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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].

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].
  • Permutationally self-dual (PSD) CSS code— For a suitable concatenation map, every level of the recursive concatenation of an \([[n_0,1,d_0]]\) PSD seed code is finite-size Kramers-Wannier self-dual under the statistical mechanical mapping of bit-flip decoding [7; Lemma 1]. If the resulting zero-rate family has a unique threshold, this self-duality constrains its optimal bit-flip threshold [7].
  • 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].
  • \([[4,2,2]]\) Four-qubit code— Concatenations of \([[4,2,2]]\) and \(C_6\) codes yield fault-tolerant quantum computation schemes [28] admitting a post-selected threshold [29,30] (see also Ref. [31]). 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 [32]. In the optimized protocol of Ref. [32], 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 [33]. The \([[4,2,2]]\) code can be concatenated with two copies of the surface code to yield the 4.6.12 color code [33].
  • \([[5,1,3]]\) 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 [5,34].
  • \([[6,2,2]]\) \(C_6\) code— Concatenations of \([[4,2,2]]\) and \(C_6\) codes yield fault-tolerant quantum computation schemes [28] admitting a post-selected threshold [29,30] (see also Ref. [31]) and the Meier-Eastin-Knill (MEK) magic-state distillation protocols [35]. 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 [32]. In the optimized protocol of Ref. [32], 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 [36]. The \([[6,4,2]]\) code can be concatenated with the surface code to yield the 6.6.6 color code [33; Appx. A].
  • \([[8,2,3]]\) Hermitian code— Applying concatenated symplectic doubling to the \([[8,2,3]]\) Hermitian code yields a \([[32,4,6]]\) self-dual CSS code [37; 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 outer \([[8,3,2]]\) blocks [26].
  • \([[9,1,3]]\) Surface-17 code— Recursively concatenating the surface-17 code yields a zero-rate family with distance \(d = n^{\log_9 3} = \sqrt{n}\) [7]. Its bit-flip phase boundary nearly coincides with that of the toric code [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 inner code with a one-logical-qubit outer quantum code preserves phantomness.
  • Quantum Logic Code (QLC)— Using the concatenation convention of the Zoo, the \(\ell>0\) members are obtained by concatenating the QLC cores (outer codes) with the Steane code (inner code), and the construction also permits any self-dual doubly-even \([[n_i,1,d_i]]\) inner code with suitable transversal Clifford gates [38].
  • 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.
  • Weakly divisible CSS code— Particular coset-divisible codes can be used as outer codes in layered constructions that implement a fault-tolerant \(T\) gate on the five-qubit or Steane code [39; Sec. V].
  • 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 [40].
  • 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 [41]. 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 [32]. In the optimized protocol of Ref. [32], 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 [42]. Quantum Hamming codes can also be concatenated with surface codes [43].
  • 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 [44–46]. This process can be viewed as an ungauging [47–56] of certain symmetries. This mapping can also be done via code concatenation [57].
  • Dense twist-defect surface code— Each column of the twist-defect dense packing is concatenated with an \([[n,n-2,2]]\) error-detecting code as the inner code, in the concatenation convention of the Zoo, which roughly doubles the distance [58]. The inner stabilizer generators are \(X^{\otimes n}\) and \(Z^{\otimes n}\), and two of the \(n\) outer code blocks serve as yoke qubits, so that the \(j\)th encoded qubit can be given the logical basis \(X_j X_{n-1}\) and \(Z_j Z_{n-2}\). This is the twist-defect analogue of the yoked surface code, which uses a column of one-qubit rotated surface-code patches in place of the dense packing. Unlike in that case, the logical operators of the dense packing generally lie in the interior, and must be moved to a boundary by twist-defect lattice surgery before the parity checks can be measured.
  • Subsystem homological product code— Concatenated CSS stabilizer codes are gauge-fixed SP codes [59; 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 [60; 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 [61].
\([[16,4,4]]\) twisted color code
Using the concatenation convention of the Zoo, the \([[16,4,4]]\) twisted color code is a block concatenation of the inner \([[8,4,2]]\) symplectic double of the \([[4,2,2]]\) code with outer \([[4,2,2]]\) blocks along a \(ZX\) duality [62].
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 [61].
Quantum parity code (QPC)
Using the concatenation convention of the Zoo, a QPC is a concatenation of an \(m_2\)-qubit phase-flip repetition code (inner code) with an \(m_1\)-qubit bit-flip repetition code (outer code).
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 [63].
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 [64] 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 [65] 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 [66].

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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.