Table 1 Probability distributions (in percentage) of measured logical states |LaLb〉 for various prepared logical states in each row, with and without stabilizers Sx or Sz applied.

The measurement basis is shown in the last column. The logical states are |00 > L … |11 > L, measured in the Z-basis, and | + + 〉L … | − − 〉L, measured in the X-basis. The very low error probability on the first logical qubit La compared to Lb shows the action of its fault-tolerant construction. We run every circuit 5000 to 6000 times. The results without stabilizer show the number of rejected runs from the parity check on the data qubits (typically ~8%) whereas the additional discard on the results with stabilizer (typically ~20%) is due to the ancilla result. The physical errors for state preparation and measurement are 0.3(1)% for states |0〉 and | + 〉, and 1.2(1)% for states |1〉 and | − 〉.

YieldMeasured logical state |LaLbMeasurement basis
|00〉|01〉|10〉|11〉Z
| + + 〉| + − 〉| − + 〉| − − 〉X
|00〉L91.1(4)98.0(2)1.7(2)0.11(6)0.18(7)Z
|00〉LSz77.8(6)97.8(2)1.7(2)0.18(8)0.3(1)Z
|00〉LSx65.2(7)97.1(3)2.4(3)0.2(1)0.3(1)Z
| + + 〉L91.1(4)95.7(3)3.9(3)0.24(8)0.22(8)X
| + + 〉LSz68.2(7)93.0(5)4.2(4)1.3(2)1.5(2)X
| + + 〉LSx72.1(6)94.3(4)4.5(4)0.5(1)0.7(2)X
| − 1〉L90.1(4)0.22(8)50.5(8)0.09(6)49.2(8)Z
| − 1〉L87.0(5)0.3(1)0.3(1)50.4(8)48.9(8)X
| − 1〉LSz79.9(6)0.15(7)50.0(8)0.10(6)49.8(8)Z
| − 1〉LSz75.5(6)0.4(1)0.3(1)50.1(8)49.2(8)X
| − 1〉LSx72.1(6)0.6(1)50.2(8)0.5(1)48.7(8)Z
| − 1〉LSx76.2(5)0.4(1)0.4(1)50.0(7)49.2(7)X
|0 + 〉L93.2(3)47.4(5)52.5(5)0.06(3)0.05(3)Z
|0 + 〉L92.4(4)50.0(8)0.04(4)49.8(8)0.09(5)X
|0 + 〉LSz81.6(6)48.3(8)51.4(8)0.17(8)0.17(8)Z
|0 + 〉LSz68.5(7)47.1(9)2.4(3)47.4(9)3.1(3)X
|0 + 〉LSx72.0(6)48.3(8)51.5(8)0.16(7)0.12(7)Z
|0 + 〉LSx70.9(7)49.4(9)0.4(1)49.7(9)0.5(1)X
|11〉LSz73.3(6)0.4(1)0.3(1)2.8(3)96.5(3)Z