In some aspects, the techniques described herein relate to a quantum method for solving a second-order cone program (SOCP) instance, the method including: defining a Newton system for the SOCP instance by constructing matrix G and vector h based on the SOCP instance; preconditioning matrix G and vector h via row normalization to reduce a condition number of matrix G; iteratively determining u until a predetermined iteration condition is met, the iterations including: causing a quantum computing system to apply matrix G and vector h to a quantum linear system solver (QLSS) to generate a quantum state; causing the quantum computing system to perform quantum state tomography on the quantum state; and updating a value of u based on a current value of u and the output of the quantum state tomography; and determining a solution to the SOCP instance based on the updated value of u.