Structure-Aware Placement and Routing of Multi-Controlled Toffoli on Bivariate Bicycle Code Architectures

Quantum error correction is essential for scaling quantum computers, with bivariate bicycle (BB) codes emerging as a promising candidate due to their favorable encoding rates and structured connectivity. However, mapping high-level logical operations like the multi-controlled Toffoli (MCT) gate onto these modular architectures presents significant compilation challenges, particularly regarding inter-module routing and the placement of magic-state factories. This paper addresses these bottlenecks by developing a structure-aware compilation strategy.

The authors leverage the hierarchical binary-tree structure inherent in optimal-Toffoli-depth MCT decompositions. By grouping interacting subtrees of the decomposition into individual BB-code modules, the compiler minimizes the need for long-range inter-module communication. The study evaluates this strategy against a baseline of sequential first-fit placement. Furthermore, the authors investigate the impact of architectural topology by comparing linear arrangements of modules against grid-based layouts, utilizing a facility-location heuristic to optimize the placement of magic-state factories.

The structure-aware placement strategy consistently outperforms the naive baseline, achieving a maximum reduction of 16.02% in inter-module instruction counts. When comparing topologies, the grid-based layout proves superior to linear architectures, with improvements in routing efficiency ranging from 6.1% to 27.7% depending on the grouping factor. The authors also integrate the bicycle-ISA error estimator to model the practical viability of these circuits, finding that for a physical error rate of 10^-3, the circuit failure threshold is reached at approximately 31 to 34 controls, depending on the configuration.

As quantum computing transitions from the NISQ era to fault-tolerant regimes, the focus of compilation must shift from physical qubit mapping to logical-level resource management. This work provides a concrete framework for optimizing complex logical gates on qLDPC-based architectures, demonstrating that architectural topology and structure-aware placement are critical factors in minimizing the overhead of fault-tolerant quantum computation.