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arXiv:2304.14392 · Journal of Mathematical Physics

Semantic embedding for quantum algorithms

Zane M. Rossi, Isaac L. Chuang

Published Dec 1, 2023• 6 citations
arXivHTMLPDF

Abstract

The study of classical algorithms is supported by an immense understructure, founded in logic, type, and category theory, that allows an algorithmist to reason about the sequential manipulation of data irrespective of a computation’s realizing dynamics. As quantum computing matures, a similar need has developed for an assurance of the correctness of high-level quantum algorithmic reasoning. Parallel to this need, many quantum algorithms have been unified and improved using quantum signal processing (QSP) and quantum singular value transformation (QSVT), which characterize the ability, by alternating circuit ansätze, to transform the singular values of sub-blocks of unitary matrices by polynomial functions. However, while the algebraic manipulation of polynomials is simple (e.g., compositions and products), the QSP/QSVT circuits realizing analogous manipulations of their embedded polynomials are non-obvious. This work constructs and characterizes the runtime and expressivity of QSP/QSVT protocols where circuit manipulation maps naturally to the algebraic manipulation of functional transforms (termed semantic embedding). In this way, QSP/QSVT can be treated and combined modularly, purely in terms of the functional transforms they embed, with key guarantees on the computability and modularity of the realizing circuits. We also identify existing quantum algorithms whose use of semantic embedding is implicit, spanning from distributed search to proofs of soundness in quantum cryptography. The methods used, based in category theory, establish a theory of semantically embeddable quantum algorithms, and provide a new role for QSP/QSVT in reducing sophisticated algorithmic problems to simpler algebraic ones.

DOI
10.1063/5.0160910

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The study of classical algorithms is supported by an immense understructure, founded in logic, type, and category theory, that allows an algorithmist to reason about the sequential manipulation of data irrespective of a computation’s realizing dynamics. As quantum computing matures, a similar need has developed for an assurance of the correctness of high-level quantum algorithmic reasoning. Parallel to this need, many quantum algorithms have been unified and improved using quantum signal processing (QSP) and quantum singular value transformation (QSVT), which characterize the ability, by alternating circuit ansätze, to transform the singular values of sub-blocks of unitary matrices by polynomial functions. However, while the algebraic manipulation of polynomials is simple (e.g., compositions and products), the QSP/QSVT circuits realizing analogous manipulations of their embedded polynomials are non-obvious. This work constructs and characterizes the runtime and expressivity of QSP/QSVT protocols where circuit manipulation maps naturally to the algebraic manipulation of functional transforms (termed semantic embedding). In this way, QSP/QSVT can be treated and combined modularly, purely in terms of the functional transforms they embed, with key guarantees on the computability and modularity of the realizing circuits. We also identify existing quantum algorithms whose use of semantic embedding is implicit, spanning from distributed search to proofs of soundness in quantum cryptography. The methods used, based in category theory, establish a theory of semantically embeddable quantum algorithms, and provide a new role for QSP/QSVT in reducing sophisticated algorithmic problems to simpler algebraic ones.
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Quantum signal processing and quantum singular value transformation are powerful tools to implement polynomial transformations of block-encoded matrices on quantum computers, and has achieved asymptotically optimal complexity in many prominent quantum algorithms. We propose a framework of quantum signal processing and quantum singular value transformation on U ( N ) , which realizes multiple polynomials simultaneously from a block-encoded input, as a generalization of those on U ( 2 ) in the original frameworks. We provide a comprehensive characterization of achievable polynomial matrices and give recursive algorithms to construct the quantum circuits that realize desired polynomial transformations. As three example applications, we propose a framework to realize bi-variate polynomial functions, demonstrate N -interval decision achieving O ( d ) query complexity with a log 2 ⁡ N improvement over iterative U ( 2 ) -QSP requiring O ( d log 2 ⁡ N ) queries, and present a quantum amplitude estimation algorithm achieving the Heisenberg limit without adaptive measurements.
Quantum algorithms for basic numerical linear algebraic tasks have proven essential for translating diverse problems to a unified quantum computational context. Many of these tasks---e.g., applying a function to the spectrum of a matrix embedded in a unitary process (a so-called block encoding), or taking linear combinations of block encodings---are well-addressed by techniques like quantum singular value transformation or linear combination of unitaries. However, there exist useful matrix transforms whose realization is unclear or inefficient. In this work we construct improved quantum algorithms for some of these transforms, the simplest of which is a polynomial function applied element-wise. We show the space required to compute quantum element-wise transforms (QEWTs) can be reduced exponentially in the degree of the applied function compared to prior work, and rectify errors in previous constructions. We prove novel space and query lower bounds for QEWTs, construct useful function families nearly saturating these bounds, and establish the hardness of QEWT dequantization. This represents the first sound theoretical treatment of QEWTs, and brings their resource requirements on par with other practical block encoding techniques. We present our algorithms alongside concrete applications to machine learning inference and signal processing.
Quantum singular value transformation (QSVT) enables the application of polynomial functions to the singular values of near arbitrary linear operators embedded in unitary transforms, and has been used to unify, simplify, and improve most quantum algorithms. QSVT depends on precise results in representation theory, with the desired polynomial functions acting simultaneously within invariant two-dimensional subspaces of a larger Hilbert space. These two-dimensional transformations are largely determined by the related theory of quantum signal processing (QSP). While QSP appears to rely on properties specific to the compact Lie group SU(2), many other Lie groups appear naturally in physical systems relevant to quantum information. This work considers settings in which SU(1,1) describes system dynamics and finds that, surprisingly, despite the non-compactness of SU(1,1), one can recover a QSP-type ansatz, and show its ability to approximate near arbitrary polynomial transformations. We discuss various experimental uses of this construction, as well as prospects for expanded relevance of QSP-like ansätze to other Lie groups.
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.