Adaptive homodyne phase discrimination and qubit measurement

Fast and accurate measurement is a highly desirable, if not vital, feature of quantum computing architectures. In this work we investigate the usefulness of adaptive measurements in improving the speed and accuracy of qubit measurement. We examine a particular class of quantum computing architectures, ones based on qubits coupled to well-controlled harmonic oscillator modes (reminiscent of cavity QED), where adaptive schemes for measurement are particularly appropriate. In such architectures, qubit measurement is equivalent to phase discrimination for a mode of the electromagnetic field, and we examine adaptive techniques for doing this. In the final section we present a concrete example of applying adaptive measurement to the particularly well-developed circuit-QED architecture.

Figure 1

(Color online) The typical method for measuring the phase of a field mode: a dyne measurement. The input mode (for the measurement) is the output of the cavity, and in the cavity-QED scenario we are considering, it will have an unknown phase $\left(\varphi \right)$ due to the pull of qubit(s) in the cavity. The local oscillator (LO) phase $\Phi \left(t\right)$ is constant for a homodyne measurement and rapidly oscillating for a heterodyne measurement.

Figure 2

(Color online) An adaptive scheme for measuring the phase of a field mode. The local oscillator phase is dynamically modified according to the measured values of the photocurrent.

Figure 3

A phase space diagram showing the possible phases in a phase discrimination task with six hypotheses.

Figure 4

A phase space diagram showing the possible phases in a phase discrimination task with two hypotheses. As shown in the main body, the optimal local oscillator phase is static and in quadrature with the bisector of the two candidate phases.

Figure 5

(Color online) Figures contrasting the evolution of the average a posteriori probability of the correct hypothesis $\left(\overline{{\mathcal{P}}_{corr}}\right)$ under the adaptive and static schemes described in Sec. 3B1. In all the figures, the solid (red) line shows the evolution under the adaptive scheme and the dashed (blue) line shows evolution under the static heterodyne scheme. (a) Time evolution of the $\overline{{\mathcal{P}}_{corr}}$ for an example fixed probe signal magnitude (i.e., fixed SNR, $\alpha$). (b) The $\overline{{\mathcal{P}}_{corr}}$ as a function of probe signal magnitude (SNR) $\alpha$ at two fixed times. The lines are interpolations between data points at integer $\left|\alpha \right|$ values to aid the eye. See main text for comments on the error bars. (c) The time taken for the $\overline{{\mathcal{P}}_{corr}}$ to reach 0.6 as a function of probe signal magnitude (SNR) $\alpha$. The lines are interpolations between data points at integer $\left|\alpha \right|$ to aid the eye.

Figure 6

(Color online) Example of a single run of the measurement in the circuit QED example of Sec. 3B1. The red (solid) line is the evolution of the a posteriori probability of the correct state $\left({\mathcal{P}}_{corr}\right)$ under the adaptive scheme and the blue (dotted) line is its evolution under the static heterodyne scheme.

Figure 7

The phase constellation (for $\left|\alpha \right|=1$) for the discrimination task described in Sec. 3B2. (The dots simply show the centers of the coherent states; the uncertainties in phase space are not shown).

Figure 8

(Color online) Contrasting the performance of the adaptive and static schemes for discriminating the 16-element constellation of Sec. 3B2. This figure shows the average a posteriori probability of the correct hypothesis as a function of probe signal magnitude (SNR) $\alpha$ at three fixed times. The dotted (blue) lines plot results for the static scheme and the solid (red) lines plot results for the adaptive scheme. As in Fig. 5b, the error bars at each data point show the standard error of the mean values plotted.