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Quantum-Inspired Trainable and Parameter-Efficient Tensor Networks for Image Inpainting

QUANTUM-INSPIRED TRAINABLE AND PARAMETER-EFFICIENT TENSOR NETWORKS FOR IMAGE INPAINTING Shiwen An iD and Konstantinos Slavakis iD Institute of Science Tokyo, Department of Information and Communications Engineering, Yokohama, Japan

arXiv:2609.17298v1 [eess.IV] 15 Sep 2026

ABSTRACT This work introduces quantum-inspired tensor-network circuits as trainable transforms for image inpainting. Among the proposed architectures, the diagonal quantum Fourier transform (QFT) relaxation is invertible with O(N 2 log N ) computational cost for N × N images, inherently preserving minimum coherence throughout training via its circuit structure and eliminating the need for explicit coherence penalties. Unconstrained gradient-based phase optimization (Riemannian-optimization free) enables efficient learning from randomly sampled training data, allowing the learned transform to generalize to test images observed through fixed sampling masks. Numerical tests show that the learned models outperform fixed transforms and per-image optimization while matching the performance of much larger unitary architectures, yet with far fewer parameters. Index Terms— Quantum, image inpainting, transforms, tensor network. 1. INTRODUCTION Image inpainting reconstructs a digital image from a partial set of observed entries, inferring missing values by assuming a prior model of the image’s structure. Sparse recovery exploits transformdomain sparsity models, using wavelets [1] or learned dictionaries [2, 3], with recovery guarantees [4–6]. Low-rank matrix completion capitalizes on row-column correlations in the image matrix, with theoretical guarantees under incoherence and sampling conditions [7, 8]. Deep-learning approaches include trained inpainting networks [9], diffusion-based methods [10], and plug-and-play solvers with learned denoising priors [11]. Other methods optimize representations directly on individual test images, including deep image priors [12] and tensor-train approaches, such as low-rank decomposition [13] and coarse-to-fine refinement [14]. For transform-based recovery, sparsity alone does not determine which transform to use. A basis atom (column vector) of the transform localized to a few pixels does not appear in the transformdomain signal representation whenever samples miss its support. Coherence µ(·) measures the squared maximum modulus of correlation between basis atoms and the standard basis—see (2)— quantifying the localization of atoms in the image domain. The complexity of recovery is known to grow with coherence [15]— a fundamental tension in compressed sensing [16]. Empirically, wavelets (highly localized bases, thus high coherence) achieve sparser approximations than the DFT, yet produce worse inpainting results under the same solver—see Sec. 4. This gap motivates the design of trainable transforms with low coherence. Since sparsity also varies across bases, coherence is an indicative but not the sole factor determining inpainting performance. Tensor networks offer a quantum-inspired and compact way to parameterize transforms—see Fig. 1. Writing pixel indices in binary represents an image as a tensor with one mode per index bit [17, 18]. A tensor network of unitary gates on these modes defines a basis shared across images. The basis retains two key properties of

the FFT: fast application with O(N 2 log N ) for N × N images, and invertibility via its adjoint. The quantum Fourier transform (QFT) circuit [19] achieves both through a factorization into Hadamard and controlled-phase gates. This circuit relaxes the diagonal gates while retaining one Hadamard per bit. Every resulting matrix yields coherence µ = 1, ensuring minimum coherence and unitarity under adaptation. Learnable butterfly factorizations [20] also adapt a fast transform, but their relaxed blocks do not guarantee minimum coherence. Building on QFT-gate relaxations developed for compression [21], where tensor networks were trained via Riemannian optimization, this work extends those circuits into image inpainting. More specifically, the proposed contributions are threefold. (a) First application to image inpainting. Quantum-inspired tensor-network circuits are adapted to image inpainting via gradientbased phase optimization for both phase-only and diagonal relaxations (Riemannian-optimization free unlike [21])—see Sec. 3. (b) Guaranteed coherence. The diagonal QFT relaxation and its phase-only subfamily guarantee minimum coherence throughout training via the circuit topology, without requiring an explicit coherence penalty. This structural guarantee distinguishes the proposed tensor networks from both unconstrained learned dictionaries and from QFT circuits with free Hadamards, whose coherence can exceed the minimum bound (Prop. 1). (c) Parameter-efficiency and performance. The proposed learned diagonal model outperforms fixed transform baselines and per-image optimization schemes while matching the performance of much larger unitary architectures, yet with far fewer parameters (Sec. 4). 2. THE IMAGE-INPAINTING PROBLEM An image X ∈ RN ×N is observed partially, with entries known only on a subset of pixel indices Ω. The observation operator PΩ : RN ×N → RN ×N models this by Y := PΩ X := MΩ ⊙ X, where MΩ ∈ {0, 1}N ×N is a binary mask with ones at observed locations Ω and zeroes elsewhere, and ⊙ denotes the Hadamard product. The sampling mask MΩ is viewed as a random variable (RV) to account for all possible sampling patterns. Typically, the entries of MΩ are modeled as independent Bernoulli RVs with p being the probability of success (appearance of 1s), also called sampling rate, so that pN 2 pixels are observed on average. Image inpainting recovers X from the partial observations Y. This work introduces tensor networks T (·)—see Fig. 1—for transform-based image inpainting. Unlike fixed classical transforms such as the DCT, the parameters θ of T (θ) are learned from training data D by minimizing the loss L(·): θ⋆ ∈ arg minθ L(θ), n o 1 X b K (θ; Y, Ω) − X∥2F , (1) EΩ N12 ∥X L(θ) := X∈D |D| where expectation EΩ {·} averages over random sampling masks b K (θ; Y, Ω) denotes the K-step recovered image from MΩ , and X

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Fig. 1: The construction of Sec. 3.1 for n = 4 per axis, depicted as four tensor networks progressing from specific to general (left to right), each applied to a copy of the vectorized image x := vec(X). (a) The separable 2D DFT, (FN ⊗ FN ) x, from Step 1a. (b) One Cooley–Tukey level: the radix-2 decomposition (3) of Step 1b. (c) The circuit T (θ) of (4) up to the bit reversal B, with trainable diagonals (red) and fixed Hadamards (blue). It is unitary by (5) and satisfies µ = 1 at every θ by Prop. 1. (d) The analysis map Aθ of Step 1d.

Algorithm 1. To manage computational budgets, stochastic/online gradient descent solves (1) by sampling a small batch of training images per step (to approximate the sum over all training data) and randomly drawing a sampling mask per image (to approximate EΩ {·})—see Algorithm 1. Once θ⋆ is learned, the network T (θ⋆ ) inpaints a test image Xtest ∈ / D observed through a fixed sampling mask Ωtest via the Recover(PΩtest Xtest , Ωtest , θ⋆ ) function of Algorithm 1. As explained in Sec. 1, coherence µ(·) quantifies the localization of basis atoms. For U = [uij ] ∈ U (N ) (where U (N ) is the set of complex-valued N × N unitary matrices), coherence is defined as: µ(U) := N max |eTi uj |2 = N max |uij |2 ∈ [1, N ] , (2) i,j

i,j

N where {ei }N and uj is the jth column i=1 is the standard basis of R

of U. Under sparsity and suitable sampling conditions, the sample complexity for recovery grows linearly with µ [15], motivating the design of transforms with low coherence. For images—which have both row and column structure—the 2D-coherence µ2D is defined as the product of coherences along each dimension. Correspondingly, the tensor-network parameter vector θ = (θr , θc ) factorizes into row and column components, respecting this 2D structure. 3. THE TENSOR NETWORKS This section writes the DFT as a quantum Fourier circuit (Sec. 3.1), frees its gates one structure at a time (Sec. 3.2), trains the result through the recovery iteration (Sec. 3.3), and shows the two properties the design rests on, isometry at every parameter value (Sec. 3.4) and minimum coherence (Sec. 3.5), which single out QFT (diagonals) as the recommended model. 3.1. An FFT-factorized isometric family Read the index of a length-(N = 2n ) axis (n ≥ 2) as an n-bit string b0 b1 · · · bn−1 , b0 the most significant bit, and draw one wire per bit; the n wires of an axis form a register. The network acts on the vectorized image x = vec(X) x , the rows of X stacked, and a transform is a network of small unitary tensors on those wires [19, 21]. Throughout, ⊗ is the Kronecker product, and every product of gates is the ordinary matrix product, written in the order applied, the rightmost first. Fig. 1 draws the four steps. Step 1a. The separable 2D DFT attaches one dense FN FN , √ (FN )jk = e2πijk/N / N , to each register: X 7→ FN XFTN , i.e., (FN ⊗ FN ) x. No gate joins the two. Step 1b. The Hadamard  1 1  on wire q is Hq = I2q ⊗ H ⊗ I2 n−1−q 1 H , with H := √ . The controlled phase on the wire pair 2 1 −1 (p, q), Φpq (ξ) Φ , is the phase gate diag(1, 1, 1, eiξ ) on wires p and q and I2 on every other wire. One level of the Cooley–Tukey

recursion factors each dense block into a Hadamard on the most significant bit, a controlled phase coupling it to each remaining bit p at the angle π/2, π/4, . . . , 2π/2n in turn, and a half-size FN/2 on the rest, the angle of the pair (p, q) being ξpq = 2π/2 p−q+1 . In matrix form this level is the classical radix-2 identity [22] n−1 Y FN = S (I2 ⊗ FN/2 ) Φp0 (ξp0 ) H0 , (3) p=1

with S the permutation matrix that moves bit 0 of the index to the last place, b0 b1 · · · bn−1 7→ b1 · · · bn−1 b0 ; the Φ factors are diagonal and commute. Step 1c. Recursing until F2 = H leaves nothing dense. Level Qn−1 q contributes Hq followed by its controlled phases Dq = p=q+1 Φpq (ξpq ), and its move of bit q to the last place, Sq = I2q ⊗ S with S on the trailing n − q bits; the moves compose into the bit reversal B = S0 S1 · · · Sn−2 . The levels compose in circuit order, the q = 0 factors rightmost and applied first: FN = B (Dn−1 Hn−1 ) · · · (D0 H0 ), the quantum Fourier circuit [19]. Step 1d. Freeing the angle of every controlled phase, ξpq 7→ θpq , gives the trainable network

T (θ) := B Dn−1 (θ)Hn−1 · · · D0 (θ)H0 , (4) Qn−1 with Dq (θ) = p=q+1 Φpq (θpq ), θ the parameter vector of Sec. 2 (its entries per model in Sec. 3.2), and T (θ 0 ) = FN at θpq = ξpq . Every factor is unitary at every parameter value, so T (θ)H T (θ) = I , ∀θ . (5) The analysis map applies T (θ)H on each register (Fig. 1(d)); with separate angles per axis, T Aθ (X) := T (θr )H X T (θc ) , A−1 θ (C) = T (θr ) C T (θc ) , (6) the second being the exact inverse of the first by (5).

3.2. A ladder of relaxations Each model frees one structure of (4), and θ collects the free parameters of whichever model is meant:

QFT (phases) QFT (diagonals) QFT (rotations) pair diagonal (1, 1, 1, eiξ ) (eiθ1 , . . . , eiθ4 ) (eiθ1 , . . . , eiθ4 ) wire gate H H V ∈ U (2) count per axis n(n − 1)/2 2n(n − 1) 2n(n − 1) + 4n QFT (phases) is (4) as written. QFT (diagonals) frees the other three phases of each pair, one per value (bp , bq ) of the two bits Φ , initialized at (0, 0, 0, ξpq ), and pinning them at zero recovers QFT (phases). QFT (rotations) frees the Hadamards too, each Vq initialized at H and held on U (2) by a Cayley retraction [23] as in [21],

Algorithm 1: Recovery and training for the phase-only and diagonal models Input: D, p, k, K, T (Sec. 4) 1 function Recover(Y, Ω, θ): 2 X(0) ← Y 3 for κ = 0, . . . , K − 1 do 4 Update X(κ+1) by (7) b K (θ; Y, Ω) := X(K) 5 return X 0 6 θ←θ // the DFT 7 for τ = 1, . . . , T do 8 Draw X ∈ D and a fresh mask as in Sec. 2 b K ← Recover(PΩ X, Ω, θ) 9 X b K − X∥2F /N 2 ) // support fixed 10 θ ← Adam(θ, ∇θ ∥X Output: θ⋆ ← θ, with µ2D = 1 (Prop. 1) and can leave the complex Hadamard set. Every replacement is unitary, so (5) holds for all three, and all start at the DFT and minimize (1), each adjacent comparison in Table 1’s last block isolating one freedom. Tying the phase vectors of QFT (diagonals) by gate distance, θpq = ψp−q as in the DFT rule ξpq of Step 1b, gives 4(n − 1) parameters per axis and, with the DFT phases at new distances, a basis at every resolution within Prop. 1’s hypothesis (Sec. 3.5). 3.3. Recovery as an unrolled map Define the hard-thresholder Hk to retain coefficients at or above the kth largest magnitude, including ties at the cutoff. Starting from X(0) = Y, the iterative update is: (κ) )) ) } . (7) X(κ+1) := PΩ Y + (Id − PΩ ) Re{ A−1 θ ( Hk (Aθ (X

This update alternates transform-domain hard thresholding with data consistency on the observed pixels [5, 6]. The output matches observed entries but is not constrained to remain k-sparse. The Kth b K (θ; Y, Ω) from (1). Training requires derivatives iterate yields X with respect to θ; since these arise from the iteration rather than a closed form, the iteration itself is differentiated. Fixing K unfolds the iteration into a finite computation graph—a piecewise differentiable map from gate angles to images, which defines algorithm unrolling [24]. Away from support changes, Hk (C) = Mk ⊙ C for a fixed binary mask Mk , with derivative DHk (C)[∆C] = Mk ⊙ ∆C. Differentiation propagates through the retained support values via reverse mode over K iterations, rematerializing at each step. 3.4. Isometry without manifold optimization The fully relaxed network searches the product manifold U (2)n ×
n(n−1)/2 U (1)4 per axis [21]. The diagonal relaxation pins the U (2) factors at H and leaves a torus, parameterized by the periodic, redundant angles through θ 7→ eiθ . Every Adam step [25] on these angles preserves gate unitarity, with no need for Riemannian tangent-space projections and retractions (Algorithm 1). The diagonal gates at each stage commute and combine into one diagonal, so applying the n Hadamards and n diagonals costs O(N 2 log N ) per image, for the map or its adjoint, with no dense T formed. 3.5. Coherence is pinned Proposition 1. Let U = B GL · · · G1 , where B is a permutation matrix and every factor Gl is either a unitary diagonal matrix or the Hadamard Hq of Step 1b, each √ wire carrying exactly √ one Hadamard factor. Then |(U)ij | = 1/ N , ∀(i, j), so that N U is a complex

Hadamard matrix [26] with µ(U) = 1. In particular µ = 1 on QFT (diagonals) and its phase-only subfamily, tied (Sec. 3.2) or untied. Proof. The proof is omitted due to lack of space. Prop. 1 guarantees minimum coherence throughout training via the circuit topology alone, without requiring an explicit coherence penalty. Unconstrained learned dictionaries lack such guarantees. The QFT (rotations) architecture falls outside this guarantee because its free parameters Vq replace the prescribed Hadamards, allowing coherence to exceed the minimum bound of 1—as observed in Sec. 4.4. 4. NUMERICAL TESTS AND DISCUSSION 4.1. Setup Numerical tests use DIV2K [27], grayscale 512 × 512 crops: 750 for training, 50 for validation, and 100 test images. Each test image carries a single mask, drawn once randomly and then fixed, shared by every method and every budget in this paper, and there is one training seed throughout. Each method is evaluated at its per-image budget optimum over sparsity levels k/m ∈ {0.015, 0.03, 0.0625, 0.125, 0.25, 0.5}, where m is the observed pixel count. All transforms are evaluated at K = 300 iterations; the QFT models train at K = 100, T = 200 steps over mini-batches of two images, k = m/8. Learning rates are tuned via validation-set PSNR over seven candidates, QFT (rotations) also over five Cayley steps. In Table 1, µ denotes coherence; “Fitted on” indicates what each method uses to train its parameters (nothing, test images, or training images). All baselines are implemented in JAX [28]. 4.2. Comparison with fixed transforms and per-image methods QFT (diagonals) outperforms the DFT by 2.02dB and the best fixed transform by 1.59dB in PSNR, and leads both in PSNR and MSSSIM on all 100 test images. Fig. 2 shows all methods on a representative test image. The bases split by coherence into global (DFTlike) and localized (wavelet-like) families, the global family leading on PSNR and MS-SSIM [31], with single-scale SSIM [32] favoring the wavelets. Per-image methods are not transforms. The tensor train [14] trails by 0.60dB in PSNR but leads by 0.005 in MS-SSIM. Among Table 1: Completion on 100 DIV2K test images at 512×512 and p = 10%: fixed bases, per-image fits, trained transforms, and proposed QFT networks

Method DFT = T (θ 0 ) DCT-II Haar [1] Daubechies-4 [1] Symlet-8 [1]

Trainable Fitted params on 0 0 0 0 0

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Tensor train [14] [34–402]k test img Nuclear norm [7, 29, 30] [1–109]k test img U (2) butterfly [20] Transform learning [3] QFT (phases) QFT (diagonals) QFT (rotations)

MSµ PSNR SSIM SSIM 1 20.72 0.443 0.677 4 21.15 0.454 0.693 65536 18.70 0.453 0.582 68453 19.23 0.472 0.616 95640 19.28 0.477 0.625 — 22.13 0.524 0.791 — 18.61 0.369 0.612

18432 tr. data 2.70 22.91 0.551 0.798 524288 tr. data 18642 21.03 0.453 0.691 72 tr. data 288 tr. data 360 tr. data

1 22.62 0.532 0.781 1 22.74 0.537 0.786 1.20 22.98 0.558 0.799

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Fig. 2: A DIV2K test image at sampling rate p = 10% and all methods in Table 1, each at its PSNR-optimal budget.

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Fig. 3: (a), (b) Mean PSNR and MS-SSIM over 100 test images vs. sampling rate (dotted: training rate). (c) Time per full-resolution step. (d) Reals each method carries, the transforms once, the per-image fits for every image.

the trained transforms, the butterfly factorization [20] (blocks constrained to U (2)) leads by 0.17dB at 64× the parameters, but its unconstrained variant loses 8dB under a single-budget protocol; transform learning [3] achieves 0.1dB below the DCT. Fig. 3(a, b) compares methods across sampling rates, with the transforms trained at p = 10% applied unchanged. QFT (diagonals) leads the DFT at all rates on both metrics. The tensor train leads in PSNR only at 1% and in MS-SSIM up to 10%; the butterfly leads by 0.2–0.3dB up to 10%, matches at 20%, and trails above. Fig. 3(c, d) analyzes computational cost. QFT (diagonals) scales as N 1.7 and costs two-fifths of the tensor train at 5122 (which scales as N 0.6 ; a rank cap). QFT (diagonals) requires 288 trainable reals at 5122 , compared to 18,432 (butterfly) and 524,288 (transform learning). Training costs 741s, equivalent to ∼ 570 tensor-train fits.

4.3. The coherence-sparsity trade-off Theory provides a fundamental trade-off: a k-sparse object is recovered from m random pixels when m ≳ µ(U) k log N [15]. At fixed sampling, workable sparsity decreases linearly with coherence. Wavelets achieve superior sparsity but suffer from high coherence— resulting in worse inpainting PSNR than the less-sparse but lowcoherence DFT. This reversal (wavelets win in compression, lose in inpainting) cannot be fully attributed to coherence alone, since sparsity and coherence vary together across bases. Isolating coherence’s effect requires trainable transforms rather than fixed bases. Standard coherence-reduction strategies also modify the sampling law [16], which i.i.d. masks preclude. The comparison uses fixed-k hard-thresholding (7), applied uniformly to all methods. Soft thresholding improves all bases—the DFT gains 1.0dB at 10% and 0.9dB at 60% sampling; Symlet-8’s high-rate deficit narrows from 3.0 to 2.3dB at 60%—but does not reverse the two families’ relative order. 4.4. Freeing the QFT circuit: performance vs. structure Table 1’s final block of rows compares architectures with increasing degrees of freedom. The phase-only model learns diagonal phase parameters while keeping Hadamards fixed. Freeing the diagonal gates adds +0.11dB over phase-only updates. Freeing the Hadamards as well (replacing them with learnable rotations, with both learning rates tuned) adds +0.24dB and 0.013 MS-SSIM, but at a cost: coherence rises with the Cayley step from 1.09 to 1.31, requiring Riemannian retractions at each iteration. At the largest step attempted, training diverges (µ = 697, 12.5dB). In contrast, the diagonal subfamily (phases + fixed Hadamards) maintains µ = 1 unconditionally with only 0.24dB loss, using plain Adam without retractions. 5. CONCLUSION The tension between sparsity and coherence in image inpainting motivated the design of trainable transforms adapted to image structure. This work introduced quantum-inspired tensor-network circuits as trainable transforms for the first time in this application. The diagonal QFT model achieved low coherence through circuit topology alone, guaranteeing minimum coherence at every parameter value without explicit penalties. This structural guarantee enabled efficient training with plain Adam and yielded competitive performance with far fewer parameters than alternative learned transforms, demonstrating that principled architectural choices can replace costly optimization constraints while establishing a new paradigm of quantuminspired methods to image reconstruction.

ACKNOWLEDGMENT S. An’s work was supported by JST SPRING, Japan, Grant Number JPMJSP2180. The authors thank Jin-Guo Liu, Zhongyi Ni and Huanhai Zhou of the Hong Kong University of Science and Technology (Guangzhou) for discussions.

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