A393197 - OEIS

%I #22 Mar 12 2026 22:27:38

%S 1,4,5,10,11,16,17

%N Least number of qubits such that a quantum error correcting code with distance n and one logical qubit exists.

%C a(8) = 22, 23, or 24. a(9) = 24 or 25. a(10) = 28. a(11) = 29. 34 <= a(12) <= 42.

%C In 1995, Shor constructed the first true quantum error correcting code, a [[9,1,3]] code, showing a(3) <= 9.

%H Charles Henry Bennett, David Peter DiVincenzo, John A. Smolin, and William Kent Wootters, <a href="https://arxiv.org/abs/quant-ph/9604024">Mixed-state entanglement and quantum error correction</a>, Physical Review A 54.5 (1996), pp. 3824-3851.

%H A. R. Calderbank, E. M. Rains, P. W. Shor and N. J. A. Sloane, <a href="http://arXiv.org/abs/quant-ph/9608006">Quantum error correction via codes over GF(4)</a>, arXiv:quant-ph/9608006, 1996-1997; IEEE Trans. Inform. Theory, 44 (1998), 1369-1387.

%H Martianus Frederic Ezerman, Markus Grassl, San Ling, Ferruh Özbudak, and Buket Özkaya, <a href="https://arxiv.org/abs/2405.15057">Characterization of Nearly Self-Orthogonal Quasi-Twisted Codes and Related Quantum Codes</a>, arXiv preprint (2024). arXiv:2405.15057 [quant-ph]

%H Markus Grassl, <a href="https://codetables.de/">Bounds on the minimum distance of linear codes and quantum codes.</a>

%H Markus Grassl, <a href="https://doi.org/10.1007/978-3-540-37634-7">Searching for linear codes with large minimum distance.</a> in: Wieb Bosma and John Cannon (Eds.), Discovering Mathematics with Magma, Springer, 2006.

%H Raymond Laflamme, Cesar Miquel, Juan Pablo Paz, and Wojciech Hubert Zurek, <a href="https://arxiv.org/abs/quant-ph/9602019">Perfect quantum error correcting code</a>, Physical Review Letters 77.1 (1996), pp. 198-201.

%H Eric M. Rains, <a href="https://arxiv.org/abs/quant-ph/9703048">Nonbinary quantum codes</a>, IEEE Transactions on Information Theory 45.6 (1999), pp. 1827-1832.

%H Eric M. Rains, <a href="https://arxiv.org/abs/quant-ph/9611001">Quantum shadow enumerators</a>, IEEE transactions on information theory 45.7 (1999), pp. 2361-2366.

%H Peter W. Shor, <a href="https://www.cs.miami.edu/home/burt/learning/Csc670.052/pR2493_1.pdf">Scheme for reducing decoherence in quantum computer memory</a>, Physical review A 52.4 (1995), pp. R2493-R2496.

%F a(n) ~ 2n. a(n) >= 2n - 1 from the quantum Singleton bound.

%e a(1) = 1 from the trivial [[1,1,1]] code (no encoding).

%e a(2) = 4 from the [[4,1,2]] code, a subcode of the [[4,2,2]] error-detecting code.

%e a(3) = 5 from the perfect [[5,1,3]] stabilizer code (Laflamme, Miquel, Paz, & Zurek and Bennett, DiVincenzo, Smolin, & Wootters).

%e a(4) = 10 from a [[10,1,4]] code, a subcode of a [[10,2,4]] additive code over GF(4).

%e a(5) = 11 from a [[11,1,5]] additive code over GF(4).

%e a(6) = 16 from a [[16,1,6]] code, a subcode of a [[16,2,6]] additive code over GF(4).

%e a(7) = 17 from a [[17,1,7]] additive code over GF(4).

%Y Cf. A090899, A094927.

%K nonn,hard,more

%O 1,2

%A _Charles R Greathouse IV_, Mar 11 2026