\([[4,2,2]]\) CSS code[1,2] 

Also known as \(C_4\) code.

Description

Four-qubit CSS stabilizer code is the smallest qubit stabilizer code to detect a single-qubit error.

Admits generators \(\{XXXX, ZZZZ\} \) and codewords \begin{align} \begin{split} |\overline{00}\rangle = (|0000\rangle + |1111\rangle)/\sqrt{2}~{\phantom{.}}\\ |\overline{01}\rangle = (|0011\rangle + |1100\rangle)/\sqrt{2}~{\phantom{.}}\\ |\overline{10}\rangle = (|0101\rangle + |1010\rangle)/\sqrt{2}~{\phantom{.}}\\ |\overline{11}\rangle = (|0110\rangle + |1001\rangle)/\sqrt{2}~. \end{split} \tag*{(1)}\end{align} This code is the smallest instance of the toric code, and its various single-qubit subcodes are small planar surface codes. It is the unique code for its parameters, up to local equivalence [3; Thm. 8].

The subcode \(\{|\overline{00}\rangle,|\overline{01}\rangle\}\) is a \([[4,1,2]]\) code [4], whose \(\pm\)-basis codewords can be written as \begin{align} |\overline{\pm}\rangle = \frac{1}{2}(|00\rangle \pm |11\rangle)^{\otimes 2}~.\tag*{(2)}\\ \end{align} This code can be thought of as a concatenation of a two-qubit bit-flip with a two-qubit phase-flip code.

Protection

Detects a single-qubit error [1] or single erasure [2]. Not able to correct arbitrary single-qubit errors because \( \lfloor \frac{d-1}{2} \rfloor =0 \).

The \([[4,1,2]]\) subcodes \(\{|\overline{00}\rangle,|\overline{10}\rangle\}\) [4] and \(\{|\overline{01}\rangle,|\overline{11}\rangle\}\) [5] approximately correct a single amplitude damping error, with the latter being a constant excitation code.

Magic

Various magic-state distillation protocols exist for the \([[4,2,2]]\) qubit code and the \([[6,2,2]]\) \(C_6\) code in what are known as Meier-Eastin-Knill (MEK) protocols [6]. For example, the magic-state scaling exponent is \(\gamma = \log_2 5 \approx 2.322\) for a protocol using the \([[10,2,2]]\) code [7; Box 2]; see also [8; Table IV].

Transversal Gates

Transversal Pauli, Hadamard, and two-qubit \(S=\sqrt{Z}\) [9] (see also [10]).A transversal \(CZ\) gate is realized by the rotation \(\sqrt{Z}\otimes\sqrt{Z}^{\dagger}\otimes\sqrt{Z}^{\dagger}\otimes\sqrt{Z}\).Adding \(XYZI\) to the stabilizer group produces a \([[4,1,2]]\) subcode that admits weight-two transversal logical Pauli operations [11].

Fault Tolerance

Preparation of certain states, both magic and non-magic, along with transversal gates can be performed fault-tolerantly, but requires post-selection because the code cannot correct errors [9]. Magic states can be injected into surface and color codes since the code is a small instance of both [12].Concatenations of \([[4,2,2]]\) and \([[6,2,2]]\) \(C_6\) codes yield fault-tolerant quantum computation schemes [13].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 [14,15].

Realizations

\([[4,1,2]]\) subcode implemented using four-qubit graph state of photons [16].Trapped-ion device by IonQ [17].Logical state preparation and flag-qubit error correction realized in superconducting-circuit devices by IBM [1821].The subcode \(\{|\overline{00}\rangle,|\overline{10}\rangle\}\) [22] and \(\{|\overline{00}\rangle,|\overline{01}\rangle\}\) [23], treated as a planar surface code, has been realized in superconducting-circuit devices.Logical gates between two copies of the subcode \(\{|\overline{10}\rangle,|\overline{11}\rangle\}\), interpreted as lattice surgery between planar surface codes, realized in superconducting circuits [24].Logical gates for the \(\{|\overline{00}\rangle,|\overline{11}\rangle\}\) subcode, treated as a planar code, realized in superconducting circuits [25].The CZ magic state has been realized on an IBM heavy-hex superconducting circuit device [12].

Parents

  • Kitaev surface code — The \([[4,2,2]]\) code is the smallest toric code. Concatenating the \([[4,2,2]]\) code with the surface code is equivalent to removing stabilizer generators from the 4.8.8 color code [26].
  • 2D color code — The \([[4,2,2]]\) code can be interpreted as a 2D color code on a square of the 4.8.8 or 4.6.12 tilings, or on a trapezoidal patch that makes up two-thirds of a hexagon of the 6.6.6 tiling [12,27]. Concatenating the \([[4,2,2]]\) code with the surface code is equivalent to removing stabilizer generators from the 4.8.8 color code [26]. Concatenating the \([[4,2,2]]\) code with two copies of the surface code yields the 4.6.12 color code [26]. A small 6.6.6 color code is a \([[4,1,2]]\) subcode with three weight-three stabilizer generators [28; Fig. 7].
  • \([[2^D,D,2]]\) hypercube code — The \([[4,2,2]]\) code is a hypercube code for \(D=2\).
  • \([[2m,2m-2,2]]\) error-detecting code — The \([[2m,2m-2,2]]\) error-detecting code for \(m=2\) reduces to the \([[4,2,2]]\) code.
  • \([[4,2,2]]_{G}\) four group-qudit code — The four group-qudit code reduces to the four-rotor code for \(G= \mathbb{Z}_2\).
  • Quantum polar code — \([[4,2,2]]\) code is a small quantum polar code [29].

Cousins

  • Rotated surface code — The subcodes \(\{|\overline{10}\rangle,|\overline{11}\rangle\}\) [24], \(\{|\overline{00}\rangle,|\overline{10}\rangle\}\) [22], \(\{|\overline{00}\rangle,|\overline{01}\rangle\}\) [23], and \(\{|\overline{00}\rangle,|\overline{11}\rangle\}\) [25] of the \([[4,2,2]]\) code are small planar rotated surface codes.
  • Bacon-Shor code — The error-detecting \([[4,1,1,2]]\) Bacon-Shor code, which reduces to a subcode of the \([[4,2,2]]\) code for a particular gauge configuration, has gauge operators \(\{XXII,IIXX,ZIZI,IZIZ\}\).
  • Quantum parity code (QPC) — 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.
  • Five-qubit perfect code — The \([[4,2,2]]\) code can be derived from the five-qubit code using a protocol that converts an \([[n,k,d]]\) code into an \([[n-1, k+1, d-1]]\) code [30; Sec. 3.5].
  • Approximate quantum error-correcting code (AQECC) — The \([[4,1,2]]\) subcodes \(\{|\overline{00}\rangle,|\overline{10}\rangle\}\) [4] and \(\{|\overline{01}\rangle,|\overline{11}\rangle\}\) [5] approximately correct a single amplitude damping error, with the latter being a constant excitation code.
  • Constant-excitation (CE) code — The \([[4,1,2]]\) subcodes \(\{|\overline{00}\rangle,|\overline{10}\rangle\}\) [4] and \(\{|\overline{01}\rangle,|\overline{11}\rangle\}\) [5] approximately correct a single amplitude damping error, with the latter being a constant excitation code. An \(((8,1))\) constant-excitation code correcting a single amplitude damping error can be obtained from concatenating that subcode with the dual-rail code [31].
  • Binomial code — \([[4,1,2]]\) subcode consisting of \(|\overline{00}\rangle\) and any other codeword reduces to the \(0,2,4\) binomial code when the basis labels in each codeword are written as in base-ten. Such a mapping can be generalized [32].
  • Heavy-hexagon code — Magic states prepared using the \([[4,1,2]]\) subcode can be injected into the heavy-hex code [12,21].
  • Dual-rail quantum code — An \(((8,1))\) constant-excitation code correcting a single amplitude damping error can be obtained from concatenating the \(\{|\overline{01}\rangle,|\overline{11}\rangle\}\) [5] subcode with the dual-rail code [31].
  • Concatenated quantum 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 [13]. 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 [14,15].' Concatenating the \([[4,2,2]]\) code with the surface code is equivalent to removing stabilizer generators from the 4.8.8 color code [26]. The \([[4,2,2]]\) code can be concatenated with two copies of the surface code to yield the 4.6.12 color code [26]. An \(((8,1))\) constant-excitation code correcting a single amplitude damping error can be obtained from concatenating the \(\{|\overline{01}\rangle,|\overline{11}\rangle\}\) [5] subcode with the dual-rail code [31].
  • Four-qubit single-deletion code — A basis of codewords for the four-qubit single-deletion code consists of the \(|\overline{00}\rangle\) and \(|\overline{01}\rangle+|\overline{10}\rangle+|\overline{11}\rangle\)states of the four-qubit code.
  • \([[6,2,2]]\) \(C_6\) code — Concatenations of \([[4,2,2]]\) and \([[6,2,2]]\) \(C_6\) codes yield fault-tolerant quantum computation schemes [13] and the Meier-Eastin-Knill (MEK) magic-state distillation protocols [6]. 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 [14,15].
  • \([[6,4,2]]\) error-detecting code — The \([[6,4,2]]\) error-detecting code can be constructed out of two \([[4,2,2]]\) codes in the quantum Lego code framework.
  • \([[2^r-1, 2^r-2r-1, 3]]\) 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].
  • Heavy-hexagon code — The \(d=2\) heavy-hexagonal code is closely related to the \([[4,1,2]]\) subcode.

References

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Zoo Code ID: stab_4_2_2

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\([[4,2,2]]\) CSS code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/stab_4_2_2
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@incollection{eczoo_stab_4_2_2, title={\([[4,2,2]]\) CSS code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/stab_4_2_2} }
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