Random stabilizer code

Random stabilizer code [1–3]

Alternative Names: 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 asymptotically saturate the non-degenerate quantum Hamming bound, and hence achieve the quantum GV bound, because a typical random stabilizer has negligible degeneracy [3; Sec. 7.6]. See notes [4] and the original random-coding argument [1]. CSS codes from random weakly self-dual codes attain the lower rate \(R\geq 1-2h(d/n)\), where \(h\) is the binary entropy function [2; Sec. 5].

Cousins

Random-circuit code— Random stabilizer codes can be constructed by sampling random Clifford circuits.
Monitored random-circuit code— An important sub-family of monitored random-circuit codes is the Clifford monitored random-circuit family, 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].
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 [9; 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 of order \(\Omega( n^{3/5}polylog(n) )\) and rate of order \(\Omega( n^{-2/5}polylog(n) )\).
Homological product code— Random homological codes are asymptotically good with high probability [10; Thm. 3].
Qubit QLDPC code— Random qubit QLDPC codes found by solving certain constraint satisfaction problems (CSPs) practically achieve the capacity of the erasure channel [11].
Qubit CSS code— CSS codes from random weakly self-dual codes \(C\subseteq C^\perp\) attain linear distance at rate \(R\geq 1-2h(d/n)\), where \(h\) is the binary entropy function [2; Sec. 5].
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.
Modular-qudit dynamical code— Dynamical codes admit instantaneous stabilizer groups, and dynamical code state initialization, logical gates, and error correction are done by a sequence of different (usually weight-two) stabilizer measurements.
Galois-qudit GRS code— Concatenations of Galois-qudit GRS codes and random stabilizer codes can achieve the quantum GV bound [12].

Member of code lists

Primary Hierarchy

Quantum Domain

Group quantum Kingdom

Group-based quantum codeQECC Quantum

Stabilizer codeHamiltonian-based QECC Quantum

Parents

Stabilizer code

Random stabilizer code

Children

Log-depth geometrically local Clifford-circuit code

Log-depth Clifford circuits on a 1D geometry yield approximate codes whose encoding rate achieves the hashing bound for Pauli noise and the channel capacity for erasure errors [13,14].

Crystalline-circuit qubit code

Random polylog-depth Clifford-circuit code

Nested HA-MN CSS code

Nested HA-MN CSS codes are sampled from the configuration-model ensemble of regular sparse binary matrices [15; Def. 2.5].

References

[1]
A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, “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, S. Nezami, X.-L. Qi, N. Thomas, M. Walter, and Z. Yang, “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]
J. Haah, “Towers of generalized divisible quantum codes”, Physical Review A 97, (2018) arXiv:1709.08658 DOI
[10]
S. Bravyi and M. B. Hastings, “Homological Product Codes”, (2013) arXiv:1311.0885
[11]
M. Tremblay, G. Duclos-Cianci, and S. Kourtis, “Finite-rate sparse quantum codes aplenty”, Quantum 7, 985 (2023) arXiv:2207.03562 DOI
[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
[13]
G. Liu, Z. Du, Z.-W. Liu, and X. Ma, “Approximate Quantum Error Correction with 1D Log-Depth Circuits”, PRX Quantum 7, (2026) arXiv:2503.17759 DOI
[14]
T. Kroll and J. Helsen, “Error correction with brickwork Clifford circuits”, (2026) arXiv:2602.20900
[15]
K. Kasai, “Finite-Degree Quantum LDPC Codes Reaching the Gilbert-Varshamov Bound”, (2026) arXiv:2603.24588

Page edit log

Victor V. Albert (2026-11-03) — most recent
Victor V. Albert (2026-06-08)
Victor V. Albert (2024-06-30)

Cite as:

“Random stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/random_stabilizer, arXiv:2606.11484

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