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arXiv:2201.08309

Lecture Notes on Quantum Algorithms for Scientific Computation

Lin Lin

Published Jan 20, 2022• 28 citations
arXivHTMLPDF

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.

DOI
10.48550/arxiv.2201.08309

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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.
Eigenproblem arises in a large number of disciplines of sciences and engineering. Combining quantum algorithms to solve linear differential equations, quantum singular value estimation and quantum phase estimation, we propose quantum algorithms to compute the eigenvalues of diagonalizable matrices of two types: matrices that only have real eigenvalues and normal matrices. The quantum algorithms return a quantum state such that the first register stores eigenvalues and the second register stores the corresponding eigenvectors. For normal matrices, the complexity to obtain this quantum state is dominated by the complexity to perform quantum singular value estimation. For the other case, the complexity is determined by the solving of a linear system of differential equations. Under certain conditions (e.g. sparse), the complexities are polylog on the size nnn of the given matrix. If we perform measurements, then we can obtain all the eigenvalues classically. If MMM is sss sparse, then the complexity is O~(sn∥M∥max⁡κ2/ϵ2)\widetilde{O}(sn\|M\|_{\max}\kappa^2/\epsilon^2)O(sn∥M∥max​κ2/ϵ2), where κ\kappaκ is the condition number of the matrix of eigenvectors.
This is a set of lecture notes for a graduate-level course on quantum algorithms, with an emphasis on quantum optimization algorithms. It is developed for applied mathematicians and engineers, and requires no previous background in quantum mechanics. The main topics of this course, in addition to a rigorous introduction to the computational model, are: input/output models, quantum search, the quantum gradient algorithm, matrix manipulation algorithms, the mirror descent framework for semidefinite optimization (including the matrix multiplicative weights update algorithm), adiabatic optimization. This is a preprint for personal use only. Please refer to the printed version of the material.
Significant developments made in quantum hardware and error correction recently have been driving quantum computing towards practical utility. However, gaps remain between abstract quantum algorithmic development and practical applications in computational sciences. In this Perspective article, we propose several properties that scalable quantum computational science methods should possess. We further discuss how block-encodings and polynomial transformations can potentially serve as a unified framework with the desired properties. Recent advancements on these topics are presented including construction and assembly of block-encodings, and various generalizations of quantum signal processing (QSP) algorithms to perform polynomial transformations. The scalability of QSP methods on parallel and distributed quantum architectures is also highlighted. Promising applications in simulation and observable estimation in chemistry, physics, and optimization problems are presented. We hope this Perspective serves as a gentle introduction of state-of-the-art quantum algorithms to the computational science community, and inspires future development on scalable quantum computational science methodologies that bridge theory and practice.
We present a comprehensive and pedagogical study of two core quantum algorithms in quantum linear algebra: the Hadamard test (HT) and quantum phase estimation (QPE).Building on the lecture notes on Quantum Algorithms for Scientific Computing [1] by Lin, we detail their mathematical formulations, circuit implementations, and convergence behavior.Both algorithms are benchmarked on simulators and IBM's 127-qubit Eagle processor (ibm rensselaer) using a simple eigenvalue problem.While QPE performs well in simulation, its hardware performance degrades beyond t = 3 ancilla qubits due to circuit depth and two-qubit gate error.In contrast, HT shows stable samplingbased convergence and greater resilience to hardware noise.These results provide a realistic baseline for implementing phase estimation on noisy intermediate-scale quantum (NISQ) era devices.
Partial differential equations are a promising application area for fault-tolerant quantum algorithms, but the subject lies between two communities with different languages: numerical analysis and quantum computation. These lecture notes provide a numerically grounded introduction for readers entering from either field. Block encoding is the organizing principle: once a discretized differential operator is embedded in a unitary, primitives such as quantum singular value transformation, Hamiltonian simulation, linear combinations of unitaries, amplitude amplification, and measurement can be assembled into algorithms for elliptic, hyperbolic, and parabolic PDEs. Each chapter begins with a standard finite difference or finite element discretization and follows the full pipeline from the continuous PDE to quantum encoding, transformation, and extraction of a quantity of interest. Particular attention is paid to the factors governing end-to-end performance, including discretization error, state preparation, normalization, postselection, and measurement cost. A final chapter introduces nonlinear problems through Carleman and Koopman-von Neumann linearizations. The aim is not a comprehensive survey or a claim of universal quantum advantage, but a mathematically transparent entry point and a shared vocabulary for researchers in both communities.
We present a comprehensive and pedagogical study of two core quantum algorithms in quantum linear algebra: the Hadamard test (HT) and quantum phase estimation (QPE). Building on Prof. Lin's lecture notes on Quantum Algorithms for Scientific Computing, we detail their mathematical formulations, circuit implementations, and convergence behavior. Both algorithms are benchmarked on simulators and IBM's 127-qubit Eagle processor (ibm rensselaer) using a simple eigenvalue problem. While QPE performs well in simulation, its hardware performance degrades beyond t = 3 ancilla qubits due to circuit depth and two-qubit gate noise. In contrast, HT shows stable, sampling-based convergence and greater resilience to hardware noise. We further apply QPE to the transversefield Ising model (TFIM), demonstrating the importance of ansatz quality for accurate ground state energy estimation. These results provide a realistic baseline for implementing phase estimation on noisy intermediate-scale quantum (NISQ) era devices.