Constant depth magic state cultivation with Clifford measurements by gauging

TL;DR

This paper introduces a constant-depth measurement protocol for magic state cultivation in color codes, improving practicality for larger code distances by using gauging techniques and repeated measurements.

Contribution

It develops a constant-depth logical measurement method via gauging, enabling scalable magic state cultivation with lower overhead and comparable error rates.

Findings

Achieves logical error rates of 10^{-12} at 0.05% physical error rate for d=7.

Uses repeated gauging measurements with post-selection, simplifying the protocol.

Maintains over 1% of shots after cultivation, indicating efficiency.

Abstract

Magic states are a scarce resource for two-dimensional qubit stabilizer codes. Magic state cultivation was recently proposed to reduce the cost of magic state preparation by measuring the transversal Clifford operator of the color code. Cultivation achieves $\sim 10^{-9}$ logical error rates for the $d=5$ color code, with substantially lower space-time overhead than magic state distillation. However, due to the $\mathcal{O}(d)$ depth of the Clifford measurement circuit, magic state cultivation becomes impractical for $d>5$. Here, we perform logical $XS^\dagger$ measurements on the color code by gauging a transversal Clifford gate, resulting in a constant-depth logical measurement circuit. We employ repeated gauging measurements with post-selection rather than performing error correction on the Clifford stabilizer code that emerges during the gauging protocol, thus gaining simplicity at the cost of scalability. Our protocol requires a regular square grid connectivity and yields logical error rates comparable to magic state cultivation. The $d=7$ version of our protocol gives access to the $10^{-12}$ logical error rate regime at $0.05\%$ physical error rate while retaining more than $1\%$ of the shots after the equivalent of the cultivation stage.