THESISONDERWERPEN QUANTUM GROUP

A framework for topological subsystem codes beyond Paulis

Topic: Quantum Information

Promoter(s): Jacob Bridgeman

Supervisor(s): Jacob Bridgeman, Lander Burgelman

Context

There are many groups, both academic and commercial, working towards building a quantum computer. Despite this, they remain rather elusive. One of the most challenging aspects is protecting the fragile quantum information from the environment. This is the realm of quantum error correction and mitigation [1,2].

Typically, the approach is to encode quantum information into a physical system in a way such that the noise cannot easily corrupt. This is done via quantum error correcting codes. There are several families of codes, with the most studied being those based on the stabilizer formalism [3].

Closely related to stabilizer codes, but with more quantum computational power, are Pauli subsystem codes [4]. Some such codes have a special property that protects against defects in the experimental devices, as well as environmental noise [5,6,7]. Subsystem codes beyond the small class of Pauli codes are poorly understood, something we hope to rectify in this project.

Goal

One of the reasons stabilizer codes are so well understood is their clear mathematical description. The central goal of this project is to devise an analogous formalism for topological subsystem codes.

The student would begin with gaining an understanding of the stabilizer formalism, and how it is used in both many-body quantum systems and quantum error correction. They would then learn about Pauli subsystem codes.

From here, the student could follow several paths. They may choose to remain in the Pauli regime, and construct/study novel codes. Alternatively, they could run numerical simulations on some candidate beyond-Pauli codes to gain a better understanding of their computational power. Finally, they may choose to contribute to the framework for beyond-Pauli subsystem codes.

1. B. M. Terhal, Quantum error correction for quantum memories, Reviews of Modern Physics 87, 307, arXiv:1302.3428 (2015)
2. B. J. Brown, D. Loss, J. K. Pachos, C. N. Self, and J. R. Wootton, Quantum memories at finite temperature, Reviews of Modern Physics 88, 045005, arXiv:1411.6643 (2016)
3. D. Gottesman, Stabilizer Codes and Quantum Error Correction, PhD Thesis, Caltech, arXiv:quant-ph/9705052 (1997)
4. D. Poulin, Stabilizer Formalism for Operator Quantum Error Correction, Physical Review Letters 95, 230504, arXiv:quant-ph/0508131 (2005)
5. H. Bombin, Topological subsystem codes, Physical Review A 81, 032301, arXiv:0908.4246 (2010)
6. H. Bombin, Gauge color codes: optimal transversal gates and gauge fixing in topological stabilizer codes, New Journal of Physics 17, 083002, arXiv:1311.0879 (2015)
7. A. Kubica and M. Vasmer, Single-shot quantum error correction with the three-dimensional subsystem toric code, Nature Communications 13, 6272, arXiv:2106.02621 (2022)