Brown-Fawzi random Clifford-circuit code[1]
Description
An \([[n,k]]\) stabilizer code whose encoder is a random Clifford circuit of depth \(O(\log^3 n)\).
An \(n\)-qubit quantum encoding circuit with \(O(n^2 \log n)\)) random two-qubit Clifford gates yields a code with distance \(d\) with probability \(1 - \Omega(1/n^8)\), granted that \begin{equation} \frac{k}{n} < 1 - \frac{d}{n}\log_2 3 - h\left(\frac{d}{n}\right)~, \tag*{(1)}\end{equation} where \(h\) is the entropy function. Noting that two gates acting on disjoint qubits could be executed simultaneously, the depth of a circuit with such a size is typically of order \(O(\log^3 n)\).'
Rate
The achievable distance of these codes is asymptotically the same as a code whose encoder is a random (not necessarily log-depth) general Clifford unitary [1].
Encoding
Random \(\log^3\)-depth Clifford circuit.
Decoding
Minimum-weight decoding via using tropical tensor networks [2].
Fault Tolerance
Fault-tolerant state preparation [2].
Parents
Cousin
- Circuit-to-Hamiltonian approximate code — Circuit-to-Hamiltonian approximate codes are constructed by converting the encoding circuit of a Brown-Fawzi random Clifford-circuit code into a Hamiltonian using the spacetime circuit-to-Hamiltonian construction [3,4].
References
- [1]
- W. Brown and O. Fawzi, “Short random circuits define good quantum error correcting codes”, 2013 IEEE International Symposium on Information Theory (2013) arXiv:1312.7646 DOI
- [2]
- J. Nelson, G. Bentsen, S. T. Flammia, and M. J. Gullans, “Fault-Tolerant Quantum Memory using Low-Depth Random Circuit Codes”, (2023) arXiv:2311.17985
- [3]
- A. Mizel, D. A. Lidar, and M. Mitchell, “Simple Proof of Equivalence between Adiabatic Quantum Computation and the Circuit Model”, Physical Review Letters 99, (2007) arXiv:quant-ph/0609067 DOI
- [4]
- N. P. Breuckmann and B. M. Terhal, “Space-time circuit-to-Hamiltonian construction and its applications”, Journal of Physics A: Mathematical and Theoretical 47, 195304 (2014) arXiv:1311.6101 DOI
Page edit log
- Victor V. Albert (2022-01-20) — most recent
- Srilekha Gandhari (2021-12-14)
Cite as:
“Brown-Fawzi random Clifford-circuit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/nonlocal_lowdepth