Universal fault-tolerant gates on concatenated stabilizer codes

TL;DR

This paper demonstrates the existence of non-transversal, fault-tolerant logical gates on small stabilizer codes, enabling universal quantum computation without relying solely on transversal gates or magic state distillation.

Contribution

It introduces a method to perform universal fault-tolerant gates directly on small stabilizer codes using intermediate error correction, expanding the scope of fault-tolerant quantum computing.

Findings

Constructed fault-tolerant Toffoli and controlled-controlled-Z gates on 5- and 7-qubit codes.

Showed any nondegenerate stabilizer code with fault-tolerant Clifford gates has a universal gate set.

Presented a method for code switching across different stabilizer codes.

Abstract

It is an oft-cited fact that no quantum code can support a set of fault-tolerant logical gates that is both universal and transversal. This no-go theorem is generally responsible for the interest in alternative universality constructions including magic state distillation. Widely overlooked, however, is the possibility of non-transversal, yet still fault-tolerant, gates that work directly on small quantum codes. Here we demonstrate precisely the existence of such gates. In particular, we show how the limits of non-transversality can be overcome by performing rounds of intermediate error-correction to create logical gates on stabilizer codes that use no ancillas other than those required for syndrome measurement. Moreover, the logical gates we construct, the most prominent examples being Toffoli and controlled-controlled-Z, often complete universal gate sets on their codes. We detail such universal constructions for the smallest quantum codes, the 5-qubit and 7-qubit codes, and then proceed to generalize the approach. One remarkable result of this generalization is that any nondegenerate stabilizer code with a complete set of fault-tolerant single-qubit Clifford gates has a universal set of fault-tolerant gates. Another is the interaction of logical qubits across different stabilizer codes, which, for instance, implies a broadly applicable method of code switching.