Efficient entanglement purification based on noise guessing decoding

TL;DR

Efficient entanglement purification is critical for scalable quantum networks. The authors introduce PGRAND, a hashing-based one-way purification method that leverages QGRAND-style noise guessing to identify and correct the most likely error patterns, dramatically reducing qubit counts and computational overhead compared with traditional hashing. The approach yields high-fidelity Bell pairs from modest ensembles (e.g., $n=16$ with $p\approx 0.1$ noise) and enables both gate-based and measurement-based implementations, with the latter offering robustness against gate errors and a measured resource-state noise threshold $q_{\min} \approx 0.0859$. Compared to recurrence protocols, PGRAND can achieve favorable effective yields in low-entropy regimes, while still approaching capacity limits in larger ensembles; the measurement-based variant broadens practical applicability for near-term quantum networks.

Abstract

In this paper, we propose a novel bipartite entanglement purification protocol built upon hashing and upon the guessing random additive noise decoding (GRAND) approach recently devised for classical error correction codes. Our protocol offers substantial advantages over existing hashing protocols, requiring fewer qubits for purification, achieving higher fidelities, and delivering better yields with reduced computational costs. We provide numerical and semi-analytical results to corroborate our findings and provide a detailed comparison with the hashing protocol of Bennet et al. Although that pioneering work devised performance bounds, it did not offer an explicit construction for implementation. The present work fills that gap, offering both an explicit and more efficient purification method. We demonstrate that our protocol is capable of purifying states with noise on the order of 10% per Bell pair even with a small ensemble of 16 pairs. The work explores a measurement-based implementation of the protocol to address practical setups with noise. This work opens the path to practical and efficient entanglement purification using hashing-based methods with feasible computational costs. Compared to the original hashing protocol, the proposed method can achieve some desired fidelity with a number of initial resources up to one hundred times smaller. Therefore, the proposed method seems well-fit for future quantum networks with a limited number of resources and entails a relatively low computational overhead.

Figures (17)

• Figure 1: Quantum circuit for the PGRAND applied to $n$ Bell pairs distributed between Alice (A) and Bob (B) in order to obtain $k$ purified pairs. The error is described by a quantum channel $\mathcal{C}$. In the circuit, the measurements performed by Alice are depicted changing the ancilla qubits, but in reality only a virtual syndrome update needs do be done. The same goes to the recovery procedure: if it consists only of Pauli strings, then $\mathcal{R}$ can be performed virtually (in software), without the need to apply extra gates.
• Figure 2: Simulation results of the protocol probability of error $p_{e}$ as a function of the yield, for a) 32 and b) 128 Bell pairs, assuming $1\%$ of LDN. While on plot a) we simulated the procedure for errors with a weight up to $t = 5$, on b) we restricted to $t = 4$. The larger white dots represent the values obtained by combining expressions \ref{['eq:theoretical_fraction']}, and \ref{['eq:theoretical_bler']}. The simulation results show that the probability of error closely agrees with the values obtained via the theoretical expression when using a relatively large number of gates for the encoding (120 gates for plot a) and 1000 for plot b)).
• Figure 3: Maximum achievable yield to purify $n$ Bell pairs with an initial fidelity $F_{i}$ with a probability of error inferior to $p_{e}$. The initial fidelity is equal to a) $F_{i}=0.90$, b) $F_{i}=0.95$, c) $F_{i}=0.975$ and d) $F_{i}=0.99$. For each graph, the blue line represents the maximum yield at which purification is achievable. The dashed red line represents the bound given by \ref{['eq:DC_capacity']}. Applying the protocol to ensembles with more than one hundred pairs allows for a substantial increase in the efficiency of the protocol.
• Figure 4: Contour plot of the number of possible error patterns with weight equal or inferior to $t$ on an ensemble of $n$ Bell pairs. Here it is possible to see that the number of error patterns increases exponentially with their weight, but sub-exponentially with the size of the ensemble. The yellowish zone marks the frontier between what is computationally feasible with the actual standards.
• Figure 5: Minimum initial fidelity $F_{\textrm{min}}$ required to achieve purification as a function of the number of pairs in Werner form when attempting to correct errors with a weight up to $t \in \{3,5,7,9,12\}$. These results were obtained by considering that only one purified Bell pair is obtained $n$-to-one. If no constraints were imposed on $t$, the value of $F_{\text{min}}$ would strictly decrease with the number of pairs. However, if one limits the number of precomputed syndromes, a saturation point is reached.