Single-qubit process tomography ($R_{ij}=\frac{1}{2}\mathrm{Tr}({\sigma }_{i}V{\sigma }_j{V}^{†})$)

To identify the physical channel $\rho \mapsto V\rho V^{†}$ of a single-qubit unitary gate, prepare each of the three positive Pauli matrix eigenstates and estimate each output Pauli expectation. Their nine means areThey determine the Bloch vector rotation $\mathbf{r}\mapsto R\mathbf{r}$, hence the channel on all states. If two gates have the same channel, their relative unitary commutes with all three Pauli matrices: commuting with $Z$ forces it diagonal and commuting with $X$ forces equal diagonal entries. Thus they differ by a scalar global phase, proving uniqueness up to that phase. This procedure needs no controlled gate or eigenstate of the unknown gate. Estimating its bounded measurement means to error $ϵ$ with fixed confidence uses order ${ϵ}^{-2}$ repetitions. It determines the relative eigenphases but cannot determine an overall eigenvalue phase from uncontrolled gate access alone.