Quanvolution: Quantum-Assisted Convolution

• Quanvolution is a hybrid quantum-classical technique that encodes local patches into quantum states and processes them with quantum circuits.
• It acts as a quantum-assisted preprocessing layer analogous to the first convolutional stage in CNNs, preserving locality and generating feature maps.
• The approach balances computational efficiency and performance, with applications in image classification, segmentation, and signal analysis.

Quanvolution, short for quantum convolution, denotes a family of hybrid quantum-classical operators in which local input patches are encoded into quantum states, processed by a quantum circuit, and measured to produce classical feature maps for downstream learning. In the dominant usage across quantum machine learning, quanvolution plays a role analogous to the first convolutional stage of a CNN, but it typically functions as a quantum-assisted preprocessing stage rather than as a fully quantum network. Locality is preserved, the same filter is applied repeatedly over patches, and the outputs are handed to classical components such as MLPs, CNN backends, Attention U-Nets, clustering pipelines, or AutoML systems (Sebastianelli et al., 2024, Alam et al., 2021, Russo et al., 18 Jul 2025).

1. Core concept and formal interpretations

A standard formalization writes the quanvolutional response for a local patch $\mathbf{u}_x$ as

$f_x = d\big(q(e(\mathbf{u}_x))\big),$

where $e(\cdot)$ is the encoding into a quantum state, $q(\cdot)$ is the unitary quantum transformation, and $d(\cdot)$ is the decoding stage, i.e. measurement and post-processing (Dou et al., 2021). In this sense, quanvolution is the quantum analogue of a classical convolutional filter: it operates on local patches, uses multiple filters, and produces transformed feature maps, but the filter is a quantum circuit rather than a classical kernel.

Several papers emphasize that the analogy to classical convolution is structural rather than algebraic. Classical convolution computes local similarity through multiplication and summation. Quanvolution instead encodes each patch into a state such as $\ket{\phi(x)}$, applies a circuit $U_W(\vartheta)$, and derives features from expectation values of observables. One common expression is

$$f(x,\vartheta) = \bra{\phi(x)}U^{\dagger}_{W}(\vartheta)\hat{O}U_W(\vartheta)\ket{\phi(x)},$$

which makes the feature value an observable expectation after quantum evolution (Sebastianelli et al., 2024).

A more explicitly interpretable line of work reformulates quanvolution as quantum overlap estimation. In Quantum Adjoint Convolution Operation (QACO), both a local patch and a kernel are amplitude-encoded, and their overlap is shown to be equivalent to the quantum normalization of convolution based on the Frobenius inner product. Quantum Adjoint Convolutional Layer (QACL) extends this primitive by Quantum Phase Estimation (QPE) so that all patch-wise Frobenius inner products can be computed in parallel (Zhao et al., 2024). This interpretation shifts quanvolution from a black-box variational circuit toward a similarity-based operator with a direct classical analogue.

The concept also appears in one-dimensional settings. In spectrum peak-finding, quanvolution is used as the quantum analogue of the convolutional feature-extraction stage in a 1D CNN: a local window is encoded, processed by a small PQC, and measured to produce output channels, while the rest of the network remains classical (Bischof et al., 15 Dec 2025).

2. Patchwise pipeline and measurement semantics

Most quanvolutional systems implement a repeated patch pipeline. A local patch is extracted, encoded into a quantum state, transformed by a circuit, measured, and written back as a classical feature tensor. In the Tunis building-segmentation study, the workflow is explicit: Sentinel-1 SAR GRD data are acquired; the image is split into $256 \times 256$ patches with stride 128; each patch is encoded into a quantum circuit using angle encoding; a quanvolutional operator is applied; the circuit output is measured to produce classical feature maps; and the resulting maps are fed to a classical Attention U-Net for pixel-wise building segmentation (Russo et al., 18 Jul 2025).

The embedding step is often written as

$$U_{\phi}(\mathbf{x})\ket{0}^{\otimes n} = \ket{\phi(\mathbf{x})} = \ket{\psi},$$

with a feature map $f_x = d\big(q(e(\mathbf{u}_x))\big),$0. The patch values are therefore transformed into a state $f_x = d\big(q(e(\mathbf{u}_x))\big),$1 in a Hilbert space of dimension $f_x = d\big(q(e(\mathbf{u}_x))\big),$2 (Russo et al., 18 Jul 2025). In many implementations, measurement returns one scalar per qubit, and the resulting number of output features is directly tied to the number of qubits available in the circuit (Sebastianelli et al., 2024).

The measurement semantics vary. Some works use Pauli-$f_x = d\big(q(e(\mathbf{u}_x))\big),$3 expectation values on each qubit, arranged spatially to form a new image-like tensor (Alam et al., 2021). Others decode with the expectation of $f_x = d\big(q(e(\mathbf{u}_x))\big),$4, or average repeated measurement outcomes into a scalar output (Bosco et al., 2024). In the 1D spectrum architecture, a 32-point local window is encoded on 5 qubits, one $f_x = d\big(q(e(\mathbf{u}_x))\big),$5-expectation is measured per qubit, and each window therefore yields 5 channels; with stride $f_x = d\big(q(e(\mathbf{u}_x))\big),$6 over a 200-point spectrum, the quantum convolution output has shape $f_x = d\big(q(e(\mathbf{u}_x))\big),$7 (Bischof et al., 15 Dec 2025).

The role of the classical backend is correspondingly clear: quanvolution usually does not perform the final task by itself. In Earth Observation pipelines, quantum feature maps are handed to BIRCH, AutoML, or AutoDL (Sebastianelli et al., 2024). In image classification studies, they are passed to an MLP or a fully connected classifier (Alam et al., 2021, Mattern et al., 2021). In segmentation, they are placed before an Attention U-Net rather than replacing it (Russo et al., 18 Jul 2025).

3. Encoding strategies and circuit families

The design of a quanvolutional layer depends strongly on how classical patches are encoded into quantum states. Several distinct encoding families have been studied.

Angle or rotational encoding maps pixel values to qubit rotation angles. In the Tunis segmentation work, the input patch is represented using angle encoding, and a design constraint is stated as

$f_x = d\big(q(e(\mathbf{u}_x))\big),$8

ensuring enough qubits to represent the local patch (Russo et al., 18 Jul 2025). In the rotational-encoding baseline examined elsewhere, a $f_x = d\big(q(e(\mathbf{u}_x))\big),$9 patch requires exactly $e(\cdot)$0 qubits, so the qubit count grows directly with kernel size (Bosco et al., 2024).

Threshold encoding, FRQI, and NEQR represent a more explicit comparison among image encodings. Threshold encoding uses one qubit per pixel and stores only thresholded binary values; FRQI encodes intensity and position as

$e(\cdot)$1

using only $e(\cdot)$2 qubits for a $e(\cdot)$3 image but with gate complexity $e(\cdot)$4; NEQR stores pixel values in basis states, uses $e(\cdot)$5 qubits, and is dominated by multi-controlled operations (Mattern et al., 2021). These three encodings were compared specifically to determine how encoding interacts with variational quanvolution.

Amplitude embedding appears in the spectrum peak-finding architecture. There, a 32-point local window is normalized and embedded into the $e(\cdot)$6-dimensional state space of a 5-qubit circuit, so the data vector becomes a superposition over the computational basis (Bischof et al., 15 Dec 2025). Amplitude encoding is also central to QACO/QACL, where both patch and kernel are encoded as amplitudes so that overlap reproduces the Frobenius inner product (Zhao et al., 2024).

Integrated encoding is a more recent modification of the quanvolution pipeline. Instead of separating encoding and processing into two distinct circuit stages, the method merges them by injecting classical features directly into entangling gates of the form

$e(\cdot)$7

The tested angle mappings are Simple, RndMul, and RndLin, and the number of gates $e(\cdot)$8 is chosen to satisfy $e(\cdot)$9, with experiments using $q(\cdot)$0 (Bosco et al., 2024). This removes the hard link between patch size and qubit count: the integrated model uses only $q(\cdot)$1 qubits even for filters as large as $q(\cdot)$2, whereas rotational encoding would require 25 qubits for that patch size (Bosco et al., 2024).

Circuit families also vary. The Tunis segmentation study compares Strongly Entangled Circuit, Random Circuit, and Basic Entangled Circuit, all with 9 qubits, 2 layers, $q(\cdot)$3 kernel, and 9 filters (Russo et al., 18 Jul 2025). Another image-classification framework uses repeated CRZ entangling gates followed by RY rotations, taken as “Circuit 13” from Sim et al. (Alam et al., 2021). The spectrum peak-finding work evaluates a strongly entangling ansatz, a simplified Two-Design ansatz, and a random ansatz in which all rotation gates are trainable (Bischof et al., 15 Dec 2025).

4. Training regimes, parameterization, and resource trade-offs

A major divide in the literature concerns whether quantum filters are fixed, unsupervisedly selected, or fully trainable.

Many NISQ-oriented systems adopt frozen quantum kernels. Quanv4EO explicitly uses a lazy training regime in which quantum parameters are not trained and only classical stages are optimized (Sebastianelli et al., 2024). The Tunis segmentation model also treats quanvolution as a fixed feature extractor placed before a classical Attention U-Net (Russo et al., 18 Jul 2025). This regime is motivated by reduced computational burden, avoidance of difficult quantum optimization, and better compatibility with shallow circuits on limited hardware.

A more structured alternative is unsupervised filter selection. In the fault-detection study, candidate circuit ansätze are measured repeatedly, their output distributions are embedded as vectors in $q(\cdot)$4, and K-means clustering is used to choose circuits near cluster centers. The stated goal is to maximize the difference of quantum properties such as expressibility and entangling capability, so that the final quanvolution filters are representative and diverse. Once the filters are fixed, features are extracted for the entire training set just once, and the downstream classifier is trained classically (Dou et al., 2021).

By contrast, several papers argue for trainable quanvolution. One framework makes a case for trainable quantum filters and reports that, after 10 epochs, trainable filters achieved on average 15.98% lower training loss, 7.49% lower validation loss, 3.46% higher training accuracy, and 3.32% higher validation accuracy than non-trainable filters, although training required about 195 seconds per epoch versus 57 seconds per epoch for non-trainable filters on a Core i7-10750H machine with 16 GB RAM (Alam et al., 2021). Another study shows that trainability interacts strongly with encoding choice: for FRQI with a $q(\cdot)$5 filter, test accuracy improved from 0.806 to 0.854, whereas for Threshold encoding with a $q(\cdot)$6 filter, the untrainable model at 0.828 exceeded the trainable one at 0.802 (Mattern et al., 2021). In spectrum peak-finding, trainable quanvolution layers improved final training loss by about 15–25%, but at higher computational cost (Bischof et al., 15 Dec 2025).

Parameter and execution efficiency are therefore central. In building segmentation over Tunis, Standard Attention U-Net has 34.8 million trainable parameters, whereas Quanvolution + Attention U-Net has 2.1 million trainable parameters, a reduction of over 93%; the trade-off is a drop in OA from 0.9491 to 0.9343–0.9384, with Basic Entangled Circuit performing best at 0.9384 (Russo et al., 18 Jul 2025). Quanv4EO reports 42k parameters + 16 frozen qubits on both MNIST and Fashion MNIST (Sebastianelli et al., 2024).

Execution cost can also be reduced at preprocessing time. Flexible quantization with memoization maps pixel values into an arbitrary number of levels $q(\cdot)$7, stores previously computed outputs for identical quantized patches, and reduces circuit executions to the number of unique quantized patches actually present in the dataset. For the main classification experiments on MiraBest and LArTPC, $q(\cdot)$8 yielded execution reductions of about 96% and 97%, respectively, while keeping MSE below $q(\cdot)$9 (Bosco et al., 2024).

5. Architectural compositions and application domains

Quanvolution has been applied across classification, segmentation, filtering, and signal-analysis settings, but almost always as the first feature-extraction stage in a hybrid architecture.

In image classification, a common pattern is a single quanvolutional layer followed by a classical classifier. One framework uses a quanvolutional layer followed by two fully connected classical layers (Alam et al., 2021). Another uses a quanvolutional layer and a classical fully connected layer trained with cross-entropy on MNIST, comparing Threshold, FRQI, and NEQR encodings (Mattern et al., 2021). A more hierarchical design stacks multiple quanvolutional layers with max-pooling at each level and ends with a three-layer fully connected neural network (Dou et al., 2021).

In Earth Observation, quanvolution is used both for standard benchmark datasets and for remote-sensing tasks. Quanv4EO reports accuracy = 0.9984 on MNIST and 96.81% on Fashion MNIST with 42k + 16 frozen qubits. On EuroSAT, quanvolutional features improve mean accuracy from 0.94 to 0.96 under BIRCH clustering, from 0.89 to 0.91 with AutoSklearn, and from 0.88 to 0.93 with AutoKeras. The same work also reports Sentinel-1 SAR speckle-filtering improvements from PSNR 19.21 → 21.72 and SSIM 0.75 → 0.81 (Sebastianelli et al., 2024).

In SAR building segmentation over Tunis, quanvolution is explicitly inserted before a classical Attention U-Net. The motivation is that dense urban SAR scenes are structurally complex, noisy, and high dimensional, and the claimed benefit is more informative feature maps that capture structural details while reducing the burden on the downstream network. The reported outcome is a strong accuracy-efficiency trade-off rather than absolute superiority over the standard model (Russo et al., 18 Jul 2025).

In classification tasks designed to probe resource usage, integrated encoding changes the comparison with both classical CNNs and rotational quanvolution. On MiraBest, the CNN baseline achieves about 71.0% accuracy, Rot peaks around 77.3% at $d(\cdot)$0, and Int with Simple or RndMul reaches about 80.8% at $d(\cdot)$1. On LArTPC, the CNN reaches about 56.8%, Rot peaks near 52.9%, and Int with Simple or RndMul reaches about 58.5% at $d(\cdot)$2 (Bosco et al., 2024).

In one-dimensional signal analysis, quanvolution has been used for bearing fault detection and spectrum peak-finding. The fault-detection study uses 299 samples, each a $d(\cdot)$3 time series, with a two-level hierarchy of learned quantum feature extractors and reports competitive results (Dou et al., 2021). The spectrum study uses synthetic NMR-inspired spectra and reports that, on the dedicated hard dataset, the strongly entangling ansatz yields roughly 10.9% improvement in F1 for peak counting and 29.8% lower MAE for peak position estimation relative to the classical CNN; the abstract summarizes the result as an 11% improvement in F1 score and a 30% reduction in mean absolute error (Bischof et al., 15 Dec 2025).

6. Conceptual debates, misconceptions, and adjacent usages

A recurrent misconception is that quanvolution is itself the final predictive model. In most of the cited work, this is incorrect: quanvolution is a front-end preprocessing layer or feature-extraction layer, while classification, clustering, segmentation, or regression remains classical (Russo et al., 18 Jul 2025, Sebastianelli et al., 2024). A second misconception is that quanvolution necessarily means fixed random quantum filters. The literature now includes frozen kernels, unsupervisedly selected filter banks, and end-to-end trainable quantum filters, so “quanvolution” names a broader design space rather than a single training regime (Dou et al., 2021, Alam et al., 2021, Bischof et al., 15 Dec 2025).

Another important point is that reported gains are conditional rather than universal. Several papers present quanvolution as a favorable accuracy-efficiency trade-off, not as a uniformly dominant replacement for classical convolution. Standard Attention U-Net still has the highest reported OA = 0.9491 in the Tunis study, exceeding all quantum-assisted variants (Russo et al., 18 Jul 2025). In the image-encoding comparison, the best $d(\cdot)$4 result is Threshold untrainable rather than a trainable model (Mattern et al., 2021). In the integrated-encoding study, the Rot baseline remains below the CNN on LArTPC, while the Int model exceeds it only for some settings (Bosco et al., 2024). In spectrum peak-finding, the authors explicitly avoid claiming hardware-level quantum advantage, and noisy simulations remain worse than ideal ones (Bischof et al., 15 Dec 2025).

Interpretability and mathematical grounding are also active themes. QACO/QACL argue that many quantum convolutional layers are black-box structures and replace them with a Frobenius-inner-product interpretation based on amplitude encoding and overlap estimation (Zhao et al., 2024). A plausible implication is that part of the field’s development is moving from heuristic circuit feature maps toward more explicitly analyzable operators.

Two adjacent usages further delimit the term. Monarq is conceptually close to quanvolution because it performs front-end quantum data processing for convolution, DFT, squared gradients, and edge detection, but it is not mainly a variational feature extractor; it is a non-variational, deterministic transform engine built from QCrank and EHands, and the paper explicitly states that it is not “quanvolution” in the usual ML sense (Balewski et al., 3 Mar 2026). By contrast, QUAN uses “quanvolution” to refer to quaternion convolution, not quantum convolution: RGB inputs are embedded into quaternion space and processed by an approximate Hamilton-product decomposition implemented with real-valued convolutions (Grant et al., 5 Sep 2025). For technical writing, this distinction is essential.

Taken together, the literature presents quanvolution as a patchwise quantum feature-extraction paradigm that remains experimentally heterogeneous. Its central variables are the encoding map, circuit family, readout rule, and training regime; its strongest empirical cases tend to involve difficult local structure, limited parameter budgets, or NISQ-motivated resource constraints; and its most robust claims concern representational flexibility and architecture-level trade-offs rather than universal accuracy gains.