Quantum process tomography with unknown single-preparation input states: Concepts and application to the qubit pair with internal exchange coupling

Quantum process tomography (QPT) methods aim at identifying, i.e., estimating, a given quantum process. QPT is a major quantum information processing tool, since it especially allows one to characterize the actual behavior of quantum gates, which are the building blocks of quantum computers. However, usual QPT procedures are complicated, since they set several constraints on the quantum states used as inputs of the process to be characterized. In this paper, we extend QPT so as to avoid two such constraints. On the one hand, usual QPT methods require one to know, hence to very precisely control (i.e., prepare), the specific quantum states used as inputs of the considered quantum process, which is cumbersome. We therefore propose a blind, or unsupervised, extension of QPT (i.e., BQPT), which means that this approach uses input quantum states whose values are unknown and arbitrary, except that they are requested to meet some general known properties (and this approach exploits the output states of the considered quantum process). On the other hand, usual QPT methods require one to be able to prepare many copies of the same (known) input state, which is constraining. In contrast, we propose “single-preparation BQPT methods” (SBQPT) , i.e., methods which can operate with only one instance of each considered input state. These two concepts are here illustrated with practical (S)BQPT methods which are numerically validated, in the case when (i) random pure states are used as inputs and their required properties are especially related to the statistical independence of the random variables that define them and (ii) the considered quantum process is based on cylindrical-symmetry Heisenberg spin coupling. As a benchmark, we moreover introduce nonblind QPT methods dedicated to the considered Heisenberg process, we analyze their theoretical behavior (this requires the tools developed in this paper for random input states), and we numerically test their sensitivity to systematic and nonsystematic errors, which are most likely to occur in practice. This shows that, even for very low preparation errors (especially systematic ones), these nonblind QPT methods yield much lower performance than our SBQPT methods. Our blind and single-preparation QPT concepts may be extended, e.g., to a much wider class of processes and to SBQPT methods based on other quantum state properties, as outlined in this paper.

Figure 1

Normalized root mean square error (NRMSE) of estimation of parameter $v$ vs number $K$ of preparations of each of the $N$ used states.

Figure 2

NRMSE of estimation of parameter $w_1$ vs number $K$ of preparations of each of the $N$ used states.

Figure 3

NRMSE of estimation of parameter $w_2$ vs number $K$ of preparations of each of the $N$ used states.

Figure 4

Mean relative error of estimation of matrix $M$ vs number $K$ of preparations of each of the $N$ used states.

Figure 5

Mean relative error of estimation of matrix $M$ vs total number $NK$ of preparations. Dashed black line: blind method. Solid color line: nonblind method receiving input states with systematic errors defined by $f_b$.

Figure 6

Mean relative error of estimation of matrix $M$ vs total number $NK$ of preparations. Dashed black line: blind method. Solid color line: nonblind method receiving input states with nonsystematic errors defined by $f_s$.