# Local Haar-random circuit qubit code[1]

## Description

An \(n\)-qubit code whose codewords are a pair of approximately locally indistinguishable states produced by starting with any two orthogonal \(n\)-qubit states and acting with a random unitary circuit of depth polynomial in \(n\). Two states are locally indistinguishable if they cannot be distinguished by local measurements. A single layer of the encoding circuit is composed of about \(n/2\) two-qubit nearest-neighbor gates run in parallel, with each gate drawn randomly from the Haar distribution on two-qubit unitaries.

The above circuit elements act on nearest-neighbor qubits arranged in a line, i.e., a one-dimensional geometry (\(D=1\), while codes for higher-dimensional geometries require \(O(n^{1/D})\)-depth circuits [1]. Follow-up work [2] revealed that optimal code properties require only \(O(\sqrt{n})\)-depth circuits for that case, and \(O(\sqrt{n})\)-depth circuits for a two-dimensional square-lattice geometry. This result has in turn to other types of Pauli noise [3], with the previous result holding for erasure noise only.

## Protection

## Encoding

## Parents

## Cousins

- Topological code — Local Haar-random codewords, like topological codewords, are locally indistinguishable [1].
- Approximate quantum error-correcting code (AQECC)
- \(t\)-design — Local Haar-random circuits of polynomial depth form approximate unitary designs [4].
- Haar-random qubit code — Approximating the random projections through \(t\)-designs is necessary in order to make the protocol practical. Replacing with random Clifford gates is especially convenient since the Clifford group forms a unitary 2-design and produces stabilizer codes.

## References

- [1]
- F. G. S. L. Brandão, A. W. Harrow, and M. Horodecki, “Local Random Quantum Circuits are Approximate Polynomial-Designs”, Communications in Mathematical Physics 346, 397 (2016) DOI
- [2]
- M. J. Gullans et al., “Quantum Coding with Low-Depth Random Circuits”, Physical Review X 11, (2021) arXiv:2010.09775 DOI
- [3]
- A. S. Darmawan et al., “Low-depth random Clifford circuits for quantum coding against Pauli noise using a tensor-network decoder”, Physical Review Research 6, (2024) arXiv:2212.05071 DOI
- [4]
- F. G. S. L. Brandão, A. W. Harrow, and M. Horodecki, “Local Random Quantum Circuits are Approximate Polynomial-Designs”, Communications in Mathematical Physics 346, 397 (2016) arXiv:1208.0692 DOI

## Page edit log

- Victor V. Albert (2021-12-14) — most recent
- Jonathan Kunjummen (2021-12-07)

## Cite as:

“Local Haar-random circuit qubit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021. https://errorcorrectionzoo.org/c/local_haar_random