Iterative tensor network transformations for element-wise evaluation of elementary and filtering functions

Xiao Wang; Tomohiro Hashizume; Pia Siegl; Dieter Jaksch

cs.LGAug 17, 2026

Iterative tensor network transformations for element-wise evaluation of elementary and filtering functions

Authors: Xiao Wang, Tomohiro Hashizume, Pia Siegl, Dieter Jaksch

Organizations: Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom · The Hamburg Centre for Ultrafast Imaging, Luruper Chaussee 149, Hamburg 22761, Germany. · Institute for Quantum Physics, University of Hamburg, Luruper Chaussee 149, Hamburg 22761, Germany · Institute of Software Methods for Product Virtualization, German Aerospace Center (DLR), Nöthnitzer Straße 46b, 01187 Dresden, Germany

Abstract

Tensor networks are powerful formats for compressing large-scale data. However, their application to general data processing has been limited by the difficulty of performing nonlinear operations. Here, we introduce iterative tensor network transformations (ITNTs), a general algorithmic framework for the element-wise evaluation of elementary and nonlinear filtering functions on data encoded as tensor trains (TTs), a class of tensor networks. Our approach operates entirely in the compressed domain, enabling efficient computation on exponentially large datasets while maintaining a controlled computational cost. We demonstrate its power in two key areas: (I) evaluating highly nonlinear elementary and filtering functions on a 3D reactive flow field, enabling high-fidelity reaction rate computation and region filtering, and (II) finding extrema in complex optimization problems, such as solving Max-SAT instances on spaces up to $2^{70}$ configurations. These results establish ITNT as a foundational tool that provides tensor network methods with the capability for general-purpose data science and large-scale optimization.

Figures & tables

Figure 1: Overview of the results. (a) Schematic of the Iterative tensor network transformations (ITNTs) methodology. The framework enables the evaluation of highly complex nonlinear transformations, $\mathcal{T}$ , directly on TT-encoded data $\Psi$ defined on array indices $\bm{j}\in\{0,1,\ldots,N-1=2^{n}-1\}$ . This is achieved by leveraging $2^{n}$ parallel evaluations of the iteratively converging Newton-Raphson method (bottom-right panels). In these panels, the red curve represents the nonlinear function being iteratively solved, while the blue and green lines illustrate the successive tangent-linear approximations during the iterative process, starting from the blue dot. This convergence allows the transformation to be applied while maintaining the data in a compressed format. (b) Validation of the framework on a simulated methane/air jet flame ( 26 ) . The ITNT method is used to compute the highly nonlinear Arrhenius reaction rate ( $\exp(-E_{c}/T)$ ). The plot shows the iso-rate surface at 10% of the maximum reaction rate, computed directly from a compressed temperature field $T$ [K] using a normalized reaction energy of $E_{c}=17865.2$ [K]. The comparison uses a TT bond dimension of $\chi=200$ , requiring only 0.4The results demonstrate high-fidelity reconstruction, with ITNT achieving enclosed volume relative error approximately 100 times lower than that of TCI, representing a significant improvement in capturing the flame’s topological features. (c) Schematic of combinatorial optimization utilizing ITNT. The framework uses iterative nonlinear operations, specifically self-multiplication, to amplify and sharpen dominant configurations, such as the global maximum ( $E_{\mathrm{opt}}$ ) and sub-optimal peaks ( $E_{\mathrm{sub}}$ ), within the problem-encoded energy landscape. Simultaneously, TT truncation (depicted by the lightning bolt) prunes away the rugged bulk spanned by the low-energy configurations by discarding the subspace corresponding to smaller singular values. This allows the algorithm to isolate optimal solutions while maintaining the data in a compressed TT format.

Figure 2: Nonlinear post processing with ITNTs. (a) Convergence comparison between ITNT and TCI for the Arrhenius equation (Eq. ( 3 )) computation. While TCI suffers from instability around $\chi=200$ , ITNT demonstrates a drop in error in the same region, consistently converging towards the exact result with increasing $\chi$ . (b) Relative $\ell_{2}$ error of the $1/T$ field computed exactly ( $\Psi_{\mathrm{exact}}$ ) from the original field and by using the ITNT algorithm ( $\Psi_{\mathrm{TT}}$ ) for $\chi=50$ , $100$ , $200$ , $300$ , and $500$ (light to dark, left panel), and the same error at the 10th iteration for different bond dimensions. In the right panel, the blue dashed line indicates $1/\chi$ decay as a guide to the eye. (c) The position of the maximum temperature is indicated by a red dot and the orange region represents the isothermal surface at $90$ % of the maximum temperature ( $0.9\max\{T\}$ ). (d) Relative error of the maximum temperature, $1-\max\{T_{\mathrm{TT}}\}/\max\{T_{\mathrm{exact}}\}$ , found using ITNT for $\chi=50$ , $100$ , $200$ , $300$ , and $500$ (light to dark). (e) Relative error of the volume of the region where $T>0.9\max\{T\}$ for $\chi=50$ , $100$ , $200$ , $300$ , and $500$ (light to dark). Here, the value of $0.9\max\{T\}$ is computed with the maximum-finding ITNT and the volume is determined by constructing the filter with sign-transformation ITNT.

Figure 3: Solving Max-3-SAT problems with ITNTs. (a) Truncation error at each bond in the zip-up self-multiplication algorithm. In order to solve 70 variables 700 clauses Max-3SAT problem, 10 self-multiplications are performed, and for each self-multiplication is computed with the zip-up algorithm with on-site truncation, while fixing $\chi=700$ . Each color corresponds to a different self-multiplication iteration (iterations 1 to 10, light to dark) (b) Deflation result for the max 3-SAT problem s3v70c700-1 . Three configurations with an energy of $E$ with $E=679$ are found after the $i=1655$ th , the $i=6442$ nd , and the $i=30186$ th extremum-finding ITNT steps. (c) Scaling of the minimum bond dimension for successful optimization and solution certification. Here, we generate 100 random $n$ -variable, $M=\alpha n$ -clause Max-3SAT problems for $n=10$ to $20$ , at clause densities $\alpha=8$ and $\alpha=3$ . Orange dashed lines show the minimum bond dimension required for the maximum search to return the correct optimum after $P=\left\lceil\log_{2}(nM)\right\rceil+1$ self-multiplication steps. The blue line shows the minimum bond dimension required for the truncation budget in Method V.7 to be obeyed after $P$ steps. The required bond dimension grows with $n$ , saturating the bond dimension $\sim 2^{n/2}$ (black dotted line).

Figure 4: ITNTs for element-wise nonlinear operations in TT representation. (a) Extrema detection with TT. i Starting from a TT representation of

\Psi

(black curve), repeated element-wise self-multiplication sharpens the state into a multi-peak profile

\Phi

, where its integral is dominated by extremal configurations (red curve). ii A peak is then located by a binary fixation procedure. This involves integrating out all unfixed sites and choosing the local configuration that maximizes the summed value. Starting from every index not being fixed, in the leftmost panel, a comparison between

j_{2}=0

(orange) and

j_{2}=1

(cyan) is performed, identifying

j_{2}=1

as a candidate for the optimal configuration. In the subsequent next panel,

j_{1}=0

(orange) and

j_{1}=1

(cyan) are compared and

j_{1}=0

is selected, while the integrated out region from the fixation of

j_{2}=0

is indicated in gray. Continuing these steps with a random sequence of local tensors identifies the index

I

of the maximum (orange, right most panel). iii The deflation process then removes the identified peak to ensure that subsequent applications find the true maximum. (b) Element-wise evaluation of sign function. A highly nonlinear, non-analytic element-wise filter is realized through the composition of iterative applications of the functions

F

(Eq. ( 1 )) and

G

(Eq. ( 16 )) that produce an element-wise sign function

\mathrm{Sign}\{f\}^{[j]}=\mathrm{Sign}\{f^{[j]}\}

. As an illustrative example, the blue curve shows the output of ITNT for the input field

f(x)=x

(black dashed line) after applying

G

twice followed by repeated iterations of

F

. The red shaded region marks the regime of input values where the composed iteration

F(\ldots F(G(G(f))))

does not converge. The curves are evaluated in the full-array representation solely to visualize the mechanism of the procedure.

Panel of Figure 5 — Figure S1 : Result of the Sign-function transformation via ITNT of the function defined in Eq. S8 for different number of roots $N_{\text{root}}$ and different qubit numbers $n$ . For all examples we consider $N_{\text{iter}}=20$ and an SVD tolerance of $10^{-8}$ . Sign transformation of the function with the following properties: (a) $n=14$ , $N_{\text{root}}=7$ , $\chi_{\text{max}}=20$ . (b) $n=14$ , $N_{\text{root}}=7$ , $\chi_{\text{max}}=25$ . (c) $n=18$ , $N_{\text{root}}=7$ , $\chi_{\text{max}}=25$ . (d) $n=18$ , $N_{\text{root}}=23$ , $\chi_{\text{max}}=75$ . For $N_{\text{root}}=23$ , a noiseless sign transformation requires $\chi_{\text{max}}=75$ .

Figure S2 : Scaling of the element-wise evaluation of sign function via ITNT (a) Element-wise evaluation of sign function $y=\mathrm{Sign}\{f_{\omega}(x)\}$ (red) using ITNT of a function $y=f_{\omega}(x)$ (black) $\omega=1.0$ . (b) Required bond dimension in dependence on the number of tensors $n$ for $\omega=0.5$ . Considered are the maximal intermediate bond-dimension $\chi_{\mathrm{peak}}$ (crosses) used by the algorithm if the bond dimension is not restricted and the SVD tolerance is fixed at $10^{-8}$ . Furthermore, we consider the final bond dimension $\chi_{\mathrm{final}}$ (circles) required to represent $y$ accurate up to the order of the SVD tolerance of $10^{-8}$ with respect to the $L^{2}$ -error. Finally, there is the cutoff bond dimension $\chi_{\mathrm{min}}$ (squares) which defines a lower cutoff bound, required to reach the result with ITNT up to this accuracy while reaching the bond dimension $\chi_{\mathrm{final}}$ within $200$ iteration steps. (c) Required bond dimensions for different numbers of roots $N_{\mathrm{roots}}$ by using $\omega=[0.5,1.0,1.5,2.0]$ corresponding to $N_{\mathrm{roots}}=[7,13,19,25]$ and $n=18$ to converge within a maximum of $200$ iteration steps. The bond dimensions are defined as for (b). (d) The number of iteration steps $N_{\mathrm{iter}}$ needed to reach the final compressed result with respect to the chosen maximal cutoff bond dimension $\chi_{\rm{max}}$ for $\omega=0.5$ for $n=14,18,22$ (light to dark). All simulations were implemented using the zip-up algorithm for point-wise multiplication.

Figure S3 : The bond dimension of the sign-transformed TT after $N_{\text{iter}}$ steps of iteration. The qubit number is $n=18$ with root number $N_{\text{root}}=7$ , and $\chi_{\text{max}}$ denotes the maximum bond dimension allowed during TT operations (e.g., element-wise products), with the SVD tolerance fixed at $10^{-8}$ . For $\chi_{\text{max}}=28$ , the convergence is reached at $N_{\text{iter}}\approx 64$ . For $\chi_{\text{max}}\gtrsim 35$ , the required iteration count reduces to $N_{\text{iter}}\approx 34$ .

$n$	$N_{\text{root}}$	$\chi_{\text{max}}$	time (sec) of ITNT-based sign tranforms	time (sec) of standard sign transform.
22	7	45	3	1.6
22	19	100	7	1.6
22	51	250	65	1.6
24	7	45	3.5	6.5
24	19	100	8.5	6.5
24	51	250	84	6.5

Table 1: Comparison of runtime performance across different qubit numbers

n

, root numbers

N_{\rm{root}}

, and chosen cutoff bond dimensions

\chi_{\rm{max}}

Figure S4 : Accuracy and scaling benchmarks. (a) Binary-discard deflation for locating extrema of a $12\times 12$ classical Ising model. The algorithm is performed for 100 runs (each including 800 extremum-finding ITNT steps $i$ ), and the cumulative best value at each step is shown, with the percentages indicating the fraction of runs that attain the corresponding value. (b) A single run of the extremum-finding ITNT steps. We fix $\chi=409$ and iteratively self-multiply $E$ that encodes the energy of all the configurations 8 times. In the resulting TT, we identified $800$ sub-leading peaks, and found the extremum $|E|_{max}\sim 33.35$ after the $i=324$ th step. Here, the configuration returned by the $i$ th deflation step is denoted as $\bm{\sigma}(i)$ with corresponding index $\sigma(i)$ , and $E^{[\sigma(i)]}$ is the corresponding energy of the found configuration.

Figure S5 : Extremum-finding ITNT without preprocessing. Shown is a single instance of the results obtained from extremum-finding ITNT steps without performing the self-multiplication preprocessing procedure. Clearly, in comparison to Fig. S4 b., the values of the found configurations are significantly lower, indicating a failure to detect the global extremum.

Figure S6 : Energy matching. Energy matching $E({\bm{\sigma}})=0$ for the $144$ -qubit model is conducted and $10^{4}$ candidate are identified. The best match, $|E({\bm{\sigma}}_{\text{best}})|\sim 1.38\times 10^{-4}$ , is found after the $i=784$ th step. The second-best match, $|E({\bm{\sigma}}_{\text{next-best}})|\sim 1.81\times 10^{-4}$ , is found after the $i=5813$ rd deflation. During the self-multiplication, the bond dimension is fixed at $818$ for $14$ iterations (each taking $\sim 10^{4}$ seconds). During the following deflation, the bond dimension is truncated to $1900$ whenever it reaches $2000$ , yielding a per-extremum-finding ITNT runtime of $\sim 60$ seconds.

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Tensor Decomposition TT Decomposition Low-Rank Compression

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We introduce the tubal tensor train (TTT) decomposition, a tensor-network model that combines the t-product algebra of the tensor singular value decomposition (T-SVD) with the low-order core structure of the tensor train (TT) format. For an order-

(N+1)

tensor with a distinguished tube mode, the proposed representation consists of two third-order boundary cores and

N-2

fourth-order interior cores linked through the t-product. As a result, for bounded tubal ranks, the storage scales linearly with the number of modes, in contrast to direct high-order extensions of T-SVD. We present two computational strategies: a sequential fixed-rank construction, called TTT-SVD, and a Fourier-slice alternating scheme based on the alternating two-cores update (ATCU). We also state a TT-SVD-type error bound for TTT-SVD and illustrate the practical performance of the proposed model on image compression, video compression, tensor completion, and hyperspectral imaging.

Abstract

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