Realization Of The Algorithm For System Of Linear Equations In Duality Quantum Computing

Solving systems of linear equations is important both in classical and quantum computing. Harrow-Hassidim-Lloyd algorithm (HHL algorithm), a quantum algorithm for solving systems of linear equations, has achieved exponentially improved performance compared with corresponding classical algorithm. We will show in this article that the HHL algorithm is actually a duality quantum algorithm with non-unitary transformation from initial to the final state. We present a detailed realization of the HHL algorithm in a duality quantum computing formalism that allows the construction of non-unitary operations. The divider structure, combinor structure, and the total quantum circuit for the HHL algorithm in the duality quantum computing formalism are given explicitly.