Fast magic state preparation by gauging higher-form transversal gates in parallel

TL;DR

This work introduces a fast, fault-tolerant scheme for parallel magic-state preparation using higher-form transversal gates on qLDPC codes. By defining higher-form transversal gates and a higher-form gauging measurement, it achieves constant-time overhead with linear qubit overhead, while preserving code-space fault tolerance and enabling simultaneous measurement of many logical Clifford operators. The approach avoids reliance on transversal non-Clifford gates or distillation, instead leveraging cohomological structure to read out magic states in parallel; it includes concrete examples based on the 3D Color Code and twisted HGGT, illustrating the potential for finite-rate magic-state generation on suitable topologies. The results motivate new quantum-code designs that support higher-form Clifford gates and invite further study of decoding and practical deployment in fault-tolerant quantum architectures.

Abstract

Magic states are a foundational resource for universal quantum computation. To survive in a realistic noisy environment, magic states must be prepared fault-tolerantly and protected by a quantum error-correcting code. The recent discovery of highly efficient quantum low-density parity-check codes, together with efficient logic gates, lays the groundwork for low-overhead fault-tolerant quantum computation. This motivates the search for fast and parallel protocols for logical magic state preparation to enable universal quantum computation. Here, we introduce a fast code surgery procedure that performs a fault-tolerant measurement of many transversal logic gates in parallel. This is achieved by performing a generalized gauging measurement on a quantum code that supports a higher-form transversal gate. The time overhead of our procedure is constant, and the qubit overhead is linear. The procedure inherits fault-tolerance properties from the base code and the structure of the higher-form transversal gate. When applied to codes that support higher-form Clifford gates our procedure achieves fast and fault-tolerant preparation of many magic states in parallel. This motivates the search for good quantum low-density parity-check codes that support higher-form Clifford gates.

Figures (6)

• Figure 1: A 1-form transversal gate with chain complex given by the cellulation of a torus shown above. Cocycles in $\ker \delta_2$ (depicted as blue dashed lines) determine the support of elements in the 1-form transversal gate. Vertices (shaded green) are generators of the logically trivial subgroup of the 1-form transversal gate $\mathop{\text{Im}} \delta_1$. Plaquettes (shaded red) generate the parity constraints on edges (red dashed lines), corresponding to $\partial_2$, that specify $\ker \delta_2$.
• Figure 2: Schematic depiction of the higher-form gauging measurement. Time runs up in this circuit diagram. The box labelled $\Pi_0$ denotes an adaptive operation that prepares the code space in constant time. The next time step shows $\ket{0}$ states being initialized on the hyperedge ancilla qubits. Next, the boxes labelled $A_{v_i}$ denote the measurement of the Gauss's law terms that deform the code. Fault-tolerance is ensured by the fact that products of $A_{v_i}$ terms in $\mathop{\text{Im}}\delta_{h}$ have eigenvalue $+1$ on the codespace. This results in detectors (inside the green shaded region) that can be used to correct measurement errors. Next, the hyperedge qubits are measured out in the $Z$ basis and a byproduct operator is applied. Finally, an adaptive operation is applied that measures the checks of the original code and applies a correction operator to restore the codespace. For simplicity of presentation, the classical measurement results and feedforward operations are not explicitly depicted in this diagram.
• Figure 3: Schematic depiction of the standard gauging measurement williamson2024low. The procedure and notation is similar to Fig. \ref{['fig:A']}. In contrast to the higher-form gauging measurement, no terms in $\mathop{\text{Im}}\delta_{h}$ are measured. This implies that $d$ rounds of syndrome extraction and error correction must be performed in the deformed code to ensure a distance against measurement errors (indicated by boxes labelled QEC) Cowtan2025Fast. The time savings of the higher-form gauging measurement are indicated by the red shaded region.
• Figure 4: A tetrahedron with 4-colored vertices $\{r,g,b,y\}$. The edges inherit a 2-coloring from the adjacent edges. The faces inherit a color given by the vertex not included in the face.
• Figure 5: A schematic depiction of the logical action of the $\hat{x}$-membrane element of the 1-form transversal Clifford gate in example \ref{['sec:EGB']} on the 3D torus. The nontrivial element of a 1-form $CZ$ gate on an $\hat{x}$ membrane (blue shaded) conjugates a logical $X$ operator on one copy of surface code (red shaded membrane) to produce a logical $Z$ operator on the other copy (green line), and vice versa.

Theorems & Definitions (12)

• Definition 1: $h$-form transversal gate
• Theorem 1: Higher-form gauging measurement
• proof
• Definition 2: Higher Cheeger constant
• Definition 3: Gauged operator
• Definition 4: Gauged code
• Theorem 2: Fault-tolerance
• proof
• Lemma 1
• proof