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Long-distance quantum communication will require the use of quantum repeaters to overcome the exponential attenuation of signal with distance. One class of such repeaters utilizes quantum error correction to overcome losses in the communication channel. Here we propose a strategy of using the bosonic Gottesman-Kitaev-Preskill (GKP) code in a two-way repeater architecture with multiplexing. The crucial feature of the GKP code that we make use of is the fact that GKP qubits easily admit deterministic two-qubit gates, hence allowing for multiplexing without the need for generating large cluster states as required in previous all-photonic architectures based on discrete-variable codes. Moreover, alleviating the need for such clique clusters entails that we are no longer limited to extraction of at most one end-to-end entangled pair from a single protocol run. In fact, thanks to the availability of the analog information generated during the measurements of the GKP qubits, we can design better entanglement swapping procedures in which we connect links based on their estimated quality. This enables us to use all the multiplexed links so that large number of links from a single protocol run can contribute to the generation of the end-to-end entanglement. We find that our architecture allows for high-rate end-to-end entanglement generation and is resilient to imperfections arising from finite squeezing in the GKP state preparation and homodyne detection inefficiency. In particular we show that long-distance quantum communication over more than 1000 km is possible even with less than 13 dB of GKP squeezing. We also quantify the number of GKP qubits needed for the implementation of our scheme and find that for good hardware parameters our scheme requires around 10^3 - 10^4 GKP qubits per repeater per protocol run. 

One-way quantum repeaters where loss and operational errors are counteracted by quantum error-correcting codes can ensure fast and reliable qubit transmission in quantum networks. It is crucial that the resource requirements of such repeaters, for example, the number of qubits per repeater node and the complexity of the quantum error-correcting operations are kept to a minimum to allow for near-future implementations. To this end, we propose a one-way quantum repeater that targets both the loss and operational error rates in a communication channel in a resource-efficient manner using code concatenation. Specifically, we consider a tree-cluster code as an inner loss-tolerant code concatenated with an outer 5-qubit code for protection against Pauli errors. Adopting flag-based stabilizer measurements, we show that intercontinental distances of up to 10,000 km can be bridged with a minimized resource overhead by interspersing repeater nodes that each specialize in suppressing either loss or operational errors. Our work demonstrates how tailored error-correcting codes can significantly lower the experimental requirements for long-distance quantum communication. 

Quantum error correction has recently been shown to benefit greatly from specific physical encodings of the code qubits. In particular, several researchers have considered the individual code qubits being encoded with the continuous variable GottesmanKitaev-Preskill (GKP) code, and then imposed an outer discrete-variable code such as the surface code on these GKP qubits. Under such a concatenation scheme, the analog information from the inner GKP error correction improves the noise threshold of the outer code. However, the surface code has vanishing rate and demands a lot of resources with growing distance. In this work, we concatenate the GKP code with generic quantum low-density parity-check (QLDPC) codes and demonstrate a natural way to exploit the GKP analog information in iterative decoding algorithms. We first show the noise thresholds for two lifted product QLDPC code families, and then show the improvements of noise thresholds when the iterative decoder – a hardware-friendly min-sum algorithm (MSA) – utilizes the GKP analog information. We also show that, when the GKP analog information is combined with a sequential update schedule for MSA, the scheme surpasses the well-known CSS Hamming bound for these code families. Furthermore, we observe that the GKP analog information helps the iterative decoder in escaping harmful trapping sets in the Tanner graph of the QLDPC code, thereby eliminating or significantly lowering the error floor of the logical error rate curves. Finally, we discuss new fundamental and practical questions that arise from this work on channel capacity under GKP analog information, and on improving decoder design and analysis. 

Abstract We propose an architecture of quantum-error-correction-based quantum repeaters that combines techniques used in discrete- and continuous-variable quantum information. Specifically, we propose to encode the transmitted qubits in a concatenated code consisting of two levels. On the first level we use a continuous-variable GKP code encoding the qubit in a single bosonic mode. On the second level we use a small discrete-variable code. Such an architecture has two important features. Firstly, errors on each of the two levels are corrected in repeaters of two different types. This enables for achieving performance needed in practical scenarios with a reduced cost with respect to an architecture for which all repeaters are the same. Secondly, the use of continuous-variable GKP code on the lower level generates additional analog information which enhances the error-correcting capabilities of the second-level code such that long-distance communication becomes possible with encodings consisting of only four or seven optical modes.