Bringing Classical LDPC Design Principles to Quantum LDPC Codes

Classical row-circulant quasi-cyclic (QC) low-density parity check (LDPC) matrices are known to generate efficient high-rate short and moderate-length QC-LDPC codes, while the comparable random structures exhibit numerous short cycles of length-4. Therefore, we conceive a general formalism for constructing nondual-containing Calderbank-Shor-Steane (CSS)-type quantum low-density parity check (QLDPC) codes from arbitrary classical row-circulant QC-LDPC matrices. We apply our proposed formalism to the classical balanced incomplete block design (BIBD)-based row-circulant QC-LDPC codes for demonstrating that our designed codes outperform their dual-containing CSS-type counterparts as well as the entanglement-assisted (EA)-QLDPC codes.Classical row-circulant quasi-cyclic (QC) low-density parity check (LDPC) matrices are known to generate efficient high-rate short and moderate-length QC-LDPC codes, while the comparable random structures exhibit numerous short cycles of length-4. Therefore, we conceive a general formalism for constructing nondual-containing Calderbank-Shor-Steane (CSS)-type quantum low-density parity check (QLDPC) codes from arbitrary classical row-circulant QC-LDPC matrices. We apply our proposed formalism to the classical balanced incomplete block design (BIBD)-based row-circulant QC-LDPC codes for demonstrating that our designed codes outperform their dual-containing CSS-type counterparts as well as the entanglement-assisted (EA)-QLDPC codes.

IEEE Communications LettersDec 10, 201518 citations

Towards practical quantum LDPC codes

Joschka Roffe

Nov 30, 20213 citations

Two-dimensional Entanglement-assisted Quantum Quasi-cyclic Low-density Parity-check Codes

Pavan Kumar, Shayan Srinivasa Garani

For any positive integer $g \ge 2$ , we derive general condition for the existence of a $2g$ -cycle in the Tanner graph of two-dimensional ( $2$ -D) classical quasi-cyclic (QC) low-density parity-check (LDPC) codes. Depending on whether $p$ is an odd prime or a composite number, we construct two distinct families of $2$ -D classical QC-LDPC codes with girth $>4$ by stacking $p \times p \times p$ tensors. Furthermore, using generalized Behrend sequences, we propose an additional family of $2$ -D classical QC-LDPC codes with girth $>6$ , constructed via a similar tensor-stacking approach. All the proposed $2\text{-D}$ classical QC-LDPC codes exhibit an erasure correction capability of at least $p \times p$ . Based on the constructed $2\text{-D}$ classical QC-LDPC codes, we derive two families of $2\text{-D}$ entanglement-assisted (EA) quantum low-density parity-check (QLDPC) codes. The first family of $2\text{-D}$ EA-QLDPC codes is obtained from a pair of $2\text{-D}$ classical QC-LDPC codes and is designed such that the unassisted part of the Tanner graph of the resulting EA-QLDPC code is free of $4$ -cycles, while requiring only a single ebit to be shared across the quantum transceiver. The second family is constructed from a single $2\text{-D}$ classical QC-LDPC code whose Tanner graph is free from $4$ -cycles. Moreover, the constructed EA-QLDPC codes inherit an erasure correction capability of $p \times p$ , as the underlying classical codes possess the same erasure correction property.For any positive integer $g \ge 2$, we derive general condition for the existence of a $2g$-cycle in the Tanner graph of two-dimensional ($2$-D) classical quasi-cyclic (QC) low-density parity-check (LDPC) codes. Depending on whether $p$ is an odd prime or a composite number, we construct two distinct families of $2$-D classical QC-LDPC codes with girth $>4$ by stacking $p \times p \times p$ tensors. Furthermore, using generalized Behrend sequences, we propose an additional family of $2$-D classical QC-LDPC codes with girth $>6$, constructed via a similar tensor-stacking approach. All the proposed $2\text{-D}$ classical QC-LDPC codes exhibit an erasure correction capability of at least $p \times p$. Based on the constructed $2\text{-D}$ classical QC-LDPC codes, we derive two families of $2\text{-D}$ entanglement-assisted (EA) quantum low-density parity-check (QLDPC) codes. The first family of $2\text{-D}$ EA-QLDPC codes is obtained from a pair of $2\text{-D}$ classical QC-LDPC codes and is designed such that the unassisted part of the Tanner graph of the resulting EA-QLDPC code is free of $4$-cycles, while requiring only a single ebit to be shared across the quantum transceiver. The second family is constructed from a single $2\text{-D}$ classical QC-LDPC code whose Tanner graph is free from $4$-cycles. Moreover, the constructed EA-QLDPC codes inherit an erasure correction capability of $p \times p$, as the underlying classical codes possess the same erasure correction property.

arXiv:2601.08927Jan 13, 20260 citations

For any positive integer

g \ge 2

, we derive general condition for the existence of a

2g

-cycle in the Tanner graph of two-dimensional (

2

-D) classical quasi-cyclic (QC) low-density parity-check (LDPC) codes. Depending on whether

p

is an odd prime or a composite number, we construct two distinct families of

2

-D classical QC-LDPC codes with girth

>4

by stacking

p \times p \times p

tensors. Furthermore, using generalized Behrend sequences, we propose an additional family of

2

-D classical QC-LDPC codes with girth

>6

, constructed via a similar tensor-stacking approach. All the proposed

2\text{-D}

classical QC-LDPC codes exhibit an erasure correction capability of at least

p \times p

. Based on the constructed

2\text{-D}

classical QC-LDPC codes, we derive two families of

2\text{-D}

entanglement-assisted (EA) quantum low-density parity-check (QLDPC) codes. The first family of

2\text{-D}

EA-QLDPC codes is obtained from a pair of

2\text{-D}

classical QC-LDPC codes and is designed such that the unassisted part of the Tanner graph of the resulting EA-QLDPC code is free of

4

-cycles, while requiring only a single ebit to be shared across the quantum transceiver. The second family is constructed from a single

2\text{-D}

classical QC-LDPC code whose Tanner graph is free from

4

-cycles. Moreover, the constructed EA-QLDPC codes inherit an erasure correction capability of

p \times p

, as the underlying classical codes possess the same erasure correction property.