Enumerating all bilocal Clifford distillation protocols through symmetry reduction

TL;DR

This paper systematically enumerates and optimizes bilocal Clifford distillation protocols for quantum entanglement, leveraging symmetry reduction to identify efficient circuits with high fidelity for multiple copies of Bell-diagonal and Werner states.

Contribution

It introduces a symmetry-based method to enumerate all bilocal Clifford distillation protocols up to 8 copies, improving on previous protocols and providing efficient circuit constructions.

Findings

All protocols for up to 8 copies of Werner states are identified.

Optimized circuits achieve high fidelity with modest depth and gate count.

The approach leverages symplectic group symmetries for protocol enumeration.

Abstract

Entanglement distillation is an essential building block in quantum communication protocols. Here, we study the class of near-term implementable distillation protocols that use bilocal Clifford operations followed by a single round of communication. We introduce tools to enumerate and optimise over all protocols for up to $n=5$ (not necessarily equal) Bell-diagonal states using a commodity desktop computer. Furthermore, by exploiting the symmetries of the input states, we find all protocols for up to $n=8$ copies of a Werner state. For the latter case, we present circuits that achieve the highest fidelity with perfect operations and no decoherence. These circuits have modest depth and number of two-qubit gates. Our results are based on a correspondence between distillation protocols and double cosets of the symplectic group, and improve on previously known protocols.