Implementing generalized measurements with superconducting qubits

We describe a method to perform any generalized purity-preserving measurement of a qubit with techniques tailored to superconducting systems. First, we consider two methods for realizing a two-outcome partial projection: using a thresholded continuous measurement in the circuit QED setup and using an indirect ancilla qubit measurement. Second, we decompose an arbitrary purity-preserving two-outcome measurement into single-qubit unitary rotations and a partial projection. Third, we systematically reduce any multiple-outcome measurement to a sequence of such two-outcome measurements and unitary operations. Finally, we consider how to define suitable fidelity measures for multiple-outcome generalized measurements.

Figure 1

Quantum circuit implementing the partial projections $D_{0,1}$ using an ancilla qubit. The ancilla is initialized in the state $|0\rangle$ and then a $Z$-controlled $Y$ rotation of $\pm \varphi$ is applied [see Eq. (8)], which creates an angular separation of $2\varphi$ between the ancilla states coupled to the qubit states of $|0\rangle$ and $|1\rangle$. Finally, the ancilla is $Y$ rotated by the angle $\epsilon -\pi /2$ and measured. For $\epsilon =0$ this corresponds to measurement in the $X$ basis; the offset $\epsilon$ allows for measurement asymmetry. The resulting partial projection of the qubit $D_{0,1}$ depends on the classical outcome 0 or 1 of the ancilla measurement.

Figure 2

Quantum circuit using a controlled-phase gate (9) to implement the same partial projections $D_{0,1}$ as in Fig. 1.

Figure 3

Quantum circuit using the standard cz gate to implement the same partial projections $D_{0,1}$ as in Fig. 1. The $R_Y\left(\varphi \right)$ rotation of the ancilla qubit, followed by the cz gate, creates an angular separation of $2\varphi$ between the ancilla states coupled to the qubit states $|0\rangle$ and $|1\rangle$. Then the ancilla qubit is measured in the slanted basis, which becomes the $X$ basis in the symmetric case (when $\epsilon =0)$.