Projected Least-Squares Quantum Process Tomography

Surawy-Stepney, Trystan; Kahn, Jonas; Kueng, Richard; Guta, Madalin

Abstract

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 d2, 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.

Citation

Surawy-Stepney, T., Kahn, J., Kueng, R., & Guta, M. (in press). Projected Least-Squares Quantum Process Tomography. Quantum, 6, 844. https://doi.org/10.22331/q-2022-10-20-844

• Journal Article Type: Article
• Acceptance Date: Oct 5, 2022
• Online Publication Date: Oct 20, 2022
• Deposit Date: Dec 9, 2022
• Publicly Available Date: Dec 9, 2022
• Journal: Quantum
• Print ISSN: 2521-327X
• Publisher: Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften
• Peer Reviewed: Peer Reviewed
• Volume: 6
• Pages: 844
• DOI: https://doi.org/10.22331/q-2022-10-20-844
• Keywords: Physics and Astronomy (miscellaneous); Atomic and Molecular Physics, and Optics
• Public URL: https://nottingham-repository.worktribe.com/output/13167020
• Publisher URL: https://quantum-journal.org/papers/q-2022-10-20-844/
• Additional Information: This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.