GKP Quantum Repeaters

• GKP quantum repeaters are quantum communication architectures that encode qubits using square-lattice bosonic codes for native Gaussian noise correction.
• They utilize deterministic Gaussian operations and syndrome extraction to efficiently mitigate photon loss and memory errors across various platforms.
• Their scalability is achieved by concatenating with higher-level codes and employing analog syndrome processing to maintain robust long-distance links.

Gottesman-Kitaev-Preskill (GKP) quantum repeaters are quantum communication protocols and network architectures that employ the GKP bosonic code to realize efficient, high-fidelity, and hardware-minimal quantum repeaters. These protocols exploit the GKP code’s ability to encode discrete-variable (qubit or qudit) information into a single bosonic mode, enabling native correction of small displacement (Gaussian) noise such as that induced by photon loss or memory errors. The core strength of GKP repeaters lies in their hardware determinism, the Gaussian nature of their entangling gates and syndrome extraction, and their capacity for simple, scalable concatenation with higher-level quantum error-correcting codes. The GKP quantum repeater paradigm has been realized in room-temperature all-optical networks, stationary matter-based platforms, and superconducting microwave circuits.

1. GKP Code Fundamentals and Error Models

The canonical square-lattice GKP code encodes a qubit (or, more generally, a qudit of dimension $D$) into a single bosonic oscillator mode, stabilized by displacement operators periodic in both quadratures. For the qubit case, the ideal logical states in the $q$ basis are

$$|\bar{0}\rangle \propto \sum_{n \in \mathbb{Z}} | q = 2n\sqrt{\pi} \rangle, \quad |\bar{1}\rangle \propto \sum_{n \in \mathbb{Z}} | q = (2n+1)\sqrt{\pi} \rangle.$$

Finite-energy, physically realized GKP states replace the Dirac comb with sharply peaked Gaussians whose width is parameterized by the squeezing. The key error model is the Gaussian random displacement channel: $$\mathcal{E}_{\sigma^2}(\rho) = \frac{1}{2\pi\sigma^2} \iint ds\, dt\, e^{-\frac{s^2+t^2}{2\sigma^2}} e^{-is\hat{p}} e^{it\hat{q}} \rho e^{-it\hat{q}} e^{is\hat{p}},$$ mapping all physical noise, including pure photon loss after suitable amplification, to Gaussian i.i.d. shifts in both $q$ and $p$ quadratures. A logical Pauli error occurs if the cumulative quadrature shift exceeds $\pm\sqrt{\pi}/2$.

Photon loss—described by a bosonic channel of transmissivity $\eta = e^{-L/L_{\mathrm{att}}}$—transforms into a Gaussian displacement channel after pre-amplification or classical post-processing. The added noise per segment is minimized using phase-sensitive (classical communication, "CC") amplification, yielding displacement-variance increments of $(1-\sqrt{\eta})/\sqrt{\eta}$ per segment, compared to the standard post-amplification value of $(1-\eta)/\eta$ (Fukui et al., 2020).

2. Deterministic Gaussian Operations and Syndrome Extraction

GKP repeaters rely on deterministic, room-temperature Gaussian operations—beamsplitters, phase-shifts, and homodyne measurements—all implementable in linear optics or with bosonic light-ensemble interfaces. Core primitives include:

• Bell-state preparation: Interfering two approximate $q$0 ("q-comb") GKP states on a 50:50 beamsplitter yields a logical Bell pair $q$1.
• Error correction: Teleportation-style syndrome extraction uses a beamsplitter plus dual quadrature (homodyne) measurements, projecting onto GKP Bell states and feeding forward a correction displacement modulo $q$2. All finite-squeezing noise can be commuted to before or after the interaction, eliminating the need for on-line squeezing or adaptive gates during the protocol (Fukui et al., 2020).

In stationary matter-based architectures (e.g., Holstein–Primakoff ensembles), collective atomic spin modes serve as effective bosonic memories, with Gaussian light–ensemble Hamiltonians engineered via pulse shaping and polarization (Häussler et al., 2024).

3. Loss-Tolerant Repeater Protocols and Amplification Strategies

Transmission loss is mapped to a correctable Gaussian shift using amplifiers:

• Phase-insensitive amplification: After loss (transmissivity $q$3), an amplifier of gain $q$4 restores quadrature means but inflates variance by $q$5 (post-amp) or $q$6 (pre-amp).
• Classical CC amplification in two-way repeaters: Rescaling and noise addition in post-processing achieves displacement noise increase $q$7, halving the penalty relative to standard amplifier-based loss conversion (Fukui et al., 2020).
• Relay-like teleamplification: Embedding loss within the teleportation Bell resource distributes the gain and achieves the minimal added Gaussian noise, as in recent parity-encoded GKP one-way repeaters (Swain et al., 10 Apr 2026).

Three principal protocol classes are distinguished (Swain et al., 10 Apr 2026):

1. Teleportation-based GKP error correction plus amplifier (Protocol I)
2. Protocol I with analog-syndrome "clipping" to discard low-confidence outcomes
3. Relay-like teleamplification in which the amplifier is distributed between repeater and resource

4. Concatenation with Higher-Level Codes, Analog Syndrome Processing, and Multiplexing

To extend range and suppress logical errors:

• Analog syndrome utilization: The continuous syndrome outcomes from GKP error correction (distribution of shifts in $q$8) allow weighting measurement confidence and discarding unreliable outcomes (HRM: "highly reliable measurement"). This produces erasures (heralded "qubit losses") that can be handled by loss-tolerant codes such as the tree code (tolerant up to $q$9 erasure) (Fukui et al., 2020).
• Path selection: Rather than discarding, multiplexed architectures select the best-matched syndrome outcomes across all links, maximizing the likelihood of low-error transmission (Rozpędek et al., 2023).
• Concatenated code architectures: GKP codes serve as inner codes, with outer error-correcting codes (e.g., Steane $$|\bar{0}\rangle \propto \sum_{n \in \mathbb{Z}} | q = 2n\sqrt{\pi} \rangle, \quad |\bar{1}\rangle \propto \sum_{n \in \mathbb{Z}} | q = (2n+1)\sqrt{\pi} \rangle.$$0) leveraging analog information from the inner layer to further suppress logical errors. Schemes using minimal-modes ($$|\bar{0}\rangle \propto \sum_{n \in \mathbb{Z}} | q = 2n\sqrt{\pi} \rangle, \quad |\bar{1}\rangle \propto \sum_{n \in \mathbb{Z}} | q = (2n+1)\sqrt{\pi} \rangle.$$1 or $$|\bar{0}\rangle \propto \sum_{n \in \mathbb{Z}} | q = 2n\sqrt{\pi} \rangle, \quad |\bar{1}\rangle \propto \sum_{n \in \mathbb{Z}} | q = (2n+1)\sqrt{\pi} \rangle.$$2) can reach thousands of kilometers with moderate squeezing (14–18 dB) and $$|\bar{0}\rangle \propto \sum_{n \in \mathbb{Z}} | q = 2n\sqrt{\pi} \rangle, \quad |\bar{1}\rangle \propto \sum_{n \in \mathbb{Z}} | q = (2n+1)\sqrt{\pi} \rangle.$$3 coupling efficiency (Rozpędek et al., 2020, Häussler et al., 21 Dec 2025, Schmidt et al., 2023).
• Multiplexing with analog information: Resource GKP entanglement is prepared in multiple parallel modes, and analog outcomes are used to rank and pair the highest-fidelity links, enabling multi-ebit throughput per protocol run (Rozpędek et al., 2023).

5. Performance Metrics, Scaling Laws, and Resource Efficiency

The end-to-end logical error over $$|\bar{0}\rangle \propto \sum_{n \in \mathbb{Z}} | q = 2n\sqrt{\pi} \rangle, \quad |\bar{1}\rangle \propto \sum_{n \in \mathbb{Z}} | q = (2n+1)\sqrt{\pi} \rangle.$$4 segments with per-segment error $$|\bar{0}\rangle \propto \sum_{n \in \mathbb{Z}} | q = 2n\sqrt{\pi} \rangle, \quad |\bar{1}\rangle \propto \sum_{n \in \mathbb{Z}} | q = (2n+1)\sqrt{\pi} \rangle.$$5 is $$|\bar{0}\rangle \propto \sum_{n \in \mathbb{Z}} | q = 2n\sqrt{\pi} \rangle, \quad |\bar{1}\rangle \propto \sum_{n \in \mathbb{Z}} | q = (2n+1)\sqrt{\pi} \rangle.$$6 (Fukui et al., 2020, Swain et al., 10 Apr 2026). The net protocol success probability scales with the product of per-segment or per-node success probabilities, which depend on syndrome clipping and multiplexing strategies. The asymptotic secure key rate (for BB84-like QKD) is

$$|\bar{0}\rangle \propto \sum_{n \in \mathbb{Z}} | q = 2n\sqrt{\pi} \rangle, \quad |\bar{1}\rangle \propto \sum_{n \in \mathbb{Z}} | q = (2n+1)\sqrt{\pi} \rangle.$$7

where $$|\bar{0}\rangle \propto \sum_{n \in \mathbb{Z}} | q = 2n\sqrt{\pi} \rangle, \quad |\bar{1}\rangle \propto \sum_{n \in \mathbb{Z}} | q = (2n+1)\sqrt{\pi} \rangle.$$8 is the binary entropy function.

GKP repeaters surpass the direct-transmission Pirandola–Laurenza–Ottaviani–Banchi (PLOB) bound (Rozpędek et al., 2020, Rozpędek et al., 2023, Häussler et al., 21 Dec 2025), enabling long-distance entanglement or key rates with orders-of-magnitude fewer physical qubits than discrete-variable (single-photon) all-optical architectures, particularly when hybrid codes and analog post-selection are employed.

Sample resource trade-offs:

Scheme	Example Rate/Distance	Physical Qubits per Link	Squeezing
GKP+tree code	$$|\bar{0}\rangle \propto \sum_{n \in \mathbb{Z}} | q = 2n\sqrt{\pi} \rangle, \quad |\bar{1}\rangle \propto \sum_{n \in \mathbb{Z}} | q = (2n+1)\sqrt{\pi} \rangle.$$9 error over $$\mathcal{E}_{\sigma^2}(\rho) = \frac{1}{2\pi\sigma^2} \iint ds\, dt\, e^{-\frac{s^2+t^2}{2\sigma^2}} e^{-is\hat{p}} e^{it\hat{q}} \rho e^{-it\hat{q}} e^{is\hat{p}},$$0 km	$$\mathcal{E}_{\sigma^2}(\rho) = \frac{1}{2\pi\sigma^2} \iint ds\, dt\, e^{-\frac{s^2+t^2}{2\sigma^2}} e^{-is\hat{p}} e^{it\hat{q}} \rho e^{-it\hat{q}} e^{is\hat{p}},$$1	$$\mathcal{E}_{\sigma^2}(\rho) = \frac{1}{2\pi\sigma^2} \iint ds\, dt\, e^{-\frac{s^2+t^2}{2\sigma^2}} e^{-is\hat{p}} e^{it\hat{q}} \rho e^{-it\hat{q}} e^{is\hat{p}},$$2–$$\mathcal{E}_{\sigma^2}(\rho) = \frac{1}{2\pi\sigma^2} \iint ds\, dt\, e^{-\frac{s^2+t^2}{2\sigma^2}} e^{-is\hat{p}} e^{it\hat{q}} \rho e^{-it\hat{q}} e^{is\hat{p}},$$3 dB
Photonic-qubit (Azuma et al)	$$\mathcal{E}_{\sigma^2}(\rho) = \frac{1}{2\pi\sigma^2} \iint ds\, dt\, e^{-\frac{s^2+t^2}{2\sigma^2}} e^{-is\hat{p}} e^{it\hat{q}} \rho e^{-it\hat{q}} e^{is\hat{p}},$$4 error over $$\mathcal{E}_{\sigma^2}(\rho) = \frac{1}{2\pi\sigma^2} \iint ds\, dt\, e^{-\frac{s^2+t^2}{2\sigma^2}} e^{-is\hat{p}} e^{it\hat{q}} \rho e^{-it\hat{q}} e^{is\hat{p}},$$5 km	$$\mathcal{E}_{\sigma^2}(\rho) = \frac{1}{2\pi\sigma^2} \iint ds\, dt\, e^{-\frac{s^2+t^2}{2\sigma^2}} e^{-is\hat{p}} e^{it\hat{q}} \rho e^{-it\hat{q}} e^{is\hat{p}},$$6	N/A
All-photonic GKP+Steane	$$\mathcal{E}_{\sigma^2}(\rho) = \frac{1}{2\pi\sigma^2} \iint ds\, dt\, e^{-\frac{s^2+t^2}{2\sigma^2}} e^{-is\hat{p}} e^{it\hat{q}} \rho e^{-it\hat{q}} e^{is\hat{p}},$$7–$$\mathcal{E}_{\sigma^2}(\rho) = \frac{1}{2\pi\sigma^2} \iint ds\, dt\, e^{-\frac{s^2+t^2}{2\sigma^2}} e^{-is\hat{p}} e^{it\hat{q}} \rho e^{-it\hat{q}} e^{is\hat{p}},$$8 km, $$\mathcal{E}_{\sigma^2}(\rho) = \frac{1}{2\pi\sigma^2} \iint ds\, dt\, e^{-\frac{s^2+t^2}{2\sigma^2}} e^{-is\hat{p}} e^{it\hat{q}} \rho e^{-it\hat{q}} e^{is\hat{p}},$$9	$q$0–$q$1 modes per station	$q$2–$q$3 dB, $q$4

(Fukui et al., 2020, Rozpędek et al., 2020, Häussler et al., 21 Dec 2025, Rozpędek et al., 2023)

The optimal segment length is typically minimized (e.g., $q$5 m to $q$6 km), balancing per-hop noise against fused resource overhead.

6. Physical Architectures: All-Optical, Stationary, and Hybrid GKP Repeaters

GKP-based repeaters are implemented in several architectures:

• All-optical (third-generation): Passive linear optics (beamsplitters, homodyne) and offline GKP ancilla preparation enable fast, parallel, memoryless protocols with deterministic two-qubit gates (Fukui et al., 2020, Rozpędek et al., 2023, Swain et al., 10 Apr 2026).
• Stationary (second-generation): GKP qubits stored in atomic spin ensembles or high-$q$7 superconducting resonators. Gate-based microwave repeaters (GBMQR) in circuit QED platforms further demonstrate autonomous GKP error correction and deterministic entanglement-swapping via cross-Kerr or controlled-displacement interactions (Häussler et al., 2024, Khalifa et al., 22 Dec 2025).
• Hybrid (fourth-generation): Combine stationary matter and flying GKP qudits, exploiting both fast distribution and robust memory, with protocol transitions between one-way (encoded transmission) and two-way (heralded swapping) depending on resource constraints (Häussler et al., 1 Aug 2025).

Implementation requirements and bottlenecks:

Deterministic, high-fidelity, and scalable GKP-state generation at $q$8 dB squeezing is the outstanding experimental challenge. All protocols use only Gaussian optics/homodyne, with no online squeezing or high-fidelity ancilla-consumption once GKP states are supplied offline (Fukui et al., 2020, Swain et al., 10 Apr 2026).

7. Comparative Advantages, Thresholds, and Practical Outlook

Compared to traditional discrete-variable or cat-code-based repeaters, GKP-based repeaters offer:

• Deterministic, hardware-minimal, and room-temperature operation (for photonic or atomic-ensemble platforms).
• Direct conversion of photon loss to correctable Gaussian displacements.
• Robustness to moderate squeezing and finite detector efficiency, especially when concatenated with higher-level codes or when analog-syndrome information is exploited.
• Enhanced loss tolerance—up to $q$9 link loss—when paired with erasure-tolerant codes.
• Flexible scaling: from all-optical (GHz clocked, memoryless) to fully stationary architectures (seconds-scale memory) (Fukui et al., 2020, Häussler et al., 2024, Khalifa et al., 22 Dec 2025, Swain et al., 10 Apr 2026, Rozpędek et al., 2023).

Critical implementation thresholds are set by achievable squeezing (typically $p$0–$p$1 dB for practical long-distance performance), coupling losses, and memory coherence times. In future networks, combining on-demand high-squeezing GKP sources with chip-scale photonics or robust atomic-ensemble interfaces would enable the realization of a scalable, room-temperature quantum internet backbone based on GKP quantum repeaters (Fukui et al., 2020, Swain et al., 10 Apr 2026, Khalifa et al., 22 Dec 2025).