Lower bounds on non-local computation from controllable correlation

TL;DR

This paper introduces two new techniques to establish lower bounds on the entanglement cost in non-local quantum computation, applicable to various unitaries including key quantum gates, with some bounds being tight.

Contribution

The authors develop general lower bound methods based on controllable correlation and entanglement, resolving the entanglement cost for the CNOT gate and extending bounds to other unitaries.

Findings

New lower bounds for Haar random two-qubit unitaries.

Established lower bounds for CNOT, DCNOT, √SWAP, and XX gates.

CNOT lower bound is tight, fully resolving its entanglement cost.

Abstract

Understanding entanglement cost in non-local quantum computation (NLQC) is relevant to complexity, cryptography, gravity, and other areas. This entanglement cost is largely uncharacterized; previous lower bound techniques apply to narrowly defined cases, and proving lower bounds on most simple unitaries has remained open. Here, we give two new lower bound techniques that can be evaluated for any unitary, based on their controllable correlation and controllable entanglement. For Haar random two qubit unitaries, our techniques typically lead to non-trivial lower bounds. Further, we obtain lower bounds on most of the commonly studied two qubit quantum gates, including CNOT, DCNOT, $\sqrt{\text{SWAP}}$, and the XX interaction, none of which previously had known lower bounds. For the CNOT gate, one of our techniques gives a tight lower bound, fully resolving its entanglement cost. The resulting lower bounds have parallel repetition properties, and apply in the noisy setting.