Tan, Andrew K. | Engineering Papers

DOE OSTI2023

Perturbative model of noisy quantum signal processing

Recent progress in quantum signal processing (QSP) and its generalization, quantum singular value transformation, has led to a grand unification of quantum algorithms. However, inherent experimental noise in quantum devices severely limits the length of realizable QSP sequences. Here, we consider a model of QSP with generic perturbative noise in the signal processing basis and present a diagrammatic notation useful for analyzing such errors. To demonstrate our technique, we study a specific coherent error, that of under- or overrotation of the signal processing operator parametrized by ε<<1. For this coherent error model, it is shown that while Pauli Z errors are not recoverable without additional resources, Pauli X and Y errors can be arbitrarily suppressed by coherently appending a noisy recovery QSP without the use of additional resources or ancillas. Furthermore, through a careful accounting of errors using our diagrammatic tools, we provide an upper and lower bound on the length of this recovery QSP operator. We anticipate that the perturbative technique and the diagrammatic notation proposed here will facilitate future study of generic noise in QSP and quantum algorithms.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2021

Grand Unification of Quantum Algorithms

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. A number of these quantum algorithms have recently been 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 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. 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 the QSVT naturally emerges. Paralleling GSLW’19, we then employ the 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 the QSVT is a single framework comprising the three major quantum algorithms, suggesting a grand unification of quantum algorithms.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗