  Solving optimization problems is a key task for which quantum computers could
possibly provide a speedup over the best known classical algorithms. Particular
classes of optimization problems including semi-definite programming (SDP) and
linear programming (LP) have wide applicability in many domains of computer
science, engineering, mathematics, and physics. Here we focus on semi-definite
and linear programs for which the dimensions of the variables involved are
exponentially large, so that standard classical SDP and LP solvers are not
helpful for such large-scale problems. We propose the QSlack and CSlack methods
for estimating their optimal values, respectively, which work by 1) introducing
slack variables to transform inequality constraints to equality constraints, 2)
transforming a constrained optimization to an unconstrained one via the penalty
method, and 3) replacing the optimizations over all possible non-negative
variables by optimizations over parameterized quantum states and parameterized
probability distributions. Under the assumption that the SDP and LP inputs are
efficiently measurable observables, it follows that all terms in the resulting
objective functions are efficiently estimable by either a quantum computer in
the SDP case or a quantum or probabilistic computer in the LP case.
Furthermore, by making use of SDP and LP duality theory, we prove that these
methods provide a theoretical guarantee that, if one could find global optima
of the objective functions, then the resulting values sandwich the true optimal
values from both above and below. Finally, we showcase the QSlack and CSlack
methods on a variety of example optimization problems and discuss details of
our implementation, as well as the resulting performance. We find that our
implementations of both the primal and dual for these problems approach the
ground truth, typically achieving errors of order $10^{-2}$.

Quantum Physics