Quantum algorithms offer significant speed-ups over their classical counterparts in various applications. In this paper, we develop quantum algorithms for the Kalman filter widely used in classical control engineering using the block encoding method. The entire calculation process is achieved by performing matrix operations on Hamiltonians based on the block encoding framework, including addition, multiplication, and inversion, which can be completed in a unified framework compared to previous quantum algorithms for solving control problems. We demonstrate that the quantum algorithm exponentially accelerates the computation of the Kalman filter compared to traditional methods. The time complexity can be reduced from $O(n^3)$ to $O(\kappa poly\log(n/\epsilon)\log(1/\epsilon'))$, where $n$ represents the matrix dimension, $\kappa$ denotes the condition number for the matrix to be inverted, $\epsilon$ indicates desired precision in block encoding, $\epsilon'$ signifies desired precision in matrix inversion. This paper provides a comprehensive quantum solution for implementing the Kalman filter and serves as an attempt to broaden the scope of quantum computation applications. Finally, we present an illustrative example implemented in Qiskit (a Python-based open-source toolkit) as a proof-of-concept.

翻译：摘要：量子算法在多种应用中相较于经典算法展现出显著的加速优势。本文基于块编码方法，针对经典控制工程中广泛使用的卡尔曼滤波器开发了量子算法。整个计算过程通过在块编码框架下对哈密顿量执行矩阵运算（包括加法、乘法与求逆）实现，相较于以往求解控制问题的量子算法，该过程可在统一框架内完成。我们证明，与传统方法相比，该量子算法能指数级加速卡尔曼滤波的计算过程。时间复杂度可从$O(n^3)$降至$O(\kappa poly\log(n/\epsilon)\log(1/\epsilon'))$，其中$n$表示矩阵维度，$\kappa$为待求逆矩阵的条件数，$\epsilon$表示块编码的期望精度，$\epsilon'$表示矩阵求逆的期望精度。本文为卡尔曼滤波器的实现提供了全面的量子解决方案，并作为拓展量子计算应用领域的尝试。最后，我们展示了在Qiskit（基于Python的开源工具包）中实现的概念验证示例。