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## "Semi-Clifford operations, structure of Ck hierarchy, and gate complexity for fault-tolerant quantum computation", Bei Zeng, Xie Chen, Isaac Chuang, 2008

*Reviewed August 25, 2023*

*Citation:* Zeng, Bei, Xie Chen, and Isaac L. Chuang. "Semi-Clifford operations, structure of C k hierarchy, and gate complexity for fault-tolerant quantum computation." Physical Review A 77.4 (2008): 042313.

*Web:* https://arxiv.org/abs/0712.2084

*Tags:* Abelian-anyons

This paper is a good general reference for the structure of the Clifford hierarchy.
In particular, it is a good reference for the claim "the structure of the Clifford hierarchy is complicated, and not fully understood".

One interesting feature of this paper is that it also has a discussion quantifying how hard it is to implement a given gate via teleportation,
by introducing a notion of "teleportation depth".
This is nice, because on some level every discussion of the Clifford hierarchy should tie it back to the reason that the Clifford hierarchy is interesting - gate teleportation.

Another reference for the structure of Clifford hierarchy is

> Bengtsson, Ingemar, et al. "Order 3 symmetry in the Clifford hierarchy." Journal of Physics A: Mathematical and Theoretical 47.45 (2014): 455302.

This second paper also has the distinction of motivating the fact that as you look at the Clifford hierarchy over
Zp for different primes you will get different structure depending on number-theoretic properties of p. In particular, p=2 is very special.

Another paper which details the structure of the Clifford hierarchy is

> Cui, Shawn X., Daniel Gottesman, and Anirudh Krishna. "Diagonal gates in the Clifford hierarchy." Physical Review A 95.1 (2017): 012329.

which classifies all diagonal gates in the Clifford hierarchy. One more reference for the structure of the Clifford hierarchy is

> Rengaswamy, Narayanan, Robert Calderbank, and Henry D. Pfister. "Unifying the Clifford hierarchy via symmetric matrices over rings." Physical Review A 100.2 (2019): 022304.

This paper is in particular interesting because it relates the kth level of the Clifford hierarchy to algebra over Z_(2^k).
Perhaps this can be used to prove a special case of the relationship between the Clifford hierarchy on Z_2 and on Z_(2^k).