The power of block-encoded matrix powers: improved regression techniques via faster Hamiltonian simulation (1804.01973v2)

Abstract: We apply the framework of block-encodings, introduced by Low and Chuang (under the name standard-form), to the study of quantum machine learning algorithms and derive general results that are applicable to a variety of input models, including sparse matrix oracles and matrices stored in a data structure. We develop several tools within the block-encoding framework, such as singular value estimation of a block-encoded matrix, and quantum linear system solvers using block-encodings. The presented results give new techniques for Hamiltonian simulation of non-sparse matrices, which could be relevant for certain quantum chemistry applications, and which in turn imply an exponential improvement in the dependence on precision in quantum linear systems solvers for non-sparse matrices. In addition, we develop a technique of variable-time amplitude estimation, based on Ambainis' variable-time amplitude amplification technique, which we are also able to apply within the framework. As applications, we design the following algorithms: (1) a quantum algorithm for the quantum weighted least squares problem, exhibiting a 6-th power improvement in the dependence on the condition number and an exponential improvement in the dependence on the precision over the previous best algorithm of Kerenidis and Prakash; (2) the first quantum algorithm for the quantum generalized least squares problem; and (3) quantum algorithms for estimating electrical-network quantities, including effective resistance and dissipated power, improving upon previous work.

• The paper introduces block-encoding techniques to implement efficient quantum linear solvers and perform singular value estimation.
• The paper demonstrates improved Hamiltonian simulation for non-sparse matrices that reduces condition number dependency in regression tasks.
• The paper achieves exponential precision gains and a 6th power improvement in generalized least squares, showcasing significant computational advantages.

Overview of Improved Regression Techniques via Quantum Matrix Powers

This paper introduces innovative quantum machine learning techniques leveraging the framework of block-encodings and Hamiltonian simulation. The authors analyze how block-encoded matrix powers enable faster and more efficient quantum algorithms compared to classical methods. They derive general results applicable to various input models, such as sparse matrix oracles and quantum-accessible data structures. This work prominently focuses on regression techniques and related problems, including weighted least squares and generalized least squares, providing efficient quantum algorithms for these tasks.

Key Contributions

1. Block-Encoding Framework: The paper extends the block-encoding framework to offer efficient implementations of quantum linear system solvers, singular value estimation, and matrix inversion. Block-encodings allow encoding a matrix as the top-left block of a unitary matrix, providing significant computational advantages.
2. Hamiltonian Simulation: Using block-encodings, the authors show how to implement Hamiltonian simulation for non-sparse matrices. This technique directly enhances the quantum algorithms for linear algebraic tasks, providing a more scalable and precise approach.
3. Weighted and Generalized Least Squares: Quantum algorithms for these problems exhibit improvements—such as a 6-th power improvement in the condition number dependence and exponential precision improvements compared to previous algorithms. The quantum generalized least squares solver marks a novel contribution, demonstrating both practical and algorithmic advancements.
4. Quantum Singular Value Estimation: The paper presents a quantum algorithm to estimate singular values when the input is given as a block-encoding. This algorithm applies to the quantum data structure model and recovers previous results with enhanced implementation efficiency.
5. Variable-Time Amplitude Estimation: Building on techniques from Ambainis and others, the authors develop an approach to estimate success probabilities for variable-stopping-time algorithms, refining computational speed while maintaining error accuracy.

Strong Numerical Results and Claims

The paper offers strong numerical evidence for the efficiency of block-encoded matrix powers in quantum algorithms. For example, the quantum algorithm for weighted least squares achieves a complexity of $O(\kappa \mu \, \mathrm{polylog}(MN/\epsilon))$, significantly reducing the dependency on the condition number $\kappa$ and offering polylogarithmic precision improvements over classical counterparts. Such results indicate substantial theoretical advancements in quantum regression techniques.

Implications and Future Developments

The implications of this research are profound, spanning practical and theoretical impacts in quantum computing and machine learning. Practically, the algorithms can lead to faster quantum computing methods for solving linear algebraic problems, crucial for big data tasks. Theoretically, this framework sets a precedent for exploring more efficient quantum algorithms and further refining the time and precision complexities of machine learning processes.

Conclusion

The presented work represents a significant step forward in adapting quantum computing algorithms for regression tasks. The integration of block-encoding framework with Hamiltonian simulation paves the way for faster, more accurate quantum algorithms that can outperform classical strategies. This paper not only sets the stage for future research in quantum machine learning but also provides a robust platform for practical implementations. Future developments are likely to expand these findings, exploring further applications and refinements within the quantum computing field.