量子计算解线性方程：HHL algorithm for quantum linear equation

最新推荐文章于 2024-07-23 22:37:12 发布

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最新推荐文章于 2024-07-23 22:37:12 发布

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HHL算法是一种量子计算方法，用于高效解决线性方程组，尤其适用于Hermitian矩阵。相较于经典计算机的O(Nsκlog(1/ϵ))复杂度，HHL算法实现为O(log(N)s²κ²/ϵ)，提供指数级加速。算法使用三个量子寄存器，通过量子态编码问题、量子相位估计等步骤来逼近解的函数。

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Introduction

Systems of linear equations arise naturally in many real-life applications in a wide range of areas, such as in the solution of Partial Differential Equations, the calibration of financial models, fluid simulation or numerical field calculation. The problem can be defined as, given a $A\in\mathbb{C}^{N\times N}$ and a vector $\vec{b}\in\mathbb{C}^{N}$ , find $\vec{x}\in\mathbb{C}^{N}$ satisfying $A\vec{x}=\vec{b}$

For example, take N=2,

$\begin{pmatrix}1 & -1/3\\-1/3 & 1 \end{pmatrix},\quad \vec{x}=\begin{pmatrix} x_{1}\\ x_{2}\end{pmatrix}\quad \text{and} \quad \vec{b}=\begin{pmatrix}1 \\ 0\end{pmatrix}$

Then the problem can also be written as find $x_{1}, x_{2}\in\mathbb{C} such that \begin{cases}x_{1} - \frac{x_{2}}{3} = 1 \\ -\frac{x_{1}}{3} + x_{2} = 0\end{cases}$

A system of linear equations is called s-sparse if A has at most s non-zero entries per row or column. Solving an s-sparse system of size N with a classical computer requires $\mathcal{ O }(Ns\kappa\log(1/\epsilon))$ running time using the conjugate gradient method. Here $\kappa$ denotes the condition number of the system and $\epsilon$ the accuracy of the approximation.

The HHL is a quantum algorithm to estimate a function of the solution with running time complexity of $\mathcal{ O }(\log(N)s^{2}\kappa^{2}/\epsilon)$ when A is a Hermitian matrix under the assumptions of efficient oracles for loading the data, Hamiltonian simulation and computing a function of the solution. This is an exponential speed up in the size of the system, however one crucial remark to keep in mind is that the classical algorithm returns the full solution, while the HHL can only approximate functions of the solution vector.

The HHL algorithm

Some mathematical background

The first step towards solving a system of linear equations with a quantum computer is to encode the problem in the quantum language. By rescaling the system, we can assume $\vec{b}$ and $\vec{x}$ to be normalised and map them to the respective quantum states $|b\rangle$ and $|x\rangle$ . Usually the mapping used is such that $i^{th}$ component of $\vec{b}$ (resp. $\vec{x})$

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