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
tractability.
