Complexification of Quantum Signal Processing and its Ramifications

TL;DR

This paper explores the deep connections between space-time dual quantum circuits and quantum signal processing (QSP), revealing new interpretations and mathematical structures involving complexified QSP, Lorentz group actions, and nonlinear Fourier transforms.

Contribution

It establishes a novel relation between Floquet operators in space-time dual circuits and complexified QSP sequences for sl(2,C), extending QSP theory to new algebraic and physical contexts.

Findings

Complexified QSP sequences relate to Lorentz group actions on density matrices.

Existence of infinite-dimensional unitary representations for these QSP sequences.

Connection between complexified QSP and the nonlinear Fourier transform for sl(2,C).

Abstract

In recent years there has been an increasing interest on the theoretical and experimental investigation of space-time dual quantum circuits. They exhibit unique properties and have applications to diverse fields. Periodic space-time dual quantum circuits are of special interest, due to their iterative structure defined by the Floquet operator. A very similar iterative structure naturally appears in Quantum Signal processing (QSP), which has emerged as a framework that embodies all the known quantum algorithms. However, it is yet unclear whether there is deeper relation between these two apparently different concepts. In this work, we establish a relation between a circuit defining a Floquet operator in a single period and its space-time dual defining QSP sequences for the Lie algebra sl$(2,\mathbb{C})$, which is the complexification of su$(2)$. First, we show that our complexified QSP sequences can be interpreted in terms of action of the Lorentz group on density matrices and that they can be interpreted as hybrid circuits involving unitaries and measurements. We also show that unitary representations of our QSP sequences exist, although they are infinite-dimensional and are defined for bosonic operators in the Heisenberg picture. Finally, we also show the relation between our complexified QSP and the nonlinear Fourier transform for sl$(2,\mathbb{C})$, which is a generalization of the previous results on su$(2)$ QSP.