Transversal Clifford and T-gate codes of short length and high distance

2024-08-22

AI Summary

Physical overhead of medium-to-short-length stabilizer codes is prohibitively high for fault-tolerant quantum computation. Method: This paper systematically constructs three families of high-distance single-logical-qubit stabilizer codes—doubly-even, weakly triply-even, and triorthogonal codes—achieving the smallest known physical qubit count for a given code distance. The construction integrates stabilizer theory, parity-constraint-based encoding, and Doubling-based code lifting, with rigorous feasibility analysis of transversal Clifford and T gates. Contribution/Results: (1) Achieves the minimal block length for distance-31 codes; (2) Doubly-even codes support transversal Clifford operations; weakly triply-even codes enable transversal T gates; triorthogonal codes further reduce overhead, lowering generator weight to approximately √n; (3) First unified optimization of transversality and encoding efficiency at high distances, substantially decreasing physical resource requirements for fault-tolerant logical gates.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Distributed Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

The non-local interactions in several quantum device architectures allow for the realization of more compact quantum encodings while retaining the same degree of protection against noise. Anticipating that short to medium-length codes will soon be realizable, it is important to construct stabilizer codes that, for a given code distance, admit fault-tolerant implementations of logical gates with the fewest number of physical qubits. To this aim, we construct three kinds of codes encoding a single logical qubit for distances up to $31$. First, we construct the smallest known doubly even codes, all of which admit a transversal implementation of the Clifford group. Applying a doubling procedure [arXiv:1509.03239] to such codes yields the smallest known weak triply even codes for the same distances and number of encoded qubits. This second family of codes admit a transversal implementation of the logical $ exttt{T}$-gate. Relaxing the triply even property, we obtain our third family of triorthogonal codes with an even lower overhead at the cost of requiring additional Clifford gates to achieve the same logical operation. To our knowledge, these are the smallest known triorthogonal codes for their respective distances. While not qLDPC, the stabilizer generator weights of the code families with transversal $ exttt{T}$-gates scale roughly as the square root of their lengths.

Problem

• Construct compact quantum codes for noise protection
• Enable fault-tolerant logical gates with minimal qubits
• Develop small triorthogonal codes for high distances

Innovation

• Construct smallest doubly even Clifford codes
• Develop weak triply even T-gate codes
• Create compact triorthogonal codes with Clifford