Modal analysis on quantum computers via qubitization

TL;DR

This paper demonstrates how quantum algorithms, specifically qubitization, can be applied to compute natural frequencies and normal modes of vibrational systems, showcasing potential for large-scale quantum vibrational analysis.

Contribution

It introduces a qubitization-based quantum algorithm for vibrational analysis, including explicit construction of block-encoding oracles and initial state preparation methods.

Findings

Shows how to apply quantum phase estimation to vibrational problems

Provides estimates of qubits and runtime for fault-tolerant quantum computers

Demonstrates the method on simple coupled oscillator examples

Abstract

Natural frequencies and normal modes are basic properties of a structure which play important roles in analyses of its vibrational characteristics. As their computation reduces to solving eigenvalue problems, it is a natural arena for application of quantum phase estimation algorithms, in particular for large systems. In this note, we take up some simple examples of (classical) coupled oscillators and show how the algorithm works by using qubitization methods based on a sparse structure of the matrix. We explicitly construct block-encoding oracles along the way, propose a way to prepare initial states, and briefly touch on a more generic oracle construction for systems with repetitive structure. As a demonstration, we also give rough estimates of the necessary number of physical qubits and actual runtime it takes when carried out on a fault-tolerant quantum computer.