Fig. 4: Three-qubit phase error correction.
a, Schematic of the quantum circuit for three-qubit phase error correction. The phase error Z(p) is a virtual phase rotation with a rotation angle of \(2{\rm{\arcsin }}\left(\sqrt{p}\right)\), which results in an effective error rate of p. We prepare the data qubit input state \(|\psi \rangle \) by initialization to a spin-down state and a subsequent single-qubit rotation I, X/2, Y/2 or X. b, Measured process fidelities for the corrected and uncorrected cases. Each data point is obtained by averaging 2,000 experiments that are segmented into 1,000 experiments with interleaved qubit frequency calibrations. The error bars are obtained by a Monte Carlo resampling method41 and represent 1σ from the mean. Inset, calculated fidelities for the ideal cases without gate errors. c, Schematic of the quantum circuit for three-qubit dephasing error correction. The waiting time tw is the time interval between the last single-qubit rotation in U and the first single-qubit rotation in U−1. The deviation of the purple curve from the black curve reflects the gate infidelities in the encoding and decoding. d, Comparison of the state fidelities of the corrected and uncorrected qubits. In the case of the physical qubit, we perform a Ramsey measurement with varying waiting time tw between the first π/2 pulse and the pre-rotation for tomographic readout. Each data point is obtained by averaging 3,000 experiments that are segmented into 1,000 experiments with interleaved qubit frequency calibrations. The data acquisition time is the same for all traces in this figure. The solid curves are guides to the eye obtained by fitting to a general exponential decaying function42 F(t) = (1 + αexp(−(t/T2)n)/2, with α = 0.492 ± 0.005, 0.432 ± 0.008 and 0.464 ± 0.007, T2 = 1.44 ± 0.02, 1.36 ± 0.04 and 1.12 ± 0.03 μs, and n = 1.90 ± 0.08, 2.1 ± 0.2 and 1.68 ± 0.08 for the physical qubit, corrected qubit and uncorrected qubit, respectively. The errors are 1σ from the mean.
