Constant-Depth Magic State Cultivation with Clifford Measurements by Gauging

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 ∼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 O(d) depth of the Clifford measurement circuit, magic state cultivation becomes impractical for d>5. Here, we perform logical XS^{†} 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 postselection 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.