论文标题
通过量子计算机上的普通微分方程解决广义特征值问题
Solving generalized eigenvalue problems by ordinary differential equations on a quantum computer
论文作者
论文摘要
实践中引起的许多特征值问题通常是广义形式$ a \ x =λb\ x $。一个特别重要的案例是对称的，即$ a，b $是Hermitian，$ b $是正面的。解决这类本征值问题的标准算法是将它们降低到遗传学特征值问题。对于量子计算机，量子相估计是解决Hermitian特征值问题的有用技术。在这项工作中，我们为使用普通微分方程的对称广义特征值问题提出了一种新的量子算法。基于量子相估计，该算法的复杂性低于标准算法。此外，它适用于更宽的情况，而不是对称：$ b $可逆，$ b^{ - 1} a $都是可对角线的，所有特征值都是真实的。
Many eigenvalue problems arising in practice are often of the generalized form $A\x=λB\x$. One particularly important case is symmetric, namely $A, B$ are Hermitian and $B$ is positive definite. The standard algorithm for solving this class of eigenvalue problems is to reduce them to Hermitian eigenvalue problems. For a quantum computer, quantum phase estimation is a useful technique to solve Hermitian eigenvalue problems. In this work, we propose a new quantum algorithm for symmetric generalized eigenvalue problems using ordinary differential equations. The algorithm has lower complexity than the standard one based on quantum phase estimation. Moreover, it works for a wider case than symmetric: $B$ is invertible, $B^{-1}A$ is diagonalizable and all the eigenvalues are real.
