Magic state cultivation: growing T states as cheap as CNOT gates

TL;DR

The paper proposes magic state cultivation as a practical method to produce high-fidelity T states within a surface-code framework, avoiding the conventional overhead of large-scale distillation. It introduces three stages—injection, cultivation, and escape—each leveraging error-detection postselection, incremental fault-distance growth, and a grafted color-to-surface-code transition to reach large, matchable codes. End-to-end simulations under uniform depolarizing noise demonstrate order-of-magnitude reductions in qubit-rounds to achieve fault rates down to 4×10^-11, with strong sensitivity to physical noise improvements, suggesting cultivation could supersede traditional distillation in practice. The work emphasizes practical deployment, including decoding strategies, hardware-connectivity considerations, and avenues for further refinement and end-to-end gate simulations.

Abstract

We refine ideas from Knill 1996, Jones 2016, Chamberland 2020, Gidney 2023+2024, Bombin 2024, and Hirano 2024 to efficiently prepare good $|T\rangle$ states. We call our construction "magic state cultivation" because it gradually grows the size and reliability of one state. Cultivation fits inside a surface code patch and uses roughly the same number of physical gates as a lattice surgery CNOT gate of equivalent reliability. We estimate the infidelity of cultivation (from injection to idling at distance 15) using a mix of state vector simulation, stabilizer simulation, error enumeration, and Monte Carlo sampling. Compared to prior work, cultivation uses an order of magnitude fewer qubit-rounds to reach logical error rates as low as $2 \cdot 10^{-9}$ when subjected to $10^{-3}$ uniform depolarizing circuit noise. Halving the circuit noise to $5 \cdot 10^{-4}$ improves the achievable logical error rate to $4 \cdot 10^{-11}$. Cultivation's efficiency and strong response to improvements in physical noise suggest that further magic state distillation may never be needed in practice.

Figures (33)

• Figure 1: Scatter plot of historical estimates of $T|+\rangle$ cost trade-offs from fowler2012surfacecodereviewli2015fowler2018latticesurgerygidney2019autocczlitinski2019notascostlysingh2022gidney2023hookhirano2024zeroleveldistillitogawa2024zeroleveldistilldistilllee2024colordistillationgidney2024ybasis, under circuit noise with a noise strength of $10^{-3}$. Points marked with "ungrown" omit the escape stage; they don't account for the cost of growing the state to a large code distance. They represent the part of the process that's experimentally accessible today.
• Figure 2: Selected historical estimates of $T|+\rangle$ cost trade-offs, without logical distillation.
• Figure 3: To-scale spacetime defect diagrams of historical constructions for producing $|T\rangle$ states, showing improvement over time. (Limited to papers that included 3d models.) Assumes $10^{-3}$ uniform depolarizing circuit noise. $\epsilon$ annotations indicate logical error rates. Red boxes indicate output locations. Left: the braided factory from fowler2012bridge. Middle left: the lattice surgery factory from fowler2018latticesurgery. Middle right: the double-output catalyzed factory from gidney2019catalyzeddistillation. Right: our construction retrying-until-success to cultivate a magic state, with a target fault distance of 5 and a target fault distance of 3.
• Figure 4: Overview of a magic state cultivation. Time moves upward. Darker colors are color code boundaries. Lighter colors are X/Z boundaries, and X/Z stabilizers. In practice cultivation must be attempted several times before it succeeds, and multiple attempts at stage 1 and stage 2 would be run in parallel.
• Figure 5: Cost of cultivation without an escape stage. Shows the similar performance of different injection stages. Estimated by enumerating all possible logical errors up to distance 5.