We study the limitations and fast-forwarding of quantum algorithms for linear ordinary differential equation (ODE) systems with a particular focus on non-quantum dynamics, where the coefficient matrix in the ODE is not anti-Hermitian or the ODE is inhomogeneous. On the one hand, for generic homogeneous linear ODEs, by proving worst-case lower bounds, we show that quantum algorithms suffer from computational overheads due to two types of ``non-quantumness'': real part gap and non-normality of the coefficient matrix. We then show that homogeneous ODEs in the absence of both types of ``non-quantumness'' are equivalent to quantum dynamics, and reach the conclusion that quantum algorithms for quantum dynamics work best. We generalize our results to the inhomogeneous case and find that existing generic quantum ODE solvers cannot be substantially improved. To obtain these lower bounds, we propose a general framework for proving lower bounds on quantum algorithms that are amplifiers, meaning that they amplify the difference between a pair of input quantum states. On the other hand, we show how to fast-forward quantum algorithms for solving special classes of ODEs which leads to improved efficiency. More specifically, we obtain quadratic improvements in the evolution time $T$ for inhomogeneous ODEs with a negative semi-definite coefficient matrix, and exponential improvements in both $T$ and the spectral norm of the coefficient matrix for inhomogeneous ODEs with efficiently implementable eigensystems, including various spatially discretized linear evolutionary partial differential equations. We give fast-forwarding algorithms that are conceptually different from existing ones in the sense that they neither require time discretization nor solving high-dimensional linear systems.
翻译:我们研究了线性普通差分方程(ODE)系统量子算法的局限性和快速推进,特别侧重于非分子动态,即ODE的系数矩阵不是反荷米蒂或ODE的不相容。一方面,对于通用的同质线性值,我们通过证明最差的下限,表明量性算法由于两种“非量性”的计算间接费用而受到影响:系数矩阵的真正部分差距和不正常性。然后我们表明,在没有两种“非量性”动态的情况下,同质性值的数值矩阵与量性不相等,得出量性基体的系数矩阵与量性动力性不相等的结论。一方面,我们将我们的结果概括到不相容的不均匀的直线性线性线性计算法,发现现有的通用量性内值解解解解算法无法大幅改进。为了获得这些更低的界限,我们提议了一个总体框架,用来证明在量性值矩阵的改进中,就是放大,意味着它们放大了ODL的相对部分数值的数值差异,在OI值的变异性变数的变数,具体的变数上,我们在的变数性变数性变数的变数,我们在的变数性变数性变数性变数,我们很快化的变数的变数的变数的变数式的变数,我们算算算算法在另一个的变的变的变数,我们算法中,我们的变到的变的变的变的变的变数式的变数式的变数式的变的变数,我们的变数,在较快的变数式的变数式的变数式的变的变的变的变数式的变到的变的变的变的变数,我们的变数式的变的变数,我们的变数式的变的变的变的变的变的变的变的变的变的变数,我们的变的变数式的变数式的变数式的变数式的变的变的变的变的变数式的变数式的变的变的变的变数式的变数式的变数法,我们的变数式的变数式的变数式的变数,我们的变式的变式的变</s>