Kirkwood-Dirac Nonpositivity is a Necessary Resource for Quantum Computing

Abstract: Classical computers can simulate models of quantum computation with restricted input states. The identification of such states can sharpen the boundary between quantum and classical computations. Previous works describe simulable states of odd-dimensional systems. Here, we further our understanding of systems of qubits. We do so by casting a real-quantum-bit model of computation in terms of a Kirkwood-Dirac (KD) quasiprobability distribution. Algorithms, throughout which this distribution is a proper (positive) probability distribution can be simulated efficiently on a classical computer. We leverage recent results on the geometry of the set of KD-positive states to construct previously unknown classically-simulable (bound) states. Finally, we show that KD nonpositivity is a resource monotone for quantum computation, establishing KD nonpositivity as a necessary resource for computational quantum advantage.