# Hierarchical code[1]

## Description

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 efficiently on a 2D geometry at the expense of a logarithmically vanishing rate.

## Rate

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

## Decoding

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 are possible because the code parameters are such that previous bounds no longer apply [2].

## Threshold

Threshold exists for the locally decaying error model; see [1; Thm. 1.3]. However, logical error rate below threshold falls super-polynomially (as opposed to exponentially) with the code distamce.

## Parents

- Qubit stabilizer code
- Quantum low-density parity-check (QLDPC) code
- Concatenated quantum code — Hierarchical code is a concatenation of a constant-rate QLDPC code with a rotated surface code.

## Child

- Rotated surface code — Hierarchical code is a concatenation of a constant-rate QLDPC code with a rotated surface code.

## References

- [1]
- C. A. Pattison, A. Krishna, and J. Preskill, “Hierarchical memories: Simulating quantum LDPC codes with local gates”, (2023) arXiv:2303.04798
- [2]
- 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

## Page edit log

- Victor V. Albert (2023-03-12) — most recent

## Cite as:

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