Experimental Machine Learning with Classical and Quantum Data via NMR Quantum Kernels

arXiv:2412.09557 · quant-ph, cs.LG, physics.app-ph · Submitted 2026-08-15 · Read on arXiv

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Introduction to the show: ident: AI Radio. Generated commentary on the latest Artificial Intelligence papers.

Tom: Next we'll be talking about the paper "Experimental Machine Learning with Classical and Quantum Data via NMR Quantum Kernels".

Jane: The paper was written by Vivek Sabarad, Vishal Varma and T. S. Mahesh from Indian Institute of Science Education and Research Pune.

Tom: Stay tuned as we take you through the paper and discuss its implications.

Title: Tom: Welcome back to the show, everyone! Today we're digging into a paper that's got the rather grand title "Experimental Machine Learning with Classical and Quantum Data via NMR Quantum Kernels." Jane, I have to say, just reading that title out loud makes me feel like I need a physics degree.

Jane: Ha! It does sound intimidating, but honestly, it's less scary than it looks. Let me break it down. "NMR" is nuclear magnetic resonance — the same technology behind MRI machines in hospitals. The researchers are using it to build a tiny quantum computer out of molecules.

Tom: So instead of a chip with wires and transistors, they're using actual molecules as their computer?

Jane: Exactly! And "quantum kernels" — that's the clever part. In regular machine learning, a kernel is a way of measuring how similar two pieces of data are, but in a really high-dimensional space where patterns become easier to spot. Think of it like sorting photos by color — you might map each photo to a point in a three dee color space, and then similar colors cluster together.

Tom: Okay, so they're using molecules to compute these similarity measurements?

Jane: Right. And the "quantum" part means the space they're mapping into is exponentially huge — way bigger than anything a classical computer could handle directly. The paper shows they can do real machine learning tasks with this setup.

Tom: And they're not just doing classical data — the title says "classical and quantum data." What's the difference there?

Jane: Classical data is your normal stuff — numbers, coordinates, measurements. Quantum data is different; it's things like quantum states or quantum operations themselves. And that's where it gets really exciting, because classical computers struggle to even represent quantum systems as they get bigger.

Tom: So this paper is saying, "Hey, we can use quantum systems to learn about other quantum systems"?

Jane: Precisely. And they actually built it and tested it, which is the impressive part. We're not talking about theory here — they ran real experiments.

Tom: I love that. Let's get into the actual experiments in the next segment, because I want to know how they pulled this off.

Summary: Jane: So, Tom, let's talk about what these researchers actually did. They used a molecule called trimethyl phosphite, which has this beautiful star-shaped structure — one central phosphorus atom surrounded by nine hydrogen atoms.

Tom: A ten-qubit quantum computer made from one molecule. That's wild.

Jane: It is! And they used it to do two classic machine learning tasks. First, regression — that's where you try to predict a continuous value. They trained their system to learn a sine wave and a polynomial function, and it worked beautifully. We're talking under one point two percent error.

Tom: Okay, so it can learn curves. What about classification — sorting things into categories?

Jane: They did that too. They took two-dimensional datasets — one shaped like concentric circles, another shaped like two interlocking moons — and trained a support vector machine using their quantum kernel. The classification worked with very low error rates.

Tom: And this is all using the quantum kernel they extract from the NMR system?

Jane: Exactly. The kernel is computed by encoding data into quantum states and then measuring how much those states overlap. It's like asking the molecule, "How similar are these two data points?" and reading the answer off the spin of the central atom.

Tom: That's clever, but here's my question — is this actually better than just using a classical kernel on a regular computer?

Jane: For the classical data tasks, probably not dramatically better. But that's not the real point of the paper. The real point comes when they switch to quantum data.

Tom: Right, the entanglement classification. That's the part I found fascinating.

Jane: Me too. They wanted to classify quantum operations — unitary transformations — based on whether they entangle a quantum state or not. Entanglement is this weird quantum property where two particles become linked, so measuring one instantly affects the other, no matter how far apart they are.

Tom: And they trained their quantum kernel to recognize which operations create entanglement?

Jane: Yes. And here's the kicker — they trained it on operations from only one region of the parameter space, but it correctly classified operations from a completely different region that it had never seen. The classical kernel they compared it against completely failed at that.

Tom: So the quantum kernel actually understood something deeper about the structure of the problem?

Jane: That's exactly what the paper suggests. It's not just memorizing patterns from training data — it's capturing the actual physics of entanglement.

Tom: That's a big deal. Let's talk about what this means for the future in the next segment.

Improvements: Tom: Jane, before we move on, I want to bring in some of our regular guests. Lu, you're the AI researcher — what do you make of this quantum kernel approach?

Lu: I think the most exciting thing is the generalization ability. The fact that the quantum kernel could classify entangling operations in a region of parameter space with zero training points — that's not something you usually see. Classical kernels tend to interpolate, not extrapolate, especially in high-dimensional spaces.

Jane: So the quantum kernel is doing something genuinely different from a classical kernel?

Lu: It seems that way. The kernel is measuring similarity directly in the space of quantum operations, not in the space of parameters. That's a fundamentally different notion of similarity, and it turns out to be the right one for this task.

Meng: From an engineering standpoint, I'm curious about scalability. They used a ten-qubit system for the classical data tasks and a three-qubit system for the quantum data task. How does this scale?

Jane: That's a fair question. The paper acknowledges that scaling up is a challenge. But the key point is that the kernel computation itself — the part that's hard classically — scales with the size of the quantum system, not the size of the data.

Meng: So for a bigger quantum computer, you'd get a more powerful kernel?

Jane: In principle, yes. The feature space grows exponentially with the number of qubits. That's the promise.

Lu: And there's another improvement the paper hints at — they only used one arm of their double-layered star register for the experimental quantum kernel. The full register would give an even richer kernel.

Tom: Wait, they didn't use the full system for the quantum data task?

Jane: Right. The full double-layered star would have more qubits involved, but they validated the concept using a simpler molecule — dibromofluoromethane — to simulate one branch of that structure. The accuracy dropped from ninety-five percent in simulation to ninety-one percent in experiment, which is pretty good given experimental imperfections.

Meng: What about the eighty-four percent accuracy on random, non-parameterized unitaries? That seems like the most practically relevant result to me.

Jane: Why do you say that?

Meng: Because in the real world, you don't always have nice parameterized operations. If you're trying to classify arbitrary quantum operations — say, to verify that a quantum device is doing what it should — you need to handle anything. The fact that it got eighty-four percent on completely random unitaries suggests it's learning something real about entanglement, not just memorizing the training set.

Lu: And that's the direction I'd push — using this for quantum device verification and error detection. If you can train a kernel to recognize entangling operations, you could potentially train it to recognize faulty operations too.

Tom: So this isn't just academic — it could actually help build better quantum computers?

Lu: That's the hope. And it's a concrete, near-term application.

Conclusion: Tom: Alright, we're wrapping up our discussion of "Experimental Machine Learning with Classical and Quantum Data via NMR Quantum Kernels." Jane, give us the final takeaway.

Jane: The big picture is this — quantum kernels work, and they work for both classical and quantum data. The team at IISER Pune showed that you can use a molecular NMR system to compute kernels that successfully handle regression, classification, and even the quantum-specific task of entanglement classification.

Tom: And the generalization result — that's the part I'll remember. The quantum kernel recognized entangling operations in a region where it had zero training data, while the classical kernel completely missed it.

Jane: Exactly. That's the strongest evidence that quantum kernels offer something classical kernels can't — a notion of similarity that respects the actual physics of the system.

Lu: I'd add that the paper is careful and honest about limitations. They report the accuracy drop from simulation to experiment, and they discuss the challenges of scaling. That's the kind of rigor we need in this field.

Meng: From my side, the practical impact is clear. Quantum device characterization, error detection, maybe even quantum communication protocol design — these are all areas where this approach could be applied in the near term.

Tom: And the future work they mention — automated kernel construction, handling non-unitary quantum maps — those are exciting directions. This feels like one of those papers that opens a door rather than closing a debate.

Jane: Well said. It's a solid experimental demonstration that quantum kernels are more than a theoretical curiosity. They're a practical tool.

Tom: Thanks to everyone who joined us today — Lu, Meng, and of course our in-house language model Lalam, who's been quiet this episode but is always thinking.

Lalam: I have been listening, and I'll just say this — the cultural impact of making quantum machine learning experimentally accessible is that it moves us from "can we?" to "how well?" That's a shift that matters.

Tom: Love that. Alright, that's a wrap on this paper. Next up, we've got something on quantum error correction that I think is going to be a wild ride. Stay tuned!

Vivek Sabarad, Vishal Varma, T. S. Mahesh

Indian Institute of Science Education and Research Pune

quant-ph, cs.LG, physics.app-ph

Submitted: 2026-08-15

Updated: 2026-08-18

Comments: 10 pages, 6 figures

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 66/100

Key concepts

NMR
Nuclear Magnetic Resonance is the technology used in MRI machines. Researchers use NMR to build a tiny quantum computer using molecules.
Quantum Kernels
In machine learning, a kernel measures data similarity in high-dimensional space. Quantum kernels use quantum states to measure this similarity, allowing them to map data into exponentially large spaces where patterns are easier to spot.
Entanglement Classification
This is a quantum task where the researchers trained the kernel to recognize which quantum operations create entanglement. The kernel successfully classified operations from regions of parameter space it had never seen before, unlike classical kernels.

Terminology

Summary

Summary

This paper experimentally demonstrates quantum kernel methods for both classical and quantum machine learning tasks using a nuclear magnetic resonance (NMR) platform. The authors implement quantum kernels on a 10-qubit star-topology register (trimethyl phosphite, where 31P and 1H spins form central and ancillary qubits) and extend the approach to a double-layered star configuration for handling quantum data.

Theoretical framework: Quantum kernel methods extend classical kernel techniques by mapping input data into the operator space of a quantum system. The quantum feature map encodes each data point xi into a unitary transformation U(xi) acting on a reference operator A0: A(xi) = U(xi)A0U†(xi). The quantum kernel is computed using the Frobenius inner product: k(xi, xj) = Tr[A(xi)A(xj)]. For the NMR implementation, the kernel is extracted as kNMR(xi, xj) ∝ Tr(U†(xj)U(xi)ρCeqU†(xi)U(xj)IzC), which corresponds to measuring the z-magnetization of the central spin after applying the unitaries sequentially.

Classical data experiments: For one-dimensional inputs, the encoding unitary is U(xi) = e-ixiIzCUe e ixiIzC, where Ue is an entangling unitary generating multiple quantum coherences. The resulting kernel is a function of the difference between input points. Using this experimentally obtained kernel with kernel ridge regression, the authors successfully perform:

  • Regression of a sine function over one period with 15 training points (RMS error 0.88%)

  • Regression of a seventh-degree polynomial y = (x-3)(x-2)(x-1)x(x+1)(x+2)(x+3) with 40 training points (RMS error 1.15%)

For two-dimensional inputs, the encoding unitary is constructed as a product over dimensions: U(xi) = ∏j e-ixi(j)IzUe e ixi(j)Iz. Using a support vector machine (SVM) classifier with the experimental kernel, the authors perform:

  • Classification of a circular dataset (hinge loss 0.15)

  • Classification of a moons dataset (hinge loss 0.08)

Quantum data experiments: The authors propose an extended quantum kernel for handling non-parametrized unitary operator inputs. The kernel is k(Ui, Uj) = Tr(A(Ui)A(Uj)) with A(Ui) = V(Ui)A0V†(Ui) and V(Ui) = UiUeUi†. The register uses a double-layered star configuration where the central qubit C connects to A qubits, and each A qubit connects to a B qubit. The task is binary classification of whether a given unitary transformation entangles a thermal equilibrium state ρ0 = e-βHZ/4 of two spin-1/2 particles.

Numerical results show the quantum kernel achieves 95% classification accuracy on 196 test unitaries after training on only 30 operators from the lower part of the parameter space. Critically, the quantum kernel demonstrates extrapolation ability: it correctly identifies the upper entangling zone even though no training points exist there. In contrast, a classical Gaussian kernel trained on the same parameter space (θ, α) completely fails to recognize the upper entangling zone. The accuracy saturates at 95% both as a function of training set size and delay time τ in the entangling unitary.

Experimental validation uses the DBFM (Dibromofluoromethane) molecule to simulate one arm of the double-layered star register. The experimental classification achieves 91% accuracy, with the reduction attributed to imperfections in the molecule's realization of the star topology and pulse imperfections. Additionally, the quantum kernel (trained on the same set) correctly classifies 84% of 100 non-parameterized random unitary operators, demonstrating its ability to handle quantum data directly without parameterization.

The authors conclude that quantum kernels can effectively perform classical machine learning tasks (regression and classification) and, more importantly, can handle quantum data for quantum tasks such as entanglement classification, showing superior generalization compared to classical kernels. They suggest future directions including automated kernel construction approaches and encoding non-unitary quantum maps.

Improvements for AI systems

Based on the paper, here are the specific improvements I can make to AI systems and what the improved systems can do:

  • Improvement: Implement the proposed double-layered star-topology quantum kernel that accepts raw unitary operators as inputs, without requiring classical parameterization.

  • What it can do: Classify arbitrary quantum operations (entangling vs. non-entangling) directly in operator space, even for unitaries never seen during training and without any classical feature extraction.

  • Improvement: Replace classical kernels (e.g., Gaussian on parameters) with the quantum kernel that computes similarity via Frobenius inner products of encoded operators.

  • What it can do: Achieve 95% accuracy on entanglement classification when trained only on a subset of parameter space, correctly identifying entangling regions with zero training points there—a task where classical kernels fail completely (0% recognition of the upper entangling zone).

  • Improvement: Use the kernel to classify 100 random, non-parameterized unitary operators without any classical representation.

  • What it can do: Correctly classify 84% of these unseen random unitaries, demonstrating the kernel's ability to generalize beyond any parametric family.

  • Improvement: Adopt the experimental protocol that measures only the central qubit's z-magnetization after applying data-dependent unitaries, exploiting the symmetry of the kernel to reduce measurement overhead.

  • What it can do: Extract reliable kernel values with RMS errors below 1.15% for regression and hinge losses of 0.08–0.15 for classification, even on a noisy 10-qubit NMR platform.

  • Improvement: Implement the encoding unitary U(x) = exp(-ixI z) U e exp(ixI z) that maps classical data into high-order multiple-quantum coherences, expanding the effective feature space exponentially with qubit count.

  • What it can do: Perform one-dimensional regression on a 7th-degree polynomial and sine functions with <1.2% RMS error, and two-dimensional classification on circular and moons datasets with high accuracy, using only 15–42 training points.

  • Improvement: Use the proven symmetry k(x i(1), x i(2), x j(1), x j(2)) = k(x i(1)-x j(1), x i(2)-x j(2), x i(2)-x i(1), 0) to reduce the number of independent kernel evaluations from O(N2) to O(N) per dimension.

  • What it can do: Compute the full kernel matrix with significantly fewer experimental runs, making the method practical for larger datasets on resource-constrained quantum hardware.

The improved AI system can:

  • Learn from quantum data (unitaries, states) without classical preprocessing.

  • Generalize to unseen regions of input space that are categorically different from training data.

  • Perform both classical (regression, classification) and quantum (entanglement detection) tasks with a single unified kernel framework.

  • Operate on noisy, near-term quantum hardware with high accuracy and minimal measurement overhead.

Abstract

Kernel methods map data into high-dimensional spaces, enabling linear algorithms to learn nonlinear functions without explicitly storing the feature vectors. Quantum kernel methods promise efficient learning by encoding feature maps into exponentially large Hilbert spaces inherent in quantum systems. In this work, we implement quantum kernels on a 10-qubit star-topology register in a nuclear magnetic resonance (NMR) platform. We experimentally encode classical data in the evolution of multiple quantum coherence orders using data-dependent unitary transformations and then demonstrate one-dimensional regression and two-dimensional classification tasks. By extending the register to a double-layered star configuration, we propose an extended quantum kernel to handle non-parametrized operator inputs. Specifically, we set up a kernel for the classification of entangling and non-entangling operations and then validate this kernel first numerically by computing it on a double-layered star register and then experimentally by computing it on a three-qubit NMR register. Our results show that this kernel exhibits an ability to generalize well over unseen data. These results confirm that quantum kernels possess strong capabilities in classical as well as quantum machine learning tasks.

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