Computational power of matchgates with supplementary resources

TL;DR

This paper explores the classical simulation complexity of matchgate computations with various resources, revealing parallels to Clifford circuits and conditions for quantum universality.

Contribution

It provides a comprehensive analysis of how different resource combinations affect the classical simulability and universality of matchgate circuits.

Findings

Matchgate circuits with certain resources are classically efficiently simulable.

Allowing three or more lines with entangled inputs leads to quantum universality.

Adaptive measurements in non-computational bases restore quantum power.

Abstract

We study the classical simulation complexity in both the weak and strong senses, of matchgate (MG) computations supplemented with all combinations of settings involving inclusion of intermediate adaptive or nonadaptive computational basis measurements, product state or magic and general entangled state inputs, and single- or multi-line outputs. We find a striking parallel to known results for Clifford circuits, after some rebranding of resources. We also give bounds on the amount of classical simulation effort required in case of limited access intermediate measurements and entangled inputs. In further settings we show that adaptive MG circuits remain classically efficiently simulable if arbitrary two-qubit entangled input states on consecutive lines are allowed, but become quantum universal for three or more lines. And if adaptive measurements in non-computational bases are allowed, even with just computational basis inputs, we get quantum universal power again.