Optimal Quantum State Tomography with Noisy Gates

For limited scenarios, depending on projector rank and system size, optimal measurement schemes for efficient QST are known.  In the case of errorless non-degenerate measurements, using mutually unbiased bases yields the optimal QST scheme [1]. Measuring one out of N qubits becomes optimal if the measurement operators project on mutually unbiased subspaces [2]. However, in the general case, the optimal measurement scheme for efficient QST is not known and, oftentimes, it may need to be numerically approximated. This problem can be generalized as a framework for customized efficient QST. Here, we extend this framework to investigate the effect of noise on the optimal QST measurement sets using two noise models:  the depolarizing channel, and over- and under-rotation in two-qubit gates. We demonstrate the benefit of using entangling gates for the efficient QST measurement schemes for two qubits at realistic noise levels, by comparing the fidelity of reconstruction of our optimized QST measurement set to the state-of-the-art scheme using only product bases.

[1] Wootters, Fields, Ann. Phys. 191, 363 (1989)

[2] Bodmann, Haas, Proc. Amer. Math. Soc. 146, 2601 (2018)