Optimal estimation of quantum processes using incomplete information: variational quantum process tomography

The characterization of a quantum system and its dynamics is a daunting challenge. The first question which arises in this scenario is what information should we possess to characterize the dynamics. To answer this question, one needs to choose a quantum process tomography (QPT) scheme. Each procedure demands different resources and operations. There are four general types of QPT procedures: (i) standard quantum process tomography (SQPT)[2]; (ii) ancilla-assisted process tomography (AAPT)[3, 4, 5]; (iii) direct characterization of quantum dynamics (DCQD)[6, 7, 8]; (iv) selective and efficient quantum process tomography(SEQPT)[9, 10, 11]. In (i) the information is obtained indirectly, performing a set of quantum state tomographies(QST)[12, 13, 14, 15, 1, 16] of the linear independent states, which spans the HilbertSchmidt space of interest, after the action of the unknown map. The second scheme (ii) also an indirect procedure makes use of an auxiliary system. The information is then extracted by means of QST of the joint space (system and ancilla). The third one (iii) obtains the dynamical information directly by means of quantum error detection (QED)[17] concepts measuring stabilizers and normalizers. Finally, the last method (iv) which also measures the parameters directly consists in estimating averages over the entire Hilbert space of products of expectation values of two operators. For the special case of one-parameter quantum channels, there is also an interesting method developed by Sarovar and Milburn [18].