Un-Weyl-ing the Clifford Hierarchy

Tefjol Pllaha1, Narayanan Rengaswamy2, Olav Tirkkonen1, and Robert Calderbank3

1Department of Communications and Networking, Aalto University, Finland
2Department of Electrical and Computer Engineering, University of Arizona, Tucson, AZ, USA
3Department of Electrical and Computer Engineering, Duke University, NC, USA

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The teleportation model of quantum computation introduced by Gottesman and Chuang (1999) motivated the development of the Clifford hierarchy. Despite its intrinsic value for quantum computing, the widespread use of magic state distillation, which is closely related to this model, emphasizes the importance of comprehending the hierarchy. There is currently a limited understanding of the structure of this hierarchy, apart from the case of diagonal unitaries (Cui et al., 2017; Rengaswamy et al. 2019). We explore the structure of the second and third levels of the hierarchy, the first level being the ubiquitous Pauli group, via the Weyl (i.e., Pauli) expansion of unitaries at these levels. In particular, we characterize the support of the standard Clifford operations on the Pauli group. Since conjugation of a Pauli by a third level unitary produces traceless Hermitian Cliffords, we characterize their Pauli support as well. Semi-Clifford unitaries are known to have ancilla savings in the teleportation model, and we explore their Pauli support via symplectic transvections. Finally, we show that, up to multiplication by a Clifford, every third level unitary commutes with at least one Pauli matrix. This can be used inductively to show that, up to a multiplication by a Clifford, every third level unitary is supported on a maximal commutative subgroup of the Pauli group. Additionally, it can be easily seen that the latter implies the generalized semi-Clifford conjecture, proven by Beigi and Shor (2010). We discuss potential applications in quantum error correction and the design of flag gadgets.

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[2] Aritra Das and Barry C. Sanders, "Blind quantum factorization of 21", Physical Review A 106 1, 012421 (2022).

[3] Jingzhen Hu, Qingzhong Liang, and Robert Calderbank, "Designing the Quantum Channels Induced by Diagonal Gates", Quantum 6, 802 (2022).

[4] Lia Yeh, Companion Proceedings of the 7th International Conference on the Art, Science, and Engineering of Programming 90 (2023) ISBN:9798400707551.

[5] Nadish de Silva, "Efficient quantum gate teleportation in higher dimensions", Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 477 2251, 20200865 (2021).

[6] Tefjol Pllaha, Olav Tirkkonen, and Robert Calderbank, "Binary Subspace Chirps", IEEE Transactions on Information Theory 68 12, 7735 (2022).

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[8] Dripto M. Debroy and Kenneth R. Brown, "Extended flag gadgets for low-overhead circuit verification", Physical Review A 102 5, 052409 (2020).

[9] Lia Yeh, "Scaling W state circuits in the qudit Clifford hierarchy", arXiv:2304.12504, (2023).

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