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 arbitrary inner code and another qubit outer code. An inner \(C = ((n_1,K,d_1))\) and outer \(C^\prime = ((n_2,2,d_2))\) 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.

Protection

Concatenating stabilizer codes can help protect against catastrophic errors such as cosmic rays [1].

Decoding

The effective channel for a concatenation of codes is the composition of the codes' effective channels [2].Message passing algorithm for concatenated codes can be equivalent to ML decoding [3].

Fault Tolerance

Fault-tolerant message passing between devices [4].

Threshold

The first methods to achieve a fault-tolerant computational threshold use concatenated qubit stabilizer codes [5–11]; see the book [12]. Such thresholds are called concatenated thresholds. These methods require constant-space and polylogarithmic time overhead, but concatenations using quantum Hamming codes improve this to quasi-polylogarithmic time [13].

Parents

Children

Cousins

  • Hamiltonian-based code — Concatenated stabilizer code Hamiltonians have been investigated [14].
  • 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].
  • Amplitude-damping (AD) code — Concatenated quantum codes can protect against qubit AD [17].
  • \(((n,1,2))\) 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 [18; Appx. D] (cf. [19]).
  • \([[2^D,D,2]]\) 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 [20]. 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 [20].
  • \([[9,1,3]]\) 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 [21] (see also Ref. [22]). 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 [13,23].' Concatenating the \([[4,2,2]]\) code with the surface code is equivalent to removing stabilizer generators from the 4.8.8 color code [24]. The \([[4,2,2]]\) code can be concatenated with two copies of the surface code to yield the 4.6.12 color code [24]. 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 [25–27]. More generally, an \([[m^2,1,m]]\) QPC corrects \(m-1\) AD errors [28]. Recursively concatenating a \([[4,1,2]]\) subcode attains a threshold [29,30].
  • Five-qubit perfect code — The recursively concatenated five-qubit code has a measurement threshold of one [31]. Code performance against general Pauli channels has been worked out [2,32].
  • \([[6,2,2]]\) \(C_6\) code — Concatenations of \([[4,2,2]]\) and \([[6,2,2]]\) \(C_6\) codes yield fault-tolerant quantum computation schemes [21] (see also Ref. [22]) and the Meier-Eastin-Knill (MEK) magic-state distillation protocols [33]. 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 [13,23].
  • \([[6,4,2]]\) error-detecting code — Concatenations of this code with itself yield the level-\(r\) \([[6^r,4^r,2^r]]\) many-hypercube code [34]. The \([[6,4,2]]\) code can be concatenated with the surface code to yield the 6.6.6 color code [24; 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 [20].
  • 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 [35].
  • 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 [36].
  • \([[2^r-1, 2^r-2r-1, 3]]\) 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 [13].
  • \([[2^r-1, 2^r-2r-1, 3]]\) quantum Hamming code — Quantum Hamming codes can be concatened with surface codes [37].
  • 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 [31].
  • Subsystem homological product code — Concatenated CSS stabilizer codes are gauge-fixed SP codes [45; Thm. 4].

References

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Zoo Code ID: qubit_concatenated

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

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