Random stabilizer code[13] 

Also known as Random Clifford-circuit code.


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.


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




  • 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 [68].
  • 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].


A. R. Calderbank et al., “Quantum Error Correction and Orthogonal Geometry”, Physical Review Letters 78, 405 (1997) arXiv:quant-ph/9605005 DOI
A. R. Calderbank and P. W. Shor, “Good quantum error-correcting codes exist”, Physical Review A 54, 1098 (1996) arXiv:quant-ph/9512032 DOI
D. Gottesman, “Stabilizer Codes and Quantum Error Correction”, (1997) arXiv:quant-ph/9705052
J. Preskill. Lecture notes on Quantum Computation. (1997–2020) URL
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
P. Hayden et al., “Holographic duality from random tensor networks”, Journal of High Energy Physics 2016, (2016) arXiv:1601.01694 DOI
X.-L. Qi and Z. Yang, “Space-time random tensor networks and holographic duality”, (2018) arXiv:1801.05289
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
M. Tremblay, G. Duclos-Cianci, and S. Kourtis, “Finite-rate sparse quantum codes aplenty”, Quantum 7, 985 (2023) arXiv:2207.03562 DOI
J. Haah, “Towers of generalized divisible quantum codes”, Physical Review A 97, (2018) arXiv:1709.08658 DOI
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
Y. Ouyang, “Concatenated Quantum Codes Can Attain the Quantum Gilbert–Varshamov Bound”, IEEE Transactions on Information Theory 60, 3117 (2014) arXiv:1004.1127 DOI
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Zoo Code ID: random_stabilizer

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“Random stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/random_stabilizer
@incollection{eczoo_random_stabilizer, title={Random stabilizer code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/random_stabilizer} }
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“Random stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/random_stabilizer

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/properties/stabilizer/random_stabilizer.yml.