Kirkwood-Dirac Nonpositivity is a Necessary Resource for Quantum Computing

TL;DR

This paper demonstrates that Kirkwood-Dirac nonpositivity is a necessary resource for quantum computational advantage, by analyzing KD quasiprobability distributions in qubit systems and identifying classically simulable states.

Contribution

It introduces a real-quantum-bit model using KD quasiprobability, constructs new classically-simulable states, and proves KD nonpositivity as a necessary resource for quantum advantage.

Findings

KD nonpositivity is necessary for quantum computational advantage

Constructed new classically-simulable states using KD geometry

Established KD nonpositivity as a resource monotone

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.