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Comment on 'Supervised quantum machine learning models are kernel methods'
Rajiv Krishnakumar
TL;DR
The note identifies small errors in Appendix A’s proof that do not affect Theorem 1’s validity but matter for explicitly computing a quantum kernel’s Fourier representation. It supplies a corrected derivation and shows that a separate error in the cosine-kernel example cancels the first, yielding the correct result by coincidence.
Problem
Small errors in Appendix A do not invalidate Theorem 1 but can affect explicit computation of a quantum kernel’s Fourier representation.
Method
The note rederives the quantum kernel’s explicit expression using the corrected equations and combines unitary matrices into redefined W matrices.
Results
The corrected derivation demonstrates how the quantum kernel is computed, while the cosine-kernel example’s correct answer results from a second error canceling the error in Equation (A5).
Takeaways & Limitations
The appendix’s computational details require correction, even though the theorem remains valid and the worked cosine-kernel example coincidentally produces the right result.
Takeaways & Limitations
The corrections include reversed indices, an extra delta factor from an overlapping summation term, and conjugation errors in Equations (A7) and (A8).
Abstract
from arXiv · showhide
We identify a few small errors in the proof of Theorem 1 in M. Schuld, 'Supervised quantum machine learning models are kernel methods' (arXiv:2101.11020v2), Appendix A. These do not affect the validity of the theorem, but do matter if one wants to use the appendix to explicitly compute the Fourier coefficients of a quantum kernel. We give the corrected derivation, and additionally show that the paper's worked cosine-kernel example contains a second, unrelated error that happens to cancel the first, thereby coincidentally yielding the correct result.
1 Summary of note
The note identifies three small errors in the Appendix A proof of Theorem 1. They do not invalidate the theorem, but they affect explicit computation of quantum-kernel Fourier coefficients.
- 1 Summary of note: Three Appendix A errors concern reversed indices, missing Kronecker-delta factors, and incorrect complex conjugations.The corrections apply to Equations (A5)–(A8).
- 1 Summary of note: The indices in Equation (A5) should be reversed because conjugate-transposing W (i) and multiplying by a row vector changes the index order.
- 1 Summary of note: Equation (A5) requires δjN,kN, while Equations (A7) and (A8) require the additional factor δjN,kNδhN,lN.The same δjN,kN factor also propagates to Equation (A6).
- 1 Summary of note: In Equations (A7) and (A8), w∗h should be wh and wl should be w∗l.
2 Derivation of the Fourier representation of quantum kernels
The corrected derivation expands the time-evolution encoding into matrix elements and uses the resulting expression to compute the quantum kernel. It incorporates the identified corrections into the explicit Fourier representation.
- 2 Derivation of the Fourier representation of quantum kernels: The kernel calculation expands each encoding layer into eigenvalue-dependent phase factors and sums over intermediate indices.
- 2 Derivation of the Fourier representation of quantum kernels: The derivation absorbs arbitrary unitary matrices V (i) and V (i)† into neighboring W (i) matrices before expanding the kernel.
- 2 Derivation of the Fourier representation of quantum kernels: The corrected expansion includes the overlapping-index factor δjN,kN when multiplying the matrix expressions.
- 2 Derivation of the Fourier representation of quantum kernels: The resulting derivation demonstrates how to compute the quantum kernel with the corrections applied to the original equations.
3 Comments on the cosine kernel example
The cosine-kernel example reaches the correct result despite an error in Equation (A5). A second error in the example reverses the sign and accidentally cancels the first mistake.
- 3 Comments on the cosine kernel example: The example is rederived using Equation 20 with N = 1 and d = 2, then compared step by step with Example A.1.
- 3 Comments on the cosine kernel example: The second error therefore inadvertently corrects the preceding Equation (A5) error, yielding the correct cosine kernel coincidentally.
- 3 Comments on the cosine kernel example: The derivation contains a term corresponding to requiring δj1k1, producing an expression similar to Equation (A13).
- 3 Comments on the cosine kernel example: Using the replacement indicated in the paper reaches Equation (A16), but using the correct replacement produces the wrong sign and answer.