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Concatenated qubit code

Description

A concatenated code whose outer code is a qubit code. In other words, a qubit code that can be thought of as a concatenation of an inner qubit code and an outer qubit code. An inner \(C_{\text{in}} = ((n_1,K,d_1))\) and outer \(C_{\text{out}} = ((n_2,2,d_2))\) qubit code yield an \(((n_1 n_2, K, d \geq d_1d_2))\) concatenated qubit code.

Concatenating an \(((n,2,d))\) qubit code can be done recursively, with the \(r\)th level of concatenation yielding an \(((n^r,2,d^r))\) code.

For qubit CSS codes, concatenation is generalized by the mapping cone framework of quantum code embedding [1].

Protection

Any distance-three recursively concatenated code protects against an open set of errors [2]. Concatenating stabilizer codes can help protect against catastrophic errors such as cosmic rays [3].

Decoding

Adaptive syndrome extraction for a concatenation of a small error-detecting code and a high-rate, high-distance QLDPC code [4].The effective channel for a concatenation of codes is the composition of the codes’ effective channels [5].Message passing algorithm for concatenated codes can be equivalent to ML decoding [6].In the statistical mechanical mapping of maximum-likelihood decoding, recursively concatenated codes yield classical models on hierarchical lattices, and optimal decoding admits a hierarchical message-passing formulation that, in the fully postselected limit, reduces to an exact real-space renormalization group flow [7].

Fault Tolerance

Fault-tolerant message passing between devices [8].Blocklet concatenation uses concatenation and transversal gates in a way that is tailored to FBQC platforms [9].

Threshold

The first methods to achieve a concatenated threshold against local stochastic noise use concatenated qubit stabilizer codes [10–17]; see the book [18]. Generalized Kramers-Wannier self-duality of the statistical mechanical model of a \([[n,1,d]]\) CSS code whose \(X\)- and \(Z\)-type parity-check matrices are equivalent up to row and column permutations is preserved under recursive concatenation, constraining the optimal code capacity threshold of the concatenated family. Below threshold, topological and concatenated code families are instead distinguished by a tradeoff between distance scaling and the number of minimum-weight logical operators, which produces a finite-size crossover in physical overhead [7].

Cousins

  • Hamiltonian-based code— Concatenated stabilizer code Hamiltonians have been investigated [19].
  • Fusion-based quantum computing (FBQC) code— Blocklet concatenation uses concatenation and transversal gates in a way that is tailored to FBQC platforms [9].
  • Gauss’ law code— The Gauss’ law code can be concatenated to form a stabilizer code for fault-tolerant quantum simulation of the underlying gauge theory [20,21].
  • Amplitude-damping (AD) code— Using the dual-rail code as an outer code with an inner \([[n,k,d]]\) qubit code yields an \([[2n,k]]\) code correcting \(d-1\) qubit AD errors [22].
  • EA qubit stabilizer code— There exist concatenated EA qubit stabilizer codes that saturate the EA quantum Griesmer and Plotkin bounds [23].
  • \(((n,2,2))\) Bravyi-Lee-Li-Yoshida PI code— The Bravyi-Lee-Li-Yoshida PI code can be concatenated to yield codes that have higher distance and that admit codewords with vanishing entanglement [24; Appx. D] (cf. [25]).
  • \([[2^D,D,2]]\) hypercube quantum code— The hypercube quantum code can be concatenated with a two-qubit quantum repetition code to yield a \([[2^{D+1},D,4]]\) error-detecting code family [26]. It can also be concatenated with \(D\) distance-two \(D\)-dimensional toric/surface-code blocks to yield a \([[2^D(2^D+1),D,4]]\) error-correcting code family that admits a transversal implementation of the logical \(C^{D-1}Z\) gate [26].
  • \([[15,1,3]]\) quantum RM code— The concatenated \([[15,1,3]]\) code has a measurement threshold less than one [27].
  • \([[16,4,4]]\) twisted color code— The \([[16,4,4]]\) code is obtained by concatenating the \([[4,2,2]]\) code with the symplectic double of the \([[4,2,2]]\) code along a \(ZX\)-duality [28].
  • \([[20,2,6]]\) B&C phantom code— The \([[20,2,6]]\) code is obtained by concatenating each qubit pair of the \([[10,2,3]]\) binarized Galois-qudit code with the \([[4,2,2]]\) code [29].
  • \([[4,1,2]]\) Leung-Nielsen-Chuang-Yamamoto (LNCY) code— The \([[4,1,2]]\) LNCY code is the smallest QPC, i.e., a concatenation of a two-qubit bit-flip with a two-qubit phase-flip repetition code. An \([[8,1,2]]\) QPC correcting a single AD error is equivalent to a concatenation of its constant-excitation version with the dual-rail code [30–32]. More generally, an \([[m^2,1,m]]\) QPC corrects \(m-1\) AD errors [22]. Recursively concatenating a \([[4,1,2]]\) LNCY subcode attains a threshold [33,34].
  • \([[4,2,2]]\) Four-qubit code— Concatenations of \([[4,2,2]]\) and \(C_6\) codes yield fault-tolerant quantum computation schemes [35] admitting a post-selected threshold [36,37] (see also Ref. [38]). Concatenating quantum Hamming codes on top of the \([[4,2,2]]\) and \(C_6\) codes yields fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads [39]. In the optimized protocol of Ref. [39], a level-five \(C_4/C_6\) code underlies concatenated quantum Hamming codes \(\mathcal{Q}_5,\mathcal{Q}_6,\mathcal{Q}_7,\mathcal{Q}_7\), yielding a \(2.5\%\) threshold and space overheads \(162\) and \(373\) physical qubits per logical qubit at physical error rate \(0.1\%\) for logical CNOT error rates \(10^{-10}\) and \(10^{-24}\), respectively. Concatenating the \([[4,2,2]]\) code with the surface code is equivalent to removing stabilizer generators from the 4.8.8 color code [40]. The \([[4,2,2]]\) code can be concatenated with two copies of the surface code to yield the 4.6.12 color code [40].
  • \([[5,1,3]]\) Five-qubit perfect code— The recursively concatenated five-qubit code has a measurement threshold of one [27]. Code performance against general Pauli channels has been worked out [5,41].
  • \([[6,2,2]]\) \(C_6\) code— Concatenations of \([[4,2,2]]\) and \(C_6\) codes yield fault-tolerant quantum computation schemes [35] admitting a post-selected threshold [36,37] (see also Ref. [38]) and the Meier-Eastin-Knill (MEK) magic-state distillation protocols [42]. Concatenating quantum Hamming codes on top of the \([[4,2,2]]\) and \(C_6\) codes yields fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads [39]. In the optimized protocol of Ref. [39], a level-five \(C_4/C_6\) code underlies concatenated quantum Hamming codes \(\mathcal{Q}_5,\mathcal{Q}_6,\mathcal{Q}_7,\mathcal{Q}_7\), yielding a \(2.5\%\) threshold and space overheads \(162\) and \(373\) physical qubits per logical qubit at physical error rate \(0.1\%\) for logical CNOT error rates \(10^{-10}\) and \(10^{-24}\), respectively.
  • \([[6,4,2]]\) error-detecting code— Concatenations of this code with itself yield the level-\(r\) \([[6^r,4^r,2^r]]\) many-hypercube code [43]. The \([[6,4,2]]\) code can be concatenated with the surface code to yield the 6.6.6 color code [40; Appx. A].
  • \([[8,2,3]]\) Hermitian code— Applying the BLT mapping to the \([[8,2,3]]\) Hermitian code and concatenating each qubit pair with the \([[4,2,2]]\) code yields a \([[32,4,6]]\) self-dual CSS code [44; Corr. 2].
  • \([[8,3,2]]\) Smallest interesting color code— Concatenating \([[8,3,2]]\) blocks with triples of qubits drawn from three cyclically rotated 3D surface/toric codes yields a 3D toric/color family with parameters \([[8n,3,2d]]\) and transversal logical \(CCZ\) implemented by physical \(T\) gates on the inner \([[8,3,2]]\) blocks [26].
  • \([[9,1,3]]\) Shor code— The Shor code is a concatenation of a three-qubit bit-flip with a three-qubit phase-flip repetition code.
  • \([[9,1,3]]\) Surface-17 code— The phase boundary of the statistical mechanical model of the recursively concatenated surface-17 code under bit-flip noise nearly coincides with that of the toric code, and its optimal code capacity threshold is constrained by a generalized Kramers-Wannier self-duality shared by all zero-rate \(em\)-symmetric codes [7].
  • Mapping cone code— The mapping cone framework can be regarded as a generalization of code concatenation for CSS codes [1].
  • Phantom code— Concatenating a phantom outer code with a one-logical-qubit inner quantum code preserves phantomness.
  • Quantum Logic Code (QLC)— The \(\ell>0\) members are obtained by concatenating the QLC cores with the Steane code, and the construction also permits any self-dual doubly-even \([[n_i,1,d_i]]\) inner code with suitable transversal Clifford gates [45].
  • Layer code— Each pair of surface-code squares in a layer code is fused (or quasi-concatenated) with perpendicular surface-code squares via lattice surgery.
  • Quantum divisible code— A fault-tolerant \(T\) gate on the five-qubit or Steane code can be obtained by concatenating with particular quantum divisible codes [46].
  • Raussendorf-Bravyi-Harrington (RBH) cluster-state code— Concatenation of the RBH code with small codes such as the \([[2,1]]\) repetition code, \([[4,1,1,2]]\) subsystem code, or Steane code can improve thresholds [47].
  • Hypergraph product (HGP) code— There is a fault-tolerant universal computation scheme for hypergraph-product codes concatenated with the \([[4,2,2]]\) code in which the full syndrome measurement on the lower hypergraph product code is performed only if an error is detected at the upper four-qubit code [4].
  • \([[2^r-1, 2^r-2r-1, 3]]\) quantum Hamming code— Concatenating a growing sequence of quantum Hamming codes yields fault-tolerant quantum computation with constant space overhead and quasi-polylogarithmic time overhead [48]. Concatenating quantum Hamming codes on top of the \([[4,2,2]]\) and \(C_6\) codes yields fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads [39]. In the optimized protocol of Ref. [39], a level-five \(C_4/C_6\) code underlies concatenated quantum Hamming codes \(\mathcal{Q}_5,\mathcal{Q}_6,\mathcal{Q}_7,\mathcal{Q}_7\), yielding a \(2.5\%\) threshold and space overheads \(162\) and \(373\) physical qubits per logical qubit at physical error rate \(0.1\%\) for logical CNOT error rates \(10^{-10}\) and \(10^{-24}\), respectively. A modified tower of interleaved quantum Hamming codes with reserved qubits and recursive hookless Pauli-product measurements yields fault-tolerant quantum computation on a 1D nearest-neighbor qubit line with asymptotic rate above \(5\%\), constant space overhead, quasi-polylogarithmic time overhead, and a threshold [49]. Quantum Hamming codes can also be concatenated with surface codes [50].
  • 3D color code— On closed 3-manifolds, the 3D color code is equivalent to multiple decoupled copies of the 3D surface code via a local constant-depth Clifford circuit [51–53]. This process can be viewed as an ungauging [54–63] of certain symmetries. This mapping can also be done via code concatenation [64].
  • Subsystem homological product code— Concatenated CSS stabilizer codes are gauge-fixed SP codes [65; Thm. 4].

Member of code lists

Primary Hierarchy

Quantum Domain
Qubit Kingdom
Qubit codeQECC Quantum
Parents
Qubit code
Concatenated quantum codeQECC Quantum
Concatenated qubit code
Children
\([[12,1,3]]\) CE CSS code
This code is obtained by dual-rail concatenation of the \([[6,1,2]]\) CSS code [66; ID 50].
\([[14,3,3]]\) CE phantom code
This code is a concatenation of the \([[7,3,2]]\) punctured hypercube code with the two-qubit phase-flip repetition code [29].
Quantum turbo code
Binarized-and-concatenated (B&C) phantom code
Each qubit pair obtained by binarizing one \(\mathbb{F}_4\) qudit is concatenated with the \([[4,2,2]]\) code [29].
Auxiliary qubit mapping (AQM) code
Concatenated Steane code
The combination of the concatenated Steane code and QLDPC codes with non-vanishing rate yields fault-tolerant quantum computation with constant space and polylogarithmic time overheads, even when classical computation time is taken into account [67].
Hierarchical code
Using the concatenation convention of the Zoo, hierarchical codes are concatenations of constant-rate QLDPC (inner) codes with rotated surface (outer) codes. The cited paper [68] uses the opposite inner/outer terminology. The block length of the outer code is picked to grow logarithmically with the block length of the inner code.
Yoked surface code
Using the concatenation convention of the Zoo, a yoked surface code is a concatenation of a QMDPC code (inner code) with a rotated surface code (outer code). The cited paper [69] uses the opposite inner/outer terminology.
XYZ\(^2\) hexagonal stabilizer code
The XYZ\(^2\) hexagonal stabilizer code can be viewed as a concatenation of the \(YZZY\) surface code with one of the possible \([[2,1]]\) repetition codes, with the case of the bit-flip repetition code yielding a concatenation of the surface code with the dual-rail code [70].

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Zoo Code ID: qubit_concatenated

Cite as:
“Concatenated qubit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/qubit_concatenated, arXiv:2606.11484
BibTeX:
@incollection{eczoo_qubit_concatenated,
title={Concatenated qubit code},
booktitle={The Error Correction Zoo},
year={2026},
editor={Albert, Victor V. and Faist, Philippe},
eprint={2606.11484},
doi={10.48550/arXiv.2606.11484},
url={https://errorcorrectionzoo.org/c/qubit_concatenated}
}
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“Concatenated qubit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/qubit_concatenated, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/qubit_concatenated.yml.