A Technical Journey Through Quantum Computing and Post-Quantum Cryptography

This paper is the third member of a three-paper series on post-quantum cryptography for African financial institutions, alongside Post-Quantum Migration Readiness for African Finance (Paper 1) and A Crypto-Agile Post-Quantum Architecture for Banking (Paper 2). Paper 1 asks whether and how African financial institutions should begin migrating to post-quantum cryptography; Paper 2 asks where post-quantum cryptography belongs inside a bank's own systems. This paper asks what the research team studied, built, and tested to reach the positions Paper 1 and Paper 2 take, and what mathematics and mechanics sit underneath them. It is written as a technical narrative rather than a formal specification. The account starts with the quantum-mechanical foundations of the work (qubits, superposition, entanglement, the gate model of computation) and the phase-estimation and Hamiltonian-simulation ideas that go beyond the team's own circuits. It then covers the mathematics of the quantum threat, with Shor's and Grover's algorithms worked against concrete RSA-2048 and secp256k1 parameters, and the mathematics of the post-quantum replacements: Module-LWE, Module-SIS, and the ML-KEM and ML-DSA constructions built on them. It also explains why NIST has standardized SLH-DSA and is standardizing HQC and FN-DSA alongside them. From there the document turns to the hybrid cryptography mechanism that Paper 2's architecture assumes, a zero-knowledge compliance layer built on STARKs rather than discrete-logarithm proofs, and the team's own validation of quantum circuits on real IBM quantum hardware, reported with the job identifiers and measured data those runs produced. It does not duplicate either paper. It documents the research and experimentation underneath both, at a technical depth that neither paper's narrative form carries. The series follows a zero-fabrication discipline, and this paper does too. Every mathematical claim traces to a cited primary source or to one of two companion technical documents produced during this research (a mathematical-foundations reference and a STARK-integration architecture report), and every experimental claim traces to a specific, named quantum-hardware job. Nothing is presented as executed that was not run. The one area with no hands-on execution, Google's quantum-computing platform, is covered from published literature only and is marked as such.