对数行列式梯度的量子算法
Quantum algorithm for the gradient of a logarithm-determinant
AI总结：
本文提出多变量量子梯度算法，用于高效计算对数行列式的导数，从而确定稀疏秩输入算子的逆；在理想情形下复杂度为O(k/ε²)，实际实现约为O((k log₂N)/ε²)，相比经典方法O(N)具有优势，并讨论了其在基于核的量子机器学习中的应用。
摘要：
对数行列式是物理学和计算机科学众多领域中广泛出现的一种运算。对数行列式的导数可用于计算统计物理模型、量子场论中的物理相关量，以及矩阵的逆。本文提出了一种多变量版本的量子梯度算法，用于计算对数行列式的导数。由此，可以高效地确定稀疏秩输入算子的逆。通过测量量子态的期望值——而非输入算子的全部 $N^2$ 个元素——在理想情况下，对于输入矩阵的 $k$ 个相关本征向量，可在 $O(k/\varepsilon^2)$ 时间内以精度 $\varepsilon$ 完成。所需算子的实际实现可能需要 $\log_2N$ 的开销，从而给出整体复杂度 $O((k\log_2 N)/\varepsilon^2)$。该方法适用范围广，且当条件数较高时在 $k$ 上呈超线性收敛。我们所知的最佳经典方法复杂度为 $N$。在与其他算法相同的资源假设下，即可以高效获得本征向量的等权叠加态时，该算法在实际情形下的复杂度评估为 $O(\log_2 N/\varepsilon^2)$。输出通过 $O(1)$ 次预言机查询给出，该预言机在此被显式给出，并且仅依赖于可以以任意小误差实现的时间演化算子。该算法设想用于完全纠错的量子计算机，但可能可以在近期设备上实现。我们讨论了该算法如何用于基于核的量子机器学习。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
