Error mitigation and performance trade-offs in scaling variational quantum algorithms

Quantum computers promise significant computational advantages across domains such as optimization, cryptography, and quantum simulation. However, the potential of current devices is constrained by noise arising from imperfect hardware implementations. By combining the potential for noise resilience of parameterized circuits with the power of classical optimizers, Variational Quantum Algorithms are among the most promising in this context. In this Thesis, two practical applications of this framework are explored. In the first, we study quantum phase transitions in the one-dimensional J1–J2 Heisenberg spin chain. This is achieved by mapping the presence of a phase transition into the crossing of low-lying excitations, and subsequently using level spectroscopy to identify the crossings. More specifically, by leveraging the system's symmetries, we constrain the optimization to selected symmetry sectors, enabling efficient low-energy computations and accurate estimation of the critical point. Moreover, the adaptation of Quantum Error Mitigation techniques, such as Zero-Noise Extrapolation, to this setting allows reliable results even in the presence of realistic noise. The second application introduces a variational approach to correcting gate calibration errors. By exploiting structured states, such as stabilizer states, we isolate errors arising from miscalibrations and compensate for them robustly under additional sources of noise. Despite its potential, the cost concentration or barren plateau phenomenon remains one of the main limitations of this framework, severely limiting its scalability, especially in the presence of noise. The primary contribution of this Thesis is to provide a general mathematical formulation describing cost concentration under arbitrary noise processes. In particular, it unifies and extends existing models, elucidating how circuit structure influences trainability and noise resilience, an essential step for the development of scalable and reliable variational algorithms. Towards this goal, it identifies a class of architectures termed Quantum Residual Networks, that are able to mitigate cost concentration, thus preserving meaningful gradients at scale.