Quantum process tomography via optimal design of experiments

Quantum process tomography, a primitive in many quantum information processing tasks, can be cast within the framework of the theory of design of experiments (DOE), a branch of classical statistics that deals with the relationship between inputs and outputs of an experimental setup. Such a link potentially gives access to the many ideas of the rich subject of classical DOE for use in quantum problems. The classical techniques from DOE, however, cannot be directly applied to the quantum process tomography due to the basic structural differences between the classical and quantum estimation problems. Here we properly formulate quantum process tomography as a DOE problem and examine several examples to illustrate the link and the methods. In particular, we discuss the common issue of nuisance parameters and point out interesting features in the quantum problem absent in the usual classical setting.

Figure 1

Contour plot of the difference $\mathrm{\Delta }{J}^{-1}[e_{\text{PT}}-e\left(2\right)]\equiv {\left\{J_{{\vartheta }_{}}{\left[e_{\text{PT}}\right]}^{-1}\right\}}_{11}-{\left\{J_{{\vartheta }_{}}^{\text{SLD}}{\left[e\left(2\right)\right]}^{-1}\right\}}_{11}$ for all possible values of ${\vartheta }_1$ and ${\vartheta }_2$.

Figure 2

Contour plot of ${\lambda }^{*}$ [Eq. (51)] as a function of ${\vartheta }_1$ and ${\vartheta }_2$.

Figure 3

Plot of the ratio $V_{\mathrm{static}}/V_{\mathrm{adapt}}$ against ${log}_{10}({\theta }_1/{\theta }_2)$ for numerical simulation of the adaptive procedure for the two-parameter family of Pauli channels, over a uniform grid of ${\theta }_1$ and ${\theta }_2$ values, with a step size of 0.01. Here $N=200$ and $K=10$. The colors show the dependence of the MSE ratio on the noise strength $1-{\vartheta }_2$; a plot of only the data points for the low-noise regime of $1-{\vartheta }_2\le 0.5$ is given in Fig. 4 below. That the adaptive strategy shows a clear benefit in the regime of high asymmetry is evidenced by the points that lie above the $V_{\mathrm{static}}/V_{\mathrm{adapt}}=1$ horizontal line, which occur only when ${\theta }_1/{\theta }_2$ is far from 1.

Figure 4

Effect of a runway on the MSE ratios. We have a total of $N=200$ uses of the channel, and only the $1-{\vartheta }_2\le 0.5$ data are shown here. (a) $K=10$, no runway; (b) $K=10$, a runway of length 100; (c) $K=5$, no runway; and (d) $K=5$, a runway of length 100. The runway clearly reduces the loss in accuracy due to the adaptation when ${\theta }_1/{\theta }_2$ is close to 1.