Constant-Overhead Fault-Tolerant Bell-Pair Distillation using High-Rate Codes

TL;DR

The paper tackles the challenge of scalable, fault-tolerant Bell-pair distillation with constant overhead by leveraging high-rate quantum LDPC codes. It shows that distillation can be performed deterministically in a one-way fashion while keeping the output Bell pairs encoded, achieving an overhead that equals the code rate and thresholds around $p_{ ext{bell}}'\approx 0.10$ for several code families. By analyzing depolarizing noise and circuit errors, the work develops effective one-sided noise models (e.g., $p_{ ext{bell}}' \approx 2p_{ ext{bell}}$ and $p' \approx 2p$) and validates these through full circuit-level simulations of HGP, LP, and SC codes, reporting code-rate metrics: ~4% (HGP), ~11% (LP), and ~1/3 (SC) with high fault-tolerance performance. The encoded Bell pairs at each node can be used directly for distributed quantum operations, enabling practical resource-efficient quantum networks and distributed computing with reduced unencoding overhead and robust local error suppression. Collectively, these results position qLDPC-based distillation as a viable path toward scalable quantum networking and distributed quantum information processing.

Abstract

We present a fault-tolerant Bell-pair distillation scheme achieving constant overhead through high-rate quantum low-density parity-check (qLDPC) codes. Our approach maintains a constant distillation rate equal to the code rate while requiring no additional overhead beyond the physical qubits of the code. Full circuit-level analysis demonstrates fault-tolerance for input Bell pair infidelities below a threshold $\sim 10\%$, readily achievable with near-term capabilities. Unlike previous proposals, our scheme keeps the output Bell pairs encoded in qLDPC codes at each node, eliminating un-encoding overhead and enabling direct use in distributed quantum applications through recent advances in qLDPC computation. These results establish qLDPC-based distillation as a practical route toward resource-efficient quantum networks and distributed quantum computing.

Figures (7)

• Figure 1: Constant-overhead Bell-pair distillation using qLDPC codes. Alice and Bob share $n$ initial raw Bell pairs. They measure the checks of a qLDPC code on each side using local ancilla qubits. Alice communicates her measurement results to Bob, who receives them and performs error correction, projecting the Bell pairs into the code state. This process results in a constant distillation rate equal to the rate of the qLDPC code, requiring only one-way communication. By performing logical operations on qLDPC codes, the encoded Bell pairs can be used for various applications, including distributed computation, communication via repeaters, and distributed sensing.
• Figure 2: Fault-tolerant simulation of qLDPC distillation. The Bell pairs experience input depolarizing noise with effective strength $p'_{\text{bell}}$. The circuit undergoes two-qubit gate depolarizing errors at a rate of $p' = 2 \times 0.1\%$. We simulate 14 decoding rounds with 3 cycles of syndrome extraction per round (42 cycles in total) and decode each round with BP, using BP+OSD for the final round. (a), (b), (c) Numerical results for the HGP, LP, and SC codes, respectively. We find a threshold $p_{\text{bell}}' \sim 10\%$ for the HGP and LP code families and a pseudothreshold of $p_{\text{bell}}\sim 7.5\%$ for the SC code. We calculate the logical error rate for the entire block per cycle, using $\bar{P} = 1 - (1-\bar{P}_{\text{tot}})^{1/42}$, where $\bar{P}_{\text{tot}}$ is the total error after 42 cycles. Error bars are calculated assuming a binomial distribution, with $N$ shots, giving $\sqrt{P_{\text{tot}} (1- P_{\text{tot}})/N}$.
• Figure 3: Distributed CNOT using qLDPC-distilled Bell pairs. The distributed CNOT gate can be implemented using local targeted CNOT operations between logical qubits and one of the $k$ encoded Bell pairs of the qLDPC code. These targeted CNOT gates can be performed using lattice surgery techniques or homomorphic gadgets. In product codes such as HGP codes, transversal approaches using homomorphic gadgets provide a constant-overhead implementation compatible with reconfigurable neutral atom architectures.
• Figure 4: Local and network noise in the encoding protocol. Alice and Bob initially share $n$ noisy Bell pairs, modeled by a depolarizing channel with error parameter $p_{\text{bell}}$. Syndrome extraction is performed using local ancilla qubits, with a two-qubit gate error rate $p$ at each node. To approximate the full noisy circuit (left), where both sides experience noise, we use a simplified model (right) in which Alice’s side is noiseless and Bob’s side has increased noise, with parameters $p'_{\text{bell}} \approx 2p_{\text{bell}}$ and $p' \approx 2p$.
• Figure 5: Sub-threshold data and fitted ansatz. Bell pairs have input depolarizing noise with strength $p_{\text{bell}}' = p$, and two-qubit gate depolarizing errors at $p_{\text{2q}}' = p/50$. We simulate 14 decoding rounds, each with 3 syndrome extraction cycles (42 total). Each round is decoded using BP, with the final round using BP+OSD. (a) Numerical results for the HGP codes. (b) Numerical results for the LP codes. (c) Numerical results for the SC code. Dotted lines fit the sub-threshold data to the ansatz $\bar{P} = A(p/p_{\text{th}})^{Bn^C}$, where $n$ is the code size, $p_{\text{th}}$ is the threshold error rate, and $A, B, C$ are fitting parameters. The logical error rate per cycle is $\bar{P} = 1 - (1-\bar{P}_{\text{tot}})^{1/42}$, where $\bar{P}_{\text{tot}}$ is the total error after 42 cycles. Error bars assume a binomial distribution with $N$ shots, computed as $\sqrt{P_{\text{tot}} (1 - P_{\text{tot}})/N}$.