Fault-tolerance thresholds for code conversion schemes with quantum Reed–Muller codes

Abstract

Code concatenation and conversion are two prominent methods to realize universal fault-tolerant quantum computation without the need for magic state distillation. In this paper, we analyze three quantum Reed-Muller (QRM) codes of different code lengths under code conversion schemes, and compare them with the 105-qubit concatenated code on optimized encoding circuits. The depolarizing error thresholds for the 7-, 15-, and 31-qubit QRM codes turn out to be 1.19 × 10 −3 , 5.40 × 10 −4 and 2.41 × 10 −6 respectively, all of three compared unfavorably with the improved result of 1.52 × 10 −3 for the 105-qubit code, which indicates a relatively small error-suppression capability and strict precision requirements for hardware equipment when using the code conversion scheme. While for resource requirements, code conversion schemes show obvious advantages. For the seven-qubit QRM code, the required qubit and gate overheads for level-3 gates to achieve the target logical error rate of 10 −15 are roughly 4 orders of magnitude smaller than those for the 105-qubit code.

Magic state distillation and cost analysis in fault-tolerant universal quantum computation

“…In that case, the threshold of T gate depends on the minimum threshold value of | Ā0 ⟩, CNOT and S gate. In particular, S gate can obtain a higher threshold on QRM (1,3) than CNOT [45,90,91], since there are fewer kinds of error-prone positions in the former gate circuit (with only one logical qubit involved). [44]) coding on them.…”

“…There are three mainstream models for implementing FTUQC, the magic state model [11,26,37,38], the concatenation mode [39][40][41], and the fault-tolerant conversion model [42][43][44][45]. As the earliest proposal to realize FTUQC, the magic distillation scheme has been extensively studied.…”