Entanglement boosting: Low-volume logical Bell pair preparation for distributed fault-tolerant quantum computation

TL;DR

This work introduces link-limited volume (LLV), a practical metric that combines the network throughput and local circuit volume needed to prepare high-fidelity logical Bell pairs for distributed FTQC. It then proposes entanglement boosting, a physical-to-logical Bell-pair protocol that uses code projection, simultaneous code expansion, and postselection via soft-information decoding to achieve $p_L o \mathcal{O}(10^{-10})$ with fewer than 100 physical Bell pairs inside a single rotated surface-code patch, dramatically reducing LLV. The authors further develop a pipelined entanglement distillation scheme using high-rate stabilizer codes to increase yield, enabling arbitrarily low logical error rates at the cost of larger local volumes, and discuss a crossover from boosting-dominated to distillation-enhanced regimes as network throughput $R$ varies. Together, these methods offer a principled, tunable approach to scalable distributed FTQC and broadly relevant strategies for other distributed quantum information tasks such as quantum key distribution and communication-assisted quantum computing.

Abstract

Distributed architecture is a promising route to scaling fault-tolerant quantum computing (FTQC) beyond the inherent limitations of single processors. For practical implementation of distributed FTQC, logical Bell pair preparation must be designed not only for efficient Bell pair consumption but also for the spacetime volume of the protocol; however, entanglement distillation protocols have primarily focused on minimizing the consumption of Bell pairs, often resulting in protocols that require a substantial number of local operations. To resolve this issue, we introduce a metric for characterizing the practical cost of preparing high-fidelity logical Bell pairs, link-limited volume (LLV), which is a circuit-volume metric incorporating both the cost of physical Bell pair consumption and the volume associated with local operations. Guided by this metric, we propose entanglement boosting protocol, which achieves efficient preparation of logical Bell pairs encoded in rotated surface code with LLV reduced by orders of magnitude compared to prior state-of-the-art methods. In this protocol, paralleling recent advances in magic state cultivation, we employ soft-information decoders and postselection to suppress the logical error rates of Bell pairs to practical levels in the order of $10^{-10}$ from fewer than 100 noisy physical Bell pairs while all local operations are implementable within a spatial region of a single surface code patch. We also present a pipelined implementation of entanglement distillation using high-rate quantum error-correcting codes, enabling arbitrarily low logical error rates while also maintaining physically efficient implementations. These results pave the way for the practical implementation of distributed FTQC, reinforcing the benefits of fast interconnect technologies and serving as a guiding principle for the efficient design of protocols and devices.

Figures (16)

• Figure 1: Two entanglement distillation protocols based on $[[n,k,d]]$ stabilizer code $\mathcal{C}$ used in this work. a) Auxiliary-qubit-assisted, projection-based preparation of logical Bell pairs from physical Bell pairs. Here, stabilizer checks of $n$ Bell pairs are measured by $n-k$ auxiliary qubits, allowing either postselection or error correction to achieve $k$ high-fidelity logical Bell pairs encoded in code $\mathcal{C}$. b) A decoding-based approach, which combines the protocol in a) and decoding of the encoded state, to result in $k$ output Bell pairs. The stabilizer measurements are performed by the measurement of $n-k$ qubits that do not constitute the output.
• Figure 2: The link-limited volume (LLV). LLV consists of network-related volume $\mathcal{V}_b=N^2/R$ and Bell pair factory volume $\mathcal{V}_f$ in each node (Eq. \ref{['eq:llv']}), where $N$ is the number of physical Bell pairs needed for the factory and $R$ is the throughput of the physical Bell pairs, defined by the number of Bell pairs generated in the duration a single syndrome extraction cycle.
• Figure 3: Entanglement boosting. Entanglement boosting begins with the preparation of $d_\text{Bell}^2$ physical Bell pairs, which are to be arranged in a square grid, together with qubits in $\ket{0}$ and $\ket{+}$ around them to form a $d_s\times d_s$ square grid. This is followed by $d_s$ cycles of syndrome extraction (SE) and MWPM decoding. We additionally perform decodings for complementary logical outcomes to compute the complementary gap Gidney2025, which allows efficient postselection (see Appendix \ref{['sec:complementary-gap']} for the details of complementary gap calculation).
• Figure 4: Numerical simulation results for the entanglement boosting protocol. a) Numerical simulation results of the logical error rates of the logical Bell pairs generated by the entanglement boosting protocol, for varying postselection criteria, for $d_\mathrm{Bell}= 3, 5, 7$ and 9, with $d_s$ fixed at 19 and $p_\mathrm{Bell} = 0.01$. Varying threshold values for the complementary gap results in different acceptance rate $q_0$; here, the logical error rate is shown for varying expected attempt count per kept shots, $1/q_0$. The shaded region is the standard error of estimated probability from $10^{10}$ sampling results. b-c) Numerical simulation results for the expected number of retries (colors), $1/q_0 - 1$, to achieve target error rates (vertical axis) for varying input Bell pair error rates $p_\text{Bell}$ (horizontal axis) over two orders of magnitude, for $d_\text{Bell}=3$ and 5.
• Figure 5: Scaling of the logical error rate of the Bell pairs produced with the entanglement boosting protocol. a-b) logical error rate as a function of the physical Bell pair error rate $p_{\mathrm{Bell}}$ for different $d_{\mathrm{Bell}}$, with acceptance rates $q_0$ of 100% (a) and 90% (b) based on the complementary gap. Each point is obtained from circuit-level simulations of the entanglement boosting protocol, and the dashed line represents the fitting with Eq. \ref{['eq:scaling']}. Error bars represent the standard error of estimated probability from the $10^8$ sampling results. b) the dependence of the fitted values of parameters $\alpha$, $\Gamma$ and $p_\mathrm{Bell}^\mathrm{(th)}$ on the discard rates $1-q_0$ of entanglement boosting, where the error bars represent 95% confidence intervals.