arXiv:2406.00425

Toward Quantum CSS-T Codes from Sparse Matrices

Eduardo Camps-Moreno, Hiram H. López, Gretchen L. Matthews, Emily McMillon

Published Jul 7, 2024• 4 citations

Abstract

CSS-T codes were recently introduced as quantum error-correcting codes that respect a transversal gate. A CSS-T code depends on a pair ($C_{1}, C_{2}$) of binary linear codes$C_{1}$and$C_{2}$that satisfy certain conditions. We prove that$C_{1}$and$C_{2}$form a CSS-T pair if and only if$C_{2}\subset \text{Hull}(C_{1})\cap \text{Hull}(C_{1}^{2})$, where the hull of a code is the intersection of the code with its dual. We show that if ($C_{1}, C_{2}$) is a CSS-T pair, and the code$C_{2}$is degenerated on$\{i\}$, meaning that the$i^{th}$-entry is zero for all the elements in$C_{2}$, then the pair of punctured codes ($C_{1}\vert_{i}, C_{2}\vert_{i}$) is also a CSS-T pair. Finally, we provide Magma code based on our results and quasi-cyclic codes as a step toward finding quantum LDPC or LDGM CSS-T codes computationally.

DOI: 10.1109/isit-w61686.2024.10591766

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arXiv:2406.00425

Toward Quantum CSS-T Codes from Sparse Matrices

Eduardo Camps-Moreno, Hiram H. López, Gretchen L. Matthews, Emily McMillon

Published Jul 7, 2024• 4 citations

arXiv HTML PDF

Abstract

CSS-T codes were recently introduced as quantum error-correcting codes that respect a transversal gate. A CSS-T code depends on a pair (

C_{1}, C_{2}

) of binary linear codes

C_{1}

and

C_{2}

that satisfy certain conditions. We prove that

C_{1}

and

C_{2}

form a CSS-T pair if and only if

C_{2}\subset \text{Hull}(C_{1})\cap \text{Hull}(C_{1}^{2})

, where the hull of a code is the intersection of the code with its dual. We show that if (

C_{1}, C_{2}

) is a CSS-T pair, and the code

C_{2}

is degenerated on

\{i\}

, meaning that the

i^{th}

-entry is zero for all the elements in

C_{2}

, then the pair of punctured codes (

C_{1}\vert_{i}, C_{2}\vert_{i}

) is also a CSS-T pair. Finally, we provide Magma code based on our results and quasi-cyclic codes as a step toward finding quantum LDPC or LDGM CSS-T codes computationally.

DOI: 10.1109/isit-w61686.2024.10591766

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Eduardo Camps-Moreno, Hiram H. López, Gretchen L. Matthews +3 more

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IEEE International Symposium on Information Theory (ISIT)Jun 26, 20221 citation

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cs.ITCross-listedarXiv:2601.08636IEEE International Symposium on Information Theory (ISIT)Jan 14, 20260 citations

CSS-T codes were recently introduced as quantum error-correcting codes that respect a transversal gate. A CSS-T code depends on a pair (

C_{1}, C_{2}

) of binary linear codes

C_{1}

and

C_{2}

that satisfy certain conditions. We prove that

C_{1}

and

C_{2}

form a CSS-T pair if and only if

C_{2}\subset \text{Hull}(C_{1})\cap \text{Hull}(C_{1}^{2})

, where the hull of a code is the intersection of the code with its dual. We show that if (

C_{1}, C_{2}

) is a CSS-T pair, and the code

C_{2}

is degenerated on

\{i\}

, meaning that the

i^{th}

-entry is zero for all the elements in

C_{2}

, then the pair of punctured codes (

C_{1}\vert_{i}, C_{2}\vert_{i}

) is also a CSS-T pair. Finally, we provide Magma code based on our results and quasi-cyclic codes as a step toward finding quantum LDPC or LDGM CSS-T codes computationally.

Abstract CSS-T codes were recently introduced as quantum error-correcting codes that respect a transversal gate. A CSS-T code depends on a CSS-T pair, which is a pair of binary codes

(C_1, C_2)

( C 1 , C 2 ) such that

C_1

C 1 contains

C_2

C 2 ,

C_2

C 2 is even, and the shortening of the dual of

C_1

C 1 with respect to the support of each codeword of

C_2

C 2 is self-dual. In this paper, we give new conditions to guarantee that a pair of binary codes

(C_1, C_2)

( C 1 , C 2 ) is a CSS-T pair. We define the poset of CSS-T pairs and determine the minimal and maximal elements of the poset. We provide a propagation rule for nondegenerate CSS-T codes. We apply some main results to Reed–Muller, cyclic and extended cyclic codes. We characterize CSS-T pairs of cyclic codes in terms of the defining cyclotomic cosets. We find cyclic and extended cyclic codes to obtain quantum codes with better parameters than those in the literature.

CSS-

T

codes over Binary Extension Fields and their Physical Foundations

We investigate the class of CSS-

T

codes, a family of quantum error-correcting codes that allows for a transversal

T

-gate. We extend the definition of a pair of linear codes

(C_1,C_2)

C_i\subseteq\mathbb{F}_q^n

, forming a

q

-ary CSS-

T

code over binary extension fields, and demonstrate the existence of asymptotically good sequences of LDPC CSS-

T

codes over any such field.

(u \mid u+v)

-construction, that is, the matrix-product code constructed from two constituent codes. Moreover, for cyclic constituent codes, we provide an explicit characterization in terms of the defining cyclotomic sets, which extends existing results for cyclic codes. This framework allows the construction of new and longer CSS-T codes.

\mathcal{C}

\operatorname{CSS} \left( {X,{\mathcal{C}_2};Z,\mathcal{C}_1^ \bot } \right)

, where the [n, k2] binary code

{\mathcal{C}_2}

determines the X-stabilizers in

\mathcal{C}

, and the [n, n−k1] binary code $\mathcal{C}_1^ \bot $ determines the Z-stabilizers in

\mathcal{C}

. The diagonal entries of a diagonal physical gate are indexed by binary vectors in

\mathbb{F}_2^n

. We show that a diagonal physical gate preserves the CSS codespace if and only if entries from the same coset of

{\mathcal{C}_2}

{\mathcal{C}_1}

(same X-logical) are identical. We also show that the target logical operator only specifies

{2^{{k_1}}}

out of 2ndiagonal entries of the diagonal physical gate. The remaining degrees of freedom can be used to optimize implementation of the physical gate within a particular quantum computing infrastructure. This encompasses optimization with respect to locality of the physical gate, a criterion that is essential to fault tolerance. When the target logical operator is the identity, the physical gates that preserve the CSS code represent noise operators to which the codespace is oblivious. We illustrate our method by providing several examples of code-gate pairs for which the target logical gate is a non-Clifford unitary. The framework is extended to stabilizer codes in https://arxiv.org/abs/2109.13481.