Kirkwood-Dirac nonpositivity as a resource for quantum computing (Part 1)

The Kirkwood--Dirac (KD) quasiprobability distributions provide a powerful and trending mathematical framework to represent quantum systems. For certain states, a KD distribution is a standard joint probability distribution. However, a KD distribution can signal nonclassical phenomena by admitting nonpositive entries. We use this feature to study quantum computation. To do so, we focus on a real-qubit model of quantum computation, onto which any quantum-computational model can be mapped. We show that previous results imply that classical computers can simulate this model efficiently for input states which are positive with respect to a specific KD distribution, which we construct. Thus, we establish KD nonpositivity as a necessary resource for quantum-computational advantages. Building on this observation, we define a KD resource theory for computation. This resource theory allows us to quantify the amount of KD nonpositivity needed for arbitrary quantum computations. In Part 2 of this talk, we use these results to explicitly construct new (classically simulable) bound-magic states in (multi-qubit) systems of even dimensionality.