We propose and investigate a new method of quantum process tomography (QPT)
    which we call projected least squares (PLS). In short, PLS consists of first
    computing the least-squares estimator of the Choi matrix of an unknown channel,
    and subsequently projecting it onto the convex set of Choi matrices. We
    consider four experimental setups including direct QPT with Pauli eigenvectors
    as input and Pauli measurements, and ancilla-assisted QPT with mutually
    unbiased bases (MUB) measurements. In each case, we provide a closed form
    solution for the least-squares estimator of the Choi matrix. We propose a
    novel, two-step method for projecting these estimators onto the set of matrices
    representing physical quantum channels, and a fast numerical implementation in
    the form of the hyperplane intersection projection algorithm. We provide
    rigorous, non-asymptotic concentration bounds, sampling complexities and
    confidence regions for the Frobenius and trace-norm error of the estimators.
    For the Frobenius error, the bounds are linear in the rank of the Choi matrix,
    and for low ranks, they improve the error rates of the least squares estimator
    by a factor $d^2$, where $d$ is the system dimension. We illustrate the method
    with numerical experiments involving channels on systems with up to 7 qubits,
    and find that PLS has highly competitive accuracy and computational

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