Multi-stage tomography based on eigenanalysis for high-dimensional dense unitary processes in gate-based quantum computers

A major quantum information processing tool is Quantum Process Tomography (QPT), which consists of methods that aim at identifying, i.e. estimating, a quantum process. QPT may especially be applied to quantum gates, that are widely used as the building blocks of quantum computers, in order to experimentally characterize their actual behavior. In this paper, we address unitary processes (corresponding to isolated systems), that may be dense, i.e. that do not have sparsity constraints. The proposed approaches to QPT require one to first perform Quantum State Tomography (QST). Afterwards, the core of our QPT methods applies to a significant number of qubits and thus to a high state space dimension and to complex problems. Our methods first achieve part of QPT by performing an eigenanalysis of the density matrix of a process output estimated with QST, by taking advantage of the unitarity of the process. Our first resulting class of complete algorithms uses only one such eigendecomposition and is thus "single-stage". We then extend our approach to algorithms that are "multiple-stage", i.e. that perform several eigendecompositions. This allows one to handle high-dimensional state spaces with a reduced sensitivity to the estimation errors that result from the use of an arbitrary QST algorithm as a preprocessing step of our QPT methods. Our multi-stage methods first include two-stage versions and then dichotomic versions, with a number of stages that increases with the considered state space dimension. Simulations were successively performed with different magnitudes for QST errors / noise. They provide evidence of the relevance of all proposed methods. Their single-stage and two-stage versions are efficient up to 13 qubits when executed on a very simple platform: an 8-year old PC with only 16 GB of RAM. An even higher accuracy (Normalized Mean Square Error down to $$10^{-3}$$ ) is obtained with multi-stage methods.