Table 1 Sixteen single-qubit basis operations.

Pθ=eiθ2P denotes the gate of rotation along the P axis by an angle of θ, where P = X, Y, Z. MP denotes the operation of measuring the eigenvalue of the Pauli operator P whose outcomes are ±1. MI+P2 and MP are the same operation, but outcomes are noted differently, and MI+P2 denotes the operation of measuring the eigenvalue of the operator I+P2 whose outcomes are 0 and 1. R∣ψ〉 denotes the operation of resetting the qubit state to ∣ψ〉. For composed operations, operations are implemented from left to right in sequence. These basis operations are linearly independent and complete; therefore, all single-qubit operations can be decomposed as linear combinations of basis operations. Non-unital operations, i.e., reset gates, are necessary in the basis set to efficiently decompose the non-unital part of an operation. The basis set minimizing the variance of the computation result is preferred.

 No. Operation No. Operation 1 I 9 Xπ,Y−π2 2 Xπ 10 Yπ,Xπ2 3 Yπ 11 MI+X2,R∣0+1〉 4 Zπ 12 MI+X2,R∣0−1〉 5 Xπ2 13 MI+Y2,R∣0+i1〉 6 Yπ2 14 MI+Y2,R∣0−i1〉 7 Zπ2 15 MI+Z2,R∣0〉 8 Xπ,Zπ2 16 MI+Z2,R∣1〉