Lecture Notes on Quantum Algorithms for Scientific Computation

TL;DR

The notes assemble a focused treatment of quantum algorithms relevant to scientific computation, foregrounding quantum phase estimation and its post-QPE extensions. By building from foundational quantum mechanics and circuit concepts to concrete algorithms such as HHL and Poisson-solvers, the document demonstrates how phase information and spectral processing enable efficient solutions to eigenvalue problems, linear systems, and differential equations on fault-tolerant quantum computers. Key contributions include a clear exposition of QPE variants (Hadamard test, Kitaev's method, QFT-based QPE), the HHL framework with controlled rotations, and illustrative cases like ground-state energy estimation and Poisson's equation, alongside careful complexity considerations and practical caveats. Overall, the notes map a path from theory to actionable quantum algorithms for large-scale scientific computing tasks, highlighting where quantum advantage can arise and where classical methods may still prevail.

Abstract

This is a set of lecture notes used in a graduate topic class in applied mathematics called ``Quantum Algorithms for Scientific Computation'' at the Department of Mathematics, UC Berkeley during the fall semester of 2021. These lecture notes focus only on quantum algorithms closely related to scientific computation, and in particular, matrix computation. The main purpose of the lecture notes is to introduce quantum phase estimation (QPE) and ``post-QPE'' methods such as block encoding, quantum signal processing, and quantum singular value transformation, and to demonstrate their applications in solving eigenvalue problems, linear systems of equations, and differential equations. The intended audience is the broad computational science and engineering (CSE) community interested in using fault-tolerant quantum computers to solve challenging scientific computing problems.

Figures (40)

• Figure 1.1: Copying classical information using multi-qubit CNOT gates.
• Figure 1.2: Circuit for uncomputation. The $\operatorname{CNOT}$ and $\operatorname{SWAP}$ operators indicate the multi-qubit copy and swap operations, respectively.
• Figure 2.1: Quantum circuit for Deutsch's algorithm.
• Figure 2.2: Geometric interpretation of one Grover iteration.
• Figure 2.3: Implementing $R_{\psi_0}$ for a three qubit system.

Theorems & Definitions (42)

• Proposition 1.1: Hybrid argument
• proof
• Proposition 1.2: Duhamel's principle for Hamiltonian simulation
• proof
• Theorem 1.3: Solovay-Kitaev
• Proposition 4.1: Controlled rotation given rotation angles
• proof
• Proposition 4.2: Diagonalization of tridiagonal matrices
• proof
• Proposition 4.3