In this thesis, we explore practical areas in the theory of quantum error correction that have become especially relevant in light of recent experimental advances. We address the question of whether classical decoding calculations, which are required in order to detect, track and correct errors, can be performed in real-time at a sufficient pace to keep up with the quantum clock speed. Our investigation includes consideration of practical details, such as the communication bandwidth that is required for the classical decoder to interact with the quantum control hardware. We consider the technique of post-selection, which is often used to boost logical performance in small-scale experimental demonstrations of quantum error correction. In particular, we address the common criticism that post-selection is not scalable, and investigate applications of post-selection for medium-scale state preparation circuits, which will continue to be relevant in large-scale quantum computers. Finally, we address the problem of circuit design, whereby theoretical constructs from quantum error correction are translated into a gate-level set of instructions that can be executed on hardware, and in particular we search for time-optimal circuits for detecting errors in topological codes.
In this thesis, we explore practical areas in the theory of quantum error correction that have become especially relevant in light of recent experimental advances. We address the question of whether classical decoding calculations, which are required in order to detect, track and correct errors, can be performed in real-time at a sufficient pace to keep up with the quantum clock speed. Our investigation includes consideration of practical details, such as the communication bandwidth that is required for the classical decoder to interact with the quantum control hardware. We consider the technique of post-selection, which is often used to boost logical performance in small-scale experimental demonstrations of quantum error correction. In particular, we address the common criticism that post-selection is not scalable, and investigate applications of post-selection for medium-scale state preparation circuits, which will continue to be relevant in large-scale quantum computers. Finally, we address the problem of circuit design, whereby theoretical constructs from quantum error correction are translated into a gate-level set of instructions that can be executed on hardware, and in particular we search for time-optimal circuits for detecting errors in topological codes.
The development of quantum computers requires not only experimental advances, but also theoretical efforts to tackle the bottlenecks and identify potential applications. This thesis addresses some essential elements in each stage of the development of quantum computers, from the currently available hardware to the long-term devices. The critical factor that hinders the quantum hardware from performing perfect operations is the presence of noise, however, it can be overcome by error correction codes. We first consider a large-scale quantum processor with asymmetrical noise and preferably a modular structure. By concatenating the surface code with an error-detecting code, and utilising the information from detection with a customised decoder, we find very high thresholds for the designed devices. The second topic is small error correction codes which are more realistic to be implemented in the near future. A measure `integrity' is introduced to benchmark the performance of quantum memories to demonstrate beneficial error correction, associated with four milestones. We show integrity is closely related with other commonly-used measures and can be easily assessed by experimentalists. This framework is then studied in detail in the context of an ion-trap quantum device implemented with the seven qubit code. We present a protocol based on detailed sequences of crystal-reconfiguration operations and stabiliser mappings to perform a full error correction cycle. Realistic error models and parameters from reported works are also given. While experimental efforts continue to battle against the noise, some hybrid quantum-classical algorithms have been developed to be implemented on the near-term noisy devices. The variational algorithm is one of the hybrid algorithms that have been widely applied to simulations of condensed matter physics and quantum chemistry. In the next chapter, we introduce a variational quantum compiler to automatically search for the encoding circuits of small error correction codes. By tailoring the ansatz to optimise the circuit for particular hardware systems, we show some novel circuits discovered with the compiler. The final chapter applies the variational algorithm to solve linear algebra problems including linear systems of equations and matrix-vector multiplications. We show that the linear algebra problems can be solved with near-term devices, with circuit depth linear to the condition number and super-linear to the number of qubits.
Quantum computing has the potential to transform information technology by offering algorithms for certain tasks, such as quantum simulation, that are vastly more efficient than what is possible with any classical device. But experimentally implementing practical quantum information processing is a very difficult task. Here we study two important, and closely related, aspects of this challenge: architectures for quantum computing, and quantum error correction Exquisite quantum control has now been achieved in small ion traps, in nitrogen-vacancy centres and in superconducting qubit clusters, but the challenge remains of how to scale these systems to build practical quantum devices. In Part I of this thesis we analyse one approach to building a scalable quantum computer by networking together many simple processor cells, thus avoiding the need to create a single complex structure. The difficulty is that realistic quantum links are very error prone. Here we describe a method by which even these error-prone cells can perform quantum error correction. Groups of cells generate and purify shared resource states, which then enable stabilization of topologically encoded data. Given a realistically noisy network (10% error rate) we find that our protocol can succeed provided that all intra-cell error rates are below 0.8%. Furthermore, we show that with some adjustments, the protocols we employ can be made robust also against high levels of loss in the network interconnects. We go on to analyse the potential running speed of such a device. Using levels of fidelity that are either already achievable in experimental systems, or will be in the near-future, we find that employing a surface code approach in a highly noisy and lossy network architecture can result in kilohertz computer clock speeds. In Part II we consider the question of quantum error correction beyond the surface code. We consider several families of topological codes, and determine the minimum requirements to demonstrate proof-of-principle error suppression in each type of code. A particularly promising code is the gauge color code, which admits a universal transversal gate set. Furthermore, a recent result of Bombin shows the gauge color code supports an error-correction protocol that achieves tolerance to noisy measurements without the need for repeated measurements, so called single-shot error correction. Here, we demonstrate the promise of single-shot error correction by designing a decoder and investigating its performance. We simulate fault-tolerant error correction with the gauge color code, and estimate a sustainable error rate, i.e. the threshold for the long time limit, of ~0.31% for a phenomenological noise model using a simple decoding algorithm.
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.
Quantum error correction will likely be essential for building a large-scale quantum computer, but it comes with significant requirements at the level of classical control software. In particular, a quantum error-correcting code must be supplemented with a fast and accurate classical decoding algorithm. Standard techniques for measuring the parity-check operators of a quantum error-correcting code involve repeated measurements, which both increases the amount of data that needs to be processed by the decoder, and changes the nature of the decoding problem. Knill error correction is a technique that replaces repeated syndrome measurements with a single round of measurements, but requires an auxiliary logical Bell state. Here, we provide a theoretical and numerical investigation into Knill error correction from the perspective of decoding. We give a self-contained description of the protocol, prove its fault tolerance under locally decaying (circuit-level) noise, and numerically benchmark its performance for quantum low-density parity-check codes. We show analytically and numerically that the time-constrained decoding problem for Knill error correction can be solved using the same decoder used for the simpler code-capacity noise model, illustrating that Knill error correction may alleviate the stringent requirements on classical control required for building a large-scale quantum computer.
Quantum error correction (QEC) has emerged as a cornerstone in the quest for scalable and fault-tolerant quantum technologies. While quantum systems promise exponential advantages over classical computation, their extreme susceptibility to noise and decoherence presents a formidable challenge. This review critically examines the trajectory of QEC from its theoretical foundations to recent experimental realizations. Foundational codes, including Shor’s and Steane’s constructions, laid the groundwork for the stabilizer formalism and the development of topological and bosonic codes. Over the past decade, experimental demonstrations have validated these concepts, with logical qubits surpassing the lifetimes of physical qubits on platforms such as superconducting circuits, trapped ions, and photonics. Emerging approaches, including low-density parity-check codes, hybrid mitigation–correction strategies, and artificial intelligence–driven decoders, further illustrate the field’s dynamism. By synthesizing theoretical progress, empirical validation, and cross-platform innovations, this review highlights both achievements and persistent challenges particularly the overhead of physical qubits, the speed of real-time decoding, and integration into full-stack architectures. The analysis concludes that while scalable fault-tolerant quantum computing remains a work in progress, recent breakthroughs mark a decisive transition from abstraction to practicality, establishing QEC as a central pillar in the development of quantum computation, communication, and sensing.
Quantum computers have the potential to solve certain problems exponentially faster than classical computers. However, realizing this potential in practice is a major challenge, as it requires precise control over large-scale quantum systems operating at extremely low error rates. Quantum error correction (QEC) provides a path to achieving such error rates, but its substantial resource overhead poses a significant practical barrier. In addition, identifying which computational problems offer a provable quantum advantage—and which remain fundamentally intractable—remains an open question. This thesis presents progress towards addressing both of these challenges. The first part of this thesis describes advances that substantially reduce the resource overhead of QEC. We begin by presenting realizations of logical circuits in dynamically reconfigurable arrays of neutral atoms. By jointly decoding the logical qubits, we reduce the cost of implementing such transversal Clifford circuits by a factor proportional to the code distance. We then develop new theories of fault tolerance to extend these savings to universal quantum computation with magic state inputs. Finally, we introduce techniques for fast correlated decoding, enabling practical implementations of these improvements in experimental hardware. Collectively, these advances accelerate progress toward large-scale computation and reduce the cost of QEC by over an order of magnitude. The second part of the thesis explores the relative power of quantum and classical computation in combinatorial optimization, a class of problems that is ubiquitous in science and engineering and foundational to the theory of computational complexity. We experimentally implement the optimized quantum adiabatic algorithm (QAA) in neutral atom arrays and observe evidence of a superlinear speedup over classical simulated annealing on certain hard problem instances. To interpret these results, we develop a theoretical framework to compare the performance of QAA with a broad class of classical Markov chain Monte Carlo algorithms. We identify conditions under which a quantum quadratic speedup is achievable and propose modifications to the QAA to reliably realize this advantage. Together, these contributions advance both the practical implementation and theoretical understanding of quantum algorithms.