Quantum computation over the butterfly network

TL;DR

This paper explores the limits of distributed quantum computation over a butterfly network, showing that certain unitary operations can be performed without additional entanglement resources, depending on their structure.

Contribution

It introduces protocols for implementing specific quantum operations over a restricted network without extra entanglement, identifying conditions for optimal resource use.

Findings

Unitary operations equivalent to controlled unitaries can be performed without additional entanglement.

Protocols for controlled traceless unitaries with 1-ebit resource are constructed.

Global Clifford operations can be implemented with 2-ebit resource.

Abstract

In order to investigate distributed quantum computation under restricted network resources, we introduce a quantum computation task over the butterfly network where both quantum and classical communications are limited. We consider deterministically performing a two-qubit global unitary operation on two unknown inputs given at different nodes, with outputs at two distinct nodes. By using a particular resource setting introduced by M. Hayashi [Phys. Rev. A \textbf{76}, 040301(R) (2007)], which is capable of performing a swap operation by adding two maximally entangled qubits (ebits) between the two input nodes, we show that unitary operations can be performed without adding any entanglement resource, if and only if the unitary operations are locally unitary equivalent to controlled unitary operations. Our protocol is optimal in the sense that the unitary operations cannot be implemented if we relax the specifications of any of the channels. We also construct protocols for performing controlled traceless unitary operations with a 1-ebit resource and for performing global Clifford operations with a 2-ebit resource.