Efficient Magic State Cultivation on the Surface Code

TL;DR

This work tackles efficient magic-state cultivation directly on the surface code, avoiding grafting from color codes. It introduces three protocols that use non-local transversal gates measured with a GHZ ancilla to project the code into magic-state eigenfunctions, with code distances expanding from $d_1=3$ to $d_2=11$ to achieve high fidelity. Clifford and state-vector simulations show state fidelities reaching $10^{-7}$–$10^{-9}$ with substantially higher acceptance rates than prior color-code schemes, especially under long-idling atom-like noise, yielding orders-of-magnitude reductions in qubit-round resources. The results also show that erasure-qubit bias can further improve fidelity with modest reductions in acceptance, enabling compact protocols such as a $d_1=2$ scheme with 9 physical qubits. Collectively, this work demonstrates a scalable, hardware-friendly path to zero-level distillation on the surface code with strong practical impact for near-term quantum architectures.

Abstract

Magic state cultivation is a newly proposed protocol that represents the state of the art in magic state generation. It uses the transversality of the $H_{XY}$ gate on the 2D triangular color-code, together with a novel grafting mechanism to transform the color-code into a matchable code with minimal overhead. Still, the resulting code has a longer cycle time and some high weight stabilizers. Here, we introduce three new cultivation protocols, each yielding a different magic state. These protocols avoid grafting by exploiting transversal operations on the surface code using non-local connectivity, allowing for a much lower post-selection rates in the expansion process. Through numerical simulations, we demonstrate that our protocol achieves state-of-the-art infidelities and acceptance rates for magic state generation, on par with another recent proposal on the $\mathbb{RP}^2$ code, while still preserving the local geometry of the surface code. Moreover, in platforms such as cold atoms and trapped ions, where idle error rates are lower than two-qubit gate errors, we demonstrate that cultivation exhibits an even greater advantage, yielding an additional order-of-magnitude reduction in resource requirements. Lastly, we analyze the effect of erasure qubits on cultivation and show that \emph{algorithmically-relevant} infidelities can be achieved using only 9 erasure qubits on a distance-2 surface code with a single cultivation round.

Figures (14)

• Figure 1: $H$,$H_{XY}$ and $CX$ cultivation -- Schematic diagrams of both $H/H_{XY}$ and $CX$ cultivation. Black dots label the data qubits, and orange rectangles label one instance of a transversal gate. (A) H cultivation on 2d triangular color code. A GHZ ancilla measures the H operator by transversal application of the controlled-Hadamard gate. (B) $CX$ cultivation on a pair of $d=3$ rotated surface codes. Blue and red dots label ancillas. The $CX$ operator is measured on a pair of surface codes by transversal application of the CCX (Toffoli) gate with a GHZ ancilla as one of the controls. (C) $H$ cultivation on the unrotated surface code, implemented using a transversal $CH$ on all the qubits, followed by conditional SWAP along the diagonal. (D) Mid-cycle projection of the rotated surface code using the $H_{XY}$ operator, applied using alternating $CH_{XY}$ and $C\bar{H}_{XY}$ gates along the diagonal and $CCZ$ gate across the diagonal.
• Figure 2: Logical error rate of cultivation with expansion - Uniform noise model -- Logical error rate, and expected attempt per kept shot of color code H cultivation by Gidney et al. gidney2024magic, and surface code $\text{CX}, \text{H}_{\text{XY}}$ and $\text{H}$ cultivation. The physical error rate is $p=10^{-3}$ using uniform noise model. The schemes expand into a larger surface codes with very similar performance gidney2024magic; The color code is expanded to a grafted surface code, while the rest expand to a standard surface codes, all with distance $d_2$. The different points label various thresholds for the complementary gap post-selection. Error bars represent a single standard deviation. We included the $d_1=5$$CX$ case although it has large error bars, giving only an approximate bound on its error rate.
• Figure 3: Logical error rate of cultivation with expansion - Atom noise model -- Logical error rate and acceptance rate of surface code $CX, H_{XY}$ and $H$ cultivation with Uniform and Atom noise model. In the Atom noise model, idling doesn't introduce any additional error, which approximate the effect of dynamical decoupling or shelving in clock-states. The physical error rate is $p=10^{-3}$, and the full noise model is defined in S.I. For practical application that utilize synthillation, achieving $10^{-7}$ infidelity is possible with $d_1=3$ and approximately $75\%$ rate.
• Figure 4: Logical error rate and Acceptance rate of erasure Qubits -- Logical error rate and acceptance rate of erasure qubits used for the initialization step of cultivation. The points are labeled by the residual Pauli error rate $p$. As mentioned in the main text, the logical error rate does not depend on the erasure rate but only on the residual Pauli error rate. Importantly, the color-code expansion step is considerably more expensive than for the surface code, but isn't shown in this plot (see Methods). Still, from this graph we can extract the tradeoff when converting between different erasure and Pauli rates. For example, if $p=10^{-3}$ can be converted to $e=2 \times 10^{-3}$ and $p=10^{-4}$, it would reduce the acceptance rate of $d_1=3$ cultivation by roughly a factor of $5$, but improve the fidelity by over $10^3$.
• Figure 5: Schematic description of phase-kickback and double-checking using a GHZ ancilla