Assessing Finite Scalability in Early Fault-Tolerant Quantum Computing for Homogeneous Catalysts

As quantum hardware advances toward fault-tolerant operation, an intermediate stage known as early fault-tolerant quantum computing (EFTQC) is emerging, where partial error correction enables meaningful computation. In this regime, the ability of quantum processors to scale in size and depth has become a crucial factor shaping their achievable performance. This study investigates how finite scalability influences resource requirements for simulating open-shell catalytic systems using Quantum Phase Estimation (QPE). The analysis compares hardware archetypes distinguished by fidelity or operation speed under two representative scalability models. Finite scalability increases qubit and runtime demands yet leaves overall scaling behavior intact, with high-fidelity architectures requiring lower minimum scalability to solve equally sized problems. These effects are largely independent of the chosen scalability model. Extending this framework, we examine runtime competitiveness across hardware and code configurations, incorporating surface-code and quantum Low-Density Parity-Check (LDPC)-based fault tolerance under finite scalability. The results identify operating regimes where high-fidelity architectures remain competitive despite slower gate speeds and show that LDPC codes further expand this regime by reducing space-time overhead. Together, these findings highlight the central role of scalability in quantifying performance and guiding the design of next-generation quantum hardware. Continued progress in scalable architectures will be essential for extending quantum computing to increasingly complex scientific and industrial applications.