Practical Guide to Quantum Computing: Quantum Algorithmic Foundations # 2

Abstract This guide is based on the IBM Quantum course Quantum Algorithmic Foundations and addresses a central question in quantum computation: Do quantum algorithms offer provable advantages beyond the oracle (black-box) query model, in more standard computational settings where inputs are given explicitly? Answering this question requires building a rigorous theoretical foundation. The course begins with a detailed discussion of computational cost in both classical and quantum models, focusing on how complexity scales with input size rather than on absolute execution time. The emphasis is placed on algorithmic efficiency, independent of hardware improvements. To ground the discussion, the material draws heavily from computational number theory, including problems such as: Integer arithmetic Greatest common divisor (GCD) computation Integer factorization Although number theory represents a specific application domain, it provides a clear and powerful framework for analyzing computational complexity and illustrating the principles underlying quantum speedups. A key theme of the lesson is the distinction between algorithmic scalability and hardware performance. The primary concern is how the cost of an algorithm grows as problem instances become larger — a perspective that remains valid across technological generations. The course concludes by addressing a critically important concept: implementing classical computations on quantum computers. The significance of this idea is not to replace classical machines, but to use classical algorithms as subroutines within quantum procedures. This hybrid approach leverages decades of classical algorithmic research while opening new pathways toward quantum computational advantage. Through practical experimentation on IBM Q, this work connects theoretical foundations with hands-on quantum programming, reinforcing the structural principles that underlie advanced quantum algorithm design.