A quantum Z-transform

Abstract

We investigate a quantum analog of the classical Z-transform with the aim of making it implementable on quantum computers, potentially offering a speedup over the classical method. Unlike the discrete Fourier transform, which is limited to frequency analysis, the Z-transform allows for versatile exploration of properties within the complex plane. Since the quantum Fourier transform underpins Shor’s factoring algorithm and serves as a subroutine in many other quantum algorithms, a quantum Z-transform promises broad applicability in quantum simulation, quantum machine learning, and quantum signal processing. This is especially relevant because Z-transforms generalize Fourier transforms in certain aspects. Given that quantum computers are particularly adept at performing unitary operations, we discretize the classical definition of the Z-transform and unitarize its matrix formulation to make it amenable for quantum computation. Our approach involves introducing a discrete Z-transform, mapping the input sequence to a discrete set of values to represent them as quantum states, and redefining the Z-transform as a finite summation to effectively handle the infinite summation of the classical definition. We then develop a matrix formulation for our redefined discrete Z-transform and extend our approach by unitarizing this matrix formulation through block-encoding, constructing unitary operators that meet the criteria for efficient quantum operations using standard quantum gates and subroutines. Our approach establishes the groundwork by fulfilling the mathematical foundations for the potential discovery of a quantum Z-transform and opens avenues for further exploration and implementation in quantum computing.