Generating Fault-Tolerant Cluster States from Crystal Structures

Michael Newman1, Leonardo Andreta de Castro1,2, and Kenneth R. Brown1

1Departments of Electrical and Computer Engineering, Chemistry, and Physics, Duke University, Durham, NC, 27708, USA
2Q-CTRL Pty Ltd, Sydney, NSW, Australia

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Measurement-based quantum computing (MBQC) is a promising alternative to traditional circuit-based quantum computing predicated on the construction and measurement of cluster states. Recent work has demonstrated that MBQC provides a more general framework for fault-tolerance that extends beyond foliated quantum error-correcting codes. We systematically expand on that paradigm, and use combinatorial tiling theory to study and construct new examples of fault-tolerant cluster states derived from crystal structures. Included among these is a robust self-dual cluster state requiring only degree-$3$ connectivity. We benchmark several of these cluster states in the presence of circuit-level noise, and find a variety of promising candidates whose performance depends on the specifics of the noise model. By eschewing the distinction between data and ancilla, this malleable framework lays a foundation for the development of creative and competitive fault-tolerance schemes beyond conventional error-correcting codes.

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[1] Shilin Huang, Michael Newman, and Kenneth R. Brown, "Fault-tolerant weighted union-find decoding on the toric code", Physical Review A 102 1, 012419 (2020).

[2] Rui Chao, Michael E. Beverland, Nicolas Delfosse, and Jeongwan Haah, "Optimization of the surface code design for Majorana-based qubits", arXiv:2007.00307, Quantum 4, 352 (2020).

[3] Benjamin J. Brown and Sam Roberts, "Universal fault-tolerant measurement-based quantum computation", arXiv:1811.11780, Physical Review Research 2 3, 033305 (2018).

[4] Hayata Yamasaki, Kosuke Fukui, Yuki Takeuchi, Seiichiro Tani, and Masato Koashi, "Polylog-overhead highly fault-tolerant measurement-based quantum computation: all-Gaussian implementation with Gottesman-Kitaev-Preskill code", arXiv:2006.05416.

[5] Armanda O. Quintavalle, Michael Vasmer, Joschka Roffe, and Earl T. Campbell, "Single-shot error correction of three-dimensional homological product codes", arXiv:2009.11790.

[6] Ye Wang, Mu Qiao, Zhengyang Cai, Kuan Zhang, Naijun Jin, Pengfei Wang, Wentao Chen, Chunyang Luan, Haiyan Wang, Yipu Song, Dahyun Yum, and Kihwan Kim, "Realization of two-dimensional crystal of ions in a monolithic Paul trap", arXiv:1912.04262.

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