Quantum repeaters constitute a promising platform for enabling long distance quantum communication and may ultimately serve as the backbone of a secure quantum internet, a scalable quantum network, or a distributed quantum computer. An efficient approach to encoding qubits within an error-correcting code is provided by bosonic codes, in which even a single oscillator mode can function as a sufficiently large physical system. In this work, initially we focus on the bosonic Gottesman Kitaev Preskill (GKP) code as a natural candidate for loss correction based quantum repeaters, which can be implemented at room temperature. We demonstrate that transmission loss can be suppressed across three related protocols at the expense of the introduction of logical errors. The third protocol, where a relay-like teleamplifier is applied is optimal. This approach enables medium-distance quantum communication without requiring higher level encoding. We compute the resulting secure key rates while leveraging analog syndrome information. Furthermore, we propose a concatenated Bell state measurement (CBSM) scheme with a modified parity encoding based on GKP qubits, CV measurement and a clipping method that corrects transmission loss without introducing logical errors. This significantly enhances the possible transmission distance. We find that GKP based repeaters can achieve performance comparable to approaches relying on photonic qubits, while requiring orders of magnitude fewer qubits.
Quantum repeaters constitute a promising platform for enabling long distance quantum communication and may ultimately serve as the backbone of a secure quantum internet, a scalable quantum network, or a distributed quantum computer. An efficient approach to encoding qubits within an error-correcting code is provided by bosonic codes, in which even a single oscillator mode can function as a sufficiently large physical system. In this work, initially we focus on the bosonic Gottesman Kitaev Preskill (GKP) code as a natural candidate for loss correction based quantum repeaters, which can be implemented at room temperature. We demonstrate that transmission loss can be suppressed across three related protocols at the expense of the introduction of logical errors. The third protocol, where a relay-like teleamplifier is applied is optimal. This approach enables medium-distance quantum communication without requiring higher level encoding. We compute the resulting secure key rates while leveraging analog syndrome information. Furthermore, we propose a concatenated Bell state measurement (CBSM) scheme with a modified parity encoding based on GKP qubits, CV measurement and a clipping method that corrects transmission loss without introducing logical errors. This significantly enhances the possible transmission distance. We find that GKP based repeaters can achieve performance comparable to approaches relying on photonic qubits, while requiring orders of magnitude fewer qubits.
Quantum repeaters are a promising platform for realizing long-distance quantum communication and thus could form the backbone of a secure quantum internet, a scalable quantum network, or a distributed quantum computer. Repeater protocols that encode information in single- or multiphoton states are limited by transmission losses and the cost of implementing entangling gates or Bell measurements. In this work, we consider implementing a quantum repeater protocol using Gottesman-Kitaev-Preskill (GKP) qubits. These qubits are natural elements for quantum repeater protocols, because they allow for deterministic Gaussian entangling operations and Bell measurements, which can be implemented at room temperature. The GKP encoding is also capable of correcting small displacement errors. At the cost of additional Gaussian noise, photon loss can be converted into a random displacement error channel by applying a phase-insensitive amplifier. Here we show that a similar conversion can be achieved in two-way repeater protocols by using phase-sensitive amplification applied in the postprocessing of the measurement data, resulting in less overall Gaussian noise per (sufficiently short) repeater segment. We also investigate concatenating the GKP code with higher-level qubit codes while leveraging analog syndrome data, postselection, and path-selection techniques to boost the rate of communication. We compute the secure key rates and find that GKP repeaters can achieve a comparative performance relative to methods based on photonic qubits while using orders-of-magnitude fewer qubits.
Quantum communication between two distant parties will serve as a cornerstone of the future quantum internet. However, generating enough entangled Bell pairs over long distances is a critical bottleneck. Although photons are ideal carriers of quantum information, overcoming photon loss and the exponential attenuation of signals remains a major challenge. We propose an all-photonic quantum repeater architecture that enables quantum communication over 1,000 km with an equidistant repeater spacing of 9 km. This repeater spacing is enabled by elementary entangled Bell pairs protected through the concatenation of continuous-variable and discrete-variable quantum error correction codes, namely, the bosonic Gottesman-Kitaev-Preskill (GKP) code and the [[7,1,3]] Steane code, whose combination yields a synergistic improvement in robustness against photon loss. This architecture incorporates a new ranking criterion and a multi-reflection mirror-based optical cavity as a free-space photonic memory module, which we model in terms of its length and mirror-reflection efficiency. Additionally, we propose two heuristic construction methods for the elementary entangled Bell pairs. One method introduces up to two-qubit correlated errors within each logical qubit but requires a large number of GKP qubits, while the other allows up to three-qubit correlated errors within each logical qubit but requires fewer GKP qubits. To more accurately capture realistic physical conditions during photonic resource preparation, we include switching-induced imperfections in our simulations, in addition to other standard optical imperfections. In the presence of these imperfections, our realization requires only a few thousand GKP qubits per repeater station per protocol run, a resource requirement significantly smaller than the corresponding resource requirements of prior third-generation all-photonic repeater proposals.
Quantum repeaters that incorporate quantum error correction codes have been shown to be a promising alternative compared with the original quantum repeaters that rely upon probabilistic quantum error detection depending on classical communication over remote repeater stations. A particularly efficient way of encoding qubits into an error correction code is through bosonic codes where even a single oscillator mode serves as a sufficiently large, physical system. Here we consider the bosonic Gottesman-Kitaev-Preskill (GKP) code as a natural choice for a loss-correction-based quantum repeater. However, unlike existing treatments, we focus on the excitation loss that occurs in the local, stationary memory qubits as represented by, for instance, collective atomic spin modes. We analyze and assess the performance of such a GKP-based quantum repeater where, apart from the initial state generations and distributions, all operations can be performed via deterministic linear mode transformations, as opposed to other existing memory-based quantum repeater schemes.
The Gottesman-Kitaev-Preskill (GKP) code offers the possibility to encode higher-dimensional qudits into individual bosonic modes with, for instance, photonic excitations. Since photons enable the reliable transmission of quantum information over long distances and since GKP states subject to photon loss can be recovered to some extent, the GKP code has found recent applications in theoretical investigations of quantum communication protocols. While previous studies have primarily focused on GKP qubits, the possible practical benefits of higher-dimensional GKP qudits are hitherto widely unexplored. In this paper, we carry out performance analyses for three quantum repeater protocols based on GKP qudits including concatenations with a multi-qudit quantum polynomial code. We find that the potential data transmission gains for qudits are often hampered by their decreased GKP error-correcting capabilities. However, we also identify parameter regimes in which having access to an increased number of quantum levels per mode can enhance the theoretically achievable secret-key rate of the quantum repeater. Some of our protocols share the attractive feature that local processing and complete error syndrome identification are realizable without online squeezing. Provided a supply of suitable multi-mode GKP states is available, this can be realized with a minimal set of passive linear optical operations, even when the logical qudits are composed of many physical qudits.
It is well known that the repeater node is an essential ingredient for the future global quantum network, which will enable high-rate private communication and entanglement distribution over very long distances. The near-term repeater architecture uses the measurement-based node that operate without both entanglement and quantum memory, which is the main idea of the measurement-device-independent quantum key distribution (MDI-QKD) protocol. The MDI-QKD protocol removes the trust condition from the inter repeaters, while its continuous variable (CV) version, when proposed, benefited from its deterministic nature, compatible with the classical devices, and shows a high rate for the short-range local area network (LAN). Whilst the theoretical backbone of CV-MDI-QKD protocol is well established, its secure transmission range is yet limited for practical LAN. In this study, we propose an enhanced scheme for the asymmetric CV-MDI-QKD protocol by using Gottesman-Kitaev-Preskill (GKP) oscillators-to-oscillators codes, where both loss error and operation error are suppressed to below the break-even point, without any delays caused by classical heralding signals. In particular, the proposed scheme, which correlates the noises of the data and ancilla via a pair of symplectic transforms, extracts the error syndromes from stabilizer measurements on the ancilla mode and informs the data mode for a corrective displacement. Numerical analysis shows the composable finite-size security of the protocol under the collective Gaussian attack, encompassing noiseless and noisy GKP states, with both wired (i.e., fiber-based) and wireless (i.e., free-space) configurations. In addition, we demonstrate that the residual errors of the GKP code can be further reduced by the concatenation method but has a trade-off between the layers number and the finite GKP squeezing.
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 ${10}^{3}\ensuremath{-}{10}^{4}$ GKP qubits per repeater per protocol run.