Also known as Random Clifford-circuit code.

## Description

An \(n\)-qubit, modular-qudit, or Galois-qudit stabilizer code whose construction is non-deterministic. Since stabilizer encoders are Clifford circuits, such codes can be thought of as arising from random Clifford circuits.

## Rate

Random qubit stabilizer codes achieve the quantum GV bound [1,3]; see notes [4]. In fact, sampling random CSS codes is sufficient [2].

## Parents

## Children

## Cousins

- Monitored random-circuit code — An important sub-family of monitored random-circuit codes are the Clifford monitored random-circuit codes, where unitaries are sampled from the Clifford group [5].
- Holographic tensor-network code — Random holographic tensor-network codes reproduce many aspects of holography [6–8].
- Quantum low-density parity-check (QLDPC) code — Random QLDPC codes found by solving certain constraint satisfaction problems (CSPs) practically achieve the capacity of the erasure channel [9].
- Dynamical automorphism (DA) code — DA codes admit instantaneous stabilizer groups, and DA code state initialization, logical gates, and error correction are done by a sequence of different (usually weight-two) stabilizer measurements.
- Generalized quantum divisible code — Random CSS codes [2] can be used to construct families of \([[O(d^{\nu−1}), \Omega(d), d]]\) level-\(\nu\) generalized quantum divisible codes [10; Sec. VI.A].
- Fiber-bundle code — Taking a random LDPC code as the base and a cyclic repetition code as the fiber yields, after distance balancing, a QLDPC code with distance \(\Omega(n^{3/5}\text{polylog}(n))\) and rate \(\Omega(n^{-2/5}\text{polylog}(n))\) is obtained.
- Homological product code — Random homological codes are asymptotically good with high probability [11; Thm. 1].
- Qubit CSS code — Random CSS codes asymptotically achieve linear distance with high probability, achieving the quantum Gilbert-Varshamov bound [2].
- Clifford-deformed surface code (CDSC) — Many useful CDSCs are constructed using random Clifford circuits.
- Compass code — Compass code families are constructed by randomly assigning stabilizers to plaquettes of a square lattice.
- Galois-qudit GRS code — Concatenations of quantum GRS codes and random stabilizer codes can achieve the quantum Gilbert-Varshamov bound [12].

## References

- [1]
- A. R. Calderbank et al., “Quantum Error Correction and Orthogonal Geometry”, Physical Review Letters 78, 405 (1997) arXiv:quant-ph/9605005 DOI
- [2]
- A. R. Calderbank and P. W. Shor, “Good quantum error-correcting codes exist”, Physical Review A 54, 1098 (1996) arXiv:quant-ph/9512032 DOI
- [3]
- D. Gottesman, “Stabilizer Codes and Quantum Error Correction”, (1997) arXiv:quant-ph/9705052
- [4]
- J. Preskill. Lecture notes on Quantum Computation. (1997–2020) URL
- [5]
- Y. Li, X. Chen, and M. P. A. Fisher, “Measurement-driven entanglement transition in hybrid quantum circuits”, Physical Review B 100, (2019) arXiv:1901.08092 DOI
- [6]
- P. Hayden et al., “Holographic duality from random tensor networks”, Journal of High Energy Physics 2016, (2016) arXiv:1601.01694 DOI
- [7]
- X.-L. Qi and Z. Yang, “Space-time random tensor networks and holographic duality”, (2018) arXiv:1801.05289
- [8]
- H. Apel, T. Kohler, and T. Cubitt, “Holographic duality between local Hamiltonians from random tensor networks”, Journal of High Energy Physics 2022, (2022) arXiv:2105.12067 DOI
- [9]
- M. Tremblay, G. Duclos-Cianci, and S. Kourtis, “Finite-rate sparse quantum codes aplenty”, Quantum 7, 985 (2023) arXiv:2207.03562 DOI
- [10]
- J. Haah, “Towers of generalized divisible quantum codes”, Physical Review A 97, (2018) arXiv:1709.08658 DOI
- [11]
- M. H. Freedman and M. B. Hastings, “Quantum Systems on Non-\(k\)-Hyperfinite Complexes: A Generalization of Classical Statistical Mechanics on Expander Graphs”, (2013) arXiv:1301.1363
- [12]
- Y. Ouyang, “Concatenated Quantum Codes Can Attain the Quantum Gilbert–Varshamov Bound”, IEEE Transactions on Information Theory 60, 3117 (2014) arXiv:1004.1127 DOI

## Page edit log

- Victor V. Albert (2024-06-30) — most recent

## Cite as:

“Random stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/random_stabilizer