From Quantum Algorithm Primitives to Design Patterns: A Unified Framework for NISQ and Fault‐Tolerant Regimes

ABSTRACT Existing surveys of quantum algorithms largely treat noisy intermediate‐scale quantum ( nisq ) heuristics and fault‐tolerant ( ft ) protocols in isolation. This review bridges that divide through a primitives‐to‐patterns framework, decomposing modern quantum algorithms into seven reusable building blocks, namely block‐encodings, quantum signal processing ( qsp ), quantum singular value transformation ( qsvt ), amplitude amplification, quantum walks, shadow tomography, and variational ansatz circuits, from which eight recurring design patterns are extracted. The framework supplies: (i) lifecycle mappings from problem formulations through nisq ‐feasible heuristics to asymptotically optimal ft algorithms, with particular scrutiny of the nascent early fault‐tolerant regime; (ii) multi‐dimensional algorithm comparisons by T‐count, circuit depth, qubit overhead, and classical co‐processing cost under explicit error‐correction assumptions; (iii) decision guidelines linking problem characteristics to algorithm families alongside regime‐dependent caveats; (iv) a consolidated treatment of trainability barriers, classical simulability bounds, shadow‐based certification, and error‐mitigation strategies; and (v) an evidence‐grounded appraisal of recent “Willow era” benchmarks redefining verifiable quantum utility. Validation against end‐to‐end resource estimates for quantum chemistry, combinatorial optimization, machine learning, and differential‐equation solvers exposes boundary cases and failure modes. A research roadmap highlights under‐explored primitive, problem, and hardware combinations alongside open questions that remain beyond the framework's current reach.