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arXiv:2610.30087

Injection is all you need: parallel universal logic in high-rate concatenated self-dual codes

Joseph Sullivan, David Amaro, Michael A. Perlin

quant-phSubmitted Oct 21, 2026• Announced Oct 22, 2026• 0 citations
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Abstract

We propose a quantum architecture that achieves addressable and parallel universal fault-tolerant logical in high-rate codes. The codes are obtained from the flexible concatenation of small-distance high-rate self-dual CSS codes. Addressable and parallel universal logic on the resulting $K$ logical qubits is injected by fault-tolerantly preparing and then consuming diverse logical $K$-qubit CSS stabilizer states and logical $\lvert H \rangle^{\otimes K}$ global magic states. Flag-at-origin is used to prepare the stabilizer resource states while a concatenated cultivation protocol prepares the global magic states. These resource states are consumed by Steane-QEC or Knill-QEC teleportation gadgets, which simultaneously perform quantum error correction. Any quantum algorithm with $T$-depth $D_T$ can be executed by consuming $12D_T$ or fewer resource states. We benchmark the fault-tolerant preparation of the resource states for several two-level concatenations of the $[[k+4,k,2]]$ H codes and the Hamming $[[15,7,3]]$ code. In particular, we find that the concatenated cultivation offers at least one order of magnitude spacetime reduction compared to cultivation in the surface code.

Comments
21 pages, 13 figues

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We propose a quantum architecture that achieves addressable and parallel universal fault-tolerant logical in high-rate codes. The codes are obtained from the flexible concatenation of small-distance high-rate self-dual CSS codes. Addressable and parallel universal logic on the resulting KKK logical qubits is injected by fault-tolerantly preparing and then consuming diverse logical KKK-qubit CSS stabilizer states and logical ∣H⟩⊗K\lvert H \rangle^{\otimes K}∣H⟩⊗K global magic states. Flag-at-origin is used to prepare the stabilizer resource states while a concatenated cultivation protocol prepares the global magic states. These resource states are consumed by Steane-QEC or Knill-QEC teleportation gadgets, which simultaneously perform quantum error correction. Any quantum algorithm with TTT-depth DTD_TDT​ can be executed by consuming 12DT12D_T12DT​ or fewer resource states. We benchmark the fault-tolerant preparation of the resource states for several two-level concatenations of the [[k+4,k,2]][[k+4,k,2]][[k+4,k,2]] H codes and the Hamming [[15,7,3]][[15,7,3]][[15,7,3]] code. In particular, we find that the concatenated cultivation offers at least one order of magnitude spacetime reduction compared to cultivation in the surface code.
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