Variational quantum state diagonalization with computational-basis probabilities

Abstract

In this report, we propose a novel quantum diagonalization algorithm based on the optimization of variational quantum circuits. Diagonalizing a quantum state is a fundamental yet computationally challenging task in quantum information science, especially as the system size increases. To address this challenge, we reformulate the problem as a variational optimization process, where parameterized quantum circuits are trained to transform the input state into a diagonal form. To guide the optimization, we develop two objective functions based on measurement outcomes in the computational basis. The first objective function utilizes global computational basis probabilities, with the optimized value directly yielding the purity of the input state. The second objective function, designed for enhanced experimental feasibility, is constructed solely from single-qubit probabilities. It admits an elegant and compact analytical form that significantly reduces the exponential measurement complexity, while still effectively driving the state toward a diagonal representation. Through numerical simulations and analytical insights, we demonstrate that our variational optimization framework successfully produces the diagonal form of an input quantum state, offering a scalable and practical solution for quantum state diagonalization.