Scalable Quantum Computational Science: A Perspective from Block-Encodings and Polynomial Transformations

TL;DR

This Perspective argues that scalable quantum computational science can be built around block-encoding (BE) and polynomial transformations realized via quantum signal processing (QSP) and its generalizations. BE provides robust matrix encoding while QSP/GQSP enable systematic, modular polynomial transforms that approximate complex functions on encoded data, with clear resource and error tradeoffs. The article surveys construction techniques, circuit realizations, and software tooling for BE, analyzes algorithmic-level error correction and tradeoffs, and outlines parallel and distributed approaches to scale QSP on heterogeneous architectures. It also demonstrates applications across real-time and imaginary-time evolution, expectation-value estimation, and problems in chemistry, physics, and optimization, illustrating practical pathways toward fault-tolerant quantum computational science. The authors call for co-design of hardware, software, and domain-application methods to realize near-term and fault-tolerant quantum advantages.

Abstract

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.

Figures (11)

• Figure 1: Figure illustrates an overview of block-encoding and polynomial transformation techniques as building blocks of scalable quantum computational science for various applications. Each algorithmic primitive in the middle (light grey) can be used to tackle different applications in the outer part (light blue). For example, the dark color aims to perform Hamiltonian simulation by combining modular-QSP with approximate block-encoding.
• Figure 2: Circuit to block-encode a sparse matrix $A$.
• Figure 3: Circuit to assemble the linear combination of unitaries.
• Figure 4: Circuit to implement the prepare operation $U_P$. Here the angles $\theta_i$ can be efficiently calculated for a given state.
• Figure 5: Circuit to assemble the select operation using multi-control gates.