Quantum Visual Fields with Neural Amplitude Encoding
Authors: Shuteng Wang
Christian Theobalt
Vladislav Golyanik
Venue: NeurIPS 2025 poster ()
Keywords: quantum machine learning
neural fields
Primary area: deep_learning

Abstract: Quantum Implicit Neural Representations (QINRs) have emerged as a promising paradigm that leverages parametrised quantum circuits to encode and process classical information. However, significant challenges remain in areas such as ansatz architecture design, the effective utility of quantum-mechanical properties, training efficiency, and the integration with classical modules. This paper advances the field by introducing a novel QINR architecture for 2D image and 3D geometric field learning, which we collectively refer to as Quantum Visual Field (QVF). QVF encodes classical data into quantum statevectors using neural amplitude encoding grounded in a learnable energy manifold, ensuring meaningful Hilbert space embeddings. Our ansatz follows a fully entangled design of learnable parametrised quantum circuits, with quantum (unitary) operations performed in the real Hilbert space, resulting in numerically stable training with fast convergence. QVF does not rely on classical post-processing---in contrast to the previous QINR learning approach---and directly employs measurements to extract learned signals encoded in the ansatz. Experiments on a quantum hardware simulator demonstrate that QVF outperforms existing quantum approach and competes widely used classical foundational baselines in terms of visual representation accuracy across various metrics and model characteristics. We also show applications of QVF in 2D and 3D field completion and 3D shape interpolation, highlighting its practical potential. Project page: \url{https://4dqv.mpi-inf.mpg.de/QVF/}.

Quantum Visual Fields with Neural Amplitude Encoding
Shuteng Wang
Christian Theobalt
Vladislav Golyanik"
MPI for Informatics, SIC, Germany
Abstract
Quantum Implicit Neural Representations (QINRs) have emerged as promising paradigm that leverages parametrised quantum circuits to encode and process classical information. However, significant challenges remain in areas such as ansatz architecture design, the effective utility quantum-mechanice properties, training efficiency, and the integration with classical modules. This paper advances the field by introducing a novel QINR architecture for 2D image and 3D geometric field learning, which we collectively refer to Quantum Visual Field (QVF). QVF encodes classical data into quantum statevectors using neural amplitude encoding grounded in a learnable energy manifold, ensuring meaningful Hilbert space embeddings. Our ansatz follows a fully entangled design of learnable parametrised quantum circuits, with quantum (unitary) operations performed in the real Hilbert space, resulting in numerically stable training with fast convergence. QVF does not rely on classical post-processing-l contrast the previous QINR learning approach-and directly employs measurements to extract learned signals encoded in the ansatz. Experiments
quantum hardware simulator demonstrate that QVF outperforms existing quantum approach and competes widely used classical foundational baselines in terms of visual representation accuracy across various metrics and model characteristics. We also show applications of QVF in 2D and 3D field completion and 3D shape interpolation, highlighting its practical potential. Project page: hhttp::// //bpsv mpi 1epi-ini inf mpg. de/QVF/
Introduction
Implicit neural representations (INRs) have emerged powerful framework for continuously mod- elling signals via neural networks. They are widely used in image and 3D shape synthesis, as well as 3D reconstruction, among other fields of visual computing (54). INRs map spatial (and also tempo- ral) coordinates to corresponding signal values, enabling resolution-indeeeedden, memory-efficient, and differentiable representations; the signal encoding network fo with parameters 0 is trained to minimise the reconstruction loss L(0) over sampled coordinates x: L(Ø) = ExEx [x) -((x)1 where X' denotes the sampled domain and S(x) is the signal value to be represented by fo. As remedy the growing computational, memory and energy demand required by INR, recent work has explored the integration of quantum circuits into INR as a promising alternative to classical methods, with potential advantages in model compactness and learning efficiency [57]. Quantum algorithms operate within Hilbert spaces, enabling superposition and entanglement of states that facilitate parallel processing beyond classical systems with comparable resource scales. Specifically, quantum machine learning (QML) models involving parameterised quantum circuits (PQCs) or ansatz parameterise the evolution of quantum states through unitary transformations (implemented as quantum gate sequences), requiring number of parameters that scale logarithmically with the Hilbert
"Corresponding author's email: golyanik mmpiniif.ipiiiiii.eeeee.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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space dimensionality. Recent studies further reveal an intrinsic link between PQCs and Fourier-bases learning mechanisms [4], critical feature for relieving biased learning or mitigating spectral bias common in classical neural networks for INRs 135 Together, these insights suggest pathway towards highly efficient and expressive QMI. models for visual computing.
Despite their theoretical promise, quantum implicit neural representations (QINRs) remain heav- ily underexplored. The recently introduced QIREN approach [57] is, our knowledge, among the first in the field that is closest to our work, designed for image representation, upsam- pling and generation. In detail, QIREN projects query coordinates into learnable Fourier fea- tures paired with classical network decoder, explicitly linking it to Fourier neural architectures while, at the same time, overshadow the quantum behaviour due classical post- post-processing. In response to these limitations, we propose Quantum Visual Field (QVF),
novel coordinate-based QMI model that leverages high-dimensiooal
Hilbert spaces for lightweight and spectrally unbiased implicit visual field representations; see the scheme in Fig. (a). Rather than using heuristic classical-to-gue -oouuatulto-uuantu data encoding methods [52, 39 & [36]- which (unreasonably) assume that handcrafted embeddings align with the inductive biases of quantum circuits we introduce a novel learnable energy module that encodes non-linear data priors to align classical Euclidean and quantum Hilbert feature representations. Our carefully designed quantum circuit leverages quantum state evolution and entanglement between qubits to effectively explore the optimal mraauuusueement quantum state representations. Specifically, the reachable Hilbert space constrained in QVF for stable gradient flow and relief of issues such as barren plateaus, i.e., vanishing gradients arising due to Haar randomness" without compromising expressiveness. The quantum circuit is measured to generate multi-dimennional signals, such as images or 3D geometries, or their collections (Fig. (b)) through conditioning on latent variables. In summary, the technical contributions of this paper include:
Source
Target
Figure 1: Our learnable coordinate-based QVF model can represent various visual fields: (a) Schematic diagram of the architecture; (b) Latent space interpolation of 3D signed distance fields [10]. (c) 2D image representation of a moder-
resolution (400 x 350 pixels) [5];
2
QVF, coordinate-based QML model for visual representation learning (2D images and 3D signed distance fields). The QVF approach is designed for execution on quantum machine simulators or fault-tolerant gate-based quantum computers.
A non-linear neural scheme for encoding classical data into quantum statevectors. Our neural amplitude encoding is grounded in learnable energy manifold ensuring meaningful Hilbert space embeddings.
An efficient PQC that processes entangled information within the real Hilbert subspace, explicitly designed for stable gradient feedback by bounding Haar randomness.
Unlike existing approaches 57 QVF is a lightweight architecture with the exact structural con- figuration dynamically depending on the input data. We evaluate QVF and compare it to the main competitor, i.e., prior QINR method QIREN (on 2D image representation learning), and, additionally, several foundational classical INR baselines (for 2D image and 3D shape representation learning); experiments are performed on a high-end simulator of gate-based quantum hardware (6]. We show that QVF consistently competes and outperforms QIREN and other compared techniques. Moreover, QVF supports problem scales and applications beyond the reach of prior QINR frameworks, such as image inpainting, shape completion and latent space interpolation (Fig. (c)), taking a step towards unlocking quantum models in real scenarios.
HHaar randomness refers the property of sampling quantum states uniformly at random from the Hilbert space according to Haar measure.
2

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Classical
Quantum
Energy Spectrum
Query Coordinate:
Concet
Energy Intercnce
Probebiltt
Amplttdoo Enrooling
Quantim
Measure
Ex-2
Ex-2
10)
S(0)
(A
b SDF
Latent Vecior:
10-000
B....
hote: (0.)) - (Plnr
Figure 2: Overview of the proposed QVF model, a QML framework for visual representation learning. Query coordinates @ encoded using 7 (positional encoding) concatenated with the con- ditioning latent code are used to infer the energy spectrum E of a quantum system, associated with Boltzmann-regulated statistical uncertainty P. The inferred statistical property is leveraged in encoding the classical data into quantum statevectors, which are subsequently processed by the parametrised quantum circuit S(O). Field properties are decoded probabilistically from projective circuit measurements.
2 Related Work
Classical Neural 2D/3D Scene Representation. Neural networks serve basis for modern implicit scene representations, employing continuous function approximations to circumvent the constraints of discrete grid-based methodologies [33 50 12 (25). Initial breakthroughs utilised multi-layer perceptrons (MLPs) to establish coordimate-o-aattibutt mappings, as demonstrated in Chen et al.' [12] continuous implicit model for arbitrary-scale super-resolution. The paradigm has since been extended to 3D representations, supplanting conventional voxel- and mesh-based approaches: DeepSDF 34] achieves geometrically coherent surface reconstruction via learned signed distance fields (SDFs), while neural radiance fields (NeRF) [I] introduce a volumetric scene representation parameterized by coordinate-bast neural mappings of spatial coordinates and viewing directions to radiance and density, enabling photorealistic novel view synthesis. Gate-based quantum computing offers great potential for fundamentally enhancing INRs.
Quantum-enhanee Computer Vision. Growing interest in quantum computing for computer vision
has established quantum-enhaneed computer vision (QeCV) as emerging research frontier. Current literature predominantly explores quantum annealers for combinatorial optimisation [16] 7 55 14
5 29), while tunable quantum circuits remain underexplored. Early works introduced foundational concepts such as quantum image denoising via localized convolutional operations 444 and quantum convolutional neural networks (QCNNs) with mid-circuit measurements emulate translational cquivariance [3]. Subsequent advances include hybrid quantum-classiet classical architectures for 3D point cloud classification through voxelization and quantum feature processing 22, well quantum
autoencoders for classical data compression via hand-crafted amplitude embeddings (36].
Our work is inspired by 3D-QAE B6), who developed hand-crafted quantum amplitude embeddings
for encoding 3D point clouds. How-
ever, their method, as acknowledged by their authors, suffers from lim-
Characteristic
Ours
QIREN 57
3D-QAE 36
No heavy post- pett-poccesinggg
X
ited scalability and underperforms classical models. Another related work is QIREN by Zhao et al. [57]. which employs the sandwich struc- ture, i.e., a quantum circuit layer placed between classical pre- and post- processing. They leverage circuit's Fourier connections to project queries in the Fourier basis, fol- lowed by a classical dense layer for inference. This draws parallels to classical positional or Fourier encodings, with the Fourier spectrum size growing exponentially. While theoretically motivated, its practical utility is debated as heavy classical postprocessing reduces the quantum component to feature generator. In contrast, our framework avoids such post-procet post-processing and more heavily relies on the ansatz; see an algorithmic comparison in Table I We use learnable, energy- based Boltzmann reguulated amplitude encoding, which is critical step towards unlocking the potential of quantum computing as demonstrated empirically in Sec. 5 At the same time, our carefully designed ansatz evolves the encoded data and ensures robust gradient feedback.
Data encoding
Neural AE
Neural Angular
AE (hand-crafted)
Qubit budget
logarithmic
linear
logarithmic
Supported dimensions
2D/3D
2D
3D Simulator
Quantum hurdware
Simulator
Simulator
Table 1:
Comparative algorithmic analysis of related work.
"AE" denotes amplitude encoding.
3

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3 Review: QML, its Unitary Nature and Fourier Structure
This paper assumes familiarity with quantum computing and its notations. For convenience, provide a refresher in App. A QML leverages parametrized unitary quantum operations on en- coded data (xb())) Σ, Vy ()(r))
learn functions typically expressed expectation values
f()) (x)))) (()()) where Ô is a Hermitian observable (Ò = Ö+). The unitary nature of quantum evolution (U UU = 1) preserves inner products and norms, ensuring that the spectral components of the encoded data are transformed by quantum eircuits. The spectral decomposition of
Ó = Ax (ex) reveals Fourier-like structure in f(z) Σx A& (ex/ (x=))) where the pro- jections (es (x))) act as Fourier coefficients and the eigenvalues λ relate to accessible frequencies. This fundamental property is consequence of quantum mechanics: unitary transformations preserve spectral components, ensuring that even complex quantum circuits inherently operate in a frequency domain. As a result, the expressivity of QML models directly linked to their accessible frequency components, influencing their ability to generalize and learn structured data representations.
4 Our QVF Approach
This section introduces the proposed QVF model, i..., QINR for learning visual representations and their collections; see Fig. 2 for its architecture. QVF takes query coordinates O and an optional latent variable (in the case more than one visual field needs to be represented) as inputs and produces 2D or 3D field properties s. We introduce encoding classical data into quantum states using neural amplitude encoding in Sec. [.]] while the quantum circuit design and measurement are detailed in Sec. 4.3 provides training details and applications supported by QVE.
Sec. 4.2
4.1
Amplitude Encoding with Neural Embeddings
Our parametrised energy-based embedding of classical data into quantum states [e): (1r) gener-
alises widely used hand-crafted amplitude encoding (AE) 136 40) AE enables an exponentially compact encoding of N = 2" classical values into probability amplitudes of qubits by leveraging -mny quantum superposition. Notably, this implies that AE induces exponentially-nan -many random Fourier features due to the inherent periodicity of quantum state phases. The fundamental limitation of hand-crafted AE stems from its a priori, possibly biased prepared quantum states, which poses a risk of misalignment with subsequent quantum evolution or suboptimal utilisation of task-specift data Fourier priors. Hence, we propose a data-driven approach for AE that learns the optimal quantum state density (ar) directly from data, i.e., for QINR in our case. We restrict the process on pure quantum states satisfying Tr(p(x)²)= 1. Drawing upon the fundamental svergy-probability duality inherent in physical systems (e.g., in statistical and quantum mechanics), we infer the conditional energy spectrum E of a given visual representation and transform it into a probability distribution P subsequently encoded as qubit state amplitudes a; E C residing in the complex Hilbert space H. Our encoding introduces non-linearity into the quanium evolution while reserving the full repertoire of B] quantum processing and mcasurements For energy inference, we employ a minimal dense MLP f(x = { z)) (()) x + activated by ReLU; we positional encoding to accelerate learning [35 31):
(I) @ denotes the field query coordinate while z represents the latent code in the case of learning visual field collections. The inferred E (r)))), is leveraged to derive the Boltzmann-regulater P of the quantum system; Boabs-BBolzzmann framework 5 serves as an inductive embedding bias for encapsulating thermodynamic uncertainty, enabling the realisation of Gibbs quantum states [13. We next formulate P = [P:]. i (l...., .) through the construction of discretised energy landscape E, derived from the Gibbs canonical ensemble:
(0) ( sin(22-1 ~0)) cos(22-1 :
p exp (-3E (EO,))) z))
where
Z
2 z= / exp (- (E de
exp exp (-,))
'tranffrmations in quantum circuits before measurement are linear, which is often seen as a key limitation ansatz expressivity

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with 8 = (kgT)- representing the inverse temperature. The quantum state amplitudes a, E C residing in the complex Hilbert space H. are characterised by their complex phases Φ; arg which are arbitrary within the interval [0, 2x);; it is subjected to the normalisation condition ||as|12
P. with Σ> P. 1. Finally, the quantum states | (O,z): encoding the query coordinates for our input fields are prepared as follows:
N
|in ( z))
Σ 1=1
v (Peetarg(a))
(4)
((,))) Win ( (0,z)) (Vii O z)]
Σ ana lobe)) (1by)-
(1b)) is the computational basis, (( z) is the density distribution of I ( 2)) and "(+)*" denotes the adjoint. We then theoretically analyse data encoding effects on the model expressiveness.
Lemma Energy inference exhibits functional equivalence to determining optimal non-linear input- dependent frequency spectrum embedded within variational quantum circuits, defining model's inherent expressiveness.
As demonstrated by Schuld et al. 44 variational circuits of the form U(x) W²g(x)WW admit truncated Fourier-type expansion when measuring circuit expectation values (M):
(M) (0) w Mg(x))" 10) C tww
where W and 2 are arbitrary unitary matrices. The effective measurement operator is defined as: M = MW2 while g(x) serves as data encoding modules applied to the physical system. Notably, unlike Schuld et al. [43] where encoding gate analysis are restricted to Pauli gates, g(x) compass more general quantum operations. Prepared imput quantum state | (I:) can be equivalently expressed as
[m)) = (( 10),
establishing direct correspondence between inferred energy landscape and the multi-dimensional frequency spectrum S2 with dependencies encoded the learnable energy inference framework.
The optimal effective measurement basis, given by MaR W M/22 opt along with the heuristic learnable circuit design Š(0), which approximates W opu will be introduced in the next sections.
4.2 Our Parametrised Quantum Circuit
Once the classical data is encoded into (0. ()). it is processed by our learnable PQC or ansatz S(0) within a high-dimensional Hilbert space. Our goal is a compact and expressive PQC for QINR learning. As unrestricted traversal of the Hilbert space can induce training instabilities, we therefore constrain the ansatz S(0) to the manifold of real-valued unitaries, constructed from Pauli-Y rotations and entangling gates to ensure efficient training of QVF. This design choice avoids the imaginary components introduced by Pauli-X and -Z rotations, which would otherwise permit unconstrained exploration of the full, complex Hilbert space and hinder training. Our PQC architecture is analogous to classical bensely-connecterr neural networks in the sense that it contains alternating layers of parametrised single-qubit Pauli-Y Y rotations and entangling operations, supporting highly correlated, lon-l quantum states that cannot be decomposed into a tensor product of individual qubits which enables parallel information processing. This design provides critical benefits for QINR learning: (1) confines state evolution the manifold of real-valued unitaries, eliminating redundant parameter dimensions that facilitate scrambled quantum states and barren plateaus; (2) it naturally discards complex phase information while preserving all measurement-rellee relevann quantities, i.e., universality, for certain basis observations such as Pauli-Z observable the complex phase factors cancel out and become irrelevant when computing Z-basis probabilities; and (3) maintains full expressivity while significantly simplifying the optimisation landscape. Additionally, our design necessitates
Figure 3: Repre- sentative pure qubit state transitions on the Bloch sphere.
5

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the enforcement of a zero complex phase in our encoded quantum data, formally expressed as arg(a;) = 0. with schematic Bloch sphere dynamies depicted in Fig. 3 Once (e,))) has been transformed by S(0), we extract the visual field attributes encoded in our QINR using projective measurements of the final quantum states.
Nulti-dimensional Measurement. To extract an m-dimensional representation (mSn) from S()). we implement local Pauli projective measurements on the first m qubits, effectively tracing out the remaining n- qubits. The corresponding family of local measurement operators {O:} is formally defined over the rt-qubit Hilbert space as:
Ô (s of (8i=+114) i (l.....m), m)),
(7)
where Læ denotes the identity operator acting on the k-th qubit, preserving its quantum state within the tensor product. The operator of = [0) (0, [1) (11, represents the Pauli-Z observable applied to the i-th qubit. Local measurements help guarantee robust gradient feedback [9 48) for the circuit. The output of (()) is defined as the expectation value of finite-shot circuit measurements (App. D analyses the influence of circuit measurements on the extracted image quality). This expectation value can be expressed as Vint, which is defined as the number of shots approaches infinity in the asymptotic limit:
Vin (+) = (r)) ())) Š
( ÓŠ()..
(8)
M (0) represents a parametrised measurement basis employed to approximate the optimal measure- ment basis Mopt via unitary quantum evolutions of our fully-entangled circuit S(0). Similar to classical universal approximation theory, quantum circuits with sufficient depth can approximate arbitrary unitary transformations. The Solovay-Kitaev Theorem provides rigorous theoretical upper bound on the number of quantum gates required to approximate an arbitrary unitary operation to a given precision 6, given by (4" log' log' (1) Circuit depth J J-analogous to the number of layers in classical neural networks- and the corresponding total number of constituent unitary quantum gates are hyperparameters of our Š(0). The local measurement V(@); corresponding to the i-th qubit is injectively mapped the corresponding dimension of the target field, requiring the qubit number m.m. We next detail the end-to-end training protocol of our model under the Bayesian framework.
4.3 Training Details and Applications
Initialisation of (()). We incorporate established PQC initialisation strategies, i.e., identity [8] and Gaussian [56]; see Fig. 4 for the architectural implications. For identity initialisation, each circuit layer S(0); depth j E (1.....J) J} can be expressed as a sub-cireuit Sn followed by its Hermitian adjoint Fn. Then, S(0); is constructed by assigning O2 410,2m to Sp. which also initialised as FR per definition. This ensures that the composite op- eration S()); = SRFR is equivalent to a zero circuit depth (identity circuit) before training. Note that while the initial configuration enforces SR/R = T, this constraint is not maintained dur- ing optimisation [18]. For the Gaussian initialisation. trainable parameters for S(0), = Sa, are, instead, sampled from zero-mean Gaussian distribution with the variance coupled to the circuit depth, i.e., 0% ~ NO00, o² (T)) [56). The overall circuit architecture Š(0) is obtained by concatenat- ing I blocks Š(0); such that the overall unitary transformation is given by $(0) = I $(0))- Note that architectural homogeneity across blocks is maintained, preserving systematic exploration of the unitary space (1(2).
Ry
Ry
Random
RRCG
Goussian
Figure 4: QVF Initialisation: Schematic circuit module initialised with identity (top) and Gaussian (bottom) schemes.
QVF Training. Consider dataset X composed of W distinct visual fields denoted by X,, for i E (1,.., W). Each data field X; encapsulates physical field properties sy, such as pixel values in images or SDF for geometric representations, sampled at specific spatial coordinates O/; here, index j denotes the sample index per field. The relationship between spatial coordinates and physical properties is defined by a function f, such that sampled points within each field are given by:
X, ((e],s)) f(O(),j j {0, 1,.,,,..111 M
(9)

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where M is the number of samples per field. Crucially, each data field X; is associated with unique latent code 24. The training objective is to maximise conditional probability distribution (s)
represents trainable parameters of QVF:
Po s (
po(s//00 z1)(((z)) [p)(s((6/.<)))()))
(10)
Under sufficiently large number of i.i.d. quantum circuit measurement shots, the conditional likelihood (s [ zi) can be approximated by a Gaussian distribution. This statement is supported by the Central Limit Theorem (CLTT, which establishes the asymptotic normality of the sum (or average) of numerous i.i.d. random variables with finite variance. Consequently, po(s/ z;) can be approximated as
(s (0(, ss exp (-Z2(V O):
(11)
where Z(-) represents the loss function that quantifies the discrepancy between the output of the circuit (zz, 0²; and the observed physical property sy. Model training can, therefore, be formulated as maximising this conditional likelihood under a Bayesian framework. To ensure a smooth representation transition in the latent space, the prior distribution over 2; is softly penalised to follow a smooth distribution; an isotropic zero-mean multivariate Gaussian distribution is reasonable choice as adopted by Park al. [4). The loss function T. minimised via training all learned W fields {&ili = 1,..., W) with M samples per 8;, is formulated as:
L 8) E.i (C(V (=,, 0%: 0), x() + =112)
(12)
QVF undergoes end-to-end training: classical parameters are updated via gradient descent, while quantum parameters are optimised using the parameter-shift rule (32].
Usage and Applications. Once trained, we can query QVF for the encoded 2D or 3D representations in coordinate- based manner. We can also infer with partial samples, enabling applications such as image inpainting and partial shape completion through latent space optimisation. Using Maximum-a- Posteriori (MAP) estimation, identify latent code & that maximises agreement with the input partial observation X, while keeping the pre-trained model fixed:
2 = argmin
Σ
L(V(z,, 0); + + 1112/12
(13)
(G),,))
Algorithmic Summary. We summarise the QVF training protocol in Algorithm in the Appendix.
Experimental Evaluation
We experimentally evaluate QVF for learning visual field representations, encompassing both 2D images and 3D geometries, while systematically analysing its generalisation in the sense of signal interpolation and the ability handle missing and occluded regions. We use I) images from the CIFAR-10 dataset (24) and high-resolution images with rich spectral details [15], and 2) 3D shapes from the ShapeNet 10 dataset. We report widely used metrics averaged over three repetitions.
Implementation Details. We empirically evaluate the model on a noiseless high-end simulator: provided by PennyLane [6] with an A100 GPU. We employ Adam optimisation (33 with an initial learning rate of = 10-3, subject to a learning rate scheduler that triggers upon plateauing with window size of 50 epochs (scaling 1j by 0.9). The number of epochs is set 5k; and 1ady=10mm 10-3 in Eq. (1).
Reffauit.qubit.ce
Hardware and Efficiency. Absence of large-scale, fault-tolerant quantum hardware forces contem- porary QML models to rely on exponentially expensive simulators run on classical hardware; see T For circuit with n qubits of depth J. the computational complexity on a classical noiseless simulator, without acceleration, is given by O(2enn where c is a constant that depends on the specific simulation method employed.
7

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5.1 Circuit Trainability
We experimentally show that constraining quan- tum transformations S(0) to real-valued uni- tary operations (resulting in bounded Haar ran- domness) helps with gradient flow. As quantum circuit parameters are inherently periodic within [0, 2T), we evaluate the gradient flow by uni- formly sampling parameters within this range and quantifying its expectation value. Due to the zero-men pero-mean nature of the expected loss gradi- ent (see App. A.3) the vanishing gradient phe- nomenon is governed by the variance decay rate.
Sanalre
Thy
Kimber
Figure 5: (Left:) comparison of the gradient vari- ance (y-axis has log scale); (right:) visualisation of reachable quantum states. We, therefore, quantify its variance Vargrad
Vaa Var
(f
t= )=
Vargra ((,,0()))
(M)
(14)
000k
where "Var(.)" denotes the variance operator and T 500 is the number of samples to evaluate the expectation. (M e is the expectation value of S(0). and iterates over ansatz parameters. Fig, reports Vargmd for the increasing number of qubits for two ansatze, i.e., of our QVF and QIREN (57
with a strongly-entangled quantum circuit which allows scrambled (i.e., non-restricted) quantum states in the Hilbert space. We observe that ansatz with bounded Haar randomness maintains
stronger gradient flow, which is crucial for its trainability and efficient representation learning.
2D (Image) Representation Learning
We evaluate image representation learning with QVF and start with sin- gle images. We first compare QVF to a classical model that takes the archi- tecture consistent with QVF's classi- cal energy inference module; QVF has overhead of 170 parameters due to its ansatz. This implies that the dif- ferences in the learning behaviour and the final representation accuracy are predominantly due to the inductive bias of the quantum ansatz, isolating influences from external factors. Fig. (a) visualises reconstructed images during training of QVF: Fig. (6-bb) plots the
Suatt
Cannegg
Grsice
Figure 6: (a) Reconstructed images during training: (top) our QVF, (bottom) classical model; (b) PSNR curves.
learning curves (PSNR) for the first thousand training epochs and, thus, highlights the differences in the training progression. Similar observations are made for other trials during the evaluation; QVF significantly accelerates learning high-freeqenee signals while performing on par with the classical method in the low- low-frequency regions.
We also experiment with the hand-crafted encoding strat- egy of Rathi et al. [6), which does not result in a recog- nisable representation upon convergence consistent with their results. This validates our design and,
observation
especially, the necessity of learnable energy embedding. We then benchmark QVF against QIREN (57), the most closely related QINR approach. While QIREN employs quantum ansatz sandwiched between classical layers, QVF uses classical component for data encoding only. We evaluate QVF and QIREN consistently with n = 5 qubits, and evaluate the performance 50 different images with metrics reported in Table 2 Results demonstrate that QVF with Siren outperforms QIREN by 30% on MSE.
Figure Visualization of the recon- structed images.
Figure 8: (a): Geometry represen- tation using QVF with latent-space- conditioned SDF inference; (b): Shape completion from partial inputs.
Next, we perform representation learning on image col- lections via latent variable conditioning, i.e., configure QVF to learn the 50 images simultaneously. Note that

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QIREN does not support this experimental setting and, hence, we compare QVF with widely-used classical foundational INR methods, i.e., MLPs with ReLUs, and Siren. The comparisons follow the same evaluation protocol, where the only difference between the baselines and QVF is the presence of the ansatz; the results are summarised in Table Reconstructed images conditioned on different latent codes are visualized in Fig.
We next consider deployment of QVF on future quantum hardware that could reduce the representation fidelity, such as measurement uncertainty. The visual field extracted from the ansatz can differ in its fidelity (accuracy and quality) due stochastic effects induced by finite sampling on quantum hardware. In App. D we visualise in Fig. [12 extracted image representations encoded within our pre-trained QVF across a varying number of shots Nshor showing the characteristics of the resulting fields with increasing sampling precision.
Ansatz Configuration. We next evaluate architectural variations in QVE. We investi- gate the impact of the key hyperparameters: 1) ansatz width, i.e., number of qubits n; 2) circuit depth J: and 3) latent space dimension p of our classical module
for encoding data into quantum states; see Fig. 9 for the results. Scaling up the QVF ansatz, i.e., increasing I and width n (while maintaining other pa- rameters), leads to per- formance gains in both cases. QVF scales ro- bustly and is not af- fected by substantial trainability problems, at least in our evaluated scenario. The expressivity of the classical module for integrating non-linear data priors and preparing encoded quantum states serves as the architectural cornerstone. With increasing p. observe consistent performance surges.
20
20"4
20
p-00
20
9-95
30
200
300
100
200
300
100
200
300
Epocis
Epochs
Epochs
Figure 9: Ablation study with modules influencing the model performance. From left to right: 1) circuit depth J: 2) number of qubits n; 3) hidden neuron dimension per layer p.
Parameter Scaling Analysis. Per design, QVF consists of: 1) the classical module for neural amplitude encoding; and 2) the ansatz. As we leverage tiny MLP in QVF, its parametrisation scales quadratically, i.e., (p²)) w.r.t. the latent space dimension p. Meanwhile, our quantum ansatz has parameter scaling of O(nJ) w.r.t its depth. While the ansatz contributes negligibly to the total parameter count, it improves the overall performance by large margin (see Fig. 6).
5.3 3D (Shape) Representation Learning
We next evaluate geometric representation learn- ing of 3D shapes the form of SDFs. This setting is investigated for the first time in the context of QINRs; poses challenges primarily concerning QINR scalability and the complications arising from varying shape topological structures. Simi- lar to images, we perform representation learning on 3D shape collections. We select three shapes from ShapeNet [10] and non-uniformly sample signed distances at 100k spatial points per shape, with higher near-surface sampling density for bet- ter surface detail capture. Note that while the scale of this experiment setup be considered moderate for classical models, significantly ad- vances the feasible scale of QINR models (which are nevertheless constrained by the simulator's performance) and provides valuable insights for future advancements. We inherit the experimental setting from the experiments with 2D images and report the representational accuracy upon convergence in Table 3 The bascline setups are, likewise,
Method
MSE (x10-3) L
PSNR 1
Ours(Gaussian) Ours(ldentity)
0.98 10.09
30.06 11.00
0.99 +0.09
30.02 +1.0
Ours(Gaussian
0.55 10.04
32.59 10.2
Ours(ldentiy>>+inn
0.54 +0.04
32.67 40.3
QIREN [77]
0.78 +0.05
31.03 +0.2
Table 2: Numerical results for 2D representa- tion learning between the previous QINR method QIREN [57 and our QVF.
Method
MSE (xil "
Images
PSNR 1
MAE [x10 9)_
3D Stapes
Ourst Cansoaans 4 ReL.
102 20.11
29311.11
0,99:2.07
Ounatitentty in ReLU MI P.R.PR 34]
1,0 10.00
29.521.2
L10E2.
2.17 10.13
26. 26.57.0.51
1,43:0.14
Currs Conssian) Sin Ours Mentirysi MLP+Sin [3]
0.6240-00
32.2.0.3
0.27:3.05
0.72 10.00
31,4004
0.32:-005
1.1940.08
29.240.4
0,4843.06
Table 3: Numerical results for 2D/3D representa- tion learning for our QVF and classical baselines.

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MLPs with different activation functions, including the ReLU, which corresponds to the DeepSDF approach [34]. The final meshes can be extracted from the queried signed distances of QVF using Marching Cubes 27 as visualised in Fig. (a).
5.4 Applications Supported by QVF
QVF supports applications such as visual field interpolation in the latent space, image inpainting and shape completion. Fig. (c) visualises linear latent-space interpolation of 3D shapes encoded in the converged QVF, i.e., a frequent experimental setting in the classical INR literature [34). QVF also supports image and shape completion by first sampling & and optimising its value using MAP; see details in Sec. [..] In the second step, the completion can be performed by leveraging the optimised latent code and inferring missing regions; see Fig. (b) and App. E for the qualitative results.
6 Discussion, Future Work and Conclusion
Our QVF is a novel QML framework for implicit representation learning of visual fields. In our experiments quantum hardware simulator, observe that QVF-even with minimal classical components-can achieve high representation fidelity across data modalities such as images and 3D shapes. Furthermore, QVF outperforms the previous QINR approach QIREN (of similar model scale) and, additionally, is competitive against foundational classical baselines. The ansatz configura- tion and ablation studies highlight the influence of each QVF module. Our ablative study confirms the sufficient circuit depth resulting in a balance between the ansatz depth and high representational accuracy. Upon expectations and the theoretical predictions, our QVF is efficient in learning signal details (Fig. 6). As the first among QINR methods, QVF supports joint repre- sentation learning of image and 3D shape collections, and applications such as image inpainting and 3D shape completion. Finally, we emphasise that this work focuses on the challenges of advancing QINR through fundamental methodological innovations. Hence, we do not aim to challenge classical well-engineered models in the absolute terms. Our implementation can be found on the project page.
Linitations. While QVF demonstrates substantial improvements over prior QINR methods in terms of both performance and supported size of visual fields, the current experimental scale, nevertheless, remains constrained due to the quantum hardware simulation overheads. Those, however, affect all existing applied QMI. works before the advent of fault-tolerant gate-based quantum computers.
Future Work. We see various promising avenues for follow-ups and QVF improvements. One direction is to explore the preparation of learnable quantum states following Gibbs distribution with reduced computational complexity (e.g., tensor train decomposition (30)). We also foresee that other problems with open challenges, such as 3D reconstruction and neural rendering from 2D images, could adopt QVF representation. We also believe that many tricks and further ideas from the INR literature could be adopted in the QINR context in future (e.g., space partitioning structures and non-rigid generalisations) 38 49 46 47
Acknowledgements. We thank Natacha Kuete Meli, Daniele Lizzio Bosco and Thomas Leimkuehler for helpful comments on the manuscript. The work was partially supported by the Deutsche Forschungsgemeinschatt (DFG, German Research Foundation), project number 534951134 134.
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Supplementary Material
This appendix supplements the main paper. starting in Sec. A with a detailed background gate-based quantum computing. It covers both quantum physics foundations and applications of quantum computing in machine learning and INR. also connects quantum circuit measurements and Bayesian inference. Next, we outline the full algorithmic (training) protocol; visualise the quantum ansatz architecture used in the experiments; and provide more implementation details in Sec. [ On the experimental side, we further analyse QVF performance with noisy circuits in Sec. C followed by the image representation quality in dependence on the number of measurement repetitions (shots) in Sec. D Applications supported by QVE, such as image inpainting and shape completion, are discussed Sees. [a while additional visualisations of shapes are shown Sec. G
Sec. H then discusses the development status of currently available real quantum hardware.
A Background
A.1 Preliminaries on Gate-based Quantum Computing
Qubits. The fundamental information blocks of a quantum processing unit (QPU) are qubits, i.e., the analogues of bits in classical computing. Unlike classical bits deterministically representing one possible state 1), qubits can statistically represent two distinct information states at the same time, denoted in the bra-ket notation as (0) and [1].
Superposition is a fundamental property distinguishing qubits from bits: It grants qubits the capacity to exist in a combinatorial state (18) of (0} and (1) such that:
Figure 10: Bloch sphere vi-
sualisation of qubit states.
Qubit O: (v)) v2 (10)
[1)), qubit I: (b) = 11).
(v) a 0) + 111))
(15)
with a, 3 E C and |a|2 + | 1. Qubit states (et) can be visualised Bloch spheres (see Fig. 10
expressed a vector form:
a |
10) 11) Iv0 (0) + 311)
(16)
Measurement in quantum mechanics inherently adopts statistical approach to extract numerical information. For a qubit state (e) = a |0) + 3 [1) measured with operator Ô (that must be Hermitian, i... Ó+ Ô), this implics probabilities la| and [0)3, respectively, for measuring the information (i.c., eigenvalue of the measurement operator Ô) stored in states (0) and [1]:
ò 0 < 10) and Ó 811)
(17)
where and eigenvalues of the measurement operator 10). The key aspect of measurement is the phenomenon known as wave function collapse, i.c., the projective measurement causes (1)) collapse to the operator's s eigenstate, (0) or [1), conditioned on the measurement, i.c., k or 5.
Entanglement further distinguishes quantum from classical computing. In the classical case, information stored in bits is independent, i.e., measuring one bit does not affect others. In the quantum realm, qubits can be highly correlated, exhibiting entanglement such that the information of one qubit can be interrelated with another, despite possible physical distance between them. For instance, a general information state of a 2-qubit system |0)),2 can be expressed as:
[w), = a 100) + b (01) + e [10) + d(11),
(18)
with a, b, c, d E such that |a| + |b + le|² - id| 1. The 2-qubit system considered entangled [w)) cannot be expressed as a tensor product of two qubits (eb) and [ indicating that their information cannot be independently measured without disturbing each other, i.c..
v)2 * [4),,, @ (1))...
(19)
Rotation Operators. The operators responsible for rotating quantum states (eb) of qubits along æ, y, axes on a Bloch sphere are referred to as rotation operators. Any single qubit operator ft can expressed as a combination of such rotation operators R2,,, Ry, Ri., ie., fR(0, 7, R2 (7) with angles 7 and 7:
His (0)
cos( () (
-i sinn
(r(r)
ccos( 3) sin )
-sin(}) cos( [3)
Ra(7) [e - 0 7
[ 0
(20)
cos(?)

---

The Pauli operators X, P, 2 represent specific instances of above rotation operators, i.e. rotations by radians along the xr, y, z-axes, respectively. These operators can also be expressed matrices in the computational basis (0) [1) as follows:
=
0
0
y
1
0
(21)
The Schrödinger Equation. Quantum computing involves the manipulation of information according to the principles of quantum mechanics, with its foundation rooted in the time deeeddeenndent Schrödinger equation:
d [b))) = f (t) (+))).
ih
(22)
dt
where h is Planck's constant, and (x)(t) and (b) are the quantum states after and before evolution, respec- tively: H is the Hamiltonian operator of the quantum system. Therefore, the evolution of quantum states can be
described by the following relationship:
(t)) T' 1 (()ddt (v())))
(23)
with 7' denoting the time ordering operator. This simplifies to e # (0)) for time-independent IT, Using a more compact notation, the Schrödinger equation can also be equivalently written as:
(v(t)) (FH,t) (=((0)) with (ft, = #
(24)
(25)
To perform rotation operations qubits, the system Hamiltonian ft can be set to Eb, with à E (X, 2). By substituting 17 2Et/h, hi arrive
(A.t)) = 48 =- 40
(1).
(26)
A.2 Review: Quantum Machine Learning
The potential of quantum computing to enhance machine learning algorithms leads the emergence of quantum machine learning (QML) 42. discipline employing quantum mechanical phenomena tackle classically intractable learning problems through enhanced computational paradigms 42). Central to QMI. are: 1) a feature map, which encodes classical input data into quantum states and 2) variational ansatz, which performs quantum transformation on the quantum states. PQC have been shown to be asymptotic universal function approximators 43) Several standardized QML algorithms have been explored, including quantum principal component analysis 26 quantum support vector machines [37] quantum Boltzmann machines [T. and quantum
k-means clustering
Feature Map. Integrating classical Euclidcan data x into quantum computational frameworks necessitates a non-trivial mapping to quantum states [x(x)) in a Hilbert space H. Several established encoding techniques exist, including basis encoding, amplitude encoding, Homiltonian evolution encoding, with each presenting distinct trade-offs in qubit efliciency and circuit depth complexity. However, the determination of optimal encoding schemes remains an open research challenge, as the relationship between encoding fidelity F(x) = I (tiiaa x) (eseedded (x)) 1 resource requirements, and task-specific performance metrics (e.g., classification accuracy or function approximation error e) remains poorly characterized across different problem domains.
Variational Ansatz. Quantum evolution of classical information embedded in states (x(x)) requires parameter- ized unitary ansatz (0) e C² x2" acting on n--ubit systems. Physically, the ansatz is constructed through sequential composition of such unitary transtormations, formally expressed as (() T (II'-, (i(O:). where T denotes the time-ordering operator governing gate sequence implementation. This induces Hilbert space transformation (0): H. > H that maps input states to processed output states through the operation [0(2, 0)) = (0) [(b()))
Measurement. Quantum computation culminates in statistical data extraction from evolved quantum states (o(x)) through projective measurements using Hermitian observables Ó, where the computational output formally defined as the expectation value: V(x) (ø(z) Ólø(z)). Measurements collapse quantum states according to the Born rule, thereby restricting access to the embedded classical information to statistical estimators derived from repeated measurements. The choice of observable Ô fundamentally governs both the Eeormatiinntthorette capacity of the measurement protocol and computational complexity.
Training a Variational Ansatz. Instead of constructing computational graph and performing backpropagation, training quantum circuits involves only forward evaluations (53). To minimize a measurement dewpeeedeeeeedeedeeeeteeee cost

---

Tunction 2(0), the exact gradients VC( can be evaluated through quantum circuit evaluations at shifted parameters 0 + Te; for basis vectors e, expressed
0 -
3o 4
(27)
This technique, i.c. parameter-shift rule, exploits the trigonometric structure of unitary gate generators G; (where (i().) = e =) to enable cardware-compatible gradientettimation estimation without numerical approximation persistent circuit memory a critical advantage over classical backpropagation that requires differentiable computational graphs.
A.3
Review: Barren Platcaus
Training a variational ansatz S(0) is fundamentally constrained by the barren plateau phenomenon, where random parameter initialisation induces exponential vanishing of cost function gradients across Hilbert space. As formally demonstrated by McClean
(28) through concentration measure analysis:
...for a wide class of reasonable parametrised quantum circuits, probability that the gradient along any reasonable direction non-zero to some fixed precision is exponentially small as a function of the number of qubits."
This phenomenon is also known as harren plateau, which can be expressed mathematically for a system with qubits follows:
Var E ( v 1/"h
E., [w,] 0,
(28)
where y characterises the circuit's entangling capacity. The variance bound's 's scaling establishes that gradient estimators require O(v") measurement samples to maintain constant precision, resulting in exponential resource overhead that renders practical optimisation infeasible for > 1. This poses challenges, particularly for gradient-based learning. Identified factors contributing to barren plateaus include observable locality 48). specific noise models [51] and an ansatz close to a 2-design, iee, matching Haar random unitaries up to the second moment [28, 20 Those highlight the importance of selecting appropriate initialisation protocols, quantum ansatz designs and observables.
A.4 Connection of PQCs to Bayesian Inference
As quantum circuits are inherently probabilistic models, they share conceptual parallels with Bayesian infer- ence. In Bayesian neural networks (BNNs), probabilistic outputs emerge from parameters governed by prior distributions, with training focused maximising the conditional likelihood of observed data labels while implicitly updating posterior distribution over parameters. For PQC, while they leverage deterministic parameters, they exhibit probabilistic outputs due to the stochastic nature of quantum measurements, which under sufficiently large shot counts approximate Gaussian distributions in accordance with the CLT. While the probahilistic outputs of PQC permit an interpretative lens rooted in Bayesian principles, their training does not inherently involve posterior inference over parameters unless explicitly cast within Bayesian formalism (2 This distinction underscores that the Bayesian interpretation PQCs arises from their measurement statistics rather than an intrinsic probabilistic parameter space.
B Algorithmic Protocol and Ansatz
We provide the complete training protocol QVF in Alg.
B.1 Detailed Variational Ansatz Visualisation
We also visualise QVF ansatz with bounded Haar randomness. Fig. compares the reachable Hilbert states between ours and ansatz in QIREN that not restrict the set possible operations real-valued unitaries [7] We highlight the circuit structures via their parametrised single-qubit rotations and inter-qubit entanglement patterns. Traversable quantum states are visualised on the Bloch sphere by sampling the ansatz.
B.2 Additional Implementation Details
We next provide additional implementation details on our experimental setup.
Parameterisation of QVF vs. QIREN. QVFi is designed to be a compact QINR model. Compared QIREN, QVF does not need classical post processing while maintaining high representational accuracy. The number of parameters in our experiments is 0.52 10 (corresponding to 128). vs. 0.74 10s for QVF and QIREN, respectively. The ansatz of QVF configured with J =5 and 555, and for QIREN, the default depth.

---

Algorithm QVF Training Protocol
1: Input: Training dataset X ((O,, 8: ))) number of qubits nı; epochs Nepoch's measurement
shots Nshori parameters 0 = (0 fct: inverse temperature 3.
2: for epoch
Nepoch do 3: Classical Inference (Sec. 4.1)
4: Compute energy spectrum E E(O,;
5:
Evaluate Gibbs distribution:
e Z
P
where Z EBB,
6:
Quantum State Preparation:
7:
Initialise Āu = Σi= P (i)-
8:
Quantum Evolution (Sec. 4.2
9:
Apply ansatz S(0,) IT==
20. B, obtain:
p(0) = S (t,)
10:
Measurement and Observables:
11:
Estimate (M&) Tr(p(0,) Ms] for k 1, ..... K.
12:
Gradient Computation of the Loss L (Sec. 4.3
13:
Quantum: ((M))000 via parameter-shift rule.
14:
Classical: Vo. L via automatic differentiation
through e 0z).
15:
Parameter Update:
16:
Adam optimiser step with learning rate v):
end for
18:
Output: Optimised parameters 0r o
Rysk 09/113
NER ASR
Figure 11: Visualisation of ansatz designs and their exemplary induced traversable quantum states within Hilbert space: strongly entangled ansatz (left) and QVF (right). Traversable states different circuit ansatze are visualised on the bottom right for both ansatze.
Simulation Stability in Preparing Quantum States following Gibbs distribution. When simulating the preparation of such quantum states, the partition function evaluation involves exponentiations of large and small Hamiltonian cigenvalues; see Eqs. 2 and (3) which could cause numerical instabilities. We leverage the log-sum-exp trick, a well-established numerical stabilisation technique for this problem.
C Image Representation with Noisy Circuits
Evaluation with quantum circuit noise provide valuable insights for the practical deployment of QVF on near-term quan- tum hardware. Hence, we investigate the influence of quantum gate infidelities on the performance of QVF, ie.., a dominant source of errors noise in the quantum operations. Gate operation infidelity arises
Method
no noise
a 0.01
= 0.05
m 0.1
Ours (Gaussian) ReLU
30.06
3..1
28.42
25.78
0.2
Ours (Identity) + ReLU
30.02
= 0.2
29.6
27.98 0.2
25.96 + 0.2
Ours (Gaussian) Sin
32.59
2: 0.2
32.4
30.66
27.94 + 0.2
Ours (Identity) + Sin
32.67
0.3
3288
02
00.31.0.
28.12 1 0.2
Table 4: QVF performance with noisy circuits. a is the perturbation ratio modelling quantum circuit infidelities.

---

from intrinsic control imperfections in quantum hardware, resulting in stochastic deviations of the performed gate operations from their expected behaviour. These imperfections constrain gate fidelity finite precision. which can effectively modelled as zero-mean perturbations to the gale parameters within bounded range. To simulate the impact of such noise, we introduce cero-mean Gaussian perturbations with varying standard deviations: Higher values correspond to the noise levels typical for current near-term quantum devices, while lower values reflect anticipated improvements in future hardware. The experiments are performed on selected 2D images using the same quantum hardware simulator of PennyLanee [6] as in the main matter (Sec. (5.2). We report the results with different levels of quantum gate perturbation ratio a in Table quantify the degradation in performance under various noise regimes. As expected, increasing ar leads to a decrease in 2D image PSNR.
Even with o = 0.1, we achieve a PSNR of 25dB or higher (cf. Fig. 12
D Image Representation with a Different Number of Samples (Shots)
QINR of images under finite sampling is fundamentally governed by the statistical uncertainty inherent to quantum measurement. The image quality depends on the number of shots, i.c., QINR query repetitions. We visualise the influence of the number of shots Nater (in total per image) in our ansatz in Fig. [2 through the progressive reduction of shot noise artefacts for an increasing number of shots from 100 to 10 With low measurement shots, zero-mean sampling noise dominates the representation. As number shots Naw increase and approaches 10', noise suppression becomes significant as expected according to CLT as noise variance follows o² x 1/Nam.. allowing the representa- tion to better approximate the ground truth. The observed noise patterns across the different shot numbers are char- acteristic of the proposed ansatz of QVF and will serve as reference for future research.
Figure 12: Qualitative images retrieved from pre-trained QVF under the different number of circuit shots. From left to right, shot counts are 100, 500, 10s and 10t, respectively. The rightmost images represent the ground truth.
E
Application: Image Inpainting
QIREN [57] and 3D-QAE [36] are constrained by their reliance on fixed latent representations rigid interpolation mechanisms, thereby being incapable of reconstructing complete, coherent outputs from partial or corrupted inputs. QVF addresses such lim- itations by conditioning the quantum circuit topol- ogy both the query point and dynamic latent space, enabling applications such as image inpaint- ing. Given images with occluded or corrupted pix- els, the circuit identifies a vector in the latent space that minimises the discrepancy between the predicted multi-dimensional properties learned by the quantum circuit and the observed noisy values. The optimised latent vector conditioning the quantum circuit enables recovery of missing field properties. Empirically, we masked out half of the image pixels and reconstructed the complete images via the protocol. Representative results of image inpainting with QVF pre-trained on 50 images are visualised in Fig. [3] demonstrating that QVF can deliver promising performance and accurately recover images even under such extreme sparsity, positioning QVF as a promising quantum circuit architecture for these tasks.
Graud Trust
Corugleddmagee
Figure 13: Image inpainting results with QVF.
F
Application: Shape Completion from Partial and Noisy Depth Maps
Similarly to image inpainting, QVF can be used for tasks such as 3D geometry completion given noisy depth maps. We adopt a similar setting as for images by cropping half of the samples along the depth dimension. We then study the effects of zero-mean Gaussian noise applied to the depth maps across different perturbation ratios o; see Fig. (b). Shape completion performance is quantified across incremental perturbation ratios, parameterised as a E {0, 0.005, 0.01, 0.02, 0.03), where a = 0 corresponds an idcalised noise-free scenario. This qualitative analysis reveals monotonic decline reconstruction fidelity with increasing noise, as evidenced by progressive geometric distortions and surface irregularities in Fig.
5

---

(a) No noise
(b) 0.005
(c) 0.01
(d) 0.02
(0) 0.03
Figure 12: Shape completion from partial and noisy input depth maps using QVE; is the noise ratio.
G Additional 3D Geometry Visualizations
In conjunction with the quantitative results summarized in Tab. 3 Fig. (a) provides a qualitative comparison of the reconstructed 3D geometries, contrasting the baseline the classical architectural component in our model with QVF. The baseline's numerically elevated loss values correlate with visual structural discontinuities, exemplified by the fragmented sofa leg, underscoring its propensity for topological inconsistencies during reconstruction. QVE, instead, demonstrates enhanced structural coherence, generating topologically intact geometries devnid of visible artifacts, as evidenced by its preservation of fine-grained features. Further geometric analysis, illustrated via color-encoded per-surface Hausdorff distance distributions in Fig- (3), reveals systematic geometric deviations for the baseline (top) and ours (bottom), corroborating its geometric fidelity. An interesting observation that the color distributions between models align well, meaning that QVF inherits representative expressiveness of the classical component but enhances due to inherent spectral connections to the quantum circuit.
(b)
Baseline
High
Ours
Ground Truth
Low
Figure 13: (a) Comparison of geometric representations using QVF and the classical model; ground truth is presented at the bottom; (b) representation fidelity visualized via color-encoded Hausdorff distance map: colors represent the distance to the ground truth. The rendered image employs color gradient (blue> graeeerrgeeyyllloo>>eee to indicate descending Hausdorff distance levels.
H
Existing Gate-based Quantum Hardware
Same as the prior work (77, evaluate the proposed QVF on simulator O due the immaturity of real quantum hardware for high-level and practical visual computing tasks. Existing gate-based quantum platforms- including superconducting circuits, trapped ions, neutral atoms, photonic systems, and quantum dots are in varying stages of development, with none yet achieving the maturity required for large-scale, fault-tolerant computation. Limiting factors include noise susceptibility, restricted execution time due quantum decoherence. and the necessity for correction. Nevertheless, classical machine learning systems continue expand in scale, driving unprecedented computational
energy demands, rapid advancements in quantum computing techniques and hardware [17] are anticipated to address these barriers in the foreseeable future, underscoring the need to proactively explore applications executable on emerging quantum computers such QVE.


Reviews:
[{"review_id": "3qNTA77Ws0", "reviewer": "Reviewer_DtQp", "summary": "This paper presents QVFs, a novel quantum-inspired framework for visual representation learning. The proposed approach employs a amplitude encoding scheme to effectively map classical data into quantum space. In contrast to conventional methods that rely on computationally intensive post-processing (typically through MLP layers), the proposed framework directly generates outputs through quantum circuit sampling.\n Comprehensive experiments are conducted across diverse 2D and 3D visual representation tasks, including image reconstruction and 3D geometry representation. The empirical results consistently demonstrate the superior performance and effectiveness of the proposed method.", "questions": "1. As indicated in the paper, the proposed algorithm's performance may be compromised by noise from hardware. Taking the weaknesses 3 into consideration, this raises concerns regarding the practical viability of extending quantum-based frameworks to visual representation tasks. The authors should provide a more rigorous evaluation methodology to properly assess the framework's feasibility under real-world conditions.\n\n2. The current study lacks sufficient ablation studies examining computational efficiency. To substantiate the key claim of replacing heavy post-processing (e.g., MLP layers) with the proposed quantum circuit module, the authors need to provide analyses towards computational complexity.", "limitations": "yes", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n1.\tNovel Framework: This paper proposes a quantum-based learning framework that integrates (1) an innovative amplitude encoding method to bridge classical data and quantum space, and (2) parameterized quantum circuits (PQCs) with a lightweight architecture and superior representation capability compared to previous approaches.\n\n2.\tAdvancing Applications: Alongside prior works (e.g., 3D-QAE and QIREN), the proposed QVFs is one of the three study to systematically explore quantum machine learning (QML) for visual implicit neural representations (INRs). The proposed work expands the application of QML in visual computing, offering new insights and opportunities for researchers in this field.\n\n3.  Empirical Validation:  Extensive experiments are conducted across diverse visual representation tasks, demonstrating the effectiveness and robustness of the proposed proposed method.\n\n4.\tTheoretical Rigor: Detailed mathematical derivations and analyses are provided to ensure clarity and theoretical soundness.\n\nWeaknesses\n\nThe manuscript suffers from several significant issues that need to be addressed. \n\n1. the presentation quality is subpar, with poorly formatted figures and tables that are not properly centered and use font sizes too small to be legible. Many of the captions fail to adequately explain the content; for example, Fig. 5(a) displays two rows of figures without any explanation in either the caption or main text.\n\n2, the paper lacks essential ablation studies to validate its core claims about efficiency. While the authors criticize previous methods for relying on computationally heavy post-processing like MLPs in QIREN, they provide no quantitative comparison of parameters or computational requirements between their approach and existing methods. This omission substantially weakens the paper's central argument.\n\n3, the rationale for applying quantum methods to visual representation tasks remains unconvincing. Given the current capabilities of classical approaches using CNNs or MLPs with modern GPU hardware - which already perform well on standard datasets like CIFAR10 - the paper fails to demonstrate what unique advantages quantum-based frameworks offer in this domain. The authors should provide either theoretical analysis or empirical evidence showing clear benefits of their quantum approach over classical methods under comparable conditions.", "was_revised": true, "final_justification": "All my questions have been adequately addressed. I acknowledge the academic rigor of this paper, though my main concern remains with its presentation quality. As the authors have stated they will focus on improving the presentation, and given that only presentational modifications seem likely possible, I will maintain my current score.", "rebuttal": "We sincerely thank the reviewers for their thorough evaluation, constructive feedback, and insightful comments, which will substantially enhance the clarity and quality of our work. We are also particularly encouraged by the positive reception of our contributions, as highlighted by Reviewer WDGf’s \"innovative approach with strong empirical results that addresses scalability,\" and Reviewer 4LWU’s \"well-organized, end-to-end quantum framework for learning INRs for visual data.\"\n\nWe would like to respectfully draw the attention of the reviewing committee that Reviewer 8wGH’s comments appear to pertain to a different manuscript. We have therefore carefully asked for further clarification. Below, we systematically address concerns raised by Reviewer DtQp in detail: \n\n**(Q1) The manuscript suffers from several significant issues that need to be addressed. The presentation quality is subpar, with poorly formatted figures and tables that are not properly centered and use font sizes too small to be legible. Many of the captions fail to adequately explain the content; for example, Fig. 5(a) displays two rows of figures without any explanation in either the caption or main text.**\n\nWe promise to address such mentioned issues and improve the presentation quality with proper formatting and caption clarity in the revised version to make sure it strictly sticks to the standard.\n\nFor the instance of Fig. 5(a), the correct caption should be: “Qualitative learning process visualization for (top) our QVF and (bottom) classical model.”\n\n**(Q2) The paper lacks essential ablation studies to validate its core claims about efficiency. While the authors criticize previous methods for relying on computationally heavy post-processing like MLPs in QIREN, they provide no quantitative comparison of parameters or computational requirements between their approach and existing methods. This omission substantially weakens the paper's central argument.**\n\nWe agree that such qualitative comparisons are essential and appreciate the reviewer for pointing it out. We provide further details due to the mentioned architectural differences:\n\n**Parameter comparison**: In our experiments, QVF has a total number of parameters of 0.52e5 vs. 0.74e5 in QIREN [1]. This difference mainly comes from the design of QVF, in which the post-processing MLP is not needed, as the reviewer noted.\n\n**Computational requirements:**\n\n**a) Execution time**: we compare the time required by QVF and QIREN, respectively, to perform 1) gradient calculation and trainable parameter update; and 2) inference latency, with the reported number averaged over 5 iterations. As QIREN only supports single-image representation learning, we leverage single CIFAR-10 images to perform this benchmark. For QIREN, it takes 1.42s for training for a single epoch and 1.38s for inference vs. 0.82s and 0.68s for our QVF, achieving almost 2x reduction in both cases. \n\n**(b) Quantum circuit optimization**: Another factor we considered when designing QVF is the optimization of the quantum circuit. In QIREN, the post-processing with MLPs requires constructing an additional computational graph, which needs to be combined with the parameter shift rule for optimizing the quantum circuit. QVF does not have this issue as it does not need further classical post-processing. This also contributes to the observed training efficiency, which can be more significant on future real hardware.\nThese additional results will be included in the revision.\n\n[1] Zhao et al. Quantum implicit neural representations. ICML, 2024. \n\n**(Q3) the rationale for applying quantum methods to visual representation tasks remains unconvincing. Given the current capabilities of classical approaches using CNNs or MLPs with modern GPU hardware - which already perform well on standard datasets like CIFAR10 - the paper fails to demonstrate what unique advantages quantum-based frameworks offer in this domain. The authors should provide either theoretical analysis or empirical evidence showing clear benefits of their quantum approach over classical methods under comparable conditions.**\n\nWe included a brief theoretical justification of using quantum methods for visual field representation in the paper; see L28-34. We expand upon it and provide further theoretical motivations here: \n\nVisual representation tasks require accurate capture of high-frequency details while classical methods like CNNs or MLPs tend to prioritize learning low-frequency details. In literature, learning incorporating Fourier features has been proven a promising approach for such tasks, given its ability to model fine-grained signal structures effectively [1,2,3,4]. It has been recently proven that quantum circuits exhibit a strong connection to Fourier processing, offering an exponential separation in computational efficiency compared to classical methods [5]. These reasons theoretically motivate our QVF, along with the previous work QIREN, and the use of quantum computers for exploration. \n\nOn the empirical side, the work of QIREN already shows promise compared to several classical methods. We, therefore, first show that QVF outperforms QIREN across the tasks supported and demonstrated in their work. We then show that QVF is capable of a broader set of tasks not supported by QIREN, and we leverage a consistent comparison strategy to demonstrate that QVF outperforms the commonly used classical methods in Tab. 3. \nLastly, energy consumption presents another compelling motivation for executing quantum circuits on future quantum machines in general. Studies have shown that future quantum systems can operate with energy requirements several orders of magnitude lower than their classical counterparts for certain problems [6]. Given the escalating energy demands of modern AI systems, quantum approaches can provide a viable pathway toward sustainable AI development.\n\nWhile the exploration of quantum methods for general AI remains in its nascent stages, with existing works such QIREN and our work QVF, we believe visual representation tasks serve as a principled and empirically validated starting point with support from both theoretical insights and experimental results. \n\n[1] Rahaman et al. On the spectral bias of neural networks. ICML, 2019.\n\n[2] Mildenhall et al. NeRF: Representing Scenes  as Neural Radiance Fields  for View Synthesis. Communications of the ACM, 2021.\n\n[3] Tancik et al. Fourier features let networks learn high frequency functions in low dimensional domains. Neurips, 2020.\n\n[4] Benbarka et al. Seeing Implicit Neural Representations as Fourier Series. WACV, 2022.\n\n[5] Schuld et al. The effect of data encoding on the expressive power of variational quantum machine learning models. Physical Review A, 2021.\n\n[6] Meier et al. Energy-consumption advantage of quantum computation, PRX Energy, 2025.\n\n**(Q4) As indicated in the paper, the proposed algorithm's performance may be compromised by noise from hardware. Taking the weaknesses 3 into consideration, this raises concerns regarding the practical viability of extending quantum-based frameworks to visual representation tasks. The authors should provide a more rigorous evaluation methodology to properly assess the framework's feasibility under real-world conditions.**\n\nUnder real-world conditions, the performance of our QVF—as well as any quantum algorithm executable on near-term hardware—can indeed be affected by noise, a consequence of the current developmental stage of quantum devices. Beyond our initial analysis of measurement noise provided in the paper; see L339-348, we further investigate the effects of quantum gate infidelity during real hardware execution, another dominant error channel in current devices. Gate operation infidelity arises from intrinsic control imperfections in quantum hardware, resulting in stochastic deviations of the implemented gate operations from their ideal targets. These imperfections constrain gate fidelity to finite precision, which can be effectively modeled as zero-mean perturbations to the gate parameters within a bounded range. To simulate the impact of such noise, we introduce zero-mean Gaussian perturbations with varying standard deviations: higher values correspond to the noise levels typical of current near-term quantum devices, while lower values reflect anticipated improvements in future hardware. We report the results of this analysis using PSNR metric, alongside corresponding noise-free baselines originally reported in Tab.2, to quantify the degradation in performance under different noise regimes:\n\n| Method                     | No-noise         | σ = 0.01        | σ = 0.05       | σ = 0.1        |\n|----------------------------|------------------|-----------------|----------------|----------------|\n| **Ours (Gaussian) + ReLU** | 30.06 ± 0.11     | 30.1 ± 0.13     | 28.42 ± 0.15   | 25.78 ± 0.23   |\n| **Ours (Identity) + ReLU** | 30.02 ± 0.23     | 29.6 ± 0.15     | 27.98 ± 0.18   | 25.96 ± 0.18   |\n| **Ours (Gaussian) + Sin**  | 32.59 ± 0.21     | 32.4 ± 0.22     | 30.66 ± 0.16   | 27.94 ± 0.21   |\n| **Ours (Identity) + Sin**  | 32.67 ± 0.34     | 32.8 ± 0.26     | 30.34 ± 0.22   | 28.12 ± 0.23   |\n\nNote that these results are averaged over 50 trials. It can be observed that under mild noise, the performance of QVF is barely influenced. With increasing perturbations, the representation fidelity of QVF decreases as expected. We emphasize that our QVF by design is inherently resistant to phase flip errors (see L211-213 for our quantum ansatz design), which introduce imaginary components, due to our engineered restrictions of information processing on real Hilbert subspace. These results will be later incorporated into the revised paper. \n\n**(Q5)  the authors need to provide analyses towards computational complexity.**\n\nSee reply to **Q2**. All additional analyses will be later incorporated into the revision."}, {"review_id": "eHFOhAWsEV", "reviewer": "Reviewer_8wGH", "summary": "This paper proposes a quantum-inspired multi-dimensional multi-modal fusion model to enhance EEG-based emotion recognition. The approach mimics quantum probability principles to capture higher-order correlations across spatial, temporal, and frequency dimensions from multi-modal EEG signals. Experimental results on benchmark datasets demonstrate superior performance compared to conventional deep learning fusion strategies.", "questions": "How does the quantum-inspired fusion mechanism improve over traditional tensor-based or attention-based fusion in terms of capturing non-linear relationships in EEG data, and can this approach generalize to other bio-signals like fNIRS or EMG?", "limitations": "YES.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "+Introduces a quantum-inspired framework to model complex dependencies in EEG signals.\n\n+Achieves better classification performance over state-of-the-art fusion techniques on standard datasets.\n\n+Effectively combines multiple EEG modalities, enhancing robustness and generalizability.\n\n-The quantum-inspired mechanisms lack biological or physiological interpretability for EEG practitioners.\n\n-Higher-dimensional fusion may increase training time and require more computational resources.\n\n-The method is tested on specific datasets, and its performance across different tasks or populations remains uncertain.", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely appreciate the reviewer’s time and thoughtful comments. Upon careful consideration, we note that the summary and questions raised appear to pertain to a different work addressing EEG-based emotion recognition, whereas our paper introduces Quantum-inspired Visual Field Representation (QVF)—a novel quantum machine learning framework designed to enhance visual field representations with key contributions including: (1) superior performance compared to our closest competitor QIREN while maintaining model compactness; (2) support learning of visual field collections and larger problem scales such as 3D geometries for the first time in the domain; and (3) a rigorously designed ansatz that improves trainability via constrained real Hilbert space traversal. We would be delighted to provide further clarification on any aspect of QVF during the author-reviewer discussion phase and improve the manuscript’s quality."}, {"review_id": "RAXNvrINXv", "reviewer": "Reviewer_WDGf", "summary": "This paper proposes Quantum Visual Fields, a quantum-inspired approach for learning visual representations using parameterized quantum circuits. The method avoids classical post-processing by combining Hamiltonian-driven dynamics with unitary evolution and projective measurements. The authors  show its effectiveness on visual tasks involving 2D and 3D data, outperforming several existing quantum circuit baselines.", "questions": "I am curious why the authors categorize their method as “quantum-inspired” rather than “quantum machine learning,” given that the approach still relies on quantum simulators for experiments. If it is indeed a quantum-inspired method, would it be appropriate to include comparisons with classical algorithms such as SIREN to more comprehensively evaluate its advantages and limitations? \n\nAs far as I understand, data re-uploading circuits can be expressed as Fourier series expansions. Why is it that the authors use only amplitude encoding yet still achieve a Fourier series representation? Does this imply that quantum circuit-based machine learning models are inherently Fourier-based? Furthermore, what is the difference between the expressivity of the proposed model and the Fourier expressivity of data re-uploading circuits?", "limitations": "See the Questions and weaknesses.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 3, "originality": 4, "strengths_and_weaknesses": "Strengths：\n\nThe paper proposes an innovative quantum approach (QVF) that avoids classical post-processing, which is conceptually novel and well-motivated.\n\nThe method effectively addresses scalability issues faced by existing quantum implicit neural representations, a key limitation in the current literature.\n\nEmpirical results are strong, showing superior performance over baselines across multiple visual representation tasks.\n\nWeaknesses：\n\nThe overall presentation is somewhat compact, and some parts are not clearly explained. For example, the description of Equation (1) is somewhat confusing: some symbols used in the formula are not explained, and conversely, some symbols appearing in the subsequent explanation do not appear in the equation itself.\n\nThe number of parameters is a critical factor in implicit neural representation tasks, as increased model capacity often leads to better performance. However, the paper does not provide a comparison of parameter counts between the proposed method and the baselines, which makes it difficult to assess the efficiency and fairness of the evaluation.", "was_revised": true, "final_justification": "The author's response addressed my concerns regarding quality and clarity, so I raised the corresponding score.", "rebuttal": "We sincerely thank the reviewers for their thorough evaluation, constructive feedback, and insightful comments, which will substantially enhance the clarity and quality of our work. We are particularly encouraged by the positive reception of our novel contributions, as highlighted by Reviewer DtQp’s \"novel framework with theoretical rigor that advances applications.\" and Reviewer 4LWU’s \"well-organized, end-to-end quantum framework for learning INRs for visual data.\"\n\nWe would like to respectfully draw the attention of the reviewing committee that Reviewer 8wGH’s comments appear to pertain to a different manuscript. We have therefore carefully asked for further clarification. Below, we systematically address concerns raised by Reviewer WDGf in detail: \n\n**(Q1) The overall presentation is somewhat compact, and some parts are not clearly explained. For example, the description of Equation (1) is somewhat confusing: some symbols used in the formula are not explained, and conversely, some symbols appearing in the subsequent explanation do not appear in the equation itself.**\n\nWe promise to address such mentioned issues and improve the presentation quality with proper formatting and caption clarity in the revised version to make sure it strictly sticks to the standard. For the mentioned instance of Eq.1, we here give a clearer explanation as follows:\n\n$\\gamma$(.) refers to positional encoding function; L is the maximum frequency level (also known as the bandwidth) in $\\gamma$(.).  $\\Theta$ is the field query coordinate. \n\n**(Q2) The number of parameters is a critical factor in implicit neural representation tasks, as increased model capacity often leads to better performance. However, the paper does not provide a comparison of parameter counts between the proposed method and the baselines, which makes it difficult to assess the efficiency and fairness of the evaluation.**\n\nWe agree with the reviewer regarding the parameter efficiency and provide the following clarifications here:\n\n**Quantum baseline (QIREN)**: We provide the parameter comparison with QIREN in the reply to question **Q2** from reviewer DTQP. \n\n**Classical baseline**: The only parametrization difference when comparing QVF with classical baselines comes from integrating the quantum circuits while other configurations remain the same as mentioned in Remark 3 in the paper. Across our  representation learning experiments, we adopt a quantum circuit of five qubits bringing parameter overhead of 170 additional parameters. This means parametrizations of QVF and classical baselines differ by a parameter count of only 170 which we consider as negligible compared to the classical baseline with parametrization on the order 1e5. We believe this configuration makes it a fair comparison for QVF and classical baseline for the performance difference and efficiency.  We also investigate by scaling up the latent space dimension of the classical method and found that it needs approx. 1.12e5 parametrization to have comparable performance with our QVF with parametrization of 0.52e5. This shows the effectiveness of our QVF design. The details would be clarified and incorporated in the revision.\n\n**(Q3) I am curious why the authors categorize their method as “quantum-inspired” rather than “quantum machine learning,” given that the approach still relies on quantum simulators for experiments. If it is indeed a quantum-inspired method, would it be appropriate to include comparisons with classical algorithms such as SIREN to more comprehensively evaluate its advantages and limitations?**\n\n**Quantum vs. Quantum-inspired**: We appreciate the reviewer’s thoughtful question regarding classification of QVF. Indeed, our work QVF, and previous work such as QIREN, 3D-QAE, are quantum machine learning methods and, therefore, our work should be better positioned as “quantum” or “quantum machine learning” instead of “quantum-inspired”. We will adjust the manuscript accordingly. \n\n**Comparison with Siren**: While QVF is a quantum machine learning method and our closest work and main competitor QIREN already outperforms SIREN in their work and shows promising performance, we also compared with SIREN for completeness. SIREN follows the dense multi-layer neural net structure but replaces traditional ReLU activations with periodic sinusoidal activation functions with the corresponding initialization strategy. We reported the comparison results with SIREN denoted by MLP + sin in Tab.3 and showed that our QVF outperforms SIREN as well. The comparison is also strictly controlled with parametrization consistent with other classical baselines with parameter overhead of only 170 parameters, which we believe makes the comparison convincing. We will clarify this and expand the discussion of classical comparisons in the revised manuscript.\n\n**(Q4) As far as I understand, data re-uploading circuits can be expressed as Fourier series expansions. Why is it that the authors use only amplitude encoding yet still achieve a Fourier series representation? Does this imply that quantum circuit-based machine learning models are inherently Fourier-based? Furthermore, what is the difference between the expressivity of the proposed model and the Fourier expressivity of data re-uploading circuits?**\n\n**Fourier-based learning nature of quantum circuit**: Quantum circuit-based machine learning models are indeed inherently Fourier-based; see L181-194. Specifically, the Fourier interpretability of quantum circuits arises from the unitary evolution of quantum states and the spectral decomposition of observables. The accessible frequency spectrum depends on the encoding of classical data.\n\n**Expressivity difference between our method and data-reuploading circuits**: The key distinction lies in the adaptability of the spectrum. In data-reuploading circuits, the pattern of frequency spectrum remains fixed due to handcrafted gate sequences (e.g. repeated Pauli rotations) while in QVF, our learnable energy-based encoding dynamically adjusts the frequency spectrum that automatically captures data-dependent frequency priors, allowing the quantum model to emphasize relevant spectral components while avoiding spectral rigidity of re-uploading circuits; see analysis in L181-L194. Such theoretical considerations are also reflected in the experiments where our QVF outperforms QIREN with parametrization on a similar level."}, {"review_id": "xghghLgy0H", "reviewer": "Reviewer_4LWU", "summary": "This work proposes quantum visual fields, an end-to-end trainable quantum circuit framework for learning implicit representations in computer graphics and visual-data-driven machine learning tasks. It claims that the proposed quantum visual fields consistently outperform multiple baseline methods across various tasks.", "questions": "1. is there any theoretical analysis such as theorem that explain why the proposed method can achieve better performance compared to classical way? i.e. rigorously analysis the gap between the proposed model with classical method.\n2. how to prepare the initial Gibbs state practically? I mean instead of numerical simulation, as you may know, the dimension (Pi in appendix algorithm 1) will grow exponentially with the system size, if it only classically simulates such process, it may face the stability issues.\n3. in line 221, it mentioned that we need $m$-dimension latent representation, how to choose the specific $m$?", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "strength\n- it is organized and well-written, clearly presents the core ideas.\n- the extensive numerical experiments are provided.\n\n\nweakness\n- to demonstrate the practical of the proposed method, the related experiments on noise cases are necessary.\n- it is better to provide the necessary introduction about the visual fields for the audiences that not familiar to the related fields.\n- some font size of the labels in figures are too small which is hard to read.\n- it is better to give the rigorous analysis to support the claims.", "was_revised": true, "final_justification": "Based on rebuttal, the authors basically addressed my concerns.", "rebuttal": "We sincerely thank the reviewers for their thorough evaluation, constructive feedback, and insightful comments, which will substantially enhance the clarity and quality of our work. We are particularly encouraged by the positive reception of our novel contributions, as highlighted by Reviewer DtQp’s acknowledgment of our \"novel framework with theoretical rigor that advances applications.\" and Reviewer WDGf’s appreciation for our \"innovative approach with strong empirical results that addresses scalability.\"\n\nWe would like to respectfully draw the attention of the reviewing committee that Reviewer 8wGH’s comments appear to pertain to a different manuscript. We have therefore carefully asked for further clarification. Below, we systematically address concerns raised by Reviewer 4LWU in detail: \n\n**Q1: to demonstrate the practicality of the proposed method, the related experiments on noise cases are necessary.**\n\nWe acknowledge this point and agree that noise due to imperfect hardware is necessary for evaluations of quantum machine learning methods for practical tasks such as visual field representation. As assessment of QVF’s performance under noise is also required by reviewer DTQP in **Q4**, we kindly ask the reviewer to refer to this reply, where we provide comprehensive examinations.\n\n**Q2: 1) it is better to provide the necessary introduction about the visual fields for the audiences that not familiar to the related fields; 2) some font size of the labels in figures are too small which is hard to read.**\n\nThis is extremely helpful for improving our work and its readability for the general audience. In response to the raised concerns, we will expand the introduction and include a detailed explanation of general visual fields and carefully check to ensure all font sizes meet the standard in the revision.\n\n**Q3: is there any theoretical analysis such as theorem that explain why the proposed method can achieve better performance compared to classical way? i.e. rigorously analysis the gap between the proposed model with classical method.**\n\nWe believe this question is very similar to and partially overlaps with the question **Q3** by reviewer DtQp which inquires about the motivations of using quantum methods (such as prior works such as QIREN and our QVF) over classical methods. We therefore kindly direct this reviewer to that response along with listed references. \n\nHere we briefly repeat and summarize it in terms of algorithmic performance: 1) **spectral learning bias**: high-fidelity visual field representation greatly benefits from un-biased spectral learning. Classical methods favor learning low-frequency signals, i.e. biased spectral learning. On the theoretical level, spectral learning or incorporating Fourier features have been shown to be effective in relieving this problem; 2) **computational separation between our quantum and the classical methods**: our QVF leverages quantum machine learning which inherently performs un-biased spectral learning. Additionally, due to the unique properties of quantum processing such as superposition, the expressiveness of such spectral learning grows exponentially w.r.t. quantum resources which makes it inefficient and nearly impossible to simulate classically on a large scale, establishing provable exponential separation between the proposed and classical methods. In summary, theoretically our method can enhance visual field representation by leveraging inherent un-biased spectral learning properties with only logarithmic overhead. This is also empirically verified that our QVF has significantly better performance to perform un-biased spectral learning; see Fig.5 for an intuitive visualization example. \n\n**Q4:  how to prepare the initial Gibbs state practically? I mean instead of numerical simulation, as you may know, the dimension (Pi in appendix algorithm 1) will grow exponentially with the system size, if it only classically simulates such process, it may face the stability issues.**\n\n**Stability in classical simulations**: in our classical simulation, we did leverage tricks such as log-sum-exp when preparing Gibbs states to ensure numerical stability. This would be clarified in the revision.\n\n**Preparation of Gibbs state on a quantum computer**: prior works have discussed ways to prepare Gibbs state on a quantum computer, via thermalization protocol for example; see the works that we cite in L166-L169. \n\n**Our future plans for reducing the exponentially-growing dimension**: We are also exploring more scalable ways to prepare Gibbs states. For example, we are planning to leverage tensor train decomposition with matrix product operator representations [1,2],  which can potentially reduce computational scaling to polynomial complexity while maintaining sufficient accuracy. We are also exploring the possibility to prepare Gibbs state with approximated amplitude encoding [3] as an alternative approach. The detailed development and validation will be pursued in future research as it aligns with broader efforts to develop scalable Gibbs quantum state preparation protocols.\n\n[1] Green et al. Quantum encoding of structured data with matrix product states. 2025.\n\n[2] Melnikov et al. Quantum state preparation using tensor networks. 2023.\n\n[3] Ben-dov et al. Approximate encoding of quantum states using shallow circuits. 2024.\n\n**Q5: in line 221, it mentioned that we need m-dimension latent representation, how to choose the specific m?**\n\nThe dimension m is determined by the intrinsic dimensionality of the target output—for example, m = 3 for RGB images or m =1 for scalar-valued signed distance fields (SDFs). This ensures alignment between the number of measured qubits (Eq. 6) and the required output dimensions while maintaining an injective mapping. The constraint n≥m (where n is the total qubit count) guarantees physically meaningful representations. This helps avoid unnecessary computational overhead due to either redundant measurement or classical post-processing. This will be further clarified in the revision."}]

Author rebuttal:

Meta-review: The submission introduces QVF (Quantum Visual Fields), a fully quantum implicit representation framework inspired by the unitary evolution of gate-based quantum systems. QVF distinguishes itself by eliminating classical post-processing and embedding learning directly into Hamiltonian-driven unitary dynamics. The model leverages quantum conservation laws and projective measurements for feature extraction, aiming to inherit desirable quantum properties such as spectral interpretability, numerical stability, and reversibility. The method is benchmarked on 2D image and 3D geometry representation tasks using up to six qubits, and is shown to outperform prior quantum baselines in terms of fidelity and learning speed.

All four reviewers rated the paper as borderline accept, with varying confidence levels and limited participation during the rebuttal phase. The authors were responsive during rebuttal, and the reviewers who engaged acknowledged that their concerns (primarily regarding presentation quality and clarity) were sufficiently addressed. Given the absence of detailed rebuttal engagement and the fact that all reviewers marginally leaned toward acceptance, the overall recommendation remains borderline.
Decision: Accept (poster)