Abstract
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.
Highlights
Quantum computing provides a fundamentally new approach to computation by exploiting the laws ofThe first level of the hierarchy is the Pauli group and the second level is the Clifford group, which is defined as the normalizer of the Pauli group in the unitary group
We show that every element of the group is supported on a subgroup of the Pauli group
We give a closed form description of the support of standard group elements, and show that the coefficients are determined by a quadratic form modulo 4
Summary
The first level of the hierarchy is the Pauli (or Heisenberg-Weyl ) group and the second level is the Clifford group, which is defined as the normalizer of the Pauli group in the unitary group. When a C(3) gate acts by conjugation on a Pauli matrix, the result is a Hermitian Clifford, one example being the aforementioned case of choosing a C(3) operation that is a tensor product of integer powers of T It is well-known that Clifford transvections (that is, square roots of Hermitian Pauli matrices; see (28)) form a different generating set for all Cliffords, compared to the standard Clifford gate set mentioned earlier [9, 20, 24]. Since flag gadgets are generally applied only to Clifford circuits, our result that any C(3) element is supported on a MCS of the Paulis, up to multiplication by a Clifford, provides a way to determine a Pauli that commutes with a non-Clifford element. This insight could be used to design flag gadgets beyond Clifford (subsections of) circuits
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