A quantum algorithm for the Kalman filter using block encoding

TL;DR

This paper develops a block-encoding-based quantum algorithm for the Kalman filter to achieve exponential speedups over classical methods by performing matrix operations via quantum circuits. The approach encodes all relevant matrices in a quantum-data-structure-enabled block-encoding, enabling addition, multiplication, and inversion through LCUs and QSVT. The authors provide a theoretical time bound of $O(\mathrm{polylog}(n/\epsilon)\kappa\log(1/\epsilon'))$ and demonstrate a Qiskit-based illustrative example, discuss resource requirements, and address practical limitations. The work expands quantum computation applications to large-scale state estimation problems in control theory and suggests avenues for future improvement in block-encoding techniques and hardware feasibility.

Abstract

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 $O(κpoly\log(n/ε)\log(1/ε'))$, where $n$ represents the matrix dimension, $κ$ denotes the condition number for the matrix to be inverted, $ε$ indicates desired precision in block encoding, $ε'$ 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.

Figures (20)

• Figure 1: Circuit for block encoding of $A$ using $a$ ancilla qubits.
• Figure 2: Circuit construction for block encoding of $A$ using $s$ ancilla qubits.
• Figure 3: Circuit for addition of two block encoded matrices ($a\leq b$).
• Figure 4: Circuit for product of two block encoded matrices.
• Figure 5: Circuit of $U_{P(A)}\in BE_{1,a+1}(P(A))$ when d is odd.