Efficient magic state cultivation with lattice surgery

TL;DR

This paper introduces an efficient physical-level magic state cultivation protocol that reduces overhead and simplifies implementation, making fault-tolerant quantum computing more practical with current hardware constraints.

Contribution

It proposes a new cultivation protocol that avoids grafted codes, reduces spatial and spacetime overhead, and is compatible with square-grid connectivity.

Findings

Achieves comparable logical error rates with half the spacetime overhead.

Reduces spatial overhead by avoiding complex grafted codes.

Demonstrates effectiveness with numerical simulations at realistic error rates.

Abstract

Magic state distillation plays a crucial role in fault-tolerant quantum computation and represents a major bottleneck. In contrast to traditional logical-level distillation, physical-level distillation offers significant overhead reduction by enabling direct implementation with physical gates. Magic state cultivation is a state-of-the-art physical-level distillation protocol that is compatible with the square-grid connectivity and yields high-fidelity magic states. However, it relies on the complex grafted code, which incurs substantial spacetime overhead and complicates practical implementation. In this work, we propose an efficient cultivation-based protocol compatible with the square-grid connectivity. We reduce the spatial overhead by avoiding the grafted code and further reduce the average spacetime overhead by utilizing code expansion and enabling early rejection. Numerical simulations show that, with a color code distance of 3 and a physical error probability of $10^{-3}$, our protocol achieves a logical error probability for the resulting magic state comparable to that of magic state cultivation ($\approx 3 \times 10^{-6}$), while requiring about half the spacetime overhead. Our work provides an efficient and simple distillation protocol suitable for megaquop use cases and early fault-tolerant devices.