Adaptive Interpolating Quantum Transform

• The paper introduces AIQT as a quantum-native framework that learns a trainable unitary interpolation between established transforms such as the Hadamard transform and the QFT.
• It reduces parameter overhead by leveraging a few global parameters to control the transition between simple and complex mixing, enhancing circuit efficiency.
• The framework extends to approximate amplitude encoding by concentrating key signal information into fewer coefficients, leading to improved error reduction and fidelity.

to=arxiv_search.search 大发彩票网 久久免费热在线精品{"query":"Adaptive Interpolating Quantum Transform (Budiutama et al., 20 Aug 2025) Approximate Amplitude Encoding with the Adaptive Interpolating Quantum Transform", "max_results": 5} to=arxiv_search.search 大发快三官网 ՞նչjson{"query":"\"Adaptive transform via quantum signal processing: application to signal and image denoising\" (Smith et al., 2018)", "max_results": 3} to=arxiv_search.search 彩神争霸是 to=arxiv_search.search 彩票开号 ՞նչjson{"query":"\"Quantum Amplitude Interpolation\" (Stefanski et al., 2022)", "max_results": 3} to=arxiv_search.search 天天中彩票nbajson{"query":"\"Efficient quantum interpolation of natural data\" (Ramos-Calderer, 2022)", "max_results": 3} to=arxiv_search.search 天天送ीसոնjson{"query":"\"Non-Markovian Exceptional Points by Interpolating Quantum Channels\" (Wong et al., 21 Jul 2025)", "max_results": 3} to=arxiv_search.search 天天彩票与你同行json{"query":"\"Adaptive Interpolating Quantum Transform: A Quantum-Native Framework for Efficient Transform Learning\" (Budiutama et al., 20 Aug 2025)", "max_results": 3} to=arxiv_search.search ＿老司机json{"query":"\"Adaptive Interpolating Quantum Transform\" AIQT", "max_results": 10} Adaptive Interpolating Quantum Transform (AIQT) is a quantum-native framework in which a trainable unitary interpolates between established quantum transforms, most prominently the Hadamard transform and the quantum Fourier transform (QFT), in order to increase expressiveness while controlling parameter overhead (Budiutama et al., 20 Aug 2025). In subsequent work, the same framework is used inside approximate amplitude encoding, where the objective is to learn a data-adapted basis that concentrates information into a small number of coefficients while preserving the efficient butterfly structure of the QFT circuit (Budiutama et al., 4 Mar 2026). The broader literature places AIQT at the intersection of transform learning, quantum interpolation, and quantum data loading, with conceptual links to Schrödinger-based adaptive dictionaries for denoising and to QFT/QCT interpolation methods for continuous and natural data (Smith et al., 2018, Stefanski et al., 2022, Ramos-Calderer, 2022).

1. Definitional scope and literature context

In the trainable-transform sense, AIQT denotes a framework that defines a unitary $U_{\mathrm{AIQT}(\boldsymbol{\theta})}$ interpolating between known quantum transforms by means of a small set of learnable parameters, while remaining unitary for all parameter values (Budiutama et al., 20 Aug 2025). This usage is distinct from, but related to, earlier literature on adaptive transforms derived from quantum mechanics and on quantum interpolation algorithms. The supplied corpus therefore supports a layered reading of the term: a core unitary transform-learning framework, a practical amplitude-encoding instantiation, and several antecedent or adjacent interpolation constructions.

Paper	Main object	Relation to AIQT
"Adaptive transform via quantum signal processing: application to signal and image denoising" (Smith et al., 2018)	Schrödinger-based adaptive dictionary	Conceptual antecedent
"Efficient quantum interpolation of natural data" (Ramos-Calderer, 2022)	QFT/QCT interpolation	Interpolation backdrop
"Quantum Amplitude Interpolation" (Stefanski et al., 2022)	Amplitude interpolation of continuous signals	Quantum interpolation backdrop
"Non-Markovian Exceptional Points by Interpolating Quantum Channels" (Wong et al., 21 Jul 2025)	Interpolation of CPTP channels	Distinct open-system extension
"Adaptive Interpolating Quantum Transform: A Quantum-Native Framework for Efficient Transform Learning" (Budiutama et al., 20 Aug 2025)	Trainable interpolating unitary	Canonical AIQT formulation
"Approximate Amplitude Encoding with the Adaptive Interpolating Quantum Transform" (Budiutama et al., 4 Mar 2026)	Sparse amplitude encoding with AIQT	Applied AIQT workflow

A recurring theme across these papers is interpolation as structure rather than interpolation as post hoc curve fitting. In AIQT proper, interpolation occurs in the space of quantum transforms. In amplitude-interpolation papers, interpolation occurs in quantum state amplitudes or transform domains. In the channel-interpolation paper, interpolation occurs in the space of CPTP maps. This suggests that AIQT is best understood as one member of a broader research program on structured quantum interpolation, rather than as an isolated circuit ansatz.

2. Trainable unitary construction

The central AIQT construction is a trainable unitary interpolation between known transforms $U_A, U_B, \ldots$ with limiting behavior

$$\lim_{\boldsymbol{\theta}\to \boldsymbol{\theta}_A} U_{\mathrm{AIQT}(\boldsymbol{\theta})} = U_A, \qquad \lim_{\boldsymbol{\theta}\to \boldsymbol{\theta}_B} U_{\mathrm{AIQT}(\boldsymbol{\theta})} = U_B,$$

while preserving unitarity for all values of $\boldsymbol{\theta}$ (Budiutama et al., 20 Aug 2025). The motivation given in the literature is that deep variational quantum circuits require a large number of trainable parameters that grows with both qubit count and circuit depth, often rendering training infeasible; AIQT addresses this by replacing many local parameters with one or a few global parameters.

A concrete instantiation interpolates between the Hadamard transform $U_H$ and the QFT $U_{\mathrm{QFT}}$. In the standard QFT circuit, the controlled-rotation gates $CR_k$ are modified so that each phase is governed by a trainable global parameter $\theta$,

$$CR_k(\theta) = \mathrm{diag}(1,1,1,e^{i\theta/2^k}).$$

At $\theta=0$, all such controlled rotations reduce to identity and the circuit reduces to the Hadamard transform. At $U_A, U_B, \ldots$0, the standard QFT phases are recovered, so that

$U_A, U_B, \ldots$1

The operational significance of this construction is that a single global parameter controls a continuum between simple and highly mixing transforms.

A second instantiation uses parameterized time evolution under a local Hamiltonian, for example the transverse-field Ising model,

$U_A, U_B, \ldots$2

Here $U_A, U_B, \ldots$3 functions as an evolution time and $U_A, U_B, \ldots$4 are learnable couplings. In both instantiations, the parameters modulate physically meaningful aspects of the transform—phase, time, or Hamiltonian strength—rather than arbitrary local rotations.

The framework is also presented as a modular preprocessing layer for quantum machine learning. The workflow is

$U_A, U_B, \ldots$5

followed by measurement in the computational basis (Budiutama et al., 20 Aug 2025). This places AIQT between fixed transforms and generic deep ansätze: the transform is not hard-coded, yet it is not an unconstrained $U_A, U_B, \ldots$6-parameter unitary.

3. Quantum-mechanical adaptive transforms as antecedents

Before AIQT was introduced as a trainable unitary framework, adaptive transform design had already appeared in a quantum-mechanical form. The 2018 paper "Adaptive transform via quantum signal processing: application to signal and image denoising" constructs an adaptive transform by treating the signal or image as a discrete potential in the stationary Schrödinger equation $U_A, U_B, \ldots$7, then using the eigenvectors of the resulting Hamiltonian matrix as an adaptive orthonormal basis (Smith et al., 2018). In this formulation, the potential $U_A, U_B, \ldots$8 is assigned directly from signal or image values, the Laplacian is discretized by finite differences, and the Hamiltonian matrix is assembled with nearest-neighbor couplings and zero-padding boundary conditions.

The basis functions produced by this construction are described as adaptive oscillatory functions whose local oscillation depends on the energy landscape induced by the data. Regions with lower potential admit higher local frequencies, whereas higher-potential regions favor smoother behavior. The denoising procedure projects the observed signal onto the Hamiltonian eigenbasis, applies thresholding, and reconstructs the estimate. A key practical nuance is that the Hamiltonian is formed from a smoothed version of the noisy input to avoid Anderson localization, which in a noisy potential would cause undesirable exponential localization of eigenfunctions.

The denoising motivation is specifically tied to signal-dependent noise. The supplied account states that low-signal regions yield basis functions with higher frequencies, which helps preserve subtle details, while high-signal regions yield smoother basis functions, supporting stronger denoising where noise is higher (Smith et al., 2018). The same account explicitly compares this method with AIQT and notes both similarities and differences: both use Schrödinger-equation and Hamiltonian formalism, both link the potential to the data, and both seek a basis whose local oscillatory properties are tuned by signal or image values, but the 2018 method constructs the discretized Hamiltonian directly on sampled data, whereas AIQT is described as typically beginning with a continuous formulation and only discretizing at the end. This suggests a conceptual lineage rather than terminological identity.

4. Quantum interpolation background

Two 2022 papers supply the interpolation background against which AIQT is later framed. "Quantum Amplitude Interpolation" presents a method for representing continuous signals with high precision by interpolating quantum state amplitudes, explicitly inspired by the Nyquist–Shannon sampling theorem (Stefanski et al., 2022). The key interpolation kernel is

$U_A, U_B, \ldots$9

and the interpolated value is obtained through an inner-product construction,

$$\lim_{\boldsymbol{\theta}\to \boldsymbol{\theta}_A} U_{\mathrm{AIQT}(\boldsymbol{\theta})} = U_A, \qquad \lim_{\boldsymbol{\theta}\to \boldsymbol{\theta}_B} U_{\mathrm{AIQT}(\boldsymbol{\theta})} = U_B,$$0

The paper also extends a generalized inner-product method from integer-valued to real-valued functions and uses amplitude encoding plus phase correction to obtain real amplitudes. In this line of work, interpolation is a mechanism for reconstructing or evaluating continuous-valued functions from discretely encoded quantum data.

"Efficient quantum interpolation of natural data" addresses interpolation of uploaded probability distributions and images by means of QFT and Quantum Cosine Transform (QCT) circuits (Ramos-Calderer, 2022). The QFT-based method applies the QFT to an encoded register, inserts ancillas corresponding to high-frequency padding, and applies an inverse QFT to obtain a higher-resolution distribution. The QCT line is motivated by the observation that natural images are often better represented by cosine transforms, and the paper introduces a novel quantum circuit for the DCT-II/QCT setting. It also develops a JPEG-inspired subspace QCT method in which local blocks are processed in parallel through superposition.

The paper reports image-quality metrics for interpolation: bicubic interpolation yields PSNR $$\lim_{\boldsymbol{\theta}\to \boldsymbol{\theta}_A} U_{\mathrm{AIQT}(\boldsymbol{\theta})} = U_A, \qquad \lim_{\boldsymbol{\theta}\to \boldsymbol{\theta}_B} U_{\mathrm{AIQT}(\boldsymbol{\theta})} = U_B,$$1 and SSIM $$\lim_{\boldsymbol{\theta}\to \boldsymbol{\theta}_A} U_{\mathrm{AIQT}(\boldsymbol{\theta})} = U_A, \qquad \lim_{\boldsymbol{\theta}\to \boldsymbol{\theta}_B} U_{\mathrm{AIQT}(\boldsymbol{\theta})} = U_B,$$2; QFT yields PSNR $$\lim_{\boldsymbol{\theta}\to \boldsymbol{\theta}_A} U_{\mathrm{AIQT}(\boldsymbol{\theta})} = U_A, \qquad \lim_{\boldsymbol{\theta}\to \boldsymbol{\theta}_B} U_{\mathrm{AIQT}(\boldsymbol{\theta})} = U_B,$$3 and SSIM $$\lim_{\boldsymbol{\theta}\to \boldsymbol{\theta}_A} U_{\mathrm{AIQT}(\boldsymbol{\theta})} = U_A, \qquad \lim_{\boldsymbol{\theta}\to \boldsymbol{\theta}_B} U_{\mathrm{AIQT}(\boldsymbol{\theta})} = U_B,$$4; n-QCT yields PSNR $$\lim_{\boldsymbol{\theta}\to \boldsymbol{\theta}_A} U_{\mathrm{AIQT}(\boldsymbol{\theta})} = U_A, \qquad \lim_{\boldsymbol{\theta}\to \boldsymbol{\theta}_B} U_{\mathrm{AIQT}(\boldsymbol{\theta})} = U_B,$$5 and SSIM $$\lim_{\boldsymbol{\theta}\to \boldsymbol{\theta}_A} U_{\mathrm{AIQT}(\boldsymbol{\theta})} = U_A, \qquad \lim_{\boldsymbol{\theta}\to \boldsymbol{\theta}_B} U_{\mathrm{AIQT}(\boldsymbol{\theta})} = U_B,$$6; and $$\lim_{\boldsymbol{\theta}\to \boldsymbol{\theta}_A} U_{\mathrm{AIQT}(\boldsymbol{\theta})} = U_A, \qquad \lim_{\boldsymbol{\theta}\to \boldsymbol{\theta}_B} U_{\mathrm{AIQT}(\boldsymbol{\theta})} = U_B,$$7-QCT yields PSNR $$\lim_{\boldsymbol{\theta}\to \boldsymbol{\theta}_A} U_{\mathrm{AIQT}(\boldsymbol{\theta})} = U_A, \qquad \lim_{\boldsymbol{\theta}\to \boldsymbol{\theta}_B} U_{\mathrm{AIQT}(\boldsymbol{\theta})} = U_B,$$8 and SSIM $$\lim_{\boldsymbol{\theta}\to \boldsymbol{\theta}_A} U_{\mathrm{AIQT}(\boldsymbol{\theta})} = U_A, \qquad \lim_{\boldsymbol{\theta}\to \boldsymbol{\theta}_B} U_{\mathrm{AIQT}(\boldsymbol{\theta})} = U_B,$$9 (Ramos-Calderer, 2022). Within the supplied account, this framework is said to admit generalization by adaptively selecting transforms such as QFT, QCT, or wavelet-like alternatives depending on data characteristics, and this is explicitly connected to a possible AIQT scheme. A plausible implication is that later AIQT work can be read as replacing fixed transform choice with learned transform interpolation.

5. AIQT for approximate amplitude encoding

The most concrete applied AIQT workflow in the supplied corpus is the 2026 paper "Approximate Amplitude Encoding with the Adaptive Interpolating Quantum Transform" (Budiutama et al., 4 Mar 2026). Its starting point is the observation that direct amplitude encoding scales poorly with input size and can offset any speedups in subsequent quantum processing. Earlier Fourier-based sparse amplitude encoding mitigated this by retaining only the dominant transform coefficients, but the fixed non-adaptive basis led to significant information loss. The proposed replacement is to use AIQT as a learned transform that concentrates information into a smaller number of coefficients at matched sparsity.

The pipeline consists of four stages. First, a classical input vector $\boldsymbol{\theta}$0 is transformed by the AIQT to obtain coefficients $\boldsymbol{\theta}$1. Second, the transform is sparsified by keeping the top-$\boldsymbol{\theta}$2 coefficients in squared magnitude and setting the rest to zero, then normalizing the retained vector. Third, the resulting $\boldsymbol{\theta}$3-sparse state is prepared on quantum hardware. Fourth, the inverse AIQT is applied to approximate the true amplitude-encoded state. The transform itself mirrors the QFT butterfly circuit: every Hadamard or phase gate is replaced by a trainable $\boldsymbol{\theta}$4, every fixed controlled phase is replaced by a trainable $\boldsymbol{\theta}$5, the circuit can be initialized exactly as the QFT, quantum gate cost scales as $\boldsymbol{\theta}$6, and classical evaluation remains quasilinear, $\boldsymbol{\theta}$7 (Budiutama et al., 4 Mar 2026).

Training is unsupervised and entirely classical. The paper defines normalized squared magnitudes

$\boldsymbol{\theta}$8

and optimizes the top-$\boldsymbol{\theta}$9 tail loss

$U_H$0

The reported implementation uses differentiable soft top-$U_H$1 selection with sigmoid masks and a straight-through estimator, Adam with cosine annealing, QFT initialization, and 150 training epochs. The paper also emphasizes that no labels are required and that no sampling from quantum hardware or a simulator is needed during training.

Quantitatively, on financial time-series data with $U_H$2 and $U_H$3, the energy concentration in the leading coefficients is reported as $U_H$4 for the Fourier Series Loader baseline and $U_H$5 for AIQT; the validation cRMSE is $U_H$6 for the baseline and $U_H$7 for AIQT, with fidelities $U_H$8 and $U_H$9, respectively (Budiutama et al., 4 Mar 2026). This is summarized in the paper as a 40% reduction in error at matched gate cost. For larger $U_{\mathrm{QFT}}$0, the cRMSE reduction is again stated to be approximately 40%, and the error scaling is reported as $U_{\mathrm{QFT}}$1 for AIQT versus $U_{\mathrm{QFT}}$2 for the Fourier baseline.

For image data, deeper stacks of AIQT blocks further improve the results. On MNIST with $U_{\mathrm{QFT}}$3 out of $U_{\mathrm{QFT}}$4, the baseline yields cRMSE $U_{\mathrm{QFT}}$5 and fidelity $U_{\mathrm{QFT}}$6, whereas Deep AIQT with four layers yields cRMSE $U_{\mathrm{QFT}}$7 and fidelity $U_{\mathrm{QFT}}$8, corresponding to approximately 50% error reduction and a fidelity increase greater than $U_{\mathrm{QFT}}$9 (Budiutama et al., 4 Mar 2026). On CIFAR-10 with $CR_k$0, Deep AIQT reduces cRMSE by approximately 16–17% relative to the baseline. The paper further notes that imaginary leakage remains negligible in practice, despite the absence of an explicit real-output constraint.

6. Performance claims, scope boundaries, and extensions

In the transform-learning setting, AIQT is reported to outperform both baseline QNNs and fixed-transform QNNs, especially in quantum phase classification, while using very few global parameters (Budiutama et al., 20 Aug 2025). The supplied account attributes to AIQT superior accuracy and training loss, sharper transitions and improved discriminability at phase boundaries, consistently lowest training and validation loss across increasing system sizes, order-of-magnitude parameter reduction relative to deep QNNs, and improved interpretability because learned parameters directly correspond to physically meaningful interpolation settings. The same account also states that AIQT can inherit quantum advantages from constituent transforms, such as QFT-based speedups in spectral analysis or efficient preparation of uniform superpositions in the Hadamard limit.

At the same time, the literature distinguishes AIQT from two nearby but non-identical constructions. First, AIQT is not simply the QFT with tunable noise or pruning; it is a trainable interpolation framework whose endpoints can reproduce the Hadamard transform, the QFT, or Hamiltonian time evolutions. Second, not every interpolated quantum object in the supplied corpus is a unitary transform. "Non-Markovian Exceptional Points by Interpolating Quantum Channels" studies linear interpolation between CPTP maps,

$CR_k$1

and shows that interpolating channels from different spectral phases can generate exceptional points, including experimentally confirmed second-order exceptional points and a three-channel extension exhibiting third-order exceptional points (Wong et al., 21 Jul 2025). In the supplied account, this is presented as an AIQT foundation for programmable open-system dynamics, but its mathematical object is a quantum channel rather than a trainable unitary. This distinction is important: channel interpolation targets open-system spectral phases and exceptional-point physics, whereas AIQT in the transform-learning papers targets efficient, interpretable unitary preprocessing and data loading.

Taken together, these works position AIQT as a structured alternative to deep variational circuits and to fixed transform pipelines. In its canonical form, AIQT learns along a constrained manifold of physically meaningful unitaries; in amplitude encoding, it is used to improve information concentration at nearly identical encoding gate cost to Fourier methods; and in the surrounding interpolation literature, it can be read as part of a wider effort to combine efficient quantum circuits with data-adaptive transform design (Budiutama et al., 20 Aug 2025, Budiutama et al., 4 Mar 2026).