Approximate Amplitude Encoding with the Adaptive Interpolating Quantum Transform

TL;DR

The adaptive interpolating quantum transform (AIQT) is replaced with the adaptive interpolating quantum transform (AIQT) in the sparse amplitude encoding workflow, preserving the efficiency of Fourier-based methods and removing a major bottleneck in data-driven amplitude-encoding methods.

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.

Figures (7)

• Figure 1: Sparse amplitude encoding workflows with Fourier (FSL) and AIQT transforms and their quantum circuits. a) Sparse amplitude encoding with the Fourier State Loader (FSL). A Fourier transform is first applied to the classical data $\mathbf{x}$, producing the full set of coefficients $\mathbf{y}$. We then keep the $k$ largest-magnitude coefficients and set the rest to zero, yielding a $k$-sparse vector $\tilde{\mathbf{y}}$. This vector is amplitude encoded on the quantum computer to obtain the state $\lvert\tilde{\phi}\rangle$. Finally, an approximate reconstruction of the data is obtained by applying the inverse quantum Fourier transform, producing the state $\lvert\tilde{\psi}\rangle_{\mathrm{FSL}}$. Here, the first and middle Fourier coefficients are always included, regardless of $k$, to maintain symmetry. Thus, for a given $k$, the FSL selects the top-$k$ coefficients with 2 additional coefficients. b) The proposed sparse amplitude encoding uses the Adaptive interpolating quantum transform (AIQT) and its inverse in place of the Fourier transform. The AIQT is trained to maximize the amount of information preserved during coefficient truncation, which leads to lower reconstruction error. Moreover, the AIQT does not enforce conjugate symmetry, thus we simply select the top-$k$ coefficients without any additional symmetry-preserving sampling. c) The Quantum Fourier Transform (QFT) circuit used in FSL. d) The AIQT circuit is constructed by parameterizing the QFT structure with single-qubit gates $\mathrm{U3}(\alpha,\beta,\gamma)$ and controlled-phase gates $\mathrm{CR}(\theta)$, yielding $U_{\mathrm{AIQT}}(\boldsymbol{\theta},\boldsymbol{\alpha},\boldsymbol{\beta},\boldsymbol{\gamma})$. Here, $\mathrm{U3}(\alpha,\beta,\gamma) = \bigl[\cos(\alpha/2)-e^{i\gamma}\sin(\alpha/2)e^{i\beta}\sin(\alpha/2)e^{i(\beta+\gamma)}\cos(\alpha/2)\bigr]$ and $\mathrm{CR}(\theta) = \mathrm{diag}(1, 1, 1, e^{i\theta}).$
• Figure 2: Coefficient statistics and single-sample reconstructions for the AIQT and the FSL. a) Squared magnitude of the top-256 coefficients as a function of training epoch for the AIQT (blue), compared with the fixed FSL baseline (orange). b) Distribution of squared magnitudes ($m_j$) across coefficient index $j \in \{1,\dots,1024\}$ for a representative sample for FSL (top, orange) and AIQT (bottom, blue). c) Average squared magnitude ($m_r$) as a function of rank $r$ for the AIQT (blue) and the FSL (orange). For each input, coefficients are sorted in descending order from largest ($m_1$) to smallest ($m_{1024}$), and we then average $m_r$ over the dataset. Solid lines show the dataset mean, and shaded regions indicate standard deviation.
• Figure 3: Reconstruction error (cRMSE) as a function of retained coefficients ($k$) for the FSL and the AIQT on the finance dataset. Dashed lines denote power-law fits ($\mathrm{cRMSE}=Ak^B$) obtained by ordinary least squares on log-transformed data. The coefficient of determination ($R^2$) denotes the goodness of fit for the linearized model in log space.
• Figure 4: Dataset-averaged imaginary-part norm $\overline{I}$ (blue) and dataset-averaged real-part norm $\overline{R}$ (red) of the AIQT reconstructed signal as a function of training epoch for the AIQT model with $k=256$. The model automatically reduces leakage into the imaginary part over training.
• Figure 5: The deep AIQT architecture. A depth-$D$ stack of AIQT blocks with independent parameters applied sequentially to the input state.