  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.
