ETD

• adaptive resource allocation
• measurement noise
• NISQ algorithms
• quantum computing
• sequential optimization
• variational quantum eigensolver

Variational Quantum Algorithms represent a promising class of hybrid quantum-classical methods for Noisy Intermediate Scale Quantum (NISQ) devices. Among these, the Variational Quantum Eigensolver (VQE) has emerged as a viable approach for studying quantum many-body systems. The VQE operates through an iterative loop: a parameterized quantum circuit prepares a trial state, measurements estimate the energy expectation value, and a classical optimizer updates the circuit parameters towards lower energy. The iterative nature of this process makes measurement costs a primary bottleneck: each energy evaluation requires numerous quantum circuit executions (shots).
This thesis introduces a multi-level perspective for adaptive resource allocation in VQE, organizing optimization decisions into three conceptually distinct but interconnected levels:

-Macro-Level: Parameter Selection. Given an ansatz with n parameters and a sequential parameter optimizer (i.e., an optimizer that updates one parameter or a small subset at each iteration), one must decide which subset to optimize. Not all parameters are expected to contribute equally; in layered ansätze, earlier layers might play a dominant role during the initial stages of optimization, while deeper layers may mainly serve to refine the solution. Updating all parameters uniformly could then waste resources on less relevant directions, providing minimal benefit.
The macro-level question is: Can we adaptively select parameters based on their relevance, reducing quantum evaluations without harming convergence?

-Meso-Level: Evaluation Strategy. Once a parameter subset is chosen, one must determine how to evaluate the cost function to estimate the optimal update.
The meso-level question is: How can evaluation protocols and shot distributions be chosen to minimize uncertainty for a given measurement budget?

-Micro-Level: Hamiltonian Term Allocation. The variance of the energy estimate for a parameterized quantum state depends on how measurement shots are distributed among the terms. Terms with large coefficients or high intrinsic variance require more measurements, making uniform allocation suboptimal.
The micro-level question is: How should shots be allocated among Hamiltonian terms to minimize energy-estimation variance?

Although these three questions are logically independent, information obtained at one level can inform decisions at the others. Existing VQE research typically addresses these problems in isolation, relying on heuristic choices rather than unified or theoretically grounded principles. A systematic multi-scale framework is currently lacking.

To develop and illustrate this perspective, we work within the Rotosolve optimization framework, a parameter update method based on the specific properties of rotational gates of parameterized quantum circuits, exploiting the functional behavior of expectation values of Hermitian operators. Its conceptual simplicity and parameter-wise update scheme make it particularly well suited for introducing adaptive strategies at each of the three levels aforementioned.

Numerical validation focuses on the macro-level strategies, tested on antiferromagnetic Ising models on triangular lattices with periodic boundary conditions. Results show that selecting parameters based on their historical contribution to energy reduction can improve convergence efficiency compared to updating all parameters in fixed order. This work contributes both a conceptual organization of resource allocation decisions in variational algorithms and initial methodologies at each level. The multi-level perspective, together with a future integration of adaptive strategies across all three levels, may help develop more efficient VQE optimization protocols for near-term quantum devices.