Semantic embedding for quantum algorithms

TL;DR

This paper develops a framework for semantic embedding in quantum algorithms, enabling modular reasoning and manipulation of quantum signal processing protocols through algebraic transforms, with implications for various quantum algorithms.

Contribution

It introduces a category-theoretic approach to semantic embedding in QSP/QSVT, allowing modular and algebraic manipulation of quantum algorithms.

Findings

Characterizes runtime and expressivity of semantic embedding protocols.

Shows how existing algorithms implicitly use semantic embedding.

Provides guarantees on the computability and modularity of quantum circuits.

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 ans\"atze, 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.