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Experimentally Extending Quantum Kernel Learning to Quantum Data by NMR

This paper experimentally demonstrates the superiority of quantum kernel learning over classical methods for processing quantum data by successfully implementing and benchmarking the approach on a 3-qubit NMR platform for regression, classification, and operator discrimination tasks.

Original authors: Vivek Sabarad, Vishal Varma, T. S. Mahesh

Published 2026-08-18

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Vivek Sabarad, Vishal Varma, T. S. Mahesh

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Machine learning has become a powerful tool for finding patterns in vast amounts of information, from predicting weather to recognizing faces. However, when the data itself comes from the quantum world—the realm of atoms and subatomic particles where the rules of physics behave very differently—standard computer programs often struggle. They are designed to handle classical information, like numbers and images, but they cannot easily process the complex, invisible structures of quantum systems without first translating them into a form that loses much of their unique character. Scientists have long theorized that if we could teach machines to learn directly from quantum data, using the natural properties of quantum systems to do the work, we might unlock a new level of efficiency. This idea sits at the intersection of two rapidly advancing fields: quantum information science, which studies how to use quantum mechanics for computation, and statistical learning, which teaches computers to make predictions. The challenge has been moving from theory to practice, proving that a machine can actually learn from quantum inputs without needing a massive, error-free quantum computer to do it.

A team of researchers at the Indian Institute of Science Education and Research in Pune has now taken a significant step toward making this a reality. They successfully demonstrated a method called quantum kernel learning on a physical quantum device, showing that it can not only process standard data but also learn directly from quantum operators, which are the mathematical descriptions of how quantum systems change. Their work, conducted using a liquid-state nuclear magnetic resonance system—a technique that uses the magnetic properties of atomic nuclei in molecules to act as a quantum processor—proves that these machines can identify complex patterns in quantum data that traditional methods miss. By training their system on a specific set of quantum operations, they showed it could correctly classify new, unseen operations with high accuracy, even when those new operations were completely different from the ones it had studied. This suggests that the method captures the deep, underlying structure of the quantum world, allowing it to generalize its knowledge far beyond its training.

To understand what the team achieved, it helps to look at how they approached the problem in two stages. First, they tested their system on classical data, which is the kind of information we use every day, such as numbers representing a curve or a set of points on a graph. They used a molecule called trimethyl phosphite, which contains ten atomic nuclei that act as qubits, the basic units of quantum information. In this setup, one central nucleus is connected to nine others, forming a star-like shape. The researchers encoded classical numbers into the quantum system by applying specific magnetic pulses that rotated the spins of these nuclei. They then measured how the system responded to pairs of these numbers to create a "kernel," which is essentially a measure of how similar two data points are in a high-dimensional space. The results were impressive: the system could predict the shape of a sine wave and a complex polynomial curve with over 98% accuracy, and it could separate two different types of data points in a two-dimensional space with very few errors. This confirmed that their quantum hardware could reliably perform standard machine learning tasks.

The true breakthrough, however, came when they moved beyond classical data to tackle the much harder problem of learning from quantum data itself. In this second phase, the input to their machine was not a number, but a quantum operator—a description of a transformation that changes the state of a quantum system. Specifically, they wanted to know if the system could distinguish between two types of transformations: those that create "entanglement," a unique quantum connection where particles become linked regardless of distance, and those that do not. To do this, they used a different molecule, dibromofluoromethane, which provided a three-qubit system arranged in a double-layered star structure. They fed the machine various quantum operators, some of which were known to create entanglement and others that did not, and asked the system to learn the difference.

The results of this experiment were striking and highlighted a key advantage of learning directly from quantum data. When the researchers simulated a traditional approach, where they tried to describe the quantum operators using a set of classical parameters, the system failed to generalize. It could only recognize the patterns it had seen during training and failed completely when presented with new data outside that specific range. In contrast, the quantum kernel learning method, which processed the operators directly without converting them into classical numbers, achieved an accuracy of 85% on the experimental data and 94% in numerical simulations. Crucially, the model was trained only on operators from one half of the possible range of variations, yet it successfully identified entangling operators in the other half, a region it had never seen before. This ability to generalize suggests that the quantum kernel method is not just memorizing examples but is actually learning the fundamental symmetry and structure of the space of quantum operators.

This work demonstrates that quantum machines can learn from quantum inputs in a way that classical computers cannot easily replicate. The researchers showed that by using the native language of quantum mechanics, they could compare quantum operations directly, avoiding the need for expensive and complex measurement techniques that would otherwise be required to understand the data. While the experiment was conducted on a small scale using a specialized NMR platform, the findings provide a practical roadmap for how quantum computers might eventually be used to analyze and learn from other quantum systems. This could have profound implications for understanding complex materials, simulating high-energy physics, and developing new quantum technologies, as it offers a way to extract meaningful patterns from the quantum world without losing the very information that makes it unique. The study confirms that quantum kernel learning is a viable path forward, offering a tool that can see the hidden order in quantum data where other methods fall short.

Technical Summary: Experimentally Extending Quantum Kernel Learning to Quantum Data by NMR

Problem Statement
Quantum Machine Learning (QML) aims to leverage the structure of quantum systems for data processing. While theoretical frameworks suggest that learning from "quantum data" (data native to quantum systems, such as unitary operators) offers exponential benefits and superior generalizability compared to classical methods, experimental demonstrations have been limited. Previous experimental progress in Quantum Kernel Learning (QKL) has largely focused on the "QC" paradigm (using quantum systems to process classical data) or the "CQ" paradigm (using classical algorithms on quantum data, often requiring tomography or parameterization). A significant gap remains in the "QQ" paradigm: the direct application of quantum algorithms to quantum inputs without converting them into classical representations. The challenge lies in creating expressive, trainable models that can handle operator-valued inputs directly on native hardware, avoiding the exponential scaling costs of state tomography and the information loss associated with parameterization.

Methodology
The authors implement and benchmark QKL using a liquid-state Nuclear Magnetic Resonance (NMR) platform. The study is divided into two main phases:

Benchmarking on Classical Data (QC Paradigm):
- System: A 10-qubit star-topology register realized using a trimethyl phosphite molecule, where a central $^{31}$ P spin is coupled to nine magnetically equivalent $^{1}$ H spins.
- Encoding: Classical $d$ -dimensional input vectors are encoded into unitary operators $V(x_i)$ via a feature map involving collective rotations and a fixed entangling unitary ( $U_e$ ).
- Kernel Computation: The kernel $k(x_i, x_j)$ is defined as the Hilbert-Schmidt inner product of the resulting operators. In the NMR implementation, this is measured as the expectation value of a reference observable ( $I_z^C$ ) after applying the sequence $V^\dagger(x_j)V(x_i)$ to an initial state.
- Tasks: The system was tested on one-dimensional regression (sine and 7th-degree polynomial functions) and two-dimensional classification (Circles and Moons datasets).
Extension to Quantum Data (QQ Paradigm):
- System: A double-layered star-topology register realized using a 3-spin system ( $^{13}$ C, $^{1}$ H, $^{19}$ F) in dibromofluoromethane.
- Input: The input data consists of unitary operators $U_i \in SU(4)$ acting on a two-qubit subspace. These operators are not necessarily parameterized by classical variables; they can be arbitrary quantum gates.
- Feature Map: The encoding unitary is defined as $V_i = U_i^\dagger U_e U_i$ , where $U_e$ is a fixed entangling gate. This allows the kernel to compare operators directly.
- Task: Binary classification of unitary operators into "entangling" and "non-entangling" categories based on the logarithmic negativity of the resulting state.
- Comparison: The authors compare the QQ paradigm against the CQ paradigm, where a classical Gaussian kernel is trained on the classical parameters ( $\theta, \alpha$ ) of the unitaries.

Key Contributions

Experimental Realization of QQ Paradigm: The paper provides the first experimental demonstration of QKL operating directly on quantum inputs (unitary operators) using an NMR processor, bypassing the need for classical parameterization or full state tomography.
Operator Classification: The authors successfully classify entangling versus non-entangling operators, a non-trivial problem in quantum information, using a quantum kernel.
Generalizability via Symmetry: The study demonstrates that QKL can generalize to unseen data regions by exploiting the symmetry structure of the operator space, a capability that classical kernels fail to achieve when applied to quantum data.
Scalable Protocol: The work establishes a protocol for computing kernels on native quantum hardware that avoids the exponential resource costs associated with reconstructing quantum states.

Results

Classical Data Performance:
- Regression: The NMR QKL achieved 99.1% RMS accuracy on a sine function regression with 15 training points and 98.8% on a 7th-degree polynomial with 42 points.
- Classification: The system successfully classified Circles and Moons datasets with hinge loss values of 0.15 and 0.08, respectively, confirming the reliability of the kernel for standard ML tasks.
Quantum Data Performance:
- CQ Paradigm Failure: When a classical Gaussian kernel was trained on the classical parameters of the unitaries (trained only on the "lower half" of the parameter space), it failed to generalize to the "upper half," resulting in ~50% accuracy.
- QQ Paradigm Success (Simulation): Numerical simulation of the QKL on the 3-spin system achieved 94% accuracy, correctly classifying operators from the untrained upper half of the parameter space. Accuracy saturated near 99% with larger training sets.
- QQ Paradigm Success (Experiment): The experimental NMR implementation achieved 85% accuracy. Crucially, it correctly identified entangling regions in both the trained (lower) and untrained (upper) parameter spaces, demonstrating the ability to capture the underlying structure of the quantum operator space.

Significance and Claims
The authors claim that this work confirms the superiority of QKL over classical methods for processing quantum data. By operating natively on quantum inputs, QKL avoids the "barren plateau" issues associated with variational circuits and the information loss inherent in classical parameterization. The results highlight the method's ability to:

Capture the inherent structure of quantum space (specifically symmetry in operator space).
Extend its domain of operation beyond the training data, achieving generalization that classical kernels cannot.
Provide a practical route for comparing operators on native quantum hardware without expensive tomography.

The paper positions this as a step toward general QML with potential near-term applications in quantum many-body physics, high-energy physics, and emerging quantum technologies, noting that QKL is realizable on current Noisy Intermediate-Scale Quantum (NISQ) platforms.

Technical Summary: Experimentally Extending Quantum Kernel Learning to Quantum Data by NMR

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