Efficient magic state cultivation with lattice surgery

TL;DR

This paper tackles the high spacetime cost of magic-state distillation for fault-tolerant quantum computing by introducing magic state cultivation with lattice surgery (MSC-LS). MSC-LS transfers a magic state from a color code to a rotated surface code through lattice surgery, leveraging code expansion and a lookup-table for early rejection to reduce resource overhead while maintaining competitive logical error rates at $p_{\mathrm{phys}}=10^{-3}$ with color-code distance $d_{\mathrm{color}}=3$. The approach achieves roughly half the spacetime overhead of previous MSC protocols and preserves fault distance, making it a practical option for near-term devices and megaquop workloads. The work emphasizes simplicity, compatibility with square-grid connectivity, and integration with existing software and hardware stacks, contributing a viable pathway toward more efficient, scalable magic-state distillation in fault-tolerant quantum computation.

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.

Figures (16)

• Figure 1: An example of a detector.
• Figure 2: Tanner graphs of the distance-5 rotated surface code (for $Z$ errors only). Gray, black, and purple circles indicate detectors, error locations, and detectors that detect $Z$ errors, respectively. The left and right panels show matching results that are complementary to each other, with matching paths colored orange.
• Figure 3: Magic state cultivation Gidney2024MagicStateCultivation for $d_{\mathrm{color}} = 3$. "Encode T" denotes the non-fault-tolerant encoding of a magic state into the distance-3 2D color code. Two consecutive "Check T" represent the application of two Hadamard tests. "PS" denotes postselection, and "SE" denotes syndrome extraction.
• Figure 4: Magic state cultivation with lattice surgery (MSC-LS) for $d_{\mathrm{color}} = 3$. The left area represents the spatial resource (physical qubits) used by the rotated surface code, and the right area represents those used by the color code. "Encode T" denotes the non-fault-tolerant encoding of a magic state into the distance-3 2D color code. Two consecutive "Check T" denote the application of two Hadamard tests. "LS (ZZ)" denotes the $ZZ$ measurement implemented by lattice surgery. "PS" denotes postselection, and "SE" denotes syndrome extraction. The detectors obtained from the syndrome extractions marked in red are used for early rejection via the lookup table.
• Figure 5: Physical qubit positions. Qubits $0$--$6$ are data qubits of the distance-3 2D color code, and qubits $a$--$e$ are data qubits on the top edge of the rotated surface code.