⚛️ quantum physics

Engineered Learning Problems on Noisy Quantum Processors

This paper demonstrates that engineered classification problems on noisy quantum processors with up to 45 qubits can outperform classical methods, particularly when enhanced by error mitigation, though surpassing state-of-the-art simulation techniques like matrix product states remains a significant challenge.

Original authors: Esteban Payares, Nadia Milazzo, Ghada Ben Slama, Thomas Cope

Published 2026-11-02

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Esteban Payares, Nadia Milazzo, Ghada Ben Slama, Thomas Cope

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

In the race to build machines that can solve problems beyond the reach of today's computers, scientists are exploring a field where the rules of physics themselves become the processor. This is the realm of quantum machine learning, a discipline that attempts to use the strange, counterintuitive behaviors of subatomic particles to recognize patterns and make predictions. The core idea relies on a concept called a "kernel," which is essentially a mathematical tool for measuring how similar two pieces of data are. In a classical computer, this tool works by comparing data points in a familiar way. In a quantum computer, the data is first translated into the language of quantum mechanics, where it exists in a complex, high-dimensional space that is difficult for classical machines to navigate. The hope is that by performing this translation, the quantum machine can find hidden connections in the data that classical methods simply cannot see. However, a persistent question has hung over the field: can these noisy, imperfect quantum devices actually outperform the most powerful classical supercomputers, or are they merely struggling under the weight of their own fragility?

A team of researchers set out to answer this question by designing a specific, artificial challenge where a quantum computer could theoretically hold an advantage. They did not try to solve a real-world problem like predicting the weather or diagnosing a disease, because the classical methods for those tasks are too strong and too well-understood. Instead, they created a custom puzzle. They took a standard set of images—pictures of clothing like t-shirts and dresses—and used a quantum circuit to transform the data into a new form. Then, they used the results of that transformation to assign labels to the images, effectively creating a test where the "correct" answers were defined by the quantum machine's own unique perspective. The goal was to see if a quantum computer could predict these labels better than a classical computer trying to solve the same puzzle. To ensure the results were genuine and not just a fluke of a specific machine's quirks, they ran the experiment on two different quantum processors, one with 54 qubits and another with 20, and even tested the same problem on different sections of the same chip to see if the results held up across different noise profiles.

The researchers found that when the quantum circuits were kept relatively simple, the quantum machine did indeed outperform the classical rivals. It successfully predicted the labels with a high degree of accuracy, while the best classical models struggled to keep up. This gap in performance was clear and reproducible across the different hardware setups. However, as the researchers made the quantum circuits deeper and more complex to handle more data, the advantage began to fade. The noise inherent in current quantum hardware—the inevitable errors that creep in as the machine operates—started to distort the results. At a certain point, with a circuit depth equivalent to ten steps of a specific calculation, the quantum machine's performance dropped so low that it could no longer beat the classical models. The quantum computer was no longer winning; it was just as confused as the classical one.

The team then turned to a set of techniques known as error mitigation, which are designed to clean up the noisy data after the experiment is run, effectively subtracting the known errors to reveal the true signal. They tested several methods, including one that tries to guess what the result would have been if the machine had no noise at all, and another that uses a series of rapid pulses to stabilize the qubits. They discovered that not all error correction methods were created equal. One technique, which simply adjusted the final numbers based on how the machine reads them, failed to help. Another method, which tried to extrapolate the noise-free result, managed to restore the performance gap, but only by making the puzzle itself harder for the classical computer to solve, rather than by making the quantum computer significantly better. The most effective technique, however, was a method that actively suppressed the noise during the process. This approach not only restored the quantum machine's advantage but actually improved its ability to solve the puzzle, bringing its performance much closer to what a perfect, noise-free machine would achieve.

Despite these successes, the researchers were careful to clarify what their work did not prove. They demonstrated that quantum hardware can outperform classical machine learning models on a specific, engineered task, but they also showed that this advantage is not absolute. When they simulated the quantum process using a powerful classical algorithm known as a matrix product state, which is designed to mimic quantum systems, the classical simulation performed just as well as the noisy quantum hardware. This suggests that the current quantum advantage is fragile and depends heavily on the specific type of problem and the level of noise. The classical simulation was able to keep up because the quantum circuits used in the experiment were not complex enough to break the limits of classical computing. The researchers concluded that while they have shown a path where quantum machines can win, the ultimate goal of beating the best classical simulation methods remains a significant hurdle. The path forward requires designing problems that are not just hard for classical computers to simulate, but impossible for them to mimic, even with the most advanced algorithms available today.

The study serves as a realistic map of the current landscape of quantum computing. It confirms that quantum machines can do things classical ones cannot, but only under carefully controlled conditions and with the help of error correction. It also highlights that the field is still in a phase where the "noise" of the machine is often the deciding factor between victory and defeat. By showing that different error mitigation techniques have vastly different effects, the work provides a roadmap for future experiments, suggesting that simply running a quantum algorithm is not enough; one must also choose the right tools to clean up the mess that the machine makes. The researchers did not claim to have solved the problem of quantum advantage, but they did provide strong evidence that it is an achievable goal, provided the problems are chosen with enough care to outsmart the classical simulations that are currently the biggest obstacle.

Technical Summary: Engineered Learning Problems on Noisy Quantum Processors

Problem Statement
Despite theoretical expectations of quantum advantage, demonstrating that quantum hardware outperforms classical methods in machine learning remains elusive. Rigorous speed-ups require fault-tolerant devices, while current Noisy Intermediate-Scale Quantum (NISQ) approaches, such as variational quantum circuits, face skepticism regarding their empirical performance. Furthermore, large-scale studies suggest quantum kernels often fail to outperform classical "one-size-fits-all" approaches on real-world data. This work addresses the challenge of establishing a measurable performance gap between quantum and classical models on current hardware. Rather than applying quantum kernels to industrial problems, the authors adopt an "engineered learning problem" approach: constructing a classification task specifically designed to maximize the predictive difference between a quantum kernel and a specific classical kernel.

Methodology
The study implements a protocol introduced by Huang et al. [23] on IQM superconducting quantum processors (Emerald with 54 qubits and Garnet with 20 qubits).

Quantum Kernel Construction: The authors utilize Projected Quantum Kernels (PQK). Data is encoded into a quantum circuit via the Trotterized time evolution of a 1D inhomogeneous XXX Heisenberg Hamiltonian. Instead of using the full fidelity (which suffers from exponential concentration), the kernel is constructed from local observables (single-qubit Pauli expectations), effectively creating a Radial Basis Function (RBF) kernel on a $3n$ -dimensional feature space.
Label Engineering: To create a task where the quantum model is theoretically superior, "referee" labels are generated by maximizing the complexity of a classical model (defined by an RBF kernel) while constraining the quantum model's complexity to a minimum. This is formulated as a generalized eigenvalue problem to maximize the "geometric difference" ( $g$ ) between the quantum and classical kernels.
Experimental Setup: Experiments were conducted on $n = 15, 30,$ and $45$ qubits with varying Trotter steps ( $T = 3$ to $10$). To ensure results were not artifacts of fitting to specific device noise, the authors tested reproducibility by running the "challenger" (prediction) on a different Quantum Processing Unit (QPU) or a different qubit chain on the same chip with a distinct noise profile.
Error Mitigation: Several mitigation techniques were evaluated to recover performance lost to noise:
- Zero Noise Extrapolation (ZNE): Using global unitary folding and Richardson extrapolation.
- Readout Error Mitigation (REM): Inverting a confusion matrix for single-qubit readout errors.
- Adaptive Pauli Twirling (APT): A new technique exploiting discarded information from standard twirling.
Classical Benchmarks: Performance was compared against a suite of classical circuit-agnostic models (e.g., RBF, Linear, Polynomial, MLP, Random Forest) and, crucially, a classically simulated quantum kernel using Matrix Product States (MPS).

Key Results

Performance Gap and Noise: On $n=15$ qubits, the quantum challenger initially outperformed classical baselines by a significant margin (e.g., +0.30 accuracy gap at $T=3$ ). However, as circuit depth increased (to $T=10$ ), accumulated noise degraded the quantum kernel's fidelity to the ideal, causing the performance gap to shrink and eventually disappear (no significant gap at $T=10$ ). For larger systems ( $n=30, 45$ ), the gap persisted across all depths but narrowed less dramatically than at $n=15$ .
Effectiveness of Error Mitigation:
- ZNE: Successfully restored a performance gap on hardware-generated labels. However, analysis revealed this was primarily due to the "label effect"—ZNE altered the labels engineered by the referee, making the task harder for classical models, rather than significantly improving the quantum challenger's own kernel fidelity.
- REM: Proved largely ineffective, failing to alter the performance gap or improve accuracy on ideal labels.
- APT: The most effective technique. It suppressed the residual feature error directly, moving the hardware kernel significantly closer to the ideal kernel (increasing Centered Kernel Alignment). APT restored the performance gap by genuinely improving the quantum challenger's predictive capability, not just by changing the task difficulty.
Classical Simulation Limits: A classical challenger using an MPS approximation (with bond dimension $\chi=128$ ) achieved prediction accuracy comparable to the noisy quantum hardware challenger on the $n=45, T=10$ experiment. This demonstrates that for the specific embedding used, the problem remains within the reach of state-of-the-art classical simulation.
Generalization and Data Scaling: The authors extended the label construction to allow for new test points not included in the initial training set. While this reduced the magnitude of the performance gap, the quantum advantage persisted. Additionally, increasing the dataset size ( $N$ ) continued to improve the performance gap, with no observed optimal peak within the tested range ( $N \le 600$ ).

Significance and Claims
The paper claims to provide strong evidence that superior empirical performance of quantum hardware over all classical techniques is an achievable goal in quantum machine learning, provided the problem is carefully engineered. The work demonstrates that:

A performance gap can be established and reproduced across different quantum hardware units.
Error mitigation, specifically Adaptive Pauli Twirling, can revive this gap by correcting bias in features rather than just rescaling magnitudes.
The primary obstacle to claiming "quantum advantage" in this domain is not the hardware noise itself, but the ability of classical simulators (like MPS) to approximate the quantum kernel.

The authors explicitly state they do not claim to have demonstrated quantum advantage in the strict sense, as their specific embedding can be efficiently simulated classically. They conclude that future demonstrations of advantage must rely on embeddings that are classically hard to simulate, moving beyond current 1D time-evolution models to more complex dynamics that challenge classical tensor network methods. The work highlights that for engineered problems, the "best" classical adversary is often a circuit simulator, not a standard machine learning model.

Technical Summary: Engineered Learning Problems on Noisy Quantum Processors

More like this