Quantum computation has the potential to solve certain mathematical and physical problems faster than a classical computer. However, one of the key challenges to realizing a quantum computing device is controlling systematic errors and maintaining the coherence of the quantum state. In addition, once a scalable quantum computer is built, we must be able to fault-tolerantly communicate a quantum algorithm to the physical device. In this dissertation, we take several significant steps towards the realization of a quantum computer. We develop a framework for use with any proposed technology to map a quantum algorithm into fault-tolerant machine instructions. Within this framework, we combat errors and decoherence by introducing quantum error correction and fault tolerance into a quantum circuit. We analyze specific fault-tolerant circuitry on two general architectures, nonlocal and nearest-neighbor, to determine failure thresholds of the physical circuit components. These failure thresholds allow physicists to evaluate technology proposals. Our goal is to provide design tools, failure thresholds, and fault-tolerant constructions to combat the faultiness of quantum computation.
Quantum computation has the potential to solve certain mathematical and physical problems faster than a classical computer. However, one of the key challenges to realizing a quantum computing device is controlling systematic errors and maintaining the coherence of the quantum state. In addition, once a scalable quantum computer is built, we must be able to fault-tolerantly communicate a quantum algorithm to the physical device. In this dissertation, we take several significant steps towards the realization of a quantum computer. We develop a framework for use with any proposed technology to map a quantum algorithm into fault-tolerant machine instructions. Within this framework, we combat errors and decoherence by introducing quantum error correction and fault tolerance into a quantum circuit. We analyze specific fault-tolerant circuitry on two general architectures, nonlocal and nearest-neighbor, to determine failure thresholds of the physical circuit components. These failure thresholds allow physicists to evaluate technology proposals. Our goal is to provide design tools, failure thresholds, and fault-tolerant constructions to combat the faultiness of quantum computation.
A reliable large-scale quantum computer would be able to solve problems in physics and chemistry exponentially faster than current classical processors. A large-scale quantum device has not been built because quantum systems are naturally sensitive to environmental influences which manifest as errors in memory and operations during computation. For a large-scale device to become a reality, protocols must be developed that reduce the influence of errors during computation in a manner that maintains scalability of the device. This scalability criteria requires the protocols developed to handle errors must be implemented in a way such that the size of the quantum system and number of operations grows in a tractable manner. Furthermore, the sources of errors must be modeled accurately for true assessments of the viability of these protocols. In this dissertation, we present an investigation into methods of performing reliable quantum computation in the presence of errors in small quantum systems (< 50 qubits). These methods should be considered as software primitives used to built reliable large-scale quantum algorithms and quantum memories. These methods occur in two flavors: quantum error correction and fault-tolerant operations. For quantum error correction, we perform assessments of error correction in the presence of error sources indicative of ion trap quantum computers. For fault-tolerant operations, we investigate the quantum resource cost and efficacy of implementing various techniques for performing reliable operations that would allow for a quantum advantange in a large-scale device.
Accurate control and addressability of quantum devices will come with the promise of improvement in a wide variety of theoretical and applied fields, such as chemistry, condensed matter physics, theoretical computer science, foundational physics, communications, metrology and others. Decoherence of quantum states and the loss of quantum systems have adverse effects and deter a satisfactory usage of quantum devices. This is the main problem to be overcome, which is the goal of quantum fault tolerance. In this thesis we present a series of works that contribute to some of the fields mentioned above, in the direction of fighting decoherence and loss. These works fall in two categories: on one hand, we looked at computer architectures which can be used to combat errors, using techniques of quantum error correcting codes. In a first project we found decoherence and loss probability thresholds below which quantum computing is provably possible. We assumed a very particular error model tailored specially to quantum dots as single photon sources and linear optics. Subsequently we looked at the problem of loss, both of heralded and unheralded, and devised some ways to fight it. The framework under which this work was done was used to develop theory which is currently being tested in a quantum optics experimental group and will be reported in an article later this year. On the other hand, we studied how the error probability can be reduced at the physical level, thanks exclusively to the properties of the system in which information is stored, as opposed to making use of quantum codes. We looked at a particular superconducting circuit, which is potentially very well protected against some types of decoherence. In particular, we observed that the interaction with the environment become weaker for certain values of the circuit external parameters.
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 transition from noisy intermediate-scale quantum (NISQ) processors to large-scale fault-tolerant machines requires more than incremental improvements in physical qubit fidelity. While advances in quantum error correction (QEC) theory continue to lower the theoretical overhead of quantum computing, much of this research remains abstracted from the physical realities of qubit implementations. This disconnect often results in hardware-agnostic designs that are inefficient or impractical to build. In this dissertation, I identify three primary gaps between theoretical QEC research and practical hardware: (1) the high levels of noise and temporal instability in experimental devices that impede scalable QEC; (2) the prohibitively large qubit and time costs of useful algorithms, driven largely by the limitations of the standard surface code; and (3) the failure of current resource estimates to account for hardware-level constraints, such as limited connectivity and the complexity of implementing non-local operations. To bridge these gaps, I propose a holistic, cross-stack approach where the error correction protocols and the physical hardware are designed in concert. By developing realistic device models and hardware-aware solutions at the lowest levels of the software stack, we can accelerate the path toward functional fault-tolerance. I present my work across three research thrusts: first, leveraging codesign to mitigate hardware-specific noise and reduce physical error rates; second, developing software tools for fault-tolerant quantum computers to reduce the space-time overhead of the QEC layer; and third, proposing two novel hardware-aware QEC architectures that provide practical, high-performance routes to scalable quantum computation.
Quantum computing has enormous potential, but this can only be realised if quantum errors can be controlled sufficiently to allow quantum algorithms to be completed reliably. However, quantum-error-corrected logical quantum bits (qubits) which can be said to have achieved meaningful error suppression have not yet been demonstrated. This thesis reports research on several topics related to the challenge of designing fault-tolerant quantum computers. The first topic is a proposal for achieving large-scale error correction with the surface code in a silicon donor based quantum computing architecture. This proposal relaxes some of the stringent requirements in donor placement precision set by previous ideas from the single atom level to the order of 10 nm in some regimes. This is shown by means of numerical simulation of the surface code threshold. The second topic then follows, it is the development of a method for benchmarking and assessing the performance of small error correcting codes in few-qubit systems, introducing a metric called 'integrity' - closely linked to the trace distance -- and a proposal for experiments to demonstrate various stepping stones on the way to 'strictly superior' quantum error correction. Most quantum error correcting codes, including the surface code, do not allow for fault-tolerant universal computation without the addition of extra gadgets. One method of achieving universality is through a process of distilling and then consuming high quality 'magic states'. This process adds additional overhead to quantum computation over and above that incurred by the use of the base level quantum error correction. The latter parts of this thesis report an investigation into how many physical qubits are needed in a `magic state factory' within a surface code quantum computer and introduce a number of techniques to reduce the overhead of leading magic state techniques. It is found that universal quantum computing is achievable with ∼ 16 million qubits if error rates across a device are kept below 10-4. In addition, the thesis introduces improved methods of achieving magic state distillation for unconventional magic states that allow for logical small angle rotations, and show that this can be more efficient than synthesising these operations from the gates provided by traditional magic states.
Usual scenarios of fault-tolerant computation are concerned with the fault-tolerant realization of quantum algorithms that compute classical functions, such as Shor's algorithm for factoring. In particular, this means that input and output to the quantum algorithm are classical. In contrast to stand-alone single-core quantum computers, in many distributed scenarios, quantum information might have to be passed on from one quantum information processing system to another one, possibly via noisy quantum communication channels with noise levels above fault-tolerant thresholds. In such situations, quantum information processing devices will have quantum inputs, quantum outputs or even both, which pass qubits among each other. Working in the fault-tolerant framework of [Kitaev, 1997], we show that any quantum circuit with quantum input and output can be transformed into a fault-tolerant circuit that produces the ideal circuit with some controlled noise applied at the input and output. The framework allows the direct composition of the statements, enabling versatile future applications. We illustrate this with a concrete application, namely, communication over a noisy channel with faulty encoding and decoding operations [Christandl and M{ü}ller-Hermes, 2024]. For communication codes with linear minimum distance, we construct fault-tolerant encoders and decoders for general noise (including coherent errors). For the weaker, but standard, model of local stochastic noise, we obtain fault-tolerant encoders and decoders for any communication code that can correct a constant fraction random errors.