Classically efficient regimes in measurement based quantum computation performed using diagonal two qubit gates and cluster measurements

TL;DR

This paper extends classical simulation results for measurement-based quantum computation to include all two-qubit diagonal gates, identifying regimes where quantum states are classically simulatable.

Contribution

It explicitly computes the key parameter for all two-qubit diagonal gates, broadening the class of states efficiently simulatable in measurement-based quantum computation.

Findings

Explicit computation of the parameter for all two-qubit diagonal gates.

Identification of a non-trivial classically simulatable phase for certain quantum states.

Numerical evidence that alternative operator sets can expand the efficient simulation regime.

Abstract

In a recent work arXiv:2201.07655v2 we showed that there is a constant $\lambda >0$ such that it is possible to efficiently classically simulate a quantum system in which (i) qudits are placed on the nodes of a graph, (ii) each qudit undergoes at most $D$ diagonal gates, (iii) each qudit is destructively measured in the computational basis or bases unbiased to it, and (iv) each qudit is initialised within $\lambda^{-D}$ of a diagonal state according to a particular distance measure. In this work we explicitly compute $\lambda$ for any two qubit diagonal gate, thereby extending the computation of arXiv:2201.07655v2 beyond CZ gates. For any finite degree graph this allows us to describe a two parameter family of pure entangled quantum states (or three parameter family of thermal states) which have a non-trivial classically efficiently simulatable "phase" for the permitted measurements, even though other values of the parameters may enable ideal cluster state quantum computation. The main the technical tool involves considering separability in terms of "cylindrical" sets of operators. We also consider whether a different choice of set can strengthen the algorithm, and prove that they are optimal among a broad class of sets, but also show numerically that outside this class there are choices that can increase the size of the classically efficient regime.