Quantum error correcting codes are designed to pinpoint exactly when and where errors occur in quantum circuits. This feature is the foundation of their primary task: to support fault-tolerant quantum computation. However, this feature could used as the basis of benchmarking: By analyzing the outputs of even small-scale quantum error correction circuits, a detailed picture can be constructed of error processes across a quantum device. Here we perform an example of such an analysis, using the results of small repetition codes to determine the error rate of each qubit while idle during a syndrome measurement. This provides an idea of the errors experienced by the qubits across a device while they are part of the kind of circuit that we expect to be typical in fault-tolerant quantum computers.
Quantum error correcting codes are designed to pinpoint exactly when and where errors occur in quantum circuits. This feature is the foundation of their primary task: to support fault-tolerant quantum computation. However, this feature could used as the basis of benchmarking: By analyzing the outputs of even small-scale quantum error correction circuits, a detailed picture can be constructed of error processes across a quantum device. Here we perform an example of such an analysis, using the results of small repetition codes to determine the error rate of each qubit while idle during a syndrome measurement. This provides an idea of the errors experienced by the qubits across a device while they are part of the kind of circuit that we expect to be typical in fault-tolerant quantum computers.
Quantum error correction codes are essential for fault-tolerant quantum computing. They are a method of detecting and correcting errors in a quantum computer by encoding logical qubits across multiple physical qubits. Several prominent error correction codes have been developed and larger and more complex codes are being developed continuously. However, evaluating the fault-tolerance of these larger codes is becoming exponentially more complicated. While conventional methods of proving and evaluating the fault-tolerance of these circuits have worked thus far, they are soon to be obsolete. The aim of this thesis is to develop a framework that automatically evaluates the fault-tolerance of a quantum error correction circuit across different hardware architectures. The framework works as follows: it takes a noiseless quantum error correction code; injects hardware-realistic noise into it; and evaluates the fault-tolerance of the resulting noisy circuit by estimating its logical error rate through Monte Carlo sampling. For the injection of hardware-realistic noise, this thesis considers noise stemming from five different quantum computing hardware architectures: superconducting qubits, neutral atoms, trapped ions, photonic qubits, and spin qubits. The noise model was constructed by extracting single-qubit gate, two-qubit gate and measurement error fidelities from current experimental literature. The noise model and circuit generation is implemented in Python, circuit representation and noise injection is executed in Stim and decoding and estimating the logical error rate is achieved through PyMatching. The framework is evaluated on the repetition code, the surface code, and the color code; the logical error rate is estimated across all five hardware architectures under the noise model; and the results are discussed in the context of hardware compatibility for the given codes.
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.
Building reliable quantum computers requires protecting fragile quantum states from inevitable environmental noise and operational errors. While quantum error correction codes like the Steane [[7,1,3]] code provide elegant theoretical solutions, their practical success hinges critically on how we measure errors - a process called syndrome extraction. The challenge lies in the ancilla qubits used for measurement: when they fail, errors can cascade across the entire quantum system, destroying the very information we're trying to protect. We address this fundamental problem by implementing and comparing three sophisticated syndrome measurement strategies: Shor's cat-state approach, which distributes measurements across multiple entangled ancillas achieving 85-92% preparation success; Steane's encoded-ancilla method using complete error-corrected logical qubits reaching 97.8% syndrome fidelity; and a flexible unified framework that adapts strategies based on hardware capabilities. Through extensive simulations using IBM's Qiskit platform spanning randomized benchmarking and T-heavy circuits, we demonstrate that intelligent ancilla management improves error suppression by up to 2.4× compared to standard approaches. Our implementations achieve logical error rates as low as 5.1×10−5 under realistic noise conditions with physical error rates of 10−3, while maintaining near-unity logical fidelity (0.99997) even for deep circuits. The threshold analysis reveals robust performance across distance-3 to distance-13 codes with characteristic threshold curves showing exponential error suppression below the critical physical error rate. These results provide immediately deployable tools for near-term quantum devices and establish practical design principles for scaling toward fault-tolerant quantum computers.
Benchmarking physical devices and verifying logical algorithms are important tasks for scalable fault-tolerant quantum computing. Numerous protocols exist for benchmarking devices before running actual algorithms. In this work, we show that both physical and logical errors of fault-tolerant circuits can even be characterized in-situ using syndrome data. To achieve this, we map general fault-tolerant Clifford circuits to subsystem codes using the spacetime code formalism and develop a scheme for estimating Pauli noise in Clifford circuits using syndrome data. We give necessary and sufficient conditions for the learnability of physical and logical noise from given syndrome data, and show that we can accurately predict logical fidelities from the same data. Importantly, our approach requires only a polynomial sample size, even when the logical error rate is exponentially suppressed by the code distance, and thus gives an exponential advantage against methods that use only logical data such as direct fidelity estimation. We demonstrate the practical applicability of our methods in various scenarios using synthetic data as well as the experimental data from a recent demonstration of fault-tolerant circuits by Bluvstein et al. [Nature 626, 7997 (2024)]. Our methods provide an efficient, in-situ way of characterizing a fault-tolerant quantum computer to help gate calibration, improve decoding accuracy, and verify logical circuits.
The repetition code is an important primitive for the techniques of quantum error correction. Here we implement repetition codes of at most 15 qubits on the 16 qubit ibmqx3 device. Each experiment is run for a single round of syndrome measurements, achieved using the standard quantum technique of using ancilla qubits and controlled operations. The size of the final syndrome is small enough to allow for lookup table decoding using experimentally obtained data. The results show strong evidence that the logical error rate decays exponentially with code distance, as is expected and required for the development of fault-tolerant quantum computers. The results also give insight into the nature of noise in the device.
Benchmarking physical devices and verifying logical algorithms are important tasks for scalable fault-tolerant quantum computing. Numerous protocols exist for benchmarking devices before running actual algorithms. In this work, we show that both physical and logical errors of fault-tolerant circuits can even be characterized using syndrome data. To achieve this, we map general fault-tolerant Clifford circuits to subsystem codes using the spacetime code formalism and develop a scheme for estimating Pauli noise in Clifford circuits using syndrome data. We give necessary and sufficient conditions for the learnability of physical and logical noise from given syndrome data, and show that we can accurately predict logical fidelities from the same data. Importantly, our approach requires only a polynomial sample size, even when the logical error rate is exponentially suppressed by the code distance, and thus gives an exponential advantage against methods that use only logical data such as direct fidelity estimation. We demonstrate the practical applicability of our methods in various scenarios using synthetic data as well as the experimental data from a recent demonstration of fault-tolerant circuits by Bluvstein [ , 7997 (2024)]. Our methods provide an efficient, way of characterizing a fault-tolerant quantum computer to help gate calibration, improve decoding accuracy, and verify logical circuits.