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

\([[20,2,6]]\) B&C phantom code[1,2]

Description

Self-dual CSS code on 20 physical qubits encoding two logical qubits with distance 6. The code is obtained by binarizing the \([[5,1,3]]_4\) code in the self-dual normal basis \(\{\omega,\omega^2\}\) to the \([[10,2,3]]\) binarized Galois-qudit code and then concatenating each qubit pair with the \([[4,2,2]]\) code [1].

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. Stabilizer generators of the \([[4,2,2]]\) inner code on each block are \(ZZZZ\) and \(XXXX\); the outer stabilizers are the lifted images of the eight stabilizer generators of the \([[10,2,3]]\) code using 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.

A stabilizer tableau for the code is [1] \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 1–5 are the \(Z\)-type inner stabilizers of each \([[4,2,2]]\) block; rows 6–9 are the lifted outer \(Z\)-type stabilizers; rows 10–14 are the \(X\)-type inner stabilizers; rows 15–18 are the lifted outer \(X\)-type stabilizers.

Protection

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

Encoding

Non-fault-tolerant encoding circuits [2].

Transversal Gates

Logical CNOT gates in both directions between the two logical qubits are realized by qubit permutations within a code block [1,2].Fold-diagonal logical \(SS\) gates are available from the self-duality of the outer \([[5,1,3]]_4\) code and the inner \([[4,2,2]]\) layer [1].

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 [2].

Cousins

\([[5,1,3]]\) Five-qubit perfect code— The \([[20,2,6]]\) code is obtained from the \([[5,1,3]]\) five-qubit code via the BLT mapping (Lemma 1) and concatenation with the \([[4,2,2]]\) code (Corollary 2) [2][3; Corr. 1].
\([[4,2,2]]\) 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 [1,2].
\([[10,2,3]]\) 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 [1].

Member of code lists

Primary Hierarchy

Quantum Domain

Qubit Kingdom

Qubit codeQECC Quantum

Union stabilizer (USt) codeQECC Quantum

Qubit stabilizer codeStabilizer Hamiltonian-based Qubit 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 [1,2].

Self-dual CSS codeCSS Stabilizer Hamiltonian-based Qubit QECC Quantum

The \([[20,2,6]]\) code is a self-dual CSS code obtained from the \([[5,1,3]]\) code via the BLT mapping and concatenation with \([[4,2,2]]\) [2; Corr. 2][3; Corr. 1].

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

References

[1]: 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
[2]: B. W. Reichardt, D. Aasen, and R. Chao, “Fire and ice: Partially fault-tolerant quantum computing with selective state filtering”, (2026) arXiv:2605.15344
[3]: S. Bravyi, B. M. Terhal, and B. Leemhuis, “Majorana fermion codes”, New Journal of Physics 12, 083039 (2010) arXiv:1004.3791 DOI

Page edit log

Victor V. Albert (2026-05-22) — most recent

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/small_distance/small/20/stab_20_2_6.yml.