Quantum Fourier transform computational accuracy analysis

TL;DR

This paper provides a detailed analysis of the factors affecting the accuracy of the quantum Fourier transform, linking classical sampling limits and quantum resource constraints, supported by theoretical proofs and simulation results.

Contribution

It formalizes the sources of accuracy degradation in QFT and relates minimal amplitude and eigenvalue resolution to qubit count through rigorous theorems.

Findings

Identifies three sources of accuracy degeneracy in QFT.

Proves theorems relating resolution to qubit number.

Simulation results illustrate theoretical accuracy limits.

Abstract

In this work, we present a rigorous accuracy analysis of the quantum Fourier transform (QFT), that identifies three natural sources of accuracy degeneracy: (i) discretization accuracy inherited from classical sampling theory, (ii) accuracy degeneracy due to limited resolution in eigenvalue (phase) estimation, and (iii) accuracy degeneracy resulting from finite quantum resources. We formalize these accuracy degradation sources by proving two theorems that relate the minimal amplitude and eigenvalue resolution to the number of qubits. In addition, we describe a gate-level implementation of the QFT and present simulation results on small-scale quantum systems that illustrate our theoretical findings. Our results clarify the interplay between classical signal discretization limits and quantum hardware limitations, and they provide guidelines for the resource requirements needed to achieve a desired precision.