Fault-tolerance of ancilla preparation for logical Toffoli gate of a family of Calderbank-Shor-Steane codes

TL;DR

This paper analyzes fault-tolerance issues in ancilla preparation for the Toffoli gate in Calderbank-Shor-Steane codes and proposes a correction method to improve error thresholds in quantum computation.

Contribution

It identifies error propagation problems in Shor's ancilla preparation method and introduces a correction technique that achieves comparable fault-tolerance thresholds to transversal gates.

Findings

Error propagation causes logical errors in ancilla preparation.

Bit flip error correction improves fault-tolerance.

Modified construction attains similar error thresholds as transversal gates.

Abstract

Fault-tolerant quantum computation using quantum error-correcting codes requires fault-tolerant constructions of nontransversal gates. Shor proposed a fault-tolerant construction of a nontransversal gate, i.e., the Toffoli gate for a family of the Calderbank-Shor-Steane codes which has even-weight stabilizer generators and odd-weight logical operators. In fact, as shown in this paper, error propagations in the ancilla preparation of Shor's construction lead to logical errors. This work provides a solution to this problem through the insertion of bit flip error correction. Finally, error rates of a modified construction are evaluated with use of the concatenated Steane codes, and the same accuracy threshold as transversal gates is obtained in cases where ancillary systems can be prepared just in time.