Complexity phase transitions in instantaneous quantum polynomial-time circuits

TL;DR

This paper investigates the complexity phases of a subclass of IQP circuits with varying two-qubit gate density, revealing multiple regimes including classically simulable and quantum advantage phases, and examining their learnability.

Contribution

It identifies multiple complexity phases in IQP circuits based on gate density, including regimes of classical simulability and quantum advantage, and analyzes their learnability.

Findings

Existence of multiple complexity phases in IQP circuits.

Identification of parameter regimes with classical simulability.

Quantum advantage potential even outside Porter-Thomas distribution.

Abstract

We study a subclass of the Instantaneous Quantum Polynomial-time (IQP) circuit with a varying density of two-qubit gates. In addition to a known anticoncentration regime, we identify novel parameter conditions where the model is classically simulable or the output distribution follows the Porter-Thomas distribution. By showing that those parameter regimes do not coincide, we argue the presence of more than two phases in the model. The learnability of the output distribution of this model is further studied, which indicates that an energy-based model fails to learn the output distribution even when it is not anticoncentrated. Our study reveals that a quantum circuit model can have multiple fine-grained complexity phases, suggesting the potential for quantum advantage even when the output distribution is far from the Porter-Thomas distribution.