Logical Operators and Fold-Transversal Gates of Bivariate Bicycle Codes

TL;DR

This work advances fault-tolerant quantum computation with quantum LDPC codes by developing a homological-algebraic framework for bivariate bicycle (BB) codes and related group-algebra codes. It introduces purity and principal concepts to structure logical operators, and leverages ZX dualities and code automorphisms to realize fold-transversal Clifford gates with no overhead, including explicit gate sets for symmetric BB codes. The authors construct two concrete BB codes, [[98,6,12]] and [[162,8,12]], and provide detailed analyses of their logical operators and fold-transversal gates, revealing rich group structures such as C2×PSL2(F_{2^3}) and Sp_2(F_{2^2})×(Sp_2(F_{2^2})⋊C2). The mathematical foundations connect BB codes to balanced product/Künneth-type frameworks, broaden the toolkit for designing scalable, fault-tolerant quantum memories, and open avenues for higher-dimensional generalizations and decoding implications.

Abstract

Quantum low-density parity-check (qLDPC) codes offer a promising route to scalable fault-tolerant quantum computation with constant overhead. Recent advancements have shown that qLDPC codes can outperform the quantum memory capability of surface codes even with near-term hardware. The question of how to implement logical gates fault-tolerantly for these codes is still open. We present new examples of high-rate bivariate bicycle (BB) codes with enhanced symmetry properties. These codes feature explicit nice bases of logical operators (similar to toric codes) and support fold-transversal Clifford gates without overhead. As examples, we construct $[[98,6,12]]$ and $[[162, 8, 12]]$ BB codes which admit interesting fault-tolerant Clifford gates. Our work also lays the mathematical foundations for explicit bases of logical operators and fold-transversal gates in quantum two-block and group algebra codes, which might be of independent interest.

Figures (3)

• Figure 1: A $[[98, 8, 12]]$ BB code $\mathcal{C}(c,d)$ for $c= x + y^3 + y^4$ and $d= y + x^3 + x^4$. Physical qubits correspond to vertical and horizontal edges and stabilizers to vertices and faces on a $7\times 7$-grid.
• Figure 2: The toric code as the two block group algebra code, see \ref{['sec:groupalgebracodedefinition']}, associated with $c = 1 + x$ and $d = 1 + y$ in $R = \mathbb{F}_2[x,y]/(x^3 - 1,y^3 - 1)$. It is also an example of a bivariate bicycle code, see \ref{['sec:bbcodedefinition']}. In (a), we see that the horizontal qubits (in white) and vertical qubits (in grey) are labeled by monomials (which form the group $\mathbb{Z}/\ell \times \mathbb{Z}/m$). Each monomial also induces an $X$-type stabilizer and a $Z$-type stabilizer as in \ref{['eq:checksofgroupalgebracodes']}. In (b), we depict examples of those, namely $X_{yx^2}$ and $Z_{y}$.
• Figure 3: The phase-type fold-transversal gate $\operatorname{CZ}_{\tau_0\omega}$ on the $[[98,8,12]]$ BB code, see \ref{['fig:77code']}. The gate is associated to the $Z\!X$-duality $\tau_0\omega$ exchanging $x$ with $y^{-1}$ and $y$ with $x^{-1}$. This corresponds to folding/reflecting the lattice along the diagonal passing through qubits labeled by $x^ny^{-n}.$ In this example, the fold-transversal gates generate the group $C_2\times\operatorname{Sp}_2(\mathbb{F}_{2^3}).$

Theorems & Definitions (42)

• Remark 2.1
• Definition 2.2
• Theorem 2.3
• proof
• Remark 2.4
• Corollary 2.5
• Corollary 2.6
• proof
• Corollary 2.7
• proof