A Grand Unification of Quantum Algorithms

TL;DR

This paper provides a comprehensive tutorial on quantum singular value transformation (QSVT), unifying major quantum algorithms like search, phase estimation, and Hamiltonian simulation into a single framework, highlighting their interconnectedness.

Contribution

It introduces a pedagogical overview of QSVT, demonstrating how it unifies key quantum algorithms and extends quantum signal processing to a broader eigenvalue transform.

Findings

QSVT encompasses major quantum algorithms

Quantum algorithms for search, phase estimation, and simulation are unified under QSVT

QSVT offers a single framework for diverse quantum computational tasks

Abstract

Quantum algorithms offer significant speedups over their classical counterparts for a variety of problems. The strongest arguments for this advantage are borne by algorithms for quantum search, quantum phase estimation, and Hamiltonian simulation, which appear as subroutines for large families of composite quantum algorithms. A number of these quantum algorithms were recently tied together by a novel technique known as the quantum singular value transformation (QSVT), which enables one to perform a polynomial transformation of the singular values of a linear operator embedded in a unitary matrix. In the seminal GSLW'19 paper on QSVT [Gily\'en, Su, Low, and Wiebe, ACM STOC 2019], many algorithms are encompassed, including amplitude amplification, methods for the quantum linear systems problem, and quantum simulation. Here, we provide a pedagogical tutorial through these developments, first illustrating how quantum signal processing may be generalized to the quantum eigenvalue transform, from which QSVT naturally emerges. Paralleling GSLW'19, we then employ QSVT to construct intuitive quantum algorithms for search, phase estimation, and Hamiltonian simulation, and also showcase algorithms for the eigenvalue threshold problem and matrix inversion. This overview illustrates how QSVT is a single framework comprising the three major quantum algorithms, thus suggesting a grand unification of quantum algorithms.