  We provide an algebraic framework for quantization of Hermitian metrics that
are solutions of the Hitchin equation for Higgs bundles over a projective
manifold. Using Geometric Invariant Theory, we introduce a notion of balanced
metrics in this context. We show that balanced metrics converge at the quantum
limit towards the solution of the Hitchin equation. We relate the existence of
balanced metrics to the Gieseker stability of the Higgs bundle.
