Small Quantum Codes from Algebraic Extensions of Generalized Bicycle Codes

TL;DR

This work develops small, experimentally feasible quantum LDPC codes by algebraically extending generalized bicycle codes. It introduces extended GB constructions that preserve a lower bound on dimension while enabling scalable growth of code length through polynomial extensions, and analyzes their sparsity and decoding performance. Thresholds are demonstrated with BP+OSD decoding under a depolarizing model, showing competitive performance to surface codes for small distances while encoding more logical qubits. Two scalable constructions are proposed to realize larger codes on chips with restricted connectivity, balancing sparsity and practicality for near-term quantum hardware. The results provide a path toward implementable quantum memories and highlight directions for fault-tolerant circuit-level analyses and broader code-family extensions.

Abstract

Quantum error correction is rapidly seeing first experimental implementations, but there is a significant gap between asymptotically optimal error-correcting codes and codes that are experimentally feasible. Quantum LDPC codes range from the surface code, which has a vanishing encoding rate, to very promising codes with constant encoding rate and linear distance. In this work, motivated by current small-scale experimental quantum processing units, we devise small quantum codes that are inspired by a subset of quantum LDPC codes, known as generalized bicycle (GB) codes. We introduce a code construction based on algebraic manipulation of the parity-check matrix of GB codes, rather than manipulation of Tanner graphs. Our construction leads to families of quantum LDPC codes of small size, and we demonstrate numerically that their performance scales comparably to the performance of surface codes for similar sizes under a phenomenological noise model. The advantage of our code family is that they encode many logical qubits in one code, at the expense of non-local connectivity. We then explore three variants of the code construction focusing on reducing the long-range connectivity by bringing it closer to the current experimental capabilities of short-range connectivity devices.

Figures (4)

• Figure 1: Small GB base codes.(a) Logical error rate (LER) of GB codes at ${\rm PER} = 0.010$. We include all GB codes with non-zero dimension at $\ell = 4$ and $\ell = 5$ (49 and 226 in total respectively). Codes are marked with triangles which are pointing down if the distance is less than 3 and pointing up if the distance is at least 3. It is clear that we need $\ell \geq 5$ to achieve distance 3, and only codes with distance 3 produce a low LER (below the dashed line at 0.020). (b) Maximum number of qubits participating in each parity check, equal to the sum of the polynomial weights $q = {\rm wt}\space(a) +{\rm wt}\space(b)$. The codes are ordered in increasing $q$, and we cap the weight of the codes we consider at 8 (dashed line). As $\ell$ increases beyond 5, this restricts the fraction of codes we consider more.
• Figure 2: Small qLDPC code family obtained from Algorithm \ref{['alg:gb_construction']}. We plot the performance of a family ${\mathcal{F}}_{\rm qLDPC}$ of small qLDPC codes. The $[[10, 2, 3]]$ base code has the smallest code length that guarantees distance $d \geq 3$, and it is constructed by polynomials $a(x) = 1 + x^4$ and $b(x) = 1 + x + x^2 + x^4$. The construction of the remaining codes in ${\mathcal{F}}_{\rm qLDPC}$ follows Algorithm \ref{['alg:gb_construction']} with $\kappa_m = m$ and $p^{(m)}(x) = 1$, leading to qLDPC codes with $\ell_m = 5m$ for $m = 1, \dots, 5$. We compare the performance of the first five codes in ${\mathcal{F}}_{\rm qLDPC}$ with the surface code of distances $d = 3,5,7$. The performance of codes in ${\mathcal{F}}_{\rm qLDPC}$ improves with increasing code length and the threshold point is around ${\rm PER} = 0.145$, similar to the surface code threshold that we plot.
• Figure 3: Chip architecture of the extended code of Theorem \ref{['thm:scalable']}. The chip architecture of the $(m+1)$-th family member is composed of three identical chips describing the connectivity of the $m$-th family member. Each chip consists of data and syndrome qubits represented by black dots. To avoid imposing a particular connectivity on the $m$-th family member, we have its schematic indicated by a cloudy hatching. Notice that there is connectivity between the qubits of the first and the second chips, between the qubits of the second and the third chips, and between the qubits of the first and the third chips such that the connectivity between all qubits is according to the required connectivity of the extended code.
• Figure 4: qLDPC families for different base code lengths. Each plot displays a qLDPC code family with given $[[\ell_1, k_1]]$ base code $\ell_1$ for $\ell_1 = 5,6,7,8,9,10$. The base code is defined by polynomials $a(x),b(x) \in {\mathbb F}_2^{\langle \ell_1 \rangle}$ provided in the text and the code dimension is given by $k_1 = 2\space{\rm deg}\space g(x)$ according to Proposition \ref{['prop:k']}. For each code family, we plot the smallest four codes, showing the breakeven point of each code and the threshold point. We compare them with the surface code of distances $3,5,7$.

Theorems & Definitions (15)

• Definition 1
• Proposition 2: GB code dimension Panteleev2021degenerate
• Theorem 3
• proof
• Theorem 4
• proof
• Definition 5
• Theorem 6
• Theorem 7
• Definition 8