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

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.

Parents

Cousins

  • Topological code — Local Haar-random codewords, like topological codewords, are locally indistinguishable [1].
  • Approximate quantum error-correcting code (AQECC)
  • 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”, (2022) arXiv:2212.05071
[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
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Zoo Code ID: local_haar_random

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
BibTeX:
@incollection{eczoo_local_haar_random, title={Local Haar-random circuit qubit code}, booktitle={The Error Correction Zoo}, year={2021}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/local_haar_random} }
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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

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/dynamic/random/local_haar_random.yml.