Hierarchical code[1] 


Member of a family of \([[n,k,d]]\) qubit stabilizer codes resulting from a concatenation of a constant-rate QLDPC code with a rotated surface code. Concatenation allows for syndrome extraction to be performed on a 2D geometry while maintining a threshold at the expense of a logarithmically vanishing rate. The growing syndrome extraction circuit depth allows known bounds in the literature to be weakened [2,3].


Rate vanishes as \(\Omega(1/\log(n)^2)\).


Decoding is performed as in a standard concatenated code using a decoder for the inner code and outer code. The syndrome extraction circuit depth for the outer code is optimized using a permutation routing algorithm [4]. The bilayer architecture allows for logical entangling gates between logical surface-code patches.Soft output decoding [5].

Fault Tolerance

2D geometrically local syndrome extraction circuits of depth \(O(\sqrt{n}/R)\) that utilize Clifford and SWAP gates of range \(R\) and that require order \(O(n)\) data and ancilla qubits. Such parameters (including a range of one) are possible while maintaining a threshold because of the concatenation step. This reduces the noise that would otherwise accumulate within a growing-depth syndrome extraction circuit. A key idea is that constant-depth syndrome extraction is not a necessary condition for fault-tolerance.


Threshold exists for the locally decaying error model; see [1; Thm. 1.3]. However, the logical error rate below threshold falls super-polynomially (as opposed to exponentially) with the code distance. The code family possesses a threshold equal to that of surface codes given by tuning the inner code size for any fixed physical error rate.



  • Rotated surface code — Hierarchical codes are concatenations of constant-rate QLDPC codes with rotated surface codes.


C. A. Pattison, A. Krishna, and J. Preskill, “Hierarchical memories: Simulating quantum LDPC codes with local gates”, (2023) arXiv:2303.04798
N. Delfosse, M. E. Beverland, and M. A. Tremblay, “Bounds on stabilizer measurement circuits and obstructions to local implementations of quantum LDPC codes”, (2021) arXiv:2109.14599
N. Baspin, O. Fawzi, and A. Shayeghi, “A lower bound on the overhead of quantum error correction in low dimensions”, (2023) arXiv:2302.04317
F. Annexstein and M. Baumslag, “A unified approach to off-line permutation routing on parallel networks”, Proceedings of the second annual ACM symposium on Parallel algorithms and architectures - SPAA ’90 (1990) DOI
N. Meister, C. A. Pattison, and J. Preskill, “Efficient soft-output decoders for the surface code”, (2024) arXiv:2405.07433
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Zoo Code ID: hierarchical

Cite as:
“Hierarchical code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/hierarchical
@incollection{eczoo_hierarchical, title={Hierarchical code}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/hierarchical} }
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Cite as:

“Hierarchical code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/hierarchical

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/qldpc/concatenated/hierarchical.yml.