Progress toward scalable tomography of quantum maps using twirling-based methods and information hierarchies

We present in a unified manner the existing methods for scalable partial quantum process tomography. We focus on two main approaches: the one presented in Bendersky et al. [Phys. Rev. Lett. 100, 190403 (2008)] and the ones described, respectively, in Emerson et al. [Science 317, 1893 (2007)] and López et al. [Phys. Rev. A 79, 042328 (2009)], which can be combined together. The methods share an essential feature: They are based on the idea that the tomography of a quantum map can be efficiently performed by studying certain properties of a twirling of such a map. From this perspective, in this paper we present extensions, improvements, and comparative analyses of the scalable methods for partial quantum process tomography. We also clarify the significance of the extracted information, and we introduce interesting and useful properties of the $\chi$-matrix representation of quantum maps that can be used to establish a clearer path toward achieving full tomography of quantum processes in a scalable way.

Figure 1

Circuit representation of (a) the action of a map $\Lambda$ and (b) the action of the map, now twirled by $U$.

Figure 2

Circuit representation of the twirling of $\Lambda$, approximated by sampling $M$ times over the elements that constitute the family of twirl operators (the $U$’s). Each time the system is prepared in the same initial state $|{\varphi }_0\rangle$. The average of these $M$ measurements will retrieve the desired probabilities.

Figure 3

Circuit representation of the action of a map ${\Lambda }_l$ with ${\Lambda }_l(\rho )=P_l\mathit{\Lambda }(\rho )P_l$, twirled by $U$.

Figure 4

Circuit representation of equivalent schemes to determine the largest ${\chi }_{l,l}$, by considering pairs of experiments.

Figure 5

Success probability of the two methods: $•$ ${\mathcal{P}}_{\mathrm{MUB}}$, using an MUB twirl; $$ ${\mathcal{P}}_C$, using a Clifford twirl.