Near-term $n$ to $k$ distillation protocols using graph codes

TL;DR

This paper explores graph code-based distillation protocols for quantum entanglement, optimizing circuit depth and fidelity, crucial for practical quantum internet deployment under noisy conditions.

Contribution

It establishes a correspondence between distillation protocols and graph codes, enabling the design of optimal, low-depth circuits for quantum entanglement distillation.

Findings

Identified optimal distillation protocols for key quantum internet tasks.

Developed a circuit construction recipe for efficient implementation.

Enhanced teleportation fidelity and rate using new protocols.

Abstract

Noisy hardware forms one of the main hurdles to the realization of a near-term quantum internet. Distillation protocols allows one to overcome this noise at the cost of an increased overhead. We consider here an experimentally relevant class of distillation protocols, which distill $n$ to $k$ end-to-end entangled pairs using bilocal Clifford operations, a single round of communication and a possible final local operation depending on the observed measurement outcomes. In the case of permutationally invariant depolarizing noise on the input states, we find a correspondence between these distillation protocols and graph codes. We leverage this correspondence to find provably optimal distillation protocols in this class for several tasks important for the quantum internet. This correspondence allows us to investigate use cases for so-called non-trivial measurement syndromes. Furthermore, we detail a recipe to construct the circuit used for the distillation protocol given a graph code. We use this to find circuits of short depth and small number of two-qubit gates. Additionally, we develop a black-box circuit optimization algorithm, and find that both approaches yield comparable circuits. Finally, we investigate the teleportation of encoded states and find protocols which jointly improve the rate and fidelities with respect to prior art.