Fold-transversal surface code cultivation

Abstract

Magic state cultivation is a state-of-the-art protocol to prepare ultra-high fidelity non-Clifford resource states for universal quantum computation. It offers a significant reduction in spacetime overhead compared to traditional magic state distillation techniques. Cultivation protocols involve measuring a transversal logical Clifford operator on an initial small-distance code and then rapidly growing to a larger-distance code. In this work, we present a new cultivation scheme in which we measure the fold-transversal Hadamard of the unrotated surface code, and leverage unitary techniques to grow within the surface code family. Using both stabilizer and state vector simulations we find that this approach achieves the lowest known spacetime overhead for magic state cultivation. Practical implementation of our protocol is best suited to architectures with non-local connectivity, showing the strength of architectures where such connectivity is readily available.

Figures (17)

• Figure 1: Magic state cultivation with a fault distance $\mathsf{f}=3$. The procedure interleaves unitary steps (light blue labels) and measurements on which postselection occurs (maroon labels). 1. Injection: prepare an eigenstate of $H_\mathrm{XY} = (X+Y)/\sqrt{2}$ on the distance-three rotated surface code $\mathsf{Rot}(3)$, and measure its stabilizers. 2. Cultivation: transform to the distance-three regular surface code $\mathsf{Reg}(3)$, and measure the fold-transversal $H_\mathrm{XY}$ operator twice via a GHZ ancilla. Then, measure the stabilizers of $\mathsf{Reg}(3)$. 3. Escape: transform to $\mathsf{Rot}(5)$ via unitary growth, then use stabilizer measurements to grow to $\mathsf{Rot}(7)$. Decode, and post-select on the associated complementary gap. If desired, unitarily grow to a larger final code.
• Figure 2: Cultivation under uniform depolarizing noise.(a) Curves show the expected number of cultivation attempts of fault distances $\mathsf{f}=3$ (triangles) and $\mathsf{f}=5$ (pentagons) to prepare a state with a particular logical error rate. The $\mathsf{f}=3$ protocol is described in \ref{['sec:protocol']}, and its extension to $\mathsf{f}=5$ in \ref{['app:fd5']}. Our scheme for $\ket{Y}$ state cultivation (dark pink) requires a lower number of expected attempts for any target LER compared to Refs. gidney2024magicchen2025efficient (blue and purple, respectively). The exact simulation of $\ket{H_\mathrm{XY}}$ magic state cultivation for $\mathsf{f}=3$ (green) is in good agreement with $\ket{Y}$ state results. (b) Expected spacetime volume (qubits $\times$ gate count $\times$ expected attempts) of $\ket{Y}$ state cultivation for fault distances $\mathsf{f}=3$ and $\mathsf{f}=5$, compared to Refs. gidney2024magicchen2025efficientclaes2025cultivatingtstatessurfacevaknin2025magic. For a fair comparison, the $\mathsf{f}=3$ protocols are normalized to the same final code distance $d_\mathrm{fin}=13$, and the $\mathsf{f}=5$ protocols to $d_\mathrm{fin}=15$. The expected volume in the absence of gap-based postselection (light pink) is shown, along with the component expected volumes for injection (dark pink), cultivation (purple) and escape (blue). Additional gap-based postselection is required to cultivate states with logical error rates of $2\times 10^{-6}$ for $\mathsf{f}=3$ (silver) and $10^{-9}$ for $\mathsf{f}=5$ (gold). Our scheme has a significantly lower overhead, owing to the efficient cultivation and escape steps that use nonlocal operations.
• Figure 3: Physically-motivated noise. Logical error rate of $\ket{Y}$ state cultivation under three different noise models. A standard depolarizing noise model (SD6) is compared to two physically-motivated noise models (PM1 and PM5). In PM1 the noise strength of single-qubit gates is ten times lower than that of multi-qubit gates. The same is true for PM5, but the noise strength of non-local multi-qubit gates is five times that of local multi-qubit gates. PM1 and PM5 have no idle errors.
• Figure A1: Unitary encoder for $\mathsf{Rot}(3)$. The rotation on the central qubit in the second step produces a physical $\ket{Y}$ state. In subsequent steps this is used to unitarily prepare the logical $\ket{Y}$ state. A subsequent round of stabilizer measurements (not pictured) is used to detect weight-1 and 2 errors.
• Figure A2: Growth from $\mathsf{Rot}(3)$ to $\mathsf{Reg}(3)$. This is simply the first two steps of the stabilizer measurement circuit for $\mathsf{Rot}(3)$. The ancilla qubits of $\mathsf{Rot}(3)$ become data qubits of $\mathsf{Reg}(3)$. Additional qubits are required for the $\mathsf{Reg}(3)$ code for stabilizer measurements. Depending on the hardware, these ancilla may be present beforehand (static architectures with non-local connectivity), or freshly initialized qubits may be introduced to the lattice (reconfigurable architectures).