On sets of commuting and anticommuting Paulis

Abstract

In this work we study the structure and cardinality of maximal sets of commuting and anticommuting Paulis in the setting of the abelian Pauli group. We provide necessary and sufficient conditions for anticommuting sets to be maximal, and present an efficient algorithm for generating anticommuting sets of maximum size. As a theoretical tool, we introduce commutativity maps, and study properties of maps associated with elements in the cosets with respect to anticommuting minimal generating sets. We also derive expressions for the number of distinct sets of commuting and anticommuting abelian Paulis of a given size.

A graph-based formalism for surface codes and twists

“…with size > 2n + 1 in which all Paulis anticommute with one another. However, this contradicts known results on the maximum size of such anticommuting sets [36].…”

“…A of size at least 2n + 2. But it is a well-known fact that the largest size of an anticommuting list of Pn is 2n + 1 (see for example [36] and Proposition 9 in [58]), which is a contradiction. For the last part, if ≤ 2, then choose any anticommuting list of size .…”

“…Notice that for w = w , ω(p w ) anticommutes with ω(p w ), because, written as products of Majoranas, p w and p w have odd overlap -they share edges, two Majoranas each, as well as the Majorana located at odd-degree vertex v. Thus, we have a set of M − 1 logical Paulis B = {ω(p w ) : odd-degree vertex w = v} that mutually anticommute. This implies, because the maximum size of an anticommuting set of K-qubit Paulis is 2K + 1 [36], that K ≥ (M − 2)/2. However, K = (M − 2)/2 is exactly the number of encoded qubits by Theorem 3.3, meaning B is a basis of all logical operators.…”

“…. , p } ⊆ Pn such that |H| = , we define following [36], the commutativity map with respect to H, to be the function Com…”

“…. , q −1 } with size k − rank (C) /2 which is again known to be impossible on n < k − rank (C) /2 qubits [36].…”