\([[20,2,6]]\) B&C phantom code

\([[20,2,6]]\) B&C phantom code[1–3]

Alternative Names: \(C_4\)-Helix code.

Description

Self-dual doubly-even CSS code obtained by binarizing each \(\mathbb{F}_4\) qudit of the \([[5,1,3]]_4\) Galois-qudit CSS code into two qubits and encoding each pair into a \([[4,2,2]]\) block [2]. Since that Galois-qudit code is a symplectic double of the \([[5,1,3]]\) five-qubit code [2], the code is also the concatenated symplectic double of the five-qubit code [4; Corr. 1][3; Corr. 2].

Binarization uses the self-dual normal basis \(\{\omega,\omega^2\}\) of \(\mathbb{F}_4\) and yields the \([[10,2,3]]\) binarized Galois-qudit code [2]. Physical qubits are arranged in five blocks of four, where block \(j\) corresponds to the \(j\)-th \(\mathbb{F}_4\) qudit of the \([[5,1,3]]_4\) code. Using the concatenation convention of the Zoo, the \([[10,2,3]]\) code is the inner code and the \([[4,2,2]]\) code is the outer code. Stabilizer generators of the \([[4,2,2]]\) outer code on each block are \(ZZZZ\) and \(XXXX\). The inner stabilizers are the lifted images of the eight stabilizer generators of the \([[10,2,3]]\) code. The lift uses the logical-operator map \(X^\omega\mapsto XXII\), \(X^{\omega^2}\mapsto XIXI\), \(Z^\omega\mapsto IZIZ\), \(Z^{\omega^2}\mapsto IIZZ\) within each block [2].

A stabilizer tableau for the code is [2] \begin{align} \begin{smallmatrix} Z & Z & Z & Z & I & I & I & I & I & I & I & I & I & I & I & I & I & I & I & I \\ I & I & I & I & Z & Z & Z & Z & I & I & I & I & I & I & I & I & I & I & I & I \\ I & I & I & I & I & I & I & I & Z & Z & Z & Z & I & I & I & I & I & I & I & I \\ I & I & I & I & I & I & I & I & I & I & I & I & Z & Z & Z & Z & I & I & I & I \\ I & I & I & I & I & I & I & I & I & I & I & I & I & I & I & I & Z & Z & Z & Z \\ I & Z & I & Z & I & Z & I & Z & I & Z & I & Z & I & Z & I & Z & I & I & I & I \\ I & I & Z & Z & I & I & Z & Z & I & I & Z & Z & I & I & Z & Z & I & I & I & I \\ I & I & I & I & I & Z & I & Z & I & Z & Z & I & I & I & Z & Z & I & Z & I & Z \\ I & I & I & I & I & I & Z & Z & I & Z & I & Z & I & Z & Z & I & I & I & Z & Z \\ X & X & X & X & I & I & I & I & I & I & I & I & I & I & I & I & I & I & I & I \\ I & I & I & I & X & X & X & X & I & I & I & I & I & I & I & I & I & I & I & I \\ I & I & I & I & I & I & I & I & X & X & X & X & I & I & I & I & I & I & I & I \\ I & I & I & I & I & I & I & I & I & I & I & I & X & X & X & X & I & I & I & I \\ I & I & I & I & I & I & I & I & I & I & I & I & I & I & I & I & X & X & X & X \\ X & X & I & I & X & X & I & I & X & X & I & I & X & X & I & I & I & I & I & I \\ X & I & X & I & X & I & X & I & X & I & X & I & X & I & X & I & I & I & I & I \\ I & I & I & I & X & X & I & I & X & I & X & I & I & X & X & I & X & X & I & I \\ I & I & I & I & X & I & X & I & I & X & X & I & X & X & I & I & X & I & X & I \end{smallmatrix}~. \tag*{(1)}\end{align} Rows 1 to 5 are the \(Z\)-type outer stabilizers of each \([[4,2,2]]\) block. Rows 6 to 9 are the lifted inner \(Z\)-type stabilizers. Rows 10 to 14 are the \(X\)-type outer stabilizers. Rows 15 to 18 are the lifted inner \(X\)-type stabilizers. This code is equivalent, up to qubit permutations and single-qubit Clifford gates, to three \([[20,2,6]]\) self-dual CSS codes in QECDB [5; ID 672e6e9c34d5954e0cb7d1b7][5; ID 67a388ef9d65c7b409826879][5; ID 67a3890a9d65c7b40982687b].

Protection

Detects errors on up to 5 qubits and corrects errors on up to 2 qubits.

Encoding

Non-fault-tolerant encoding circuits [3].

Transversal and Permutation-Based Gates

Logical CNOT gates in both directions between the two logical qubits are realized by qubit permutations within a code block [2,3].Fold-diagonal logical \(SS\) gates are available from the self-duality of the inner \([[5,1,3]]_4\) code and the outer \([[4,2,2]]\) layer [2].All logical Clifford gates can be realized as two-fold transversal gates, i.e., by depth-one circuits of two-local code-preserving physical gates [1,2,6,7].

Fault Tolerance

Selective state filtering (post-selection): logical error rates of \(\approx 2\times 10^{-6}\) per round of Steane-style error correction at physical error rate \(p=10^{-3}\), without being fully fault-tolerant [3].

Realizations

Trapped-ion devices: twenty rounds of error correction with an \(X\)-basis logical error rate of \(4.6\times 10^{-5}\) per logical qubit per round on the 98-qubit Helios device by Quantinuum [8]. Randomized benchmarking of the two-qubit logical Clifford group within one code block under adaptive syndrome extraction yields an error of \(2.8\times 10^{-4}\) per two-qubit logical Clifford [8]. A depth-two chain-map CNOT gate [9] between a \([[25,1,5]]\) rotated surface code and one logical qubit was used to prepare a three-logical-qubit GHZ state with a fidelity lower bound of \(99.93\%\) [8].

Cousins

Group-algebra code— The \([[20,2,6]]\) code is equivalent to a group-algebra CSS code with \(c=\{0,1,2,4,6,7,10,16\}\) [6].
\([[5,1,3]]\) Five-qubit perfect code— The \([[20,2,6]]\) code is obtained from the \([[5,1,3]]\) five-qubit code by concatenated symplectic doubling [3][4; Corr. 1][1; Table I]. The same code arises by binarizing the \([[5,1,3]]_4\) Galois-qudit CSS code and concatenating each qubit pair with the \([[4,2,2]]\) code [2]. This is because that Galois-qudit code is a symplectic double of the five-qubit code [2].
\([[4,2,2]]\) Four-qubit code— The \([[20,2,6]]\) code is obtained by concatenating each qubit pair of the \([[10,2,3]]\) binarized code with the \([[4,2,2]]\) code [2,3].
\([[10,2,3]]\) binarized Galois-qudit 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 [2].

Member of code lists

Primary Hierarchy

Quantum Domain

Qubit Kingdom

Qubit codeQECC Quantum

Union stabilizer (USt) codeQECC Quantum

Qubit stabilizer codeQubit Stabilizer Hamiltonian-based QECC Quantum

Coherent-parity-check (CPC) code

Qubit CSS codeCSS Stabilizer Hamiltonian-based QECC Quantum

Phantom code

Binarized-and-concatenated (B&C) phantom codeQubit Concatenated quantum QECC Quantum

Parents

Binarized-and-concatenated (B&C) phantom code

The \([[20,2,6]]\) code is the B&C phantom code obtained from the \([[5,1,3]]_4\) Galois-qudit CSS code [2,3].

Quantum Logic Code (QLC)Qubit CSS Stabilizer Hamiltonian-based QECC Quantum

The \(AGL(1,5)\) group-algebra core in the Quantum Logic Code construction is permutation-equivalent to the \([[20,2,6]]\) B&C phantom code [6].

\([[20,2,6]]\) B&C phantom code

References

[1]: N. Berthusen and E. Durso-Sabina, “Simple logical quantum computation with concatenated symplectic double codes”, Quantum 10, 2195 (2026) arXiv:2510.18753 DOI
[2]: J. M. Koh, A. Gong, A. C. Diaconu, D. B. Tan, A. A. Geim, M. J. Gullans, N. Y. Yao, M. D. Lukin, and S. Majidy, “Entangling logical qubits without physical operations”, (2026) arXiv:2601.20927
[3]: B. W. Reichardt, D. Aasen, and R. Chao, “Fire and ice: Partially fault-tolerant quantum computing with selective state filtering”, (2026) arXiv:2605.15344
[4]: S. Bravyi, B. M. Terhal, and B. Leemhuis, “Majorana fermion codes”, New Journal of Physics 12, 083039 (2010) arXiv:1004.3791 DOI
[5]: S. Burton, “qecdb.org: Quantum Error Correction Database”, URL
[6]: A. Holmes, “Quantum Logic Codes: Complete Transversal Logical Clifford Instruction Sets for High-Rate Stabilizer Quantum Error Correcting Codes”, (2026) arXiv:2606.13521
[7]: V. V. Albert, “Beyond transversality: structure of Clifford circuits for CSS codes”, (2026) arXiv:2608.05688
[8]: N. Berthusen et al., “Experimental validation of a compact fault-tolerant architecture for trapped ions”, (2026) arXiv:2609.03194
[9]: A. Benhemou and N. Berthusen, “Automated logical Clifford gadgets for heterogeneous architectures via chain maps”, (2026) arXiv:2607.02482

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Victor V. Albert (2026-09-26)
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Victor V. Albert (2026-06-08)
Victor V. Albert (2026-05-22)

Cite as:

“\([[20,2,6]]\) B&C phantom code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/stab_20_2_6, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/css/self_dual/stab_20_2_6.yml.