Abstract
In this paper we propose a technique for distributing entanglement in architectures in which interactions between pairs of qubits are constrained to a fixed networkG. This allows for two-qubit operations to be performed between qubits which are remote from each other inG, through gate teleportation. We demonstrate how adaptingquantum linear network codingto this problem of entanglement distribution in a network of qubits can be used to solve the problem of distributing Bell states and GHZ states in parallel, when bottlenecks inGwould otherwise force such entangled states to be distributed sequentially. In particular, we show that by reduction to classical network coding protocols for thek-pairs problem or multiple multicast problem in a fixed networkG, one can distribute entanglement between the transmitters and receivers with a Clifford circuit whose quantum depth is some (typically small and easily computed) constant, which does not depend on the size ofG, however remote the transmitters and receivers are, or the number of transmitters and receivers. These results also generalise straightforwardly to qudits of any prime dimension. We demonstrate our results using a specialised formalism, distinct from and more efficient than the stabiliser formalism, which is likely to be helpful to reason about and prototype such quantum linear network coding circuits.
Highlights
One of the most important problems to solve, in the realisation of quantum algorithms in hardware, is how to map operations onto the architecture
We demonstrate how adapting quantum linear network coding to this problem of entanglement distribution in a network of qubits can be used to solve the problem of distributing Bell states and GHZ states in parallel, when bottlenecks in G would otherwise force such entangled states to be distributed sequentially
We consider the problem of entanglement distribution in quantum architectures with constraints on the interactions between pairs of qubits, described by a network G
Summary
One of the most important problems to solve, in the realisation of quantum algorithms in hardware, is how to map operations onto the architecture. Previous work [25, 26, 29, 38] has shown that when a classical binary linear network code exists for the “multiple unicast” problem (the problem of sending signals between k pairs of sources and targets) on a classical network, there exists a quantum network code to distribute Bell states between each source–target pair in a quantum network of the same connectivity These results suppose that each “node” is a small device, hosting multiple qubits and able to perform arbitrary transformations on them before transmitting onward “messages” through the network. This allows us to reason about them more efficiently than is possible even with the stabiliser formalism This yields at least a factor 2 improvement in space and time requirements, and achieves O(n) complexity (without using sparse matrix techniques) to simulate protocols which only involve superpositions of O(1) standard basis states. Which demonstrates the way in which a QLNC circuit may be regarded as realising a linear network code on an extended network G ⊇ G
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