Quantum Non-Demolition Measurement Techniques for Gradient Evaluation and Resource Reduction in Variational Quantum Algorithms

Quantum computing has become a potential way to solve complex problems that conventional computers can’t handle, especially those involving strongly correlated quantum systems. Classical simulation of these systems becomes exponentially more challenging as the number of interacting particles rises, resulting in significant computational and memory limitations. As a result, classical computational approaches are unable to handle many significant physical and chemical systems, including molecular structures, biological molecules, and quantum materials. Quantum computers operate using the same quantum mechanical principles as these systems. Consequently, they provide a natural framework for exploring large Hilbert spaces and capturing complex quantum correlations. This thesis begins by discussing the fundamental principles of quantum mechanics that are applicable to computation, such as qubits, superposition, and entanglement. The construction of quantum circuits is discussed after a discussion of basic quantum gates and their matrix representations. These ideas are the foundation of quantum algorithms and offer the essential framework for comprehending the modeling and manipulation of complicated quantum states using quantum hardware. Building on this foundation, the thesis focuses on Variational Quantum Algorithms (VQAs), which offer a feasible hybrid quantum-classical approach to optimizing near-term quantum devices. In VQAs, a parameterized quantum circuit is used to approximate the ground-state energy of the quantum system. A classical optimizer then iteratively updates the variational parameters to minimize a cost function, usually the expectation value of the system’s Hamiltonian. A thorough analysis of the function of gradient-based optimization is conducted, encompassing the parameter-shift rule that is employed to calculate precise energy derivatives. The discussion is expanded to include major applications such as the Quantum Approximate Optimization Algorithm and developing approaches to quantum machine learning, emphasizing the importance of efficient and accurate gradient evaluation. A major challenge in the real-world application of VQAs is the high resource cost of gradient estimation using conventional direct measurement techniques. These techniques require repeated projective measurements and several executions of parameter-shifted quantum circuits, which lead to significant measurement overhead and deeper circuits. Quantum Non-Demolition Measurement (QNDM) is examined in this thesis as a potential framework for gradient evaluation to get around this limitation. Based on recent theoretical advances, QNDM introduces an ancillary detector qubit that coherently stores gradient information within its phase. Derivatives can then be estimated indirectly through detector measurements. This allows gradient information to be extracted without destroying the quantum state and with a greatly reduced number of measurements. The efficiency of QNDM is evaluated by extending the study to systems with bigger Hamiltonians and greater molecular complexity, as well as by comparing it in detail with direct measurement techniques utilizing benchmark molecular systems like LiH, Li₂, and H₂. Although both methods successfully converge to the ground-state energy, as shown by numerical results, QNDM consistently uses fewer quantum resources in terms of logical gate operations, measurement shots, and circuit executions. Additionally, as system size grows, QNDM’s resource advantage becomes more noticeable, suggesting that it has favorable scalability for bigger molecular systems. This thesis concludes that, while both direct measurement and quantum non-demolition measurement approaches allow for precise variational optimization, QNDM gives a significant decrease in quantum resources. These results establish QNDM as an effective and scalable method for developing quantum simulations of realistic physical and chemical systems and for executing variational quantum algorithms on near-term quantum hardware.