Quantum Fourier transform computational accuracy analysis

TL;DR

This work analyzes the accuracy of the quantum Fourier transform (QFT) by isolating three degradation channels: discretization errors from classical sampling, phase (eigenvalue) estimation precision, and limitations from finite quantum resources. It develops a formal framework with two theorems that tie the required qubit count $n$ to bounds on minimal signal amplitudes and eigenvalue ratios, and it provides a gate‑level QFT realization complemented by small‑scale simulations. The contributions include explicit bounds such as $N=2^n$ and $\min(a_k) \ge \frac{\sum_k |a_k|}{2^n}$, and $\lambda_{\min}/\lambda_{\max} = 1/2^{n-1}$, along with practical insights for resource budgeting using Qiskit simulations. The results clarify how classical discretization limits interact with quantum hardware constraints and offer guidelines for achieving a desired precision in QFT‑based algorithms.

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.

Figures (3)

• Figure 1: Simulation results for a single input signal (on the left)) and 16-channel signal (on the right). The x-axis shows the representation of the signal in a binary form and y-axis the measurement probability of that signal
• Figure 2: Simulation results for $3$ phases of values $\{3, 5, 7\}$ with the following relations of the amplitudes: $a_1/a_1 = 1/2$ and $a_1/a_3 = 1/4$ (on the left) and phases of values $\{2, 4.5, 7 \}$ with the same amplitudes (on the right). The non-integer phases spawn aliasing in the Fourier transform measurement
• Figure 3: Simulation results for phases of values$\{15, 17\}$. If the phase equals to $17$, the measurement gives $1$ as residue of $17/16 = 1_{10} = 0001_2$

Theorems & Definitions (2)

• Theorem 1
• Theorem 2