Grand Unification of Quantum Algorithms

John M Martyn,Andrew K Tan,Isaac L Chuang,Zane M Rossi

doi:10.1103/prxquantum.2.040203

John M Martyn, Andrew K Tan + Show 2 more

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https://doi.org/10.1103/prxquantum.2.040203

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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.

Highlights

Quantum algorithms offer significant speed-ups over their classical counterparts for a variety of problems
The essential ideas behind quantum signal processing (QSP) originate from the early days of practical control of two-level quantum systems, with nuclear magnetic resonance [9,10,11]. We present in this tutorial a pedagogical overview of the modern approach to quantum search, factoring, and simulation, focusing on how all three of these central quantum algorithms may be unified as instances of the recently developed quantum singular value transform (QSVT) algorithm [12]
We present a quantum singular value transformation (QSVT)-based algorithm for the eigenvalue threshold problem, wherein one wishes to know if a matrix has an eigenvalue above some threshold, and use this to simplify the presentation of the use of the QSVT for phase estimation

Summary

Introduction

Quantum algorithms offer significant speed-ups 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. In the seminal GSLW’19 paper on the QSVT [Gilyén et al, ACM STOC 2019], many algorithms are encompassed, including amplitude amplification, methods for the quantum linear systems problem, and quantum simulation. Paralleling GSLW’19, we employ the QSVT to construct intuitive quantum algorithms for search, phase estimation, and Hamiltonian simulation, and showcase algorithms for the eigenvalue threshold problem and matrix inversion. This overview illustrates how the QSVT is a single framework comprising the three major quantum algorithms, suggesting a grand unification of quantum algorithms

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