A Step-by-Step HHL Algorithm Walkthrough to Enhance Understanding of Critical Quantum Computing Concepts

Abstract

After learning basic quantum computing concepts, it is desirable to reinforce the learning using an important and relatively complex algorithm through which students can observe and appreciate how qubits evolve and interact with each other. Harrow-Hassidim-Lloyd (HHL) quantum algorithm, which can solve linear system problems with exponential speed-up over the classical method and is the basis of many important quantum computing algorithms, is used to serve this purpose. The HHL algorithm is explained analytically followed by a 4-qubit numerical example in bra-ket notation. Matlab code corresponding to the numerical example is available for students to gain a deeper understanding of the HHL algorithm from a pure matrix point of view. A quantum circuit programmed using qiskit is also provided for real hardware execution in IBM quantum computers. After going through the material, students are expected to have a better appreciation of the concepts such as basis transformation, bra-ket and matrix representations, superposition, entanglement, controlled operations, measurement, quantum Fourier transformation, quantum phase estimation, and quantum programming. To help readers review these basic concepts, brief explanations augmented by the HHL numerical examples in the main text are provided in the Appendix.

Keywords

• Harrow-Hassidim-Lloyd (HHL) quantum algorithm
• quantum Fourier transform (QFT)
• inverse quantum Fourier transform (IQFT)
• quantum phase estimation (QPE)
• linear system problem (LSP)
• quantum education