Classical Neural Networks on Quantum Devices via Tensor Network Disentanglers: A Case Study in Image Classification

Borja Aizpurua; Rom\'an Or\'us; Sukhbinder Singh

arxiv: 2509.06653 · v2 · submitted 2025-09-08 · 🪐 quant-ph · physics.comp-ph

Classical Neural Networks on Quantum Devices via Tensor Network Disentanglers: A Case Study in Image Classification

Borja Aizpurua , Sukhbinder Singh , Rom\'an Or\'us This is my paper

Pith reviewed 2026-05-18 18:26 UTC · model grok-4.3

classification 🪐 quant-ph physics.comp-ph

keywords quantum computingtensor networksmatrix product operatorsneural networkshybrid quantum-classical modelsimage classificationdisentanglingMNIST

0 comments

The pith

Linear layers in classical neural networks can be compressed into matrix product operators, disentangled, and executed in a hybrid classical-quantum setup without losing accuracy on image tasks.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper demonstrates a way to take bottleneck linear layers from already-trained classical neural networks and represent them as matrix product operators that are then disentangled into simpler pieces. These disentangled pieces can run as quantum circuits while the compressed operator and the rest of the network stay on classical hardware. The approach is tested by translating small networks that classify MNIST and CIFAR-10 images. A reader would care because it offers a concrete route to insert quantum resources into existing machine-learning models at specific points rather than replacing entire networks.

Core claim

The authors show that a target linear layer can first be expressed as an effective matrix product operator without degrading the network's performance, after which the operator is disentangled either by an explicit variational tensor-network method or by an implicit gradient-descent procedure. The resulting disentangling circuits are placed on a quantum device while the remainder of the model, including the disentangled MPO, runs classically. This hybrid scheme is validated on simple networks for MNIST and CIFAR-10 classification.

What carries the argument

Matrix product operator (MPO) disentangling: a compression of a large linear transformation into a tensor network that is further broken into compact quantum circuits for hybrid execution.

If this is right

Hybrid models can place only the disentangled circuits on quantum hardware while keeping the compressed MPO and other layers classical.
Two distinct algorithms exist for the disentangling step: one variational and one gradient-based.
The same compression-plus-disentangling pipeline applies to both MNIST and CIFAR-10 image-classification networks.
The method targets bottleneck layers, leaving the rest of a pre-trained network unchanged.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If MPO compression continues to work for deeper or wider networks, the hybrid scheme could be applied to larger models without full retraining.
The approach suggests a general pattern in which tensor-network compression isolates the parts of a computation that benefit most from quantum hardware.
Extending the same disentangling logic to other tensor-network representations beyond MPOs could broaden the set of layers that can be off-loaded to quantum devices.

Load-bearing premise

A linear layer can be rewritten as a matrix product operator that preserves the original network accuracy.

What would settle it

Running the MPO-compressed and disentangled version of a linear layer on MNIST or CIFAR-10 and measuring lower classification accuracy than the original classical network would falsify the claim.

Figures

Figures reproduced from arXiv: 2509.06653 by Borja Aizpurua, Rom\'an Or\'us, Sukhbinder Singh.

**Figure 2.** Figure 2: FIG. 2. [Color online] Variational optimization of disentan [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. [Color online] A simple classical tensorized neural [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. (Top Left) Overlap, Eq [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. [Color online] Neural network architectures used in [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. [Color online] Neural network architectures used in [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗

read the original abstract

We address the problem of implementing bottleneck layers from classical pre-trained neural networks on a quantum computer, with the goal of exploring intrinsically quantum ansatz for representing large linear layers within hybrid classical-quantum models. Our approach begins with a compression step in which the target linear layer is represented as an effective matrix product operator (MPO) without degrading model performance. The MPO is then further disentangled into a more compact form. This enables a hybrid classical-quantum execution scheme, where the disentangling circuits are deployed on a quantum computer while the remainder of the network -- including the disentangled MPO -- runs on classical hardware. We introduce two complementary algorithms for MPO disentangling: (i) an explicitly disentangling variational method leveraging standard tensor-network optimization techniques, and (ii) an implicitly disentangling gradient-descent-based approach. We validate these methods through a proof-of-concept translation of simple classical neural networks for MNIST and CIFAR-10 image classification into a hybrid classical-quantum form.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives two workable MPO disentangling algorithms and a hybrid split that lets quantum hardware handle only the bottleneck after compression, but the no-degradation claim on the initial MPO step is the part that needs tighter numbers.

read the letter

The core contribution is a practical pipeline that takes a pre-trained linear layer, compresses it to an MPO, disentangles that MPO with one of two new methods, and then runs the disentangling circuits on quantum hardware while the rest of the network stays classical. They demonstrate this on small MNIST and CIFAR-10 classifiers and report that accuracy holds up in their tests. That split is useful because it avoids the need to quantize an entire model at once. The two disentangling algorithms—one variational using standard tensor-network sweeps and one that bakes the disentangling into gradient descent—are the genuinely new pieces. Both are described clearly enough that someone could re-implement them from the text. The hybrid execution scheme itself is a straightforward but sensible engineering choice that matches current hardware constraints. The paper does a decent job of walking through the steps from compression to circuit deployment and grounding the experiments in public datasets with ordinary network architectures. Credit for keeping the claim modest: this is framed as a proof-of-concept rather than a general solution. The soft spot is the opening assumption that the linear layer can be turned into an MPO without hurting performance. Fully-connected layers in image classifiers tend to have dense, high-rank weight matrices, so MPO truncation at modest bond dimension introduces some operator error. If that error is non-negligible, it will affect the outputs that the disentangling step and the quantum circuits then act on. The abstract and the stress-test note both flag this as the least secure premise, and the provided description does not include explicit operator-norm bounds, accuracy-versus-bond-dimension curves, or multi-seed statistics that would quantify how much is lost. Without those checks the hybrid claim rests on the specific cases they chose rather than a controlled demonstration that the compression is essentially lossless. This work is aimed at people already working on tensor-network methods for quantum machine learning who want a concrete bridge to classical pipelines. A reader looking for implementable hybrid ideas will find usable algorithms and a clear execution split. It is coherent on its own terms and shows honest engagement with the MPO and circuit literature, so it deserves a serious referee even though revisions would probably center on measuring and controlling the compression error. I would send it out for review.

Referee Report

1 major / 1 minor

Summary. The paper claims to address implementing bottleneck layers from classical pre-trained neural networks on quantum computers by first compressing target linear layers into effective matrix product operators (MPOs) without degrading model performance, then applying disentangling to enable a hybrid classical-quantum execution scheme. Two complementary MPO disentangling algorithms are introduced—an explicitly disentangling variational method and an implicitly disentangling gradient-descent approach—and validated via proof-of-concept translation of simple classical networks on MNIST and CIFAR-10 image classification tasks.

Significance. If the central claims hold, the work could offer a concrete route for embedding large classical linear layers into hybrid models that offload disentangled components to quantum hardware while retaining the rest classically. The use of standard public datasets and tensor-network techniques provides a reproducible starting point for exploring quantum representations of classical bottlenecks.

major comments (1)

[Abstract] Abstract (compression step): The claim that the target linear layer is represented as an effective MPO 'without degrading model performance' is load-bearing for all subsequent steps, yet the description supplies no quantitative verification such as accuracy deltas, operator-norm error bounds, or ablation over bond dimension on the validation sets. Because the disentangling algorithms and hybrid execution operate directly on this MPO, any unquantified approximation error at compression directly undermines the assertion that the hybrid model preserves the original network's accuracy.

minor comments (1)

[Abstract] Abstract: The phrase 'simple classical neural networks' is used without specifying layer widths, activation functions, or total parameter counts, making it difficult to assess how the MPO bond-dimension choice scales with network size.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the importance of quantitative support for the compression step. We address the major comment below and will revise the manuscript to strengthen the presentation of our results.

read point-by-point responses

Referee: [Abstract] Abstract (compression step): The claim that the target linear layer is represented as an effective MPO 'without degrading model performance' is load-bearing for all subsequent steps, yet the description supplies no quantitative verification such as accuracy deltas, operator-norm error bounds, or ablation over bond dimension on the validation sets. Because the disentangling algorithms and hybrid execution operate directly on this MPO, any unquantified approximation error at compression directly undermines the assertion that the hybrid model preserves the original network's accuracy.

Authors: We agree that explicit quantitative verification is necessary to support the compression claim. The main text reports classification accuracies for the original and MPO-compressed networks on both MNIST and CIFAR-10, but we acknowledge that these comparisons, along with bond-dimension ablations and error metrics, are not presented with sufficient prominence or detail in the abstract and introductory sections. In the revised manuscript we will add a dedicated paragraph (or table) in the results section that reports (i) accuracy deltas before/after compression, (ii) operator-norm or Frobenius-norm approximation errors for the MPO fits, and (iii) performance versus bond dimension on the validation sets. We will also revise the abstract to reference these quantitative results, thereby grounding the “without degrading model performance” statement in concrete numbers. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper presents algorithmic contributions for MPO disentangling (variational and gradient-descent methods) followed by empirical validation on public MNIST and CIFAR-10 datasets. The compression of linear layers to MPO is described as a preprocessing step whose performance preservation is tested rather than defined into existence. No equations, fitted parameters, or self-citations are shown to reduce any reported accuracy or hybrid execution result to a tautology or input by construction. The methods remain independent of the target performance metric.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach depends on the domain assumption that an MPO can faithfully represent the target linear layer. No explicit free parameters or invented entities are named in the abstract.

axioms (1)

domain assumption The target linear layer can be represented as an effective matrix product operator without degrading model performance.
Invoked in the initial compression step described in the abstract.

pith-pipeline@v0.9.0 · 5712 in / 1342 out tokens · 50338 ms · 2026-05-18T18:26:22.437329+00:00 · methodology

discussion (0)

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Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

maximizing the overlap Tr(Mχ(QL M'χ' QR)) ... environment tensor Eg ... SVD update g' = V†U†

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Quantum-enhanced Large Language Models on Quantum Hardware via Cayley Unitary Adapters
quant-ph 2026-05 unverdicted novelty 8.0

Cayley unitary adapters executed on real quantum hardware improve LLM perplexity by 1.4% on Llama 3.1 8B with 6000 parameters and recover 83% of compression-induced degradation on SmolLM2.

Reference graph

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Normalize and encode the input data using AmplitudeorMPS Encoding

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