Abstract We construct new stabilizer quantum error-correcting codes from generalized monomial-Cartesian codes. Our construction uses an explicitly defined twist vector, and we present formulas for the minimum distance and dimension. Generalized monomial-Cartesian codes arise from polynomials inmvariables. When $m=1$ m=1 our codes are MDS, and when $m=2$ m=2 and our lower bound for the minimum distance is 3, the codes are at least Hermitian almost MDS. For an infinite family of parameters, when $m=2$ m=2 we prove that our codes beat the Gilbert–Varshamov bound. We also present many examples of our codes that are better than any known code in the literature.Abstract We construct new stabilizer quantum error-correcting codes from generalized monomial-Cartesian codes. Our construction uses an explicitly defined twist vector, and we present formulas for the minimum distance and dimension. Generalized monomial-Cartesian codes arise from polynomials inmvariables. When $$m=1$$ m=1 our codes are MDS, and when $$m=2$$ m=2 and our lower bound for the minimum distance is 3, the codes are at least Hermitian almost MDS. For an infinite family of parameters, when $$m=2$$ m=2 we prove that our codes beat the Gilbert–Varshamov bound. We also present many examples of our codes that are better than any known code in the literature.

arXiv:2307.15488Quantum Information ProcessingMar 1, 20242 citations

Low-Overhead Entangling Gates from Generalised Dehn Twists

Ryan Tiew, Nikolas P. Breuckmann

We generalise the implementation of logical quantum gates via Dehn twists from topological codes to the hypergraph and balanced products of cyclic codes. These generalised Dehn twists implement logical entangling gates with no additional qubit overhead and $\mathcal{O}(d)$ time overhead. Due to having more logical degrees of freedom in the codes, there is a richer structure of attainable logical gates compared to those for topological codes. To illustrate the scheme, we focus on families of hypergraph and balanced product codes that scale as $[[18q^2,8,2q]]_{q\in \mathbb{N}}$ and $[[18q,8,\leq 2q]]_{q\in \mathbb{N}}$ respectively. For distance 6 to 12 hypergraph product codes, we find that the set of twists and fold-transversal gates generate the full logical Clifford group. For the balanced product code, we show that Dehn twists apply to codes in this family with odd $q$ . We also show that the $[[90,8,10]]$ bivariate bicycle code is a member of the balanced product code family that saturates the distance bound. We also find balanced product codes that saturate the bound up to $q\leq8$ through a numerical search.We generalise the implementation of logical quantum gates via Dehn twists from topological codes to the hypergraph and balanced products of cyclic codes. These generalised Dehn twists implement logical entangling gates with no additional qubit overhead and $\mathcal{O}(d)$ time overhead. Due to having more logical degrees of freedom in the codes, there is a richer structure of attainable logical gates compared to those for topological codes. To illustrate the scheme, we focus on families of hypergraph and balanced product codes that scale as $[[18q^2,8,2q]]_{q\in \mathbb{N}}$ and $[[18q,8,\leq 2q]]_{q\in \mathbb{N}}$ respectively. For distance 6 to 12 hypergraph product codes, we find that the set of twists and fold-transversal gates generate the full logical Clifford group. For the balanced product code, we show that Dehn twists apply to codes in this family with odd $q$. We also show that the $[[90,8,10]]$ bivariate bicycle code is a member of the balanced product code family that saturates the distance bound. We also find balanced product codes that saturate the bound up to $q\leq8$ through a numerical search.

arXiv:2411.03302Nov 5, 20240 citations

Constructions of quasi-twisted quantum codes

Jingjie Lv, Ruihu Li, Junli Wang

arXiv:1912.05142Quantum Information ProcessingJul 30, 202010 citations

Low-Overhead Entangling Gates From Generalised Dehn Twists

Ryan Tiew, Nikolas P. Breuckmann

We generalise the implementation of logical quantum gates via Dehn twists from topological codes to the hypergraph and balanced products of cyclic codes. These generalised Dehn twists implement logical entangling gates with no additional qubit overhead andO(d)time overhead. Due to having more logical degrees of freedom in the codes, there is a richer structure of attainable logical gates compared to those for topological codes. To illustrate the scheme, we focus on families of hypergraph and balanced product codes that scale as [[18q2, 8, 2q]]q∈Nand [[18q, 8,≤ 2q]]q∈Nrespectively. For distance 6 to 12 hypergraph product codes, we find that the set of twists and fold-transversal gates generate the full logical Clifford group. For the balanced product code, we show that Dehn twists apply to codes in this family with oddq. We also show that the [[90, 8, 10]] bivariate bicycle code is a member of the balanced product code family that saturates the distance bound, and find other balanced product codes that saturate the bound up toq≤ 8 through a numerical search.We generalise the implementation of logical quantum gates via Dehn twists from topological codes to the hypergraph and balanced products of cyclic codes. These generalised Dehn twists implement logical entangling gates with no additional qubit overhead andO(d)time overhead. Due to having more logical degrees of freedom in the codes, there is a richer structure of attainable logical gates compared to those for topological codes. To illustrate the scheme, we focus on families of hypergraph and balanced product codes that scale as [[18q2, 8, 2q]]q∈Nand [[18q, 8,≤ 2q]]q∈Nrespectively. For distance 6 to 12 hypergraph product codes, we find that the set of twists and fold-transversal gates generate the full logical Clifford group. For the balanced product code, we show that Dehn twists apply to codes in this family with oddq. We also show that the [[90, 8, 10]] bivariate bicycle code is a member of the balanced product code family that saturates the distance bound, and find other balanced product codes that saturate the bound up toq≤ 8 through a numerical search.

IEEE Transactions on Information TheoryMay 19, 20253 citations

A New Description of the Additive Quantum Codes

唐黎明, 刘焕平, 吕学琴

东北数学：英文版Jan 1, 20060 citations

m=1

m=1 our codes are MDS, and when

m=2

m=2 and our lower bound for the minimum distance is 3, the codes are at least Hermitian almost MDS. For an infinite family of parameters, when

m=2

m=2 we prove that our codes beat the Gilbert–Varshamov bound. We also present many examples of our codes that are better than any known code in the literature.

\mathcal{O}(d)

time overhead. Due to having more logical degrees of freedom in the codes, there is a richer structure of attainable logical gates compared to those for topological codes. To illustrate the scheme, we focus on families of hypergraph and balanced product codes that scale as

[[18q^2,8,2q]]_{q\in \mathbb{N}}

and

[[18q,8,\leq 2q]]_{q\in \mathbb{N}}

respectively. For distance 6 to 12 hypergraph product codes, we find that the set of twists and fold-transversal gates generate the full logical Clifford group. For the balanced product code, we show that Dehn twists apply to codes in this family with odd

q

. We also show that the

[[90,8,10]]

bivariate bicycle code is a member of the balanced product code family that saturates the distance bound. We also find balanced product codes that saturate the bound up to

q\leq8

through a numerical search.