Abstract
We give a new algorithm for computing therobustness of magic- a measure of the utility of quantum states as a computational resource. Our work is motivated by themagic state modelof fault-tolerant quantum computation. In this model, all unitaries belong to the Clifford group. Non-Clifford operations are effected by injecting non-stabiliser states, which are referred to asmagic statesin this context. Therobustness of magicmeasures the complexity of simulating such a circuit using a classical Monte Carlo algorithm. It is closely related to the degree negativity that slows down Monte Carlo simulations through the infamoussign problem. Surprisingly, the robustness of magic issub- multiplicative. This implies that the classical simulation overhead scales subexponentially with the number of injected magic states - better than a naive analysis would suggest. However, determining the robustness ofncopies of a magic state is difficult, as its definition involves a convex optimisation problem in a 4n-dimensional space. In this paper, we make use of inherent symmetries to reduce the problem tondimensions. The total run-time of our algorithm, while still exponential inn, is super-polynomially faster than previously published methods. We provide a computer implementation and give the robustness of up to 10 copies of the most commonly used magic states. Guided by the exact results, we find a finite hierarchy of approximate solutions where each level can be evaluated in polynomial time and yields rigorous upper bounds to the robustness. Technically, we use symmetries of the stabiliser polytope to connect the robustness of magic to the geometry of a low-dimensional convex polytope generated by certainsigned quantum weight enumerators. As a by-product, we characterised the automorphism group of the stabiliser polytope, and, more generally, of projections onto complex projective 3-designs.
Highlights
In fault-tolerant quantum computation, each logical qubit is encoded in a non-local subspace of a number of physical qubits
We find a hierarchy of such robustness of magic (RoM) approximations by restricting to k-partite entangled stabiliser states which converges to the exact RoM
The complexity of the RoM problem can be significantly reduced by exploiting the symmetries of the problem, a procedure that we will call symmetry reduction and is well-known in convex optimisation theory, see e. g. [2]
Summary
In fault-tolerant quantum computation (for a recent review, see Ref. [8]), each logical qubit is encoded in a non-local subspace of a number of physical qubits. The starting point of these theories is the Gottesman-Knill Theorem [30] It states that quantum circuits consisting only of preparations of stabiliser states, Clifford unitaries, and computational basis measurements can be efficiently simulated on a classical computer. While the projection of the stabiliser polytope to this invariant space (Fig. 4) still has exponentially many vertices, it turns out that formulating the optimisation problem in this symmetry-reduced way leads to a super-polynomially faster algorithm. Every level of the hierarchy can be computed in polynomial time Both the exact and approximate results imply a runtime of O(20.737t) for simulating a circuit with t T gates using the RoM algorithm.
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