Modular quantum signal processing in many variables

TL;DR

This paper develops a modular framework for quantum signal processing that allows complex quantum algorithms to be built from simple, composable units called gadgets, facilitating easier design and implementation of multi-variable quantum functions.

Contribution

It introduces a theory of modular multi-input-output QSP superoperators and a Python toolkit for assembling and compiling these gadgets into quantum circuits.

Findings

Gadgets enable efficient block encoding of multivariable functions.

The framework supports a functional programming approach to quantum algorithm design.

Gadgets can be combined easily like LEGO pieces.

Abstract

Despite significant advances in quantum algorithms, quantum programs in practice are often expressed at the circuit level, forgoing helpful structural abstractions common to their classical counterparts. Consequently, as many quantum algorithms have been unified with the advent of quantum signal processing (QSP) and quantum singular value transformation (QSVT), an opportunity has appeared to cast these algorithms as modules that can be combined to constitute complex programs. Complicating this, however, is that while QSP/QSVT are often described by the polynomial transforms they apply to the singular values of large linear operators, and the algebraic manipulation of polynomials is simple, the QSP/QSVT protocols realizing analogous manipulations of their embedded polynomials are non-obvious. Here we provide a theory of modular multi-input-output QSP-based superoperators, the basic unit of which we call a gadget, and show they can be snapped together with LEGO-like ease at the level of the functions they apply. To demonstrate this ease, we also provide a Python package for assembling gadgets and compiling them to circuits. Viewed alternately, gadgets both enable the efficient block encoding of large families of useful multivariable functions, and substantiate a functional-programming approach to quantum algorithm design in recasting QSP and QSVT as monadic types.