Gate-Based Microwave Quantum Repeater Via Grid-State Encoding

TL;DR

This work introduces a gate-based microwave quantum repeater (GBMQR) that uses autonomous error-corrected bosonic grid states (GKP codes) to enable deterministic entanglement generation and all-bosonic entanglement swapping in a superconducting circuit setting. Each repeater node carries a transmon and two bosonic resonators, one as a memory encoded in a GKP grid state and the other as an information bus, enabling sequential entanglement generation via photon absorption and a CZ-based swap without joint interference. The authors quantify performance under realistic imperfections, showing substantial key-rate advantages over beamsplitter-based BSM schemes, with entanglement-generation and swapping success probabilities around 0.75 and 0.58 respectively at stationary damping times of $\kappa_{\text{damp}}^{-1} \approx 40$ ms. The results indicate GBMQR can be implemented with current superconducting microwave technology and offers practical routes for chip-to-chip secure communications and distributed quantum computing, with clear paths to further improvement through higher squeezing and improved codeword orthogonality. This approach addresses mode-mismatch losses inherent to optical-like beamsplitter BSMs and highlights the potential of bosonic grid-state encodings for scalable quantum networking.

Abstract

In autonomous quantum error correction the lifetime of a logical bosonic qubit can be extended beyond its physical constituents without feedback measurements. Leveraging autonomous error correction, we propose a second-generation gate-based microwave quantum repeater (GBMQR) with encoded bosonic grid states. Each repeater station comprises a transmon and two bosonic resonators: one resonator serving as a stationary quantum memory utilizing autonomous error correction, and the other as an information bus for entanglement generation. Entanglement is generated sequentially through the successful absorption of a microwave photon wavepacket. This method enables deterministic entanglement generation, in contrast to a probabilistic mixing of two heralding signals on a balanced beamsplitter. Furthermore, our GBMQR employs an all-bosonic entanglement swapping Bell-state measurement. This is implemented via a bosonic controlled-Z gate and two separate X-basis projective homodyne measurements on the stationary stored codewords. Our approach circumvents mode-mismatch losses associated with routing and interfering of heralding modes on a beamsplitter, and confines losses to those arising from stationary storage. We evaluate the performance of the proposed quantum repeater by calculating its secret key rate under realistic lab environments. Moreover, we explicitly demonstrate that at stationary damping rate of $κ^{-1}_{\text{damp}}=$~\SI{40}{\milli\second}, GBMQR can achieve entanglement generation and swapping success probabilities approx.~$0.75$, and $0.58$ respectively, surpassing the hallmark success probability of $1/2$ set by ideal linear beamsplitter-based Bell-state measurements. The proposed device can be implemented using currently available superconducting microwave technology and is suited for secure chip-to-chip communication and distributed quantum computing.

Figures (3)

• Figure 1: A schematic of the proposed GBMQR. The figure depicts two repeater segments, where nodes A and D are the end nodes of the repeater, while B and C are located in the same place to perform a swapping operation. Each node is equipped with a transmon-two-resonator system. One resonator acts as a quantum memory loaded with a GKP codeword, while the other is coupled to a transmission line of transmissivity $\eta$ that mediates remote sequential entanglement across each repeater segment. A controlled-displacement gate (CD-gate) is employed at the state preparation step as described in the main text. A swapping Bell-state measurement (BSM) is performed bosonically on the memory codewords via a controlled-Z operation followed by single bosonic-qubit measurements in the X-basis. This gate can be implemented either via an additional transmon or a SNAIL device. Ideally, the final repeater state is a bosonic Bell-state.
• Figure 2: Overlap between two logical GKP codewords when each is subjected to a different loss channels (Appendix \ref{['app:MisError']}). Panel (a) considers the similar overlap $\beta = {_{\Delta'}\langle \bar{0} \lvert \bar{0} \rangle_{\Delta"}}={_{\Delta'}\langle \bar{1} \lvert \bar{1} \rangle_{\Delta"}}$ as a function of $\Delta'$ and $\Delta"$, whereas in panel (b) the cross-overlap $\alpha ={_{\Delta'}\langle \bar{0} \lvert \bar{1} \rangle_{\Delta"}}={_{\Delta'}\langle \bar{1} \lvert \bar{0} \rangle_{\Delta"}}$ is considered. The initial lossless variance of the codewords was assumed $\Delta=0.3$. Ideally, when no losses are incurred, codewords are orthogonal, $\alpha=0$, and normalized, $\beta=1$. In the limit of large losses, $\alpha \rightarrow 1$, this implies that the code distance became zero. Consequently, the robustness of the code to losses, i.e. $\beta \rightarrow 0$ vanishes, as well as the code itself.
• Figure 3: A comparison between GBMQR and BSMQR secret key rates as a function of the fidelity $\mathcal{F}$ of the final entangled state. The plot considers the fidelity threshold where a quantum repeater is capable of generating a positive key rate RevModPhys.81.1301. The best performance of GBMQR is achieved when stationary damping is $\kappa^{-1}_{\rm damp}=$ 40ms (solid green). For stationary damping rate of $\kappa^{-1}_{\rm damp}=$ 25ms (solid blue) GBMQR still outperforms an ideal BSMQR at overall losses of $\eta_{1}=\eta_{2}=0.95$ (dashed purple). Realistic BSMQR generates the smallest secret key rate (solid orange).