I am working to understand quantum state tomography, specifically using the algorithm presented in PRL 108, 070502. This paper is referenced in IBMQ implementations of QST, both in old deprecated Ignis code (my primary reference so far) and in the new Qiskit Experiments. In the paper, the authors present an algorithm where the basis is changed from measurement results $m_i$ to a noisy density matrix: $$\mu=(1/d)\sum_im_i\sigma_i$$ where the $\sigma_i$'s are the Pauli matrices plus the identity. After this, their "Subproblem 1" speeds up the least squares optimization problem by transforming $\mu$ to a positive semidefinite matrix $\rho$ which is presented as the least squares solution without having to actually perform least squares minimization. However in the IBMQ code I referenced, the algorithm uses traditional least squares optimization directly without performing the basis change, then the PSD rescaling is applied to the result.

Is there a particular reason for this? My assumption is that performing least squares directly will be slower than the basis change proposed in the paper.

I am also not understanding how to produce $m_i$ corresponding to the identity matrix following the algorithm in the paper, which the IBMQ implementation seems to skip, so perhaps this is another motivation?



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