Optimization landscapes of variational quantum algorithms

Abstract

Optimization plays a central role in variational quantum algorithms, where the objective function typically takes the form $F(\boldsymbolθ)= \sum_{m=1}^{M} f_m \left(\mathrm{Tr}[U(\boldsymbolθ)ρ_m U^\dagger(\boldsymbolθ) O_m]\right)$, with $U(\boldsymbolθ)$ being a parameterized quantum ansatz. Understanding the optimization landscape of such objective functions is crucial for assessing the trainability and performance of these algorithms. For the special case $M=1$, it is known that under certain assumptions, the landscape is free of false traps (FTs), i.e., local optima that are not global. In this work, we investigate optimization landscapes of the general case $M\geq1$ and show that the landscape becomes intrinsically more complex. First, we establish a complete framework for analyzing critical features of the optimization landscape, by deriving necessary and sufficient conditions to identify and classify all critical points under some assumptions, which is also of practical importance in designing efficient algorithms independent of whether these assumptions are satisfied. Then, we show that FTs can still emerge on landscapes for $M>1$, standing in stark contrast to the $M=1$ case and further revealing that parameter sufficiency alone is not enough to guarantee a trap-free landscape. Moreover, we uncover a close connection that the emergence of FTs is necessarily attributed to the loss of distinguishability among the states and/or operators, and fundamentally, to the loss of compatibility of the spectral ordering governed by different objective terms. Our results provide a deeper understanding of the optimization complexity and practical guidance for both algorithmic and problem-setting designs.