Robust quantum computation requires encoding delicate quantum information into degrees of freedom that are hard for the environment to change. Quantum encodings have been demonstrated in many physical systems by observing and correcting storage errors, but applications require not just storing information; we must accurately compute even with faulty operations. The theory of fault-tolerant quantum computing illuminates a way forward by providing a foundation and collection of techniques for limiting the spread of errors. Here we implement one of the smallest quantum codes in a five-qubit superconducting transmon device and demonstrate fault-tolerant state preparation. We characterize the resulting code words through quantum process tomography and study the free evolution of the logical observables. Our results are consistent with fault-tolerant state preparation in a protected qubit subspace.
We propose a measurement-based fault-tolerant quantum computing (MB-FTQC) architecture for high-connectivity platforms, targeting early-FTQC devices with thousands to tens of thousands of physical qubits. Unlike earlier teleportation-based architectures that rely on code concatenation for scalability, MB-FTQC operates in a zero-concatenation regime with compact fixed-distance codes, keeping the qubit overhead within reach of such devices. MB-FTQC combines offline-verified logical resource states with Knill's error-correcting teleportation, allowing each online step to use single-round syndrome extraction and independent lookup-table decoding. Under circuit-level depolarizing noise, we study a Steane-code implementation with partially fault-tolerant analog
RZ(θ) rotations and a Golay-code implementation with higher-order zero-level distillation for fault-tolerant
T-state preparation. At a physical error rate of
10−4, our estimates give a megaquop-scale equivalent
T-count of
1.89×106 for the Steane implementation and a gigaquop-scale characteristic
T-count of
1.87×109 for the Golay implementation. With sufficient parallel resource supply, MB-FTQC achieves a short online critical path, while resource estimates including verification failures and retries show competitive spacetime costs. These results show that compact, fixed-distance codes can support fault-tolerant computation at substantial scales in the early-FTQC regime, without code concatenation.
In this thesis, we consider steps towards building a useful quantum computer in the presence of realistic errors. We start by presenting a novel scheme for fault-tolerant quantum computation with Rydberg atoms. The scheme uses topological error correction codes to provide fault-tolerance and utilises electromagnetically induced transparency to perform entangling gates between qubits. We examine the benefits of this scheme compared to alternative approaches, and we investigate the errors that are likely to occur and how these can be managed. We then consider how to perform quantum computation with a surface code based quantum computer that has permanently faulty components. We study both faulty links between qubits, which prevent qubits from interacting, and faulty qubits themselves; such faults could affect a wide variety of physical implementations, including schemes that use superconducting qubits and those that use trapped ions. The approach we propose for dealing with these faulty components requires minimal modification to the quantum operations required by the surface code and should therefore be implementable by almost all current and future proposals for surface code based quantum devices. Finally, we consider how to perform fault-tolerant quantum computation with components that fail probabilistically. This section is largely motivated by quantum computation with linear optics and therefore focuses on quantum computation with cluster states. We show that these probabilistic failures can be tolerated providing the failure rate is suitably low, and we present methods for increasing error thresholds in the presence of such errors.