Lecture notes on quantum entanglement: From stabilizer states to stabilizer channels

Abstract

We study mathematical, physical and computational aspects of the stabilizer formalism arising in quantum information and quantum computation. The measurement process of Pauli observables with its algorithm is given. It is shown that to detect genuine entanglement we need a full set of stabilizer generators and the stabilizer witness is coarser than the GHZ (Greenberger–Horne–Zeilinger) witness. We discuss stabilizer codes and construct a stabilizer code from a given linear code. We also discuss quantum error correction, error recovery criteria and syndrome extraction. The symplectic structure of the stabilizer formalism is established and it is shown that any stabilizer code is unitarily equivalent to a trivial code. The structure of graph codes as stabilizer codes is identified by obtaining the respective stabilizer generators. The distance of embeddable stabilizer codes in lattices is obtained. We discuss the Knill−Gottesman theorem, tableau representation and frame representation. The runtime of simulating stabilizer gates is obtained by applying stabilizer matrices. Furthermore, an algorithm for updating global phases is given. Resolution of quantum channels into stabilizer channels is shown. We discuss capacity achieving codes to obtain the capacity of the quantum erasure channel. Finally, we discuss the shadow tomography problem and an algorithm for constructing classical shadow is given.

Graphical abstract

Keywords

Pauli product / stabilizer state / measurement process / entanglement detection / stabilizer code / stabilizer circuit / quantum channel / tomography

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Acknowledgements

This work was performed in the Laboratory of Combinatorial and Geometric Structures, School of Applied Mathematics and Informatics, MIPT. I would like to thank Prof. Andrei M. Raigorodskii, Dr. Andrey B. Kupavskii and my colleagues.

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2024 Higher Education Press