Selective and efficient quantum process tomography for non-trace preserving maps: a superconducting quantum processor implementation

Alternatively to the full reconstruction of an unknown quantum process, the so-called selective and efficient quantum process tomography (SEQPT) allows estimating, individually and up to the required accuracy, a given element of the matrix that describes such an operation with a polynomial amount of resources. The implementation of this protocol, which depends on the possibility of building, or approximating, an especial set of quantum states (a state 2--design), has been carried out with success to characterize the evolution of a quantum system that is well described by a trace preserving quantum map. Here, we deal with a more general type of quantum process that does not preserve the trace of the input quantum state, which naturally arises in the presence of imperfect devices and system-environment interactions, in the context of quantum information science or quantum dynamics control. In that case, we show that with the aid of \emph{a priori} information on the losses structure of the quantum channel, the SEQPT reconstruction can be adapted to reconstruct the non-trace-preserving map. We explicitly describe how to implement the reconstruction in an arbitrary Hilbert space of finite dimension $d$. The method is experimentally verified on a superconducting quantum processor of the IBM Quantum services, by estimating several non trace-preserving quantum processes in two representative cases: when maps act on a Hilbert space of dimension $d=3$, where a state 2--design can be obtained from a complete set of mutually unbiased bases, and when they act on a Hilbert space of dimension $d=6$, where no such set is known to exist, and thus 2--designs must be obtained in another way. Our results show that it is possible to efficiently reconstruct non trace-preserving processes, with high precision, and with significantly higher fidelity than when the process is assumed to be trace-preserving