# Local Haar-random circuit 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.

## Protection

In a 1D geometry, the code approximately detects any error with support on a segment of length \(\leq n/4\), with deviations exponentially suppressed in \(n\).

## Encoding

Random local circuit of depth proportional to \(n^{\alpha}\), with \(\alpha\) depending on system geometry.

## Parent

## Cousins

- Topological code — Local Haar-random codewords, like topological codewords, are locally indistinguishable [1].
- Approximate quantum error-correcting code (AQECC)
- Haar-random 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.

## Zoo code information

## 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). DOI; 2010.09775

## Cite as:

“Local Haar-random circuit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/t-designs

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/qubits/t-designs.yml.