The axiomatic and the operational approaches to resource theories of magic do not coincide

TL;DR

This paper demonstrates that the operational and axiomatic approaches to resource theories of magic in quantum computing do not coincide, revealing a gap that impacts the power of simulation techniques.

Contribution

It provides a counter-example showing the difference between stabiliser operations and completely stabiliser-preserving channels in resource theory of magic.

Findings

Counter-example distinguishes CSP channels from stabiliser operations

CSP channels are strictly more powerful than Gottesman-Knill methods

Reveals a fundamental gap similar to entanglement theory

Abstract

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.