aiqbench analysis
This research presents a novel quantum gradient algorithm that efficiently computes derivatives of the logarithm-determinant, offering significant speedups over classical $O(N)$ methods for matrix inversion and pseudo-inverse calculations. By leveraging time-evolution operators to measure expectation values rather than full operator elements, the approach achieves polynomial scaling in relevant eigenvectors while maintaining applicability to near-term quantum hardware. The technique also establishes a direct pathway for enhancing kernel-based quantum machine learning models through more efficient derivative evaluations.
Quantum algorithm for the gradient of a logarithm-determinant
arXiv:2501.09413v4 Announce Type: replace Abstract: The logarithm-determinant is a widely-present operation in many areas of physics and computer science. Derivatives of the logarithm-determinant compute physically relevant quantities in statistical physics models, quantum field theories, as well as the inverses of matrices. A multi-variable version of the quantum gradient algorithm is developed here to evaluate the derivative of the logarithm-determinant. From this, the pseudo-inverse of a sparse-rank input operator may be determined efficiently. Measuring an expectation value of the quantum state--instead of all $N^2$ elements of the input operator--can be accomplished in $O(k/\varepsilon^2)$ time in the idealized case for $k$ relevant eigenvectors of the input matrix with precision $\varepsilon$. A practical implementation of the required operator will likely need $\log_2N$ overhead, giving an overall complexity of $O((k\log_2 N)/\varepsilon^2)$. The method applies widely and converges super-linearly in $k$ when the condition number is high. The best classical method we are aware of scales as $N$. Given the same resource assumptions as other algorithms, such that an equal superposition of eigenvectors is available efficiently, the algorithm is evaluated in the practical case as $O(\log_2 N/\varepsilon^2)$. The output is given in $O(1)$ queries of an oracle, which is given explicitly here and only relies on time-evolution operators that can be implemented with arbitrarily small error. The algorithm is envisioned for fully error-corrected quantum computers but may be implementable on near-term machines. We discuss how this algorithm can be used for kernel-based quantum machine-learning.
Abstract:The logarithm-determinant is a widely-present operation in many areas of physics and computer science. Derivatives of the logarithm-determinant compute physically relevant quantities in statistical physics models, quantum field theories, as well as the inverses of matrices. A multi-variable version of the quantum gradient algorithm is developed here to evaluate the derivative of the logarithm-determinant. From this, the pseudo-inverse of a sparse-rank input operator may be determined efficiently. Measuring an expectation value of the quantum state--instead of all $N^2$ elements of the input operator--can be accomplished in $O(k/\varepsilon^2)$ time in the idealized case for $k$ relevant eigenvectors of the input matrix with precision $\varepsilon$. A practical implementation of the required operator will likely need $\log_2N$ overhead, giving an overall complexity of $O((k\log_2 N)/\varepsilon^2)$. The method applies widely and converges super-linearly in $k$ when the condition number is high. The best classical method we are aware of scales as $N$. Given the same resource assumptions as other algorithms, such that an equal superposition of eigenvectors is available efficiently, the algorithm is evaluated in the practical case as $O(\log_2 N/\varepsilon^2)$. The output is given in $O(1)$ queries of an oracle, which is given explicitly here and only relies on time-evolution operators that can be implemented with arbitrarily small error. The algorithm is envisioned for fully error-corrected quantum computers but may be implementable on near-term machines. We discuss how this algorithm can be used for kernel-based quantum machine-learning.
arXiv:2501.09413 [quant-ph]
(or arXiv:2501.09413v4 [quant-ph] for this version)
From: Thomas E. Baker [view email] [v1] Thu, 16 Jan 2025 09:39:31 UTC (368 KB) [v2] Wed, 16 Apr 2025 17:16:13 UTC (370 KB) [v3] Sat, 20 Sep 2025 22:51:18 UTC (371 KB) [v4] Mon, 28 Sep 2026 03:04:28 UTC (376 KB)
Full text under the publisher's open licence · 284 words
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