Existence of Pauli-like stabilizers for every quantum error-correcting code
Jhih-Yuan Kao, Hsi-Sheng Goan
TL;DR
This work shows that every quantum error-correcting code admits stabilization by a Paulian stabilizer group, a Pauli-like, commuting set of operators acting on an extended subspace. By mapping correctable errors to binary syndromes and employing generalized CNOT measurements, the authors provide a unified framework to extract error syndromes and perform correction beyond Pauli stabilizer codes, including subsystem, concatenated, and codeword-stabilized codes. They outline a minimal and an extended construction of Paulian stabilizers, discuss measurement strategies, and illustrate with examples from Pauli-stabilized, bosonic, and CWSC families, including practical limitations with certain codes. The approach offers a pathway to realize and search for new quantum codes by leveraging Paulian operators and their syndrome structure, potentially broadening the toolbox for robust quantum information processing.
Abstract
The Pauli stabilizer formalism is perhaps the most thoroughly studied means of procuring quantum error-correcting codes, whereby the code is obtained through commutative Pauli operators and ``stabilized'' by them. In this work we will show that every quantum error-correcting code, including Pauli stabilizer codes and subsystem codes, has a similar structure, in that the code can be stabilized by commutative ``Paulian'' operators which share many features with Pauli operators and which form a \textbf{Paulian stabilizer group}. By facilitating a controlled gate we can measure these Paulian operators to acquire the error syndrome. Examples concerning codeword stabilized codes and bosonic codes will be presented; specifically, one of the examples has been demonstrated experimentally and the observable for detecting the error turns out to be Paulian, thereby showing the potential utility of this approach. This work provides a possible approach to implement error-correcting codes and to find new codes.