Optimal control of a quantum measurement

Pulses to steer the time evolution of quantum systems can be designed with optimal control theory. In most cases it is the coherent processes that can be controlled and one optimizes the time evolution toward a target unitary process, sometimes also in the presence of noncontrollable incoherent processes. Here we show how to extend the gradient ascent pulse engineering (GRAPE) algorithm in the case where the incoherent processes are controllable and the target time evolution is a nonunitary quantum channel. We perform a gradient search on a fidelity measure based on Choi matrices. We illustrate our algorithm by optimizing a phase qubit measurement pulse. We show how this technique can lead to a large measurement contrast close to $99\%$. We also show, within the validity of our model, that this algorithm can produce short $1.4$-ns pulses with $98.2\%$ contrast.

Figure 1

Sketch of the phase qubit's potential focusing on the shallow well. The wavy line indicates the $0\leftrightarrow 1$ transition frequency which is a coherent process and enters in the Hamiltonian. Controllable incoherent processes are indicated by solid straight lines whereas the uncontrollable $T_1$ relaxation process is constant.

Figure 2

$|0\rangle \leftrightarrow |1\rangle$ state mixing parameter $\eta$ as function of the bias phase. The solid line indicates numerical DVR data while the dashed line is a third-order polynomial fit to preserve analyticity when performing the gradient search.

Figure 3

Frequency of the $|0\rangle \leftrightarrow |1\rangle$ transition for the harmonic approximation and for the DVR data which takes all orders into account. The dashed line is the fit to the five-parameter function $a{(b+c{\phi }_{\mathrm{b}})}^d+e$ showing excellent agreement for the range of bias flux of concern. Note that beyond ${\phi }_{\mathrm{b}}=0.945\times 2\pi ,$ the DVR no longer finds two states in the shallow well. This is in excellent agreement with the three-level model validity condition shown in Fig. 4.

Figure 4

Parameter controlling the tunneling rates in the three-level model. The solid line shows the value of $\alpha$ as computed by Eq. (8) while the dashed line shows a second-order fit used to preserve analyticity during the gradient search. When $\alpha$ goes below the horizontal line, the three-level model is no longer valid.

Figure 5

Optimal pulses for different gate durations with their corresponding fidelities. The dashed lines show the initial guess. As can be seen the fidelity of the optimal pulses is higher for the fast pulses. This is because faster pulses allow $|1\rangle$ to tunnel into $|m\rangle$ before $T_1$ relaxes it to $|0\rangle$.

Figure 6

Time evolution of the populations for the 10-ns pulse of Fig. 5 starting from the $|1\rangle \langle 1|$ state. As can be seen the initial pulse (corresponding to the thin lines) has a nonoptimal pulse that fails to transfer population to $|m\rangle$ which would result in missed counts. Its channel fidelity and contrast are respectively ${\Phi }_{\mathrm{ch},\mathrm{i}}^{\prime }=87.0\%$ and ${\xi }_{\mathrm{i}}=37.8\%$. The optimal pulse (thick lines) corrects for this and prevents population transfers between $|0\rangle$ and $|1\rangle$. It has ${\Phi }_{\mathrm{ch},\mathrm{f}}^{\prime }=98.8\%$ and ${\xi }_{\mathrm{f}}=97.9\%$.

Figure 7

Optimization of a fast readout pulse. (a) Initial pulse sequence with fidelity ${\Phi }_{\mathrm{ch},\mathrm{i}}^{\prime }=83.8\%$ and contrast ${\xi }_{\mathrm{i}}=19.8\%$. (b) Optimized pulse shape. (c) Initial time evolution of populations. Again, the nonoptimized pulse fails to let $|1\rangle$ tunnel into $|m\rangle$. (d) Time evolution of populations after pulse optimization resulting in a high contrast of ${\xi }_{\mathrm{f}}=98.2\%$ and final fidelity ${\Phi }_{\mathrm{ch},\mathrm{f}}^{\prime }=99.2\%$.