Quantum-Inspired Tensor Network Autoencoders for Anomaly Detection: A MERA-Based Approach
Authors: Emre Gurkanli, Michael Spannowsky
Organizations: Department of Physics, Sinop University, Turkiye · Institute for Theoretical Physics, Campus Süd, Karlsruhe Institute of Technology (KIT), D-76128 Karlsruhe, Germany · Institute for Quantum Materials and Technologies, Karlsruhe Institute of Technology, Karlsruhe 76131, Germany
We investigate whether a multiscale tensor-network architecture can provide a useful inductive bias for reconstruction-based anomaly detection in collider jets. Jets are produced by a branching cascade, so their internal structure is naturally organised across angular and momentum scales. This motivates an autoencoder that compresses information hierarchically and can reorganise short-range correlations before coarse-graining. Guided by this picture, we formulate a MERA-inspired autoencoder acting directly on ordered jet constituents. To the best of our knowledge, a MERA-inspired autoencoder has not previously been proposed, and this architecture has not been explored in collider anomaly detection. We compare this architecture to a dense autoencoder, the corresponding tree-tensor-network limit, and standard classical baselines within a common background-only reconstruction framework. The paper is organised around two main questions: whether locality-aware hierarchical compression is genuinely supported by the data, and whether the disentangling layers of MERA contribute beyond a simpler tree hierarchy. To address these questions, we combine benchmark comparisons with a training-free local-compressibility diagnostic and a direct identity-disentangler ablation. The resulting picture is that the locality-preserving multiscale structure is well matched to jet data, and that the MERA disentanglers become beneficial precisely when the compression bottleneck is strongest. Overall, the study supports locality-aware hierarchical compression as a useful inductive bias for jet anomaly detection.
Figures & tables
Figure 1: Schematic overview of the MERA-based autoencoder used in this work. After locality-preserving ordering and standardisation, the encoder alternates disentanglers Uℓ and isometries Wℓ through the hierarchy 48→24→12→6→3 , followed by a projection to a latent vector z∈RB . The decoder mirrors this structure with untied tensors U~ℓ and W~ℓ and reconstructs the ordered jet constituents. The anomaly score is computed from the reconstruction error between the original jet and its reconstruction, as indicated by the direct comparison arrow into the score box.
Figure 2: Schematic action of one MERA encoder layer on neighbouring sites. A layer first applies local disentanglers U to reorganise short-range correlations across block boundaries and then applies isometries W that coarse-grain adjacent pairs into a reduced set of effective sites. This is the local operation described by the layerwise maps above: the purpose of the disentanglers is to move locally relevant information into the subspace retained by the subsequent isometries before compression to the next scale.
Figure 3: Schematic relationship between the penalised training objective and the inference-time anomaly score. On the left, the reconstruction branch and the two regularisation branches all originate from the same MERA encoder–decoder, because the disentanglers and isometries that define the model also enter the penalties that keep them close to their intended orthogonal and isometric structure. On the right, inference is shown as a separate path in which a trained MERA encoder–decoder is applied to the input jet and the anomaly score is defined only by the reconstruction term, not by the full penalised training objective.
Figure 4: ROC curves for the nominal benchmark at latent dimensions B=8,16,32 . Solid lines denote the mean over random seeds where applicable, and shaded bands indicate the corresponding ±1σ variation.
B
Model
Architecture
Params
Test AUC
AUC(std)
CI low
CI high
8
MERA
(L=4, latent=8, η=3×10−4 )
5021
0.7959
0.0065
0.7855
0.8063
8
AE
(h1=48, h2=24, bn=8, η=3×10−4 )
16808
0.7616
0.0038
0.7554
0.7677
8
PCA
(ncomp=8)
–
0.7192
0.0000
0.7192
0.7192
8
Gaussian
Mahalanobis (ridge-reg. cov)
–
0.6679
0.0000
0.6679
0.6679
8
IForest
(ntrees=100)
–
0.6726
0.0142
0.6500
0.6952
16
MERA
(L=4, latent=16, η=2×10−4 )
5173
0.7885
0.0190
0.7582
0.8189
Table 1: Test-set ROC-AUC values for all models at latent dimensions B=8,16,32 , where AUC denotes the area under the receiver operating characteristic (ROC) curve and CI denotes the confidence interval quoted for repeated runs. For the neural architectures, the corresponding number of trainable parameters is also listed.
Ordering
Mean adjacent ΔR
Retained variance (layer 0)
Retained variance (layer 1)
Locality
0.131
0.869
0.920
pT
0.361
0.857
0.890
Random
0.397±0.004
0.721
0.830
Table 2: Geometric locality and two-site compressibility of the ordered constituent representation. The retained variance fractions are computed from the optimal three-dimensional approximation to neighbouring two-site blocks on the background sample.
Figure 5: MERA test ROC-AUC as a function of input ordering for B=8,16,32 , aggregated over three independently tuned runs for each ordering. Error bars indicate the corresponding confidence intervals. The locality-preserving ordering gives the strongest MERA performance at every latent dimension, consistent with the geometric and compressibility diagnostics reported in Table 2 .
Model
Params
Test AUC
TPR@10−3
MERA
5021
0.8002±0.0052
1.37×10−3
MERA( U=I )
1781
0.7922±0.0138
1.10×10−3
Gain of full MERA
—
+0.0081
+2.71×10−4
Table 3: Focused multi-seed disentangler ablation at the strongest-compression setting B=8 . MERA( U=I ) denotes the identity-disentangler limit of the same hierarchy, in which the local disentangling maps are fixed rather than learned. Both models are evaluated on the same eight seeds after separate retuning at B=8 . The last row gives the mean matched-seed gain of the full MERA over MERA( U=I ).
Quantum machine learning offers the ability to capture complex correlations in high-dimensional feature spaces, crucial for the challenge of detecting beyond the Standard Model physics in collider events, along with the potential for unprecedented computational efficiency in future quantum processors. Near-term utilization of these benefits can be achieved by developing quantum-inspired algorithms for deployment in classical hardware to enable applications at the "edge" of current scientific experiments. This work demonstrates the use of tensor networks for real-time anomaly detection in collider detectors. A spaced matrix product operator (SMPO) is developed that provides sensitivity to a variety beyond the Standard Model benchmarks, and can be implemented in field programmable gate array hardware with resources and latency consistent with trigger deployment. The cascaded SMPO architecture is introduced as an SMPO variation that affords greater flexibility and efficiency in ways that are key to edge applications in resource-constrained environments. These results reveal the benefit and near-term feasibility of deploying quantum-inspired ML in high energy colliders.
Sagar Addepalli, Prajita Bhattarai, Abhilasha Dave +1
SLAC National Accelerator Laboratory, 2575 Sand Hill Rd, Menlo Park, 94025, CA, USA.
Multi-dimensional time series, inherently tensorial, are common in practice. Despite great progress in time series anomaly detection, most existing methods are confined to uni-/multi-variate time series. When handling multi-dimensional time series using these methods, reshaping operations are required, which inevitably break the intrinsic correlations and thus lead to performance degradation. In uni-/multi-variate time series anomaly detection, AutoEncoders (AEs) are widely adopted and generally categorized into reconstruction-based and prediction-based AEs. The reconstruction-based AE utilizes the current observation for reconstruction, while the prediction-based AE utilizes the historical information to predict the current observation. Thus, the two AEs utilize different information. To bridge the gap between reconstruction-based and prediction-based AEs, so as to fully leverage the available information and thus further enhance performance, we propose a predictive prior and incorporate it into the reconstruction-based AE. It may not be very difficult to conceive this idea, but designing the predictive prior so that it can work for tensor anomaly detection is non-trivial. Specifically, to avoid breaking the intrinsic correlations within the multi-dimensional time series, we use the tensor AE as the backbone. To incorporate the predictive prior into the reconstruction-based AE, we propose a Bayesian fusion approach and our analysis reveals that this approach can enhance the modeling capability of the model for normal data. To mitigate the over-generalization problem of AE, we incorporate physical laws, i.e. tensor low-rank decomposition rules, into the neural networks in the predictive prior, leading to the Physics-informed Predictive Prior Tensor AE (PPPTAE) framework. Experimental results on real-world datasets demonstrate the effectiveness of the proposed method.
Reconstruction-based detectors for multivariate time series anomaly detection typically learn a single entangled representation of normal behavior and flag anomalies based on reconstruction error. However, real-world anomalies are not only isolated spikes in a single channel but also disruptions in the relationships between channels, where each sensor individually appears normal, but their joint behavior deviates from the norm. Because these models compress both temporal evolution and inter-channel structure into this single space, they lack an explicit mechanism to isolate such inter-channel dependencies from normal temporal dependencies, leading to lower detection rates for cross-channel anomalies. We propose CARMA (Correlation-Aware Reconstruction for Multivariate Time Series Anomaly Detection), which is a reconstruction-based variational autoencoder with two complementary latent spaces: a temporal latent that captures sequential dynamics, and a correlation latent that is trained to reconstruct the window's pairwise Pearson correlations. Both representations are fused in the signal decoder, and anomalies are detected from the reconstruction error. We evaluate CARMA on the TSB-AD-M benchmark suite (17 datasets, 180 series) against 29 baseline methods, and show that it significantly outperforms all of them across multiple threshold-free and range-aware metrics, achieving a VUS-PR of 0.4929. Ablations show that removing the correlation branch degrades performance, that adding a second latent without supervision recovers only part of the gain, and that supervising the correlation latent with an explicit correlation target accounts for the rest and produces a sharper error response at anomalies.
Md. Johir Raihan, Zahra Zamanzadeh Darban, Yueyang Liu +3
Monash University, Melbourne, Victoria, Australia · University of New South Wales, Sydney, New South Wales, Australia