The quantum linear problem and its quantum algorithmic solutions

Abstract

In this final year project, we will attempt to quantum mechanically solve the solution vector for the system of linear equations problem, given a problem matrix and vector, by treating it as a quantum linear problem. However, solving for solution quantum state that involves Hermitian operators requires careful treatment. Intuitively, the solution state, can be interpreted as a normalised pre-collapse quantum state of the measurement of input quantum state by the Hermitian operator with normalisation factor. By employing three ideas that were implicitly developed by Aram Harrow, Avinatan Hassidim and Seth Lloyd (HHL), it is possible to construct a quantum linear solver (QLS) algorithm to obtain the solution state probabilistically, but not without implementation problems. To mitigate this, two promising version of QLS, a simplified and a full version, will be proposed and thoroughly analysed for the 2 and 4 dimensions, with an extension to the 2^n dimension. Several worked examples will be suggested and implemented by constructing and simulating its quantum circuit on an IBM Qiskit quantum simulator. The theoretical results suggest that predicting the probability of obtaining the solution state requires knowledge of the eigenvalues of M. In addition, constructing any quantum circuit version of HHL QLS algorithm requires the possession of multiple copies of controlled exponential of the Hermitian operator gate or its gate decomposition. Otherwise, its implementation would be impossible.