HHL algorithm without ancilla postselection for Toeplitz and Hankel systems

Recently, Babukhin [D. V. Babukhin, Phys. Rev. A, 107. 042408(2023)] present the Harrow-Hassidim-Lloyd (HHL) algorithm without ancilla postselection which can work without postselection of the ancilla qubit. However, the remaining questions are to find more useful problems, which satisfy the postselection free condition. In this paper, we first prove that the symmetric Toeplitz matrices and their corresponding observable satisfy the postselection free condition. Then we show that the persymmetric Hankel matrix and the selected observable also satisfy this condition. Furthermore, we provide a specific example of a Toeplitz matrix (which is also a Hankel matrix) and a given observable. Finally, we compare the expectation values of the observable under the two different outcomes of the auxiliary qubit measurements with classical results and find excellent consistency. These results can significantly extend the application scope of postselection free HHL algorithm.