Adaptive Interpolating Quantum Transform: A Quantum-Native Framework for Efficient Transform Learning

TL;DR

The paper introduces the Adaptive Interpolating Quantum Transform (AIQT), a quantum-native framework that learns a low-parameter unitary $U_{ ext{AIQT}}(oldsymbol{ heta})$ interpolating between base quantum transforms (e.g., $U_{ ext{H}}$ and $U_{ ext{QFT}}$). By embedding AIQT as a preprocessing layer before a shallow QNN, the model inherits the advantages of the constituent transforms while remaining adaptable to task structure via a compact parameter set, ensuring unitary evolution throughout. Empirical results on quantum phase classification demonstrate that AIQT–QNN outperforms fixed-transform and baseline QNNs, with TE-based AIQT variants offering additional gains due to local interaction structures and entanglement dynamics. The work suggests AIQT as a scalable, interpretable strategy for efficient quantum learning, capable of leveraging different physical priors through alternative instantiations such as QFT-based or TFIM time-evolution-based transforms.

Abstract

Machine learning on quantum computers has attracted attention for its potential to deliver computational speedups in different tasks. However, deep variational quantum circuits require a large number of trainable parameters that grows with both qubit count and circuit depth, often rendering training infeasible. In this study, we introduce the Adaptive Interpolating Quantum Transform (AIQT), a quantum-native framework for flexible and efficient learning. AIQT defines a trainable unitary that interpolates between quantum transforms, such as the Hadamard and quantum Fourier transforms. This approach enables expressive quantum state manipulation while controlling parameter overhead. It also allows AIQT to inherit any quantum advantages present in its constituent transforms. Our results show that AIQT achieves high performance with minimal parameter count, offering a scalable and interpretable alternative to deep variational circuits.

Figures (4)

• Figure 1: (a) A schematic illustration of the Adaptive Interpolating Quantum Transform (AIQT) as a low-cost adaptive unitary operator. Here, $\phi_1, \dots, \phi_{T}$ denote the trainable parameters of the QNN. (b) Parametrized QFT as an instance of AIQT. Here, a trainable global parameter $\theta$ is introduced into the QFT phase structure, enabling smooth interpolation between Hadamard ($\theta = 0$) and full QFT ($\theta = 2\pi$). The controlled-rotation gate, denoted $CR_k(\theta)$, applies a phase of $\theta/2^k$ ($CR_k(\theta) = \mathrm{diag}(1, 1, 1, e^{i \theta / 2^k})$). (c) Architecture of the AIQT-QNN used in this work, combining AIQT-based input transformation with a hardware-efficient QNN ansatz.
• Figure 2: (a) The training curves of AIQT-QNN (blue), QFT-QNN (red), and QNN (black) for a 10-qubit system. $L$ denotes the loss value as defined by the loss function in Eq. \ref{['eq:loss_function']}. The shaded region represents the standard deviation for 10 random initial parameters. Across all models, 3 layers of QNN were used. (b) The probability of being in the trivial phase (P(Trivial)) (top) and SPT (P(SPT)) (bottom) phase at $g_{zz}=0.1$ of AIQT-QNN (blue), QFT-QNN (red), and QNN (black) after 100 training epochs. (c) The phase diagram of the Hamiltonian described in Eq. (8) as reported in literature PhysRevB.96.165124PhysRevLett.120.057001smith_crossing_2022. The classification of the quantum phases using QFT-QNN (d) and AIQT-QNN (e). Here, $g_{zxz}$, $g_{zz}$, and $g_{x}$ are the parameters of the Hamiltonian described in Eq. \ref{['eq:energy_2']}.
• Figure 3: Training loss (a) and validation loss (b) as a function of the number of qubits for AIQT-QNN (blue), QFT-QNN (red), and baseline QNN (black) after 100 training epochs. The markers and whiskers show the average and standard deviation values derived from 10 random initial parameters.
• Figure 4: Validation performance in loss (a) and accuracy (b) for models using two different AIQT variants: QFT-based AIQT (blue) and Time Evolution (TE)-based AIQT with TFIM Hamiltonian (red) for a 10-qubit system, evaluated over 100 training epochs. Shaded regions indicate standard deviation across 10 training runs.