Approximate Amplitude Encoding with the Adaptive Interpolating Quantum Transform

TL;DR

This paper introduces the adaptive interpolating quantum transform (AIQT), which learns a data-adapted basis for sparse amplitude encoding, significantly reducing reconstruction error while maintaining efficiency and avoiding the need for quantum hardware sampling.

Contribution

The paper proposes AIQT, an adaptive basis method for amplitude encoding that outperforms Fourier-based methods in information retention and error reduction without increasing computational complexity.

Findings

AIQT reduces reconstruction error by up to 50% on image data.

AIQT achieves 40% lower error on financial time-series data.

The method maintains Fourier transform efficiency with quadratic gate scaling.

Abstract

Amplitude encoding of real-world data on quantum computers is often the workflow bottleneck: direct amplitude encoding scales poorly with input size and can offset any speedups in subsequent processing. Fourier-based sparse amplitude encoding lowers cost by retaining only a small subset of dominant coefficients, but its fixed, non-adaptive basis leads to significant information loss. In this work, we replace the Fourier transform with the adaptive interpolating quantum transform (AIQT) in the sparse amplitude encoding workflow. The AIQT learns a data-adapted basis that concentrates information into a small number of coefficients. Consequently, at matched sparsity, the AIQT retains more information and achieves lower reconstruction error compared to the Fourier baseline. On financial time-series data, the AIQT reduces reconstruction error by 40% relative to the Fourier baseline, and on image datasets the reduction is up to 50% at the same sparsity level, with nearly identical encoding gate cost. Crucially, the approach preserves the efficiency of Fourier-based methods: the AIQT is built on the structure of the quantum Fourier transform circuit. Its gate count scales quadratically with the number of qubits, while classical evaluation can be carried out in quasilinear time. In addition, the AIQT is trained without labels and does not require sampling from quantum hardware or a simulator, removing a major bottleneck in data-driven amplitude-encoding methods.