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Projected Least-Squares Quantum Process Tomography

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Trystan Surawy-Stepney1, Jonas Kahn2, Richard Kueng3, and Madalin Guta1

1School of Mathematical Sciences, University of Nottingham, United Kingdom
2Institut de Mathématiques de Toulouse and ANITI, Université de Toulouse, France
3Institute for Integrated Circuits, Johannes Kepler University Linz, Austria

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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 $d^2$, 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.

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[2] Ryan Levy, Di Luo, and Bryan K. Clark, “Classical Shadows for Quantum Process Tomography on Near-term Quantum Computers”, arXiv:2110.02965.

[3] Jonathan Kunjummen, Minh C. Tran, Daniel Carney, and Jacob M. Taylor, “Shadow process tomography of quantum channels”, arXiv:2110.03629.

[4] Christopher W. Warren, Jorge Fernández-Pendás, Shahnawaz Ahmed, Tahereh Abad, Andreas Bengtsson, Janka Biznárová, Kamanasish Debnath, Xiu Gu, Christian Križan, Amr Osman, Anita Fadavi Roudsari, Per Delsing, Göran Johansson, Anton Frisk Kockum, Giovanna Tancredi, and Jonas Bylander, “Extensive characterization of a family of efficient three-qubit gates at the coherence limit”, arXiv:2207.02938.

[5] Zuzana Gavorová, Matan Seidel, and Yonathan Touati, “Topological obstructions to implementing quantum if-clause”, arXiv:2011.10031.

[6] Shahnawaz Ahmed, Fernando Quijandría, and Anton Frisk Kockum, “Gradient-descent quantum process tomography by learning Kraus operators”, arXiv:2208.00812.

[7] Akshay Gaikwad, Arvind, and Kavita Dorai, “Efficient experimental characterization of quantum processes via compressed sensing on an NMR quantum processor”, arXiv:2109.13189.

[8] Guo-Dong Lu, Zhou Zhang, Yue Dai, Yu-Li Dong, and Cheng-Jie Zhang, “Not All Entangled States are Useful for Ancilla‑Assisted Quantum Process Tomography”, Annalen der Physik 534 5, 2100550 (2022).

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