Advancing Quantum Complexity Theory: Bridging NISQ Devices and Theoretical Foundations for Next-Generation Quantum Computing

Quantum computing has experienced significant advancements over recent decades, leading to a burgeoning need for a robust theoretical framework in quantum complexity theory akin to that of classical computational complexity. This theory addresses the computational limits and structural constraints that underpin algorithm development. Quantum complexity theory has evolved particularly in response to the challenges and capabilities of Noisy Intermediate-Scale Quantum (NISQ) devices. These devices represent a critical phase in quantum technology, where algorithms that blend classical and quantum computational processes are progressively being optimized. This work provides a foundational overview of quantum complexity theory, emphasizing its developmental trajectory parallel to that of quantum device engineering. It explores the integration of quantum complexity with hybrid algorithms suited for the current landscape of available quantum technologies. Moreover, this paper outlines the pivotal role of quantum complexity theory in rationalizing the operational thresholds and potentials of NISQ devices, which are crucial for the next-generation advancements in quantum computing.