# Concatenated quantum code[1]

## Description

A combination of two quantum codes, an inner code \(C\) and an outer code \(C^\prime\), where the physical subspace used for the inner code consists of the logical subspace of the outer code. In other words, first one encodes in the inner code \(C^\prime\), and then one encodes each of the physical registers of \(C^\prime\) in an outer code \(C\). An inner \(C = ((n_1,K,d_1))_{q_1}\) and outer \(C^\prime = ((n_2,q_1,d_2))\) block quantum code yield an \(((n_1 n_2, K, d \geq d_1d_2))_{q_2}\) concatenated quantum code [2].

Concatenating an \(((n,q,d))_q\) block quantum code can be done recursively, with the \(r\)th level of concatenation yielding an \(n^r,q,d^r))_q\) code.

A generalization of concatenation exists [3].

## Protection

## Decoding

## Threshold

## Notes

## Parent

- Quantum Lego code — Encoders for a concatenated codes correspond to tree tensor networks.

## Children

- Group-based QPC — A group-based QPC is a concatenation of a phase-flip group-based repetition code with a bit-flip group-based repetition code.
- Concatenated bosonic code
- Quantum turbo code
- Auxiliary qubit mapping (AQM) code
- Concatenated Steane code
- Hierarchical code — Hierarchical codes are concatenations of constant-rate QLDPC (outer) codes with (inner) rotated surface codes. The block length of the inner code is picked to grow logarithmically with the block length of the outer code.
- Yoked surface code — A yoked surface code is a concatenation of a QMDPC code (outer code) with a rotated surface code (inner code).

## Cousins

- 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 [15,16].
- Concatenated code
- Concatenated c-q code
- Rotor GKP code — The rotor GKP code can be thought of as a concatenation of a homological rotor code and a modular-qudit GKP code [17; Fig. 3].
- \([[4,2,2]]_{G}\) four group-qudit code — The \(|\overline{g_1=1,g_2}\rangle\) \([[4,1,2]]_{G}\) subcode is the smallest group-based QPC, i.e., a concatenation of a two-qubit bit-flip with a two-qubit phase-flip group-based repetition code for that group.
- Dual-rail quantum code — The KLM protocol, one of the first protocols for fault-tolerant quantum computation, utilizes concatenations of the dual-rail with a stabilizer code [18]. Concatenating the dual-rail code with an \([[n,k,d]]\) stabilizer code yields an \([[2n,k,d]]\) constant-excitation code [19] that protects against \(d-1\) AD errors [20].
- Two-mode binomial code — Two-mode binomial codes can be concatenated with repetition codes to yield bosonic analogues of QPCs [21].
- Amplitude-damping (AD) code — Concatenated quantum codes can protect against AD [22].
- EA qubit stabilizer code — EA concatenated codes have been studied [23,24].
- \(((n,1,2))\) 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 [25; Appx. D].
- \([[2^D,D,2]]\) hypercube code — The hypercube 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 a distance-two \(D\)-dimensional surface code 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].
- \([[9,1,3]]\) Shor code — The Shor code is a concatenation of a three-qubit bit-flip with a three-qubit phase-flip repetition code.
- \([[4,2,2]]\) Four-qubit code — The \(\{|\overline{00}\rangle,|\overline{01}\rangle\}\) \([[4,1,2]]\) subcode is the smallest QPC, i.e., a concatenation of a two-qubit bit-flip with a two-qubit phase-flip repetition code. Concatenations of \([[4,2,2]]\) and \([[6,2,2]]\) \(C_6\) codes yield fault-tolerant quantum computation schemes [27]. Concatenations of quantum Hamming codes with the \([[4,2,2]]\) and \([[6,2,2]]\) \(C_6\) codes yield fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads [14,28].' Concatenating the \([[4,2,2]]\) code with the surface code is equivalent to removing stabilizer generators from the 4.8.8 color code [29]. The \([[4,2,2]]\) code can be concatenated with two copies of the surface code to yield the 4.6.12 color code [29]. An \([[8,1,2]]\) QPC correcting a single AD error is equivalent to a concatenation of the \(\{|\overline{01}\rangle,|\overline{11}\rangle\}\) (constant-excitation) subcode of the \([[4,2,2]]\) code with the dual-rail code [19,30,31]. More generally, an \([[m^2,1,m]]\) QPC corrects \(m-1\) AD errors [20].
- Five-qubit perfect code — The concatenated five-qubit code has a measurement threshold of one [32]. Code performance against general Pauli channels has been worked out [5,33].
- \([[6,2,2]]\) \(C_6\) code — Concatenations of \([[4,2,2]]\) and \([[6,2,2]]\) \(C_6\) codes yield fault-tolerant quantum computation schemes [27] and the Meier-Eastin-Knill (MEK) magic-state distillation protocols [34]. Concatenations of quantum Hamming codes with the \([[4,2,2]]\) and \([[6,2,2]]\) \(C_6\) codes yield fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads [14,28].
- \([[6,4,2]]\) error-detecting code — Concatenations of this code with itself yield the level-\(r\) \([[6^r,4^r,2^r]]\) many-hypercube code [35]. The \([[6,4,2]]\) code can be concatenated with the surface code to yield the 6.6.6 color code [29; Appx. A].
- \([[8,3,2]]\) CSS code — The \([[8,3,2]]\) code can be concatenated with a 3D surface code to yield a \([[O(d^3),3,2d]]\) code family that admits a transversal implementation of the logical \(CCZ\) gate [26].
- Layer code — Each pair of surface-code squares in a layer code are 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 [36].
- Raussendorf-Bravyi-Harrington (RBH) cluster-state code — Concatenation of the RBH code with small codes such as the \([[2,1,1]]\) repetition code, \([[4,1,1,2]]\) subsystem code, or Steane code can improve thresholds [37].
- \([[2^r-1, 2^r-2r-1, 3]]\) quantum Hamming code — Concatenations of quantum Hamming codes with the \([[4,2,2]]\) and \([[6,2,2]]\) \(C_6\) codes yield fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads [14].
- 3D color code — The 3D color code is equivalent to multiple decoupled copies of the 3D surface code via a local constant-depth Clifford circuit [38–40]. This process can be viewed as an ungauging [41–43,43] of certain symmetries. This mapping can also be done via code concatenation [44].
- \([[15,1,3]]\) quantum Reed-Muller code — The concatenated \([[15,1,3]]\) code has a measurement threshold less than one [32].
- Subsystem homological product code — Concatenated CSS stabilizer codes are gauge-fixed SP codes [45; Thm. 4].
- Modular-qudit CWS code — Generalized concatenatenations of modular-qudit CWS codes yield a family of codes that have larger logical dimension than stabilizer codes and that asymptotically approach the modular-qudit Hamming bound [3].
- \([[9,1,3]]_{\mathbb{Z}_q}\) modular-qudit code — The \([[9,1,3]]_{\mathbb{Z}_q}\) modular-qudit code is a concatenation of a two-group-qudit bit-flip with a two-qubit phase-flip group repetition code for \(G=\mathbb{Z}_q\).
- Galois-qudit GRS code — Concatenations of quantum GRS codes and random stabilizer codes can achieve the quantum Gilbert-Varshamov bound [46].

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## Page edit log

- Victor V. Albert (2022-02-15) — most recent

## Cite as:

“Concatenated quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/quantum_concatenated