Finite-Degree Quantum LDPC Codes Reaching the Gilbert-Varshamov Bound

Abstract

We construct asymptotically good nested Calderbank-Shor-Steane (CSS) code pairs from Hsu-Anastasopoulos codes and MacKay-Neal codes. In the fixed-degree regime, we prove that the coding rate stays bounded away from zero and that the relative distances on both sides stay bounded away from zero with probability tending to one as the blocklength grows. Moreover, within an explicit low-degree search window, we determine exactly which even regular degree choices in our construction attain the classical Gilbert-Varshamov (GV) bound on both constituent sides, and consequently the CSS GV bound at fixed finite degree.

Figures (5)

• Figure 1: Z-side extended parity-check matrix for the illustrative example $(j_Z,k_Z,j_\Delta,k_\Delta,k)=(3,8,2,8,2)$, $n=40$, $m_Z=15$, and $m_\Delta=10$ (hence $m_X=25$): $H_Z^\prime= A_Z0BI_n \in \mathbb F_2^{55\times 80}.$ The blue block in the upper left is the $(3,8)$-regular matrix $A_Z\in\mathbb F_2^{15\times 40}$, the light-blue block in the lower left is the square $(2,2)$-regular sparse map $B\in\mathbb F_2^{40\times 40}$, the yellow block in the lower right represents $I_n$, and the upper-right block is zero. Black lines indicate block boundaries.
• Figure 2: X-side extended parity-check matrix for the same example: $H_X^\prime=[A_X^T\ B^T]=[A_Z^T\ A_\Delta^T\ B^T]\in \mathbb F_2^{40\times 65}$. From left to right, the blue block is the $(3,8)$-regular matrix $A_Z^T\in\mathbb F_2^{40\times 15}$, the red block is the $(2,8)$-regular matrix $A_\Delta^T\in\mathbb F_2^{40\times 10}$, and the light-blue block is $B^T\in\mathbb F_2^{40\times 40}$. Thus the stacked structure of $A_X=[A_Z;A_\Delta]\in\mathbb F_2^{25\times 40}$ is visualized directly. Black lines indicate block boundaries.
• Figure 3: Z-side compressed parity-check matrix for the same example: $H_Z\in\mathbb F_2^{16\times 40}.$ It is obtained by projecting a basis of the kernel of $[A_Z^T\ B^T]$ to the visible component, without taking RREF on the final visible matrix. Thus each row represents an explicit generator of $C_Z^\perp$.
• Figure 4: X-side compressed parity-check matrix for the same example: $H_X\in\mathbb F_2^{16\times 40}.$ If $K_X$ is a basis matrix of $\operatorname{Ker}(A_X)$, then this figure displays $H_X=K_XB^T$ directly, again without taking RREF on the final visible matrix. Thus each row represents an explicit generator of $C_X^\perp$.
• Figure 5: Comparison between the finite-degree numerical proxy $\widehat{\delta}^{\mathrm{lin}}$ and the GV curve for small balanced triples. The axes are drawn from the origin, and the balanced GV curve is shown over the full range $0\le R_Q^{\mathrm{des}}\le 1$. The plotted tuples exhaust the balanced triples satisfying $k\le 30$, $j_Z\le 10$, and $j_Z<k/2$; odd cases are included as well. Marker shapes distinguish representative rigorously certified finite-degree GV points, numerically near-GV points without certification, and positive-proxy but non-GV points. Tuples with $\widehat{\delta}^{\mathrm{lin}}=0$ are omitted, and each displayed point is labeled by its triple $(j_Z,j_X,k)$.

Theorems & Definitions (55)

• Definition 2.1: General Framework of the Proposed Construction
• Theorem 2.2: Nested CSS pair
• proof
• Definition 2.3: Compressed parity-check matrices
• Proposition 2.4: Dimension formulas for the actual rates
• proof
• Definition 2.5: Nested regular sparse family
• Definition 2.6: Design rates
• Proposition 2.7: Formula for the design quantum rate
• proof