Noisy intermediate-scale quantum algorithm for semidefinite programming

TL;DR

This paper introduces a NISQ algorithm for semidefinite programming that leverages convex classical optimization over a lower-dimensional space, enabling scalable solutions to complex quantum and combinatorial problems.

Contribution

It presents a novel NISQ eigensolver based on SDP formulation, which is convex and solvable in polynomial time, improving ground state energy estimation and solving large-scale quantum problems.

Findings

Successfully estimated ground state energies with scalable NISQ algorithms.

Solved large eigenvalue problems up to 2^1000 dimensions.

Applied to quantum contextuality and graph problems.

Abstract

Semidefinite programs (SDPs) are convex optimization programs with vast applications in control theory, quantum information, combinatorial optimization and operational research. Noisy intermediate-scale quantum (NISQ) algorithms aim to make an efficient use of the current generation of quantum hardware. However, optimizing variational quantum algorithms is a challenge as it is an NP-hard problem that in general requires an exponential time to solve and can contain many far from optimal local minima. Here, we present a current term NISQ algorithm for solving SDPs. The classical optimization program of our NISQ solver is another SDP over a lower dimensional ansatz space. We harness the SDP based formulation of the Hamiltonian ground state problem to design a NISQ eigensolver. Unlike variational quantum eigensolvers, the classical optimization program of our eigensolver is convex, can be solved in polynomial time with the number of ansatz parameters and every local minimum is a global minimum. We find numeric evidence that NISQ SDP can improve the estimation of ground state energies in a scalable manner. Further, we efficiently solve constrained problems to calculate the excited states of Hamiltonians, find the lowest energy of symmetry constrained Hamiltonians and determine the optimal measurements for quantum state discrimination. We demonstrate the potential of our approach by finding the largest eigenvalue of up to $2^{1000}$ dimensional matrices and solving graph problems related to quantum contextuality. We also discuss NISQ algorithms for rank-constrained SDPs. Our work extends the application of NISQ computers onto one of the most successful algorithmic frameworks of the past few decades.