论文标题
热方程式变分量子算法的深度分析
Depth analysis of variational quantum algorithms for heat equation
论文作者
论文摘要
变分量子算法是解决部分微分方程的有前途的工具。其数值解决方案的标准方法是有限差方案，可以将其简化为线性代数问题。我们考虑了三种解决量子计算机上的热方程的方法。使用直接的变分方法，我们最大程度地降低了哈密顿量的期望值，其基态是研究该问题的解决方案。通常，在哈密顿分解中的Pauli产品数量不允许实现量子速度。基于Hadamard测试的方法解决了这个问题，但是，执行的模拟显然证明ANSATZ电路相对于Qubits的数量具有多项式深度。 Ansatz树方法利用了矩阵的明确形式，这使得比经典算法获得优势是可能的。在我们的数值模拟最多$ n = 11 $ QUBITS中，此方法揭示了指数速度的提高。
Variational quantum algorithms are a promising tool for solving partial differential equations. The standard approach for its numerical solution are finite difference schemes, which can be reduced to the linear algebra problem. We consider three approaches to solve the heat equation on a quantum computer. Using the direct variational method we minimize the expectation value of a Hamiltonian with its ground state being the solution of the problem under study. Typically, an exponential number of Pauli products in the Hamiltonian decomposition does not allow for the quantum speed up to be achieved. The Hadamard test based approach solves this problem, however, the performed simulations do not evidently prove that the ansatz circuit has a polynomial depth with respect to the number of qubits. The ansatz tree approach exploits an explicit form of the matrix what makes it possible to achieve an advantage over classical algorithms. In our numerical simulations with up to $n=11$ qubits, this method reveals the exponential speed up.
