Parameterized variational quantum objective

A finite variational quantum algorithm (VQA) specifies a parameter domain $\mathrm{\Theta }\subseteq {\mathbb{R}}^p$, a supplied parameterized finite circuit $U(\theta )$, finitely many prepared density operators ${\rho }_k$, Hermitian observables $O_k$, real-valued functions $f_k$ on the corresponding expectation ranges, and the objective $C(\theta )={\sum }_{k=1}^N{f}_k\text{\hspace{0.17em}⁣}(\mathrm{tr}(O_{k}U(\theta ){\rho }_{k}U(\theta {)}^{\ast }))\mathrm{.}$ The convention is root inner product linear in the first argument and Dirac expectation $⟨\psi \mathrm{\mid }O\mathrm{\mid }\psi {⟩}_D=\mathrm{tr}(O\rho )$. State how each ${\rho }_k$ is prepared, the measurement/shot model and finite precision, the classical optimizer, its stopping rule, the output, and every circuit, inverse, controlled access, observable, measurement, arithmetic, and state-preparation cost. The definition asserts no optimizer existence, convergence, global optimality, or quantum advantage.