Entanglement Boosting: Low-Volume Logical Bell Pair Preparation for Distributed Fault-Tolerant Quantum Computation

Distributed architecture is a promising route to scaling fault-tolerant quantum computing (FTQC) beyond the inherent limitations of single processors, for which high-fidelity logical Bell pairs need to be prepared from many noisy physical Bell pairs with high efficiency. 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. A key challenge is to find an appropriate balance between these two contrasting features. To resolve this issue, we introduce a metric for characterizing the practical cost of preparing high-fidelity logical Bell pairs, (LLV), which is a circuit-volume metric incorporating, in a single quantity, both the cost of physical Bell pair consumption and the volume associated with local operations. Guided by this metric, we propose the protocol that 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, e.g. less than <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msup> <a:mn>10</a:mn> <a:mrow> <a:mo>−</a:mo> <a:mn>10</a:mn> </a:mrow> </a:msup> </a:math> from 86 noisy physical Bell pairs at 1% error, while all local operations are implementable within a spatial region of a single surface-code patch with two-dimensional local connectivity. This is a substantial reduction from other protocols, such as remote lattice surgery operations requiring nearly 1000 physical Bell pairs under the same setting. To further augment the entanglement boosting, 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.