  Physical design refers to mathematical optimization of a desired objective
(e.g. strong light--matter interactions, or complete quantum state transfer)
subject to the governing dynamical equations, such as Maxwell's or
Schrodinger's differential equations. Computing an optimal design is
challenging: generically, these problems are highly nonconvex and finding
global optima is NP hard. Here we show that for linear-differential-equation
dynamics (as in linear electromagnetism, elasticity, quantum mechanics, etc.),
the physical-design optimization problem can be transformed to a sparse-matrix,
quadratically constrained quadratic program (QCQP). Sparse QCQPs can be tackled
with convex optimization techniques (such as semidefinite programming) that
have thrived for identifying global bounds and high-performance designs in
other areas of science and engineering, but seemed inapplicable to the design
problems of wave physics. We apply our formulation to prototypical photonic
design problems, showing the possibility to compute fundamental limits for
large-area metasurfaces, as well as the identification of designs approaching
global optimality. Looking forward, our approach highlights the promise of
developing bespoke algorithms tailored to specific physical design problems.
