Queried-Convolutions (Qonvolutions) Overview

• Qonvolutions are an enriched convolution paradigm that fuses query information with classical, quantum, and algebraic techniques to enhance high-frequency signal representation.
• They enable neural networks to reconstruct high-frequency signals accurately, facilitate efficient quantum circuit implementations with exponential parallelism, and underpin robust q-polynomial convolutions.
• Empirical studies demonstrate improved super-resolution, enhanced quantum classification accuracy, and maintenance of fundamental algebraic structures across diverse applications.

Queried-Convolutions (Qonvolutions) encompass a set of operator-theoretic, algebraic, and neural architectural strategies that generalize or enhance classical convolution, quantum convolution, and finite free convolution mechanisms. Qonvolutions appear in diverse contexts: as “queried” convolutions for high-frequency signal learning in neural networks (Kumar et al., 15 Dec 2025), as quantum circuits efficiently realizing large-scale convolutional transformations (Qu et al., 11 Apr 2025, Yang et al., 16 Mar 2026), and as $q$-deformations of finite free convolution producing new algebraic structures for $q$-polynomial families (Martinez-Finkelshtein et al., 12 Jun 2026). Central to all usages is an enriched notion of convolution designed to preserve or expand classical properties, domain compatibility (e.g., queries or quantum indices), or structural algebraic features.

1. Queried-Convolutions in Neural Networks

The Queried-Convolution (“Qonvolution”) formalism introduced by Lu, Ding, and Yu is designed to enhance high-frequency signal learning by incorporating explicit “query” information into the classical convolutional paradigm (Kumar et al., 15 Dec 2025). In standard settings, a neural network often receives a low-frequency approximation $f_{low}(q_i)$ of the signal, together with a “query” $q_i$ (such as spatial or scene coordinate). Qonvolutions generalize the convolutional layer by fusing these queries with local convolutional neighborhoods:

$$\hat f(q_i) = \sum_{r \in N(0)} W_r \left [ \phi(q_{i + r}) ; f_{low}(q_{i + r}) \right ] + b$$

Here, $N(0)$ denotes a convolutional neighborhood, $W_r$ are learnable kernels, $;$ denotes concatenation, and $\phi$ is a query encoding (typically identity, but extensible to Fourier or exponential embeddings). In 2D applications, the approach fuses (e.g., via channel concatenation) spatial coordinate queries with low-frequency image base, and processes through a standard convolutional stack. This construction enables perfect reconstruction of high-frequency signals in the infinite-data/capacity setting (Theorem 3.1), surpassing MLP or vanilla CNN baselines, particularly in tasks such as super-resolution and novel view synthesis.

2. Qonvolutional Circuits in Quantum Computation

Quantum computing provides an inherent convolutional structure, as explicated by Qu et al. (Qu et al., 11 Apr 2025). In a quantum neural network (QNN), an $n$-qubit unitary $q$0, when selectively applied to contiguous blocks of $q$1-qubit amplitude-encoded data, acts identically to a classical convolution: it maps direct sums of patches to direct sums of feature blocks, with local connectivity, parameter sharing, and multi-channel support realized “for free.” The output after applying $q$2 to each $q$3-element patch:

$q$4

translates to a convolutional transformation with stride $q$5, kernel size $q$6, and $q$7 channels—all implemented in a single quantum gate invocation. This quantum parallelism enables exponentially faster application of large convolutional layers relative to classical patchwise computation. Depth and channel structure can be expanded by cascading layers and increasing register width, and parameter sharing emerges naturally from circuit block structure.

QCNN architectures exploiting this principle recover key CNN traits, achieve high accuracy in multiclass tasks (e.g., MNIST), and require orders of magnitude fewer parameters than classical or previous quantum baselines. Limitations include amplitude-shrinkage and gradient noise for large/deep models, and the challenge of native quantum nonlinearity (Qu et al., 11 Apr 2025).

3. Quantum LCU-Based Qonvolution and Hermitian Block-Encoding

Discrete circular convolution can be realized as a linear-combination-of-unitaries (LCU) circuit, with a modular adder acting as the shift operator in the quantum basis. Yang et al. (Yang et al., 16 Mar 2026) construct “Qonvolution” by representing circular convolution $q$8 over $q$9 as:

$f_{low}(q_i)$0

where shifts $f_{low}(q_i)$1 are implemented by modular addition. The asymmetric block-encoding circuit prepares the kernel state in an ancilla register, applies SELECT$f_{low}(q_i)$2, uncomputes to the uniform state, and postselects, effectively applying $f_{low}(q_i)$3 (or its Hermitian variant $f_{low}(q_i)$4 for real-valued kernels) to the data state. For real kernels, this construction is Hermitian, enabling direct quantum singular value transformation (QSVT) without condition number squaring.

A recursive, bitwise-carry implementation yields polylogarithmic depth/gate count and efficient resource scaling, making this form of Qonvolution a viable primitive for quantum machine learning and quantum signal processing. The paradigm is distinct from classical FFT-based quantum convolutions, requiring only poly($f_{low}(q_i)$5) gates for both macro-blocks and controlled increments (Yang et al., 16 Mar 2026).

4. Algebraic Qonvolutions: $f_{low}(q_i)$6-Finite Free Convolutions

Advancing from classical finite free convolution, Qonvolutions arise as $f_{low}(q_i)$7-multiplicative ($f_{low}(q_i)$8) and $f_{low}(q_i)$9-additive ($q_i$0) finite free convolutions on degree-$q_i$1 polynomials (Martinez-Finkelshtein et al., 12 Jun 2026). The $q_i$2-multiplicative convolution is coefficientwise and admits an operator form via $q_i$3-derivatives. Crucially, $q_i$4 is closed on $q_i$5-hypergeometric polynomial families—convolving two such polynomials concatenates their parameter sets, preserves (under mesh bounds) real-rootedness, and controls logarithmic mesh.

The $q_i$6-additive convolution $q_i$7, while linear and associative, does not generically preserve real-rootedness, but a corrected version (composed with a specific $q_i$8-multiplicative convolution) restores this property. This machinery generalizes and systematizes the extraction of convolution identities from classical product formulas for basic hypergeometric functions, enabling the translation of product identities directly into convolution identities for truncated $q_i$9-polynomials.

5. Comparative Summary and Domain-Specific Properties

Table: Qonvolutional Principles Across Domains

Domain	Qonvolution Mechanism	Core Properties/Advantages
Neural Networks	Conv+query fusion (spatial queries + low-freq)	Enables Hi-Freq learning, simple CNN integration, drop-in
Quantum Circuits	Unitary blocks = conv layers (ampl-encoded data)	Exponential parallelism, efficient parameter sharing
Quantum LCU	LCU+modular adder, Hermitian block-encoding	Hermiticity for QSVT, efficient multi-scale convolution
$$\hat f(q_i) = \sum_{r \in N(0)} W_r \left [ \phi(q_{i + r}) ; f_{low}(q_{i + r}) \right ] + b$$0-Polynomial Algebra	$$\hat f(q_i) = \sum_{r \in N(0)} W_r \left [ \phi(q_{i + r}) ; f_{low}(q_{i + r}) \right ] + b$$1 convolutions	Preserves hypergeometric structure, mesh/root control

In neural settings, Qonvolution yields uniform improvements in PSNR/SSIM over conventional and even Fourier-enhanced MLPs, especially important in super-resolution and novel view synthesis. In quantum computation, it enables convolutional layers with depth and channel structure that are exponentially costly classically; in algebraic domains, it preserves and extends hypergeometric polynomial families, with guaranteed root and mesh properties.

6. Limitations, Open Directions, and Generalizations

Limitations in the neural setting include dependence on the existence of low-frequency signals and a spatial grid; Qonvolution cannot apply to single-ray or fully implicit models and requires both query and neighborhood information (Kumar et al., 15 Dec 2025). In the quantum setting, circuit depth, qubit connectivity, and gradient noise present challenges as model scale increases (Qu et al., 11 Apr 2025). For $$\hat f(q_i) = \sum_{r \in N(0)} W_r \left [ \phi(q_{i + r}) ; f_{low}(q_{i + r}) \right ] + b$$2-polynomial Qonvolutions, while closure and real-rootedness are partially ensured, the additive case requires auxiliary corrections for these properties to hold (Martinez-Finkelshtein et al., 12 Jun 2026).

Open directions include extension to 3D Qonvolutions, dynamic kernel/query encodings, integration with hybrid quantum-classical pooling/activations, and comprehensive finite-sample complexity theory. Further, mapping quantum Qonvolutions to physical hardware and benchmarking on large-scale datasets remain crucial for practical impact.

7. Selected Applications and Empirical Results

Empirical studies confirm the versatility of Qonvolutions. In neural systems, they improve PSNR by $$\hat f(q_i) = \sum_{r \in N(0)} W_r \left [ \phi(q_{i + r}) ; f_{low}(q_{i + r}) \right ] + b$$3–$$\hat f(q_i) = \sum_{r \in N(0)} W_r \left [ \phi(q_{i + r}) ; f_{low}(q_{i + r}) \right ] + b$$4 dB (SR tasks), $$\hat f(q_i) = \sum_{r \in N(0)} W_r \left [ \phi(q_{i + r}) ; f_{low}(q_{i + r}) \right ] + b$$5–$$\hat f(q_i) = \sum_{r \in N(0)} W_r \left [ \phi(q_{i + r}) ; f_{low}(q_{i + r}) \right ] + b$$6 dB (2D regression), and enhance edge metrics in NVS, outperforming strong baselines at modest compute cost (Kumar et al., 15 Dec 2025). Quantum architectures leveraging unitary-based Qonvolutions achieve state-of-the-art or comparable accuracy on multiclass image classification with drastically reduced parameter budgets (e.g., 40 vs 50–5200+ parameters), a direct consequence of the parameter-sharing and exponential channelization effect (Qu et al., 11 Apr 2025). For $$\hat f(q_i) = \sum_{r \in N(0)} W_r \left [ \phi(q_{i + r}) ; f_{low}(q_{i + r}) \right ] + b$$7-polynomials, Qonvolutions generate new families of real-rooted, interlacing polynomials and systematize the algebraic translation of hypergeometric product identities, giving rise to novel functional identities in the analytic theory (Martinez-Finkelshtein et al., 12 Jun 2026).