We propose families of protocols for magic-state distillation—important components of fault-tolerance schemes—for systems of odd prime dimension. Our protocols utilize quantum Reed-Muller codes with transversal non-Clifford gates. We find that, in higher dimensions, small and effective codes can be used that have no direct analogue in qubit (two-dimensional) systems. We present several concrete protocols, including schemes for three-dimensional (qutrit) and five-dimensional (ququint) systems. The five-dimensional protocol is, by many measures, the best magic-state-distillation scheme yet discovered. It excels both in terms of error threshold with respect to depolarizing noise (36.3%) and the efficiency measure known as yield, where, for a large region of parameters, it outperforms its qubit counterpart by many orders of magnitude.
The main focus of this thesis is to explore the advantages of using higher-dimensional quantum systems (qudits) as building blocks for fault-tolerant quantum computation. In particular, we investigate the two main essential ingredients of many state-of-the-art fault-tolerant schemes [133], which are magic state distillation and topological error correction. The theory for both of these components is well established for the qubit case, but little has been known for the generalised qudit case. For magic state distillation, we first present a general numerical approach that can be used to investigate the distillation properties of any stabilizer code. We use this approach to study small threedimensional (qutrit) codes and classify, for the first time, new types of qutrit magic states. We then provide an analytic study of a family of distillation protocols based on the quantum Reed-Muller codes. We discover a particular five-dimensional code that, by many measures, outperforms all known qubit codes. For the topological error correction, we study the qudit toric code serving as a quantum memory. For this purpose we examine an efficient renormalization group decoder to estimate the error correction threshold. We find that when the qudit toric code is subject to a generalised bit-flip noise, and for a sufficiently high dimension, a threshold of 30% can be obtained under perfect decoding.
Error-correcting codes protect quantum information and form the basis of fault-tolerant quantum computing. Leading proposals for fault-tolerant quantum computation require codes with an exceedingly rare property, a transversal non-Clifford gate. Codes with the desired property are presented for d-level qudit systems with prime d. The codes use n=d-1 qudits and can detect up to ∼d/3 errors. We quantify the performance of these codes for one approach to quantum computation known as magic-state distillation. Unlike prior work, we find performance is always enhanced by increasing d.
Magic state distillation is a leading but costly approach to fault-tolerant quantum computation, and it is important to explore all possible ways of minimizing its overhead cost. The number of ancillae required to produce a magic state within a target error rate
ε is
O(logγ(ε−1)) where
γ is known as the yield parameter. Hastings and Haah derived a family of distillation protocols with sublogarithmic overhead (i.e.,
γ<1) based on punctured Reed-Muller codes. Building on work by Campbell \textit{et al.} and Krishna-Tillich, which suggests that qudits of dimension
p>2 can significantly reduce overhead, we generalize their construction to qudits of arbitrary prime dimension
p. We find that, in an analytically tractable puncturing scheme, the number of qudits required to achieve sublogarithmic overhead decreases drastically as
p increases, and the asymptotic yield parameter approaches
lnp1 as
p→∞. We also perform a small computational search for optimal puncture locations, which results in several interesting triorthogonal codes, including a
[[519,106,5]]5 code with
γ=0.99.
Despite significant overhead reductions since its first proposal, magic state distillation is often considered to be a very costly procedure that dominates the resource cost of fault-tolerant quantum computers. The goal of this work is to demonstrate that this is not true. By writing distillation circuits in a form that separates qubits that are capable of error detection from those that are not, most logical qubits used for distillation can be encoded at a very low code distance. This significantly reduces the space-time cost of distillation, as well as the number of qubits. In extreme cases, it can cost less to distill a magic state than to perform a logical Clifford gate on full-distance logical qubits.
Practical quantum computation requires high-fidelity instruction executions on qubits. Among them, Clifford instructions are relatively easy to perform, while non-Clifford instructions require the use of magic states. This makes magic state distillation a central procedure in fault-tolerant quantum computing. A magic state distillation factory consumes many low-fidelity input magic states and produces fewer, higher-fidelity states. To reach high fidelities, multiple distillation factories are typically chained together into a multi-level pipeline, consuming significant quantum computational resources. Our work optimizes the resource usage of distillation pipelines by introducing a novel dynamic pipeline architecture. Observing that distillation pipelines consume magic states in a burst-then-steady pattern, we develop dynamic factory scheduling and resource allocation techniques that go beyond existing static pipeline organizations. Dynamic pipelines reduce the qubit cost by 16%-70% for large-scale quantum applications and achieve average reductions of 26%-37% in qubit-time volume on generated distillation benchmarks compared to state-of-the-art static architectures. By significantly reducing the resource overhead of this building block, our work accelerates progress towards the practical realization of fault-tolerant quantum computers.