-tolerant logical gate set. In this thesis, we determine conditions that allow us to construct fault-tolerant protocols for code switching. We adapt these protocols to the capabilities of current hardware, thereby enabling the first experimental realization of universal fault-tolerant logic by means of code switching. However, the overall performance of these schemes is limited by slow and noisy mid-circuit measurements. To address this bottleneck, we construct a suite of measurement-free fault-tolerant universal operations that do not require any measurements during the execution of an algorithm. In this framework, error correction and logical operations are implemented coherently as part of the quantum circuit, without mid-circuit measurements or in-sequence classical control. We introduce measurement-free code-switching schemes that can be scaled to arbitrary code distances through concatenation, and which thereby provide, in principle, arbitrarily strong error suppression. We further tailor our approach to the capabilities of current trapped-ion quantum processors. New circuit constructions support the first experimental implementation of a logical quantum algorithm without mid-circuit measurements, demonstrated through Grover's search on three encoded qubits. Together, the results of this thesis expand the toolbox for early fault-tolerant quantum computation. They show how protocols for quantum error correction and universal logical gates can be designed without mid-circuit measurements with realistic hardware constraints in mind, opening new pathways towards universal quantum logic.
Fault tolerance is an indispensable prerequisite for constructing large-scale universal quantum computers. Drawing philosophies from classical computer architecture, this paper presents a hardware-agnostic three-layer high-level architectural framework for generic fault-tolerant quantum computation. Guided by the real execution workflows of fault-tolerant quantum algorithms, the proposed model is decoupled from specific physical qubit hardware platforms and quantum error correction codes, serving as a universal abstract standard rather than a platform-specific implementation scheme. Special attention is devoted to the intermediate Fault-Tolerance Layer, which serves as the architectural bridge between application-level logical programs and hardware-level execution. We systematically characterize its five internal components, the interfaces and data exchanged among them, and the execution, correction, and adaptation paths that together enable logical synthesis, fault-tolerant resources management, decoding, and runtime fault-tolerant control. An end-to-end example is further provided to illustrate the full-stack operating pipeline of fault-tolerant quantum algorithms under this framework. Given the increasing emphasis on modular, heterogeneous, and cross-layer fault-tolerant quantum systems, our architecture provides a unified foundational model for organizing such designs.