Negative Quasi-Probability as a Resource for Quantum Computation

TL;DR

This paper links quantum computational speedup to negative quasi-probability values, showing that certain mixed states cannot be distilled into universal resources, and provides an efficient simulation method for Clifford circuits.

Contribution

It establishes a connection between quantum speedup and negativity in quasi-probability representations, resolving a key open problem in magic-state distillation.

Findings

Existence of bound states outside stabilizer convex hull that cannot be distilled.

Negative quasi-probability indicates potential for quantum speedup.

Efficient simulation protocol for Clifford circuits with mixed states.

Abstract

A central problem in quantum information is to determine the minimal physical resources that are required for quantum computational speedup and, in particular, for fault-tolerant quantum computation. We establish a remarkable connection between the potential for quantum speed-up and the onset of negative values in a distinguished quasi-probability representation, a discrete analog of the Wigner function for quantum systems of odd dimension. This connection allows us to resolve an open question on the existence of bound states for magic-state distillation: we prove that there exist mixed states outside the convex hull of stabilizer states that cannot be distilled to non-stabilizer target states using stabilizer operations. We also provide an efficient simulation protocol for Clifford circuits that extends to a large class of mixed states, including bound universal states.