Quantum Circuit Design Methodology for Multiple Linear Regression

Abstract

Multiple linear regression assumes an imperative role in supervised machine learning. In 2009, Harrow et al. [Phys. Rev. Lett. 103, 150502 (2009)] showed that their HHL algorithm can be used to sample the solution of a linear system $\mathbf{Ax}=\mathbf{b}$ exponentially faster than any existing classical algorithm, with some manageable caveats. The entire field of quantum machine learning gained considerable traction after the discovery of this celebrated algorithm. However, effective practical applications and experimental implementations of HHL are still sparse in the literature. Here, we demonstrate a potential practical utility of HHL, in the context of regression analysis, using the remarkable fact that there exists a natural reduction of any multiple linear regression problem to an equivalent linear systems problem. We put forward a $7$-qubit quantum circuit design, motivated from an earlier work by Cao et al. [Mol. Phys. 110, 1675 (2012)], to solve a $3$-variable regression problem, using only elementary quantum gates. We also implement the Group Leaders Optimization Algorithm (GLOA) [Mol. Phys. 109 (5), 761 (2011)] and elaborate on the advantages of using such stochastic algorithms in creating low-cost circuit approximations for the Hamiltonian simulation. We believe that this application of GLOA and similar stochastic algorithms in circuit approximation will boost time- and cost-efficient circuit designing for various quantum machine learning protocols. Further, we discuss our Qiskit simulation and explore certain generalizations to the circuit design.

14 pages, 7 figures

References in corpus (18)

• Quantum Machine Learning
• A variational eigenvalue solver on a quantum processor
• Quantum algorithm for solving linear systems of equations
• Hardware-efficient Variational Quantum Eigensolver for Small Molecules and Quantum Magnets
• A Quantum Approximate Optimization Algorithm
• Quantum algorithm for systems of linear equations with exponentially improved dependence on precision
• Cluster-state quantum computation
• A quantum linear system algorithm for dense matrices
• A Krylov subspace algorithm for evaluating the phi-functions appearing in exponential integrators
• Experimental realization of quantum algorithm for solving linear systems of equations
• Quantum-circuit design for efficient simulations of many-body quantum dynamics
• Variable time amplitude amplification and a faster quantum algorithm for solving systems of linear equations
• Quantum linear systems algorithms: a primer
• Quantum Circuit Design for Solving Linear Systems of Equations
• Group Leaders Optimization Algorithm
• A Trotter-Suzuki approximation for Lie groups with applications to Hamiltonian simulation
• Efficient quantum algorithm for solving travelling salesman problem: An IBM quantum experience
• Exact search algorithm to factorize large biprimes and a triprime on IBM quantum computer

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