A quantum computation can be described by the evolution of a collection of quantum bits, or qubits, according to the laws of quantum mechanics, from an initial state encoding the input of a problem, to its solution represented by the final state. A machine capable of performing this task, a quantum computer, must be robust against errors caused by the inherent imperfections in the fabrication and control of such complex systems. This thesis examines the theory of fault‑tolerant quantum computation, tailored to the physical constraints of today’s leading platforms, most notably superconducting circuits and neutral‑atom arrays.A fault-tolerant quantum computation requires to correct errors in real time, at the cost of a dedicated architectural layer whose integration may introduce additional noise. A possible simplification uses a quantum memory model where information is encoded in a pair of configurations of a multi-qubit system, stabilized by the uniform application of a simple local rule fitting within the theory of cellular automata. We propose a short-term solution, the shearing-rule which protects a one-dimensional repetition code against errors with good performance but lacks a threshold. This shortcoming is addressed by adding small local classical memories, used to define the signal-rules decoders, where error correction relies on exchanging binary signals. Combined with biased-noise qubits, such as cat qubits, these local decoders define a fully local one-dimensional quantum memory.The final part of the manuscript addresses the advantage of quantum machines over their classical counterparts. We emphasize that any such demonstration must account for the presence of noise to be meaningful — for example, while exact sampling from the output distribution of circuits drawn from certain families is believed to be hard for classical computers, the task may become tractable if one allows for an approximate solution that reflects the presence of noise. In this context, we propose a constant-depth quantum circuit that solves a sampling problem in a fault-tolerant manner. Because our approach is particularly well-suited to the atom rearrangement capabilities of neutral-atom array platforms, we hope it will contribute to a future experimental demonstration of a fault-tolerant quantum advantage.
A quantum computation can be described by the evolution of a collection of quantum bits, or qubits, according to the laws of quantum mechanics, from an initial state encoding the input of a problem, to its solution represented by the final state. A machine capable of performing this task, a quantum computer, must be robust against errors caused by the inherent imperfections in the fabrication and control of such complex systems. This thesis examines the theory of fault‑tolerant quantum computation, tailored to the physical constraints of today’s leading platforms, most notably superconducting circuits and neutral‑atom arrays.A fault-tolerant quantum computation requires to correct errors in real time, at the cost of a dedicated architectural layer whose integration may introduce additional noise. A possible simplification uses a quantum memory model where information is encoded in a pair of configurations of a multi-qubit system, stabilized by the uniform application of a simple local rule fitting within the theory of cellular automata. We propose a short-term solution, the shearing-rule which protects a one-dimensional repetition code against errors with good performance but lacks a threshold. This shortcoming is addressed by adding small local classical memories, used to define the signal-rules decoders, where error correction relies on exchanging binary signals. Combined with biased-noise qubits, such as cat qubits, these local decoders define a fully local one-dimensional quantum memory.The final part of the manuscript addresses the advantage of quantum machines over their classical counterparts. We emphasize that any such demonstration must account for the presence of noise to be meaningful — for example, while exact sampling from the output distribution of circuits drawn from certain families is believed to be hard for classical computers, the task may become tractable if one allows for an approximate solution that reflects the presence of noise. In this context, we propose a constant-depth quantum circuit that solves a sampling problem in a fault-tolerant manner. Because our approach is particularly well-suited to the atom rearrangement capabilities of neutral-atom array platforms, we hope it will contribute to a future experimental demonstration of a fault-tolerant quantum advantage.
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 computation promises to revolutionize information processing by exploiting superposition and entanglement, enabling applications such as quantum simulation, combinatorial optimization, and secure quantum communication. Beyond individual processors, the quantum internet aims to connect quantum nodes through entanglement distribution, enabling distributed computation. However, quantum states are fragile: environmental interactions, imperfect gate operations, and measurement errors can quickly degrade encoded information. Reliable quantum computation therefore requires carefully designed quantum error-correcting codes and fault-tolerant protocols. Unlike classical redundancy schemes, quantum error correction must preserve delicate phase relationships and relies on indirect error detection through syndrome extraction with ancillary qubits, enabling error identification without collapsing encoded states. Topological codes, in particular, emerged early as the preferred choice for many architectures due to their planar structure, local interactions, and natural support for fault-tolerant logical operations. Color codes, for instance, allow transversal implementation of all Clifford gates, whereas non-Clifford operations require magic-state injection. More recent constructions, including quantum low-density parity-check codes, offer higher encoding rates and reduced resource overhead, but require long-range qubit connectivity and more complex protocols. Despite experimental progress, several challenges remain. Many platforms exhibit biased noise, motivating codes that exploit channel asymmetry. Real-time decoding remains a critical bottleneck, as fault-tolerant computation requires low-latency processing of rapidly generated syndrome data. Fault-tolerant logical state initialization also remains resource-intensive and scales poorly with code distance. Addressing these challenges, this thesis develops a unified framework spanning theory and experiment. It provides precise performance analysis of quantum error-correcting codes in the low physical error-rate regime, introduces codes tailored to biased noise, develops fast low-complexity decoding algorithms for real-time operation, and presents modular and efficient protocols for fault-tolerant state preparation of CSS codes. The latter is demonstrated experimentally through the preparation of a logical zero state of the Golay code on the Quantinuum H2 trapped-ion quantum computer.
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 Müller-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 of random errors.
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.
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.
Quantum computers have the potential to solve several interesting problems in polynomial \ntime for which no polynomial time classical algorithms have been found. However, \none of the major challenges in building quantum devices is that quantum systems are very \nsensitive to noise arising from undesired interactions with the environment. Noise can lead \nto errors which can corrupt the results of the computation. Quantum error correction is \none way to mitigate the effects of noise arising in quantum devices. \nWith a plethora of quantum error correcting codes that can be used in various settings, \none of the main challenges of quantum error correction is understanding how well various \ncodes perform under more realistic noise models that can be observed in experiments. \nThis thesis proposes a new decoding algorithm which can optimize threshold values of \nerror correcting codes under different noise models. The algorithm can be applied to \nany Markovian noise model. Further, it is shown that for certain noise models, logical \nClifford corrections can further improve a code's threshold value if the code obeys certain \nsymmetries. \n \nSince gates and measurements cannot in general be performed with perfect precision, \nthe operations required to perform quantum error correction can introduce more errors \ninto the system thus negating the benefits of error correction. Fault-tolerant quantum \ncomputing is a way to perform quantum error correction with imperfect operations while \nretaining the ability to suppress errors as long as the noise is below a code's threshold. \nOne of the main challenges in performing fault-tolerant error correction is the high resource \nrequirements that are needed to obtain very low logical noise rates. With the use of \nflag qubits, this thesis develops new fault-tolerant error correction protocols that are applicable \nto arbitrary distance codes. Various code families are shown to satisfy the requirements \nof flag fault-tolerant error correction. We also provide circuits using a constant number of \nqubits for these codes. It is shown that the proposed flag fault-tolerant method uses fewer \nqubits than previous fault-tolerant error correction protocols. \n \nIt is often the case that the noise afflicting a quantum device cannot be fully characterized. \nFurther, even with some knowledge of the noise, it can be very challenging to use \nanalytic decoding methods to improve the performance of a fault-tolerant scheme. This \nthesis presents decoding schemes using several state of the art machine learning techniques \nwith a focus on fault-tolerant quantum error correction in regimes that are relevant to near \nterm experiments. It is shown that even in low noise rate regimes and with no knowledge \nof the noise, noise can be further suppressed for small distance codes. Limitations of machine \nlearning decoders as well as the classical resources required to perform active error \ncorrection are discussed. \n \nIn many cases, gate times can be much shorter than typical measurement times of \nquantum states. Further, classical decoding of the syndrome information used in quantum \nerror correction to compute recovery operators can also be much slower than gate times. \nFor these reasons, schemes where error correction can be implemented in a frame (known \nas the Pauli frame) have been developed to avoid active error correction. In this thesis, we \ngeneralize previous Pauli frame schemes and show how Clifford frame error correction can \nbe implemented with minimal overhead. Clifford frame error correction is necessary if the \nlogical component of recovery operators were chosen from the Clifford group, but could \nalso be used in randomized benchmarking schemes.