Kirkwood-Dirac Nonpositivity Is a Necessary Resource for Quantum Computing

We elucidate the boundary between classical and quantum computation by constructing qubit Clifford circuits with nonstabilizer inputs that can be efficiently simulated classically. We do so by casting the quantum circuits realizable by defect braiding in the surface code in terms of a Kirkwood-Dirac (KD) quasiprobability distribution, a generalization of a joint probability distribution. If this distribution remains a proper (positive) probability distribution throughout a circuit, then a classical algorithm can simulate the circuit efficiently. By leveraging recent results on the geometry of KD-positive states, we construct bound-magic states. Classical computers can efficiently simulate these bound-magic states’ evolutions under the circuits, although other magic states enable universal quantum computation when inputted. Furthermore, we show that KD nonpositivity is a resource monotone in this model. Thus, we establish KD nonpositivity as a necessary resource for quantum-computational advantages.