  Stabiliser operations occupy a prominent role in fault-tolerant quantum
computing. They are defined operationally: by the use of Clifford gates, Pauli
measurements and classical control. These operations can be efficiently
simulated on a classical computer, a result which is known as the
Gottesman-Knill theorem. However, an additional supply of magic states is
enough to promote them to a universal, fault-tolerant model for quantum
computing. To quantify the needed resources in terms of magic states, a
resource theory of magic has been developed. Stabiliser operations (SO) are
considered free within this theory, however they are not the most general class
of free operations. From an axiomatic point of view, these are the completely
stabiliser-preserving (CSP) channels, defined as those that preserve the convex
hull of stabiliser states. It has been an open problem to decide whether these
two definitions lead to the same class of operations. In this work, we answer
this question in the negative, by constructing an explicit counter-example.
This indicates that recently proposed stabiliser-based simulation techniques of
CSP maps are strictly more powerful than Gottesman-Knill-like methods. The
result is analogous to a well-known fact in entanglement theory, namely that
there is a gap between the operationally defined class of local operations and
classical communication (LOCC) and the axiomatically defined class of separable
channels.
