Stabilizer operations (SO) 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. 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 stabilizer-preserving (CSP) channels, defined as those that preserve the convex hull of stabilizer 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 stabilizer-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 and the axiomatically defined class of separable channels.
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A similar analysis can also be done for qudits, which is, however, more evolved.
This can, in principle, be done by computing the Choi states of the corresponding channels and then finding a hyperplane that separates them from the stabilizer polytope SP2n.
