Selective and efficient quantum process tomography in arbitrary finite dimension

The characterization of quantum processes is a key tool in quantum information processing tasks for several reasons: on one hand, it allows one to acknowledge errors in the implementations of quantum algorithms; on the other, it allows one to characterize unknown processes occurring in nature. Bendersky, Pastawski, and Paz [A. Bendersky, F. Pastawski, and J. P. Paz, Phys. Rev. Lett. 100, 190403 (2008); Phys. Rev. A 80, 032116 (2009)] introduced a method to selectively and efficiently measure any given coefficient from the matrix description of a quantum channel. However, this method heavily relies on the construction of maximal sets of mutually unbiased bases (MUBs), which are known to exist only when the dimension of the Hilbert space is the power of a prime number. In this article, we lift the requirement on the dimension by presenting two variations of the method that work on arbitrary finite dimensions: one uses tensor products of maximal sets of MUBs, and the other uses a dimensional cutoff of a higher prime power dimension.

Figure 1

Circuit depicting the protocol for SEQPT. When averaging over all the states $\left|{\psi }_i\right.〉$ from a state 2-design, the mean value of ${\sigma }_x$ (${\sigma }_y$) conditioned to the survival of state $\left|{\psi }_i\right.〉$ yields the real (imaginary) part of ${\chi }_{ij}$.

Figure 2

Circuit that estimates the real part of ${\chi }_{j_1{j}_2}^{i_1{i}_2}$. The initial states $\left|{\psi }_1\right.〉$ and $\left|{\psi }_2\right.〉$ are uniformly sampled from the set of states that belongs to the 2-designs $X_1$ and $X_2$, respectively. A measurement of the expectation value of ${\sigma }_x$ (${\sigma }_y$) conditioned to the results of the measurements at the other two systems allows one to estimate the $Re\left({\chi }_{j_1{j}_2}^{i_1{i}_2}\right)$ $[Im\left({\chi }_{j_1{j}_2}^{i_1{i}_2}\right)]$.

Figure 3

Circuit that estimates the real part of ${\chi }_j^i$ for a process that acts on a system of arbitrary dimension $d={\prod }_{a=1}^N{p}_a^{n_a}$. The initial states $\left|{\psi }_i\right.〉$ are uniformly sampled from the set of states that belongs to the 2-designs $X_1,\cdots ,X_N$. A measurement of the expectation value of ${\sigma }_x$ (${\sigma }_y$) conditioned to the results of the measurements at the other systems allows one to estimate the $Re\left({\chi }_j^i\right)$ $[Im\left({\chi }_j^i\right)]$.