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Comment on 'Supervised quantum machine learning models are kernel methods'

This paper corrects minor errors in the proof of Theorem 1 and the worked cosine-kernel example from Schuld's 2021 work on supervised quantum machine learning, clarifying that while the original theorem remains valid, the derivations needed for explicitly computing Fourier coefficients were flawed and coincidentally masked by a second error in the example.

Original authors: Rajiv Krishnakumar

Published 2026-08-27

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Rajiv Krishnakumar

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 rapidly evolving field of quantum computing, researchers are constantly searching for ways to make machines that harness the strange rules of the subatomic world to solve complex problems. One of the most promising applications is machine learning, where computers learn from data to make predictions or recognize patterns. A central tool in this process is the "kernel," a mathematical function that measures how similar two pieces of data are to one another. In the classical world, these functions are well-understood, but when scientists move these ideas into the quantum realm, the landscape becomes far more intricate. The core question facing the community is whether the powerful models built on quantum computers are truly new, unique entities, or if they are simply existing mathematical tools wearing a different disguise. Understanding this distinction is vital because it determines how researchers should design their algorithms and what kind of advantages they can realistically expect from quantum hardware.

A significant step in answering this question was taken in a previous study by Maria Schuld, which proposed that supervised quantum machine learning models are fundamentally a type of kernel method. This idea suggested that the complex operations performed by a quantum computer could be described using a specific mathematical structure known as a Fourier series, which breaks down complicated waves into simpler, repeating components. The original paper provided a proof to support this claim and included a worked example to show how the math worked in practice. However, a new note by Rajiv Krishnakumar has carefully examined that original proof and found that while the main conclusion remains correct, the path taken to get there contained a few small but important errors. These mistakes were not fatal to the overall theory, but they would lead to incorrect results if a researcher tried to use the original steps to calculate specific numbers for a real-world application.

Krishnakumar's work focuses on correcting the derivation of how these quantum kernels are built. The original proof attempted to show how a quantum circuit, which processes information by rotating and shifting quantum states, translates into a sum of waves. In doing so, the original author made a subtle mistake in the way the indices of the matrices were arranged, essentially swapping the order of certain terms in a way that would scramble the calculation. Additionally, the original derivation missed a necessary condition that ensures the mathematical terms overlap correctly, and it incorrectly handled the complex conjugate values of certain coefficients. These errors meant that if someone followed the original instructions to compute the specific frequencies of the quantum kernel, they would arrive at the wrong answer. The new note provides a corrected, step-by-step reconstruction of the proof, ensuring that the indices are in the right order and that all necessary mathematical terms are included to accurately describe the quantum system.

What makes this correction particularly interesting is a curious coincidence found in the original paper's example. The original author tried to demonstrate the theory using a specific case involving a cosine function, a common wave shape. In that example, the author made a second, unrelated error: they used the wrong value for one of the matrix elements. Remarkably, this second mistake happened to perfectly cancel out the first error. As a result, the final answer in the original paper was correct, even though the steps taken to get there were flawed. It was as if two wrong turns on a map accidentally led the traveler back to the right destination. Krishnakumar's analysis untangles this knot, showing that the correct result was achieved by luck rather than by the logic presented in the original text. By fixing both errors, the new derivation confirms that the quantum kernel does indeed follow the expected Fourier structure, but it does so with a mathematically rigorous path that will allow future researchers to compute these values accurately without relying on accidental cancellations.

This clarification reinforces the validity of the original theorem that quantum machine learning models are kernel methods, but it serves as a crucial reminder of the precision required in theoretical physics. The work does not overturn the field or suggest that quantum machine learning is fundamentally different from what was thought; rather, it tightens the mathematical foundation upon which these ideas rest. By providing the correct formulas and pointing out where the previous logic faltered, the note ensures that the community can move forward with a clear and accurate understanding of how quantum circuits translate into the language of kernels. For scientists building the next generation of quantum algorithms, this means they can now rely on a corrected map to navigate the complex terrain of quantum data, confident that their calculations will reflect the true behavior of the quantum systems they are trying to harness.

Problem
The paper addresses a technical discrepancy found in the proof of Theorem 1 within M. Schuld's work, "Supervised quantum machine learning models are kernel methods" [1]. While Theorem 1 correctly establishes that supervised quantum machine learning models with time-evolution encoding embeddings can be expressed as kernel methods with a specific Fourier series structure, the derivation provided in Appendix A of the original paper contains errors. These errors do not invalidate the theorem itself but compromise the utility of the appendix for explicitly computing the Fourier coefficients of a quantum kernel. Furthermore, the original paper's worked example of a cosine kernel, while yielding the correct final result, relied on a second, unrelated error that coincidentally canceled out the first.

Methodology
The author, Rajiv Krishnakumar, performs a rigorous re-derivation of the Fourier representation of quantum kernels starting from the definition of the time-evolution encoding embedding $S(x)$ . The methodology involves:

Step-by-Step Re-derivation: Explicitly calculating the expectation value $\langle 0| S(x')S(x) |0\rangle$ by expanding the unitary operators and diagonalizing the generating Hamiltonians.
Error Identification: Comparing the corrected derivation against the steps in Appendix A of [1] to pinpoint specific algebraic mistakes regarding matrix indices, Kronecker delta terms, and complex conjugation.
Case Study Analysis: Re-evaluating the specific cosine-kernel example from [1] using the corrected general formula (Equation 20) to demonstrate how the original derivation contained two distinct errors that masked each other.

Key Contributions
The paper identifies and corrects three specific errors in the original proof:

Index Reversal in Equation (A5): The terms involving unitary matrices $W^{(i)}$ had reversed indices. The correct form requires $W^{(1)}_{k_1 1} \dots W^{(N)}_{k_N k_{N-1}}$ rather than the originally stated $W^{(1)}_{1 k_1} \dots W^{(N)}_{k_{N-1} k_N}$ . This arises from the correct handling of the conjugate transpose when multiplying a vector from the left.
Missing Kronecker Delta Terms: The original derivation omitted necessary overlap terms. Specifically, Equation (A5) requires an extra $\delta_{j_N, k_N}$ term. Consequently, Equations (A6), (A7), and (A8) must include $\delta_{j_N, k_N}$ and $\delta_{h_N, l_N}$ to correctly account for the summation overlaps when taking the expectation of matrix multiplications.
Complex Conjugation Errors: In Equations (A7) and (A8), the terms $w^*_h$ and $w_l$ were incorrectly assigned. The derivation shows that due to the absolute value squared operation and the conjugation of specific matrix terms, the indices must be switched such that $w^*_h$ should be $w_h$ and $w_l$ should be $w^*_l$ .

Additionally, the paper provides the fully corrected derivation leading to Equation (20) and (21), which explicitly defines the Fourier coefficients $c_{st}$ with the proper index constraints and conjugation rules.

Results
The re-derivation confirms that the quantum kernel can indeed be written in the form $k(x, x') = \sum_{s,t \in \Omega} e^{-isx} e^{itx'} c_{st}$ , but with corrected expressions for the coefficients.

The corrected general formula (Equation 20) includes the necessary Kronecker delta $\delta_{j_N k_N}$ , ensuring the summation indices align correctly at the final layer of the circuit.
In the analysis of the cosine-kernel example, the author demonstrates that the original paper's Equation (A13) implicitly relied on the $\delta_{j_1 k_1}$ term. However, the subsequent substitution of matrix elements contained a sign error (using $V^*_{12}$ as indicated in the paper instead of the correct $V^*_{21}$ ). This second error inadvertently canceled the first, leading to the correct final cosine result despite the flawed intermediate steps. Using the corrected Equation (20) yields the correct cosine kernel $\cos^2(\frac{x-x'}{2})$ without relying on coincidental cancellations.

Significance
The significance of this note is primarily corrective and foundational. It clarifies the mathematical details required to explicitly compute the Fourier coefficients of quantum kernels, a task that was previously hindered by the errors in the original appendix. The author emphasizes that while the validity of Theorem 1 remains intact, the corrected derivation is essential for researchers who wish to use the appendix to perform explicit calculations of kernel representations. The note serves to ensure the reproducibility and mathematical rigor of the connection between supervised quantum machine learning models and kernel methods.

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