Constant Overhead Entanglement Distillation via Scrambling

Abstract

High-fidelity quantum entanglement enables key quantum networking capabilities such as secure communication and distributed quantum computing, but long-distance entanglement distribution is limited by noise and loss. Entanglement distillation protocols address this problem by extracting high-fidelity Bell pairs from multiple noisy ones. The primary objective is minimizing the resource overhead: the number of noisy input pairs needed to distill each high-fidelity output pair. While protocols achieving optimal overhead are known in theory, they often require complex decoding operations that make practical implementation challenging. We circumvent this challenge by introducing protocols that use quantum scrambling -- the spreading of quantum information under chaotic dynamics -- through random Clifford operations. Based on this scrambling mechanism, our protocol maintains asymptotically \emph{constant} overhead, independent of the desired output error rate $\bar{\varepsilon}$, and can be implemented with shallow quantum circuits of depth $O(\mathrm{poly} \log \log \bar{\varepsilon}^{-1})$ and memory $O(\mathrm{poly} \log \bar{\varepsilon}^{-1})$. Our protocol remains effective even with noisy quantum gates. By incorporating error correction, our protocol achieves state-of-the-art performance: starting with pairs of 10% initial infidelity, we require only 7 noisy inputs per output pair to distill a single Bell pair with infidelity $\bar{\varepsilon}=10^{-12}$, substantially outperforming existing schemes. We demonstrate the utility of our protocols for quantum repeater networks.

Figures (5)

• Figure 1: An initial error (indicated in red) gets scrambled into an easily detectable global error under a scrambling unitary $C$, allowing a small number of measurements to detect the presence of the error via the syndrome $\vec{s}$. In the error-detection ('passive') setting, any nonzero syndrome $\vec{s}$ (i.e., disagreement of the measurement outcomes $\vec{x}$ and $\vec{y}$) means the protocol must be rerun, while in the error-correction ('active') setting, $\vec{s}$ is used to infer, then correct, the error.
• Figure 2: (a) Performance of \ref{['alg:random_bilocal_Clifford']} in the passive setting in terms of the fidelity $\bar{f}^k$ of the output state (top) and the single-pair infidelity $\bar{\varepsilon}$ (bottom). The initial single-pair fidelity is set to $f=1-\varepsilon=0.8$. The transition at the fraction $m/n=-\log_2 f$ of measured qubits is shown with a dashed line. (b) A three-layer concatenated protocol, consisting of a $6 \to 4$ protocol (green), an $8 \to 5$ protocol (blue), followed by a $10 \to 6$ protocol (purple).
• Figure 3: Overhead of our entanglement distillation protocols, with $L$ layers of concatenation, as a function of the initial single-pair infidelity $\varepsilon_0$. Dashed lines indicate the passive setting of our \ref{['alg:random_bilocal_Clifford']}. Solid lines indicate our protocol with the first layer having active error correction (with a maximum $E=3 \times 10^6$), assuming IID depolarizing noise. The parameters $n$ and $m$ at each layer of the protocol are optimized to achieve the lowest expected overhead, subject to the constraint that the output infidelity $\bar{\varepsilon}$ is below $10^{-12}$. We compare with pattison2024constoverheaddistillation and protocols using lattice surgery fowler2010surfaceramette2023faultsinclair2024fault. The solid black line indicates a lower bound for the best achievable overhead given by the Rains bound on distillable entanglement rains1999boundRains01.
• Figure 4: Random bilocal Clifford protocols applied to: (a) overheads for long-baseline interferometry; and (b) secret-key rates for QKD. The initial infidelity in (a) is $\varepsilon_0=0.0035$. The target infidelity in both cases is $\bar{\varepsilon}=10^{-9}$. In (b), each level halves the communication distance via intermediate repeater stations. The dashed line indicates our protocol with $n \leq 12$.
• Figure 5: The achieved infidelity of our random bilocal Clifford protocol with a finite number $G$ of gates and varying two-qubit gate depolarizing strengths $\lambda$, with the noiseless case shown in the dashed line. We set the number of noisy Bell pairs to $n=30$ and the initial infidelity to $\varepsilon=0.02$. Note the infidelity for local noise rates $\lambda$ asymptotes to $\sim\!\lambda$.

Theorems & Definitions (38)

• Theorem 1
• Theorem 2
• Lemma 1: Acceptance probabilities of \ref{['alg:random_bilocal_Clifford']}, passive setting
• proof
• Corollary 1: Output fidelity of \ref{['alg:random_bilocal_Clifford']}, passive setting
• proof
• Definition 1: Entanglement distillation
• Remark 1
• Definition 2: Overhead of an entanglement distillation protocol
• Remark 2: Non-IID states