Abstract We present a detailed numerical study of an alternative approach, named quantum non-demolition measurement (QNDM) (Solinas et al 2023 Eur. Phys. J. D 77 76), to efficiently estimate the gradients or the Hessians of a quantum observable. This is a key step and a resource-demanding task when we want to minimize the cost function associated with a quantum observable. In our detailed analysis, we account for all the resources needed to implement the QNDM approach with a fixed accuracy and compare them to the current state-of-the-art method (Mari et al 2021 Phys. Rev. A 103 012405; Schuld et al 2019 Phys. Rev. A 99 032331; Cerezo et al 2021 Nat. Rev. Phys. 3 625). We find that the QNDM approach is more efficient, i.e. it needs fewer resources, in evaluating the derivatives of a cost function. These advantages are already clear in small dimensional systems and are likely to increase for practical implementations and more realistic situations. A significant outcome of our study is the implementation of the QNDM method in Python, provided in the supplementary material (Caletti and Minuto 2024 https://github.com/ simonecaletti/qndm-gradient ). Given that most variational quantum algorithms (VQA) can be formulated within this framework, our results can have significant implications in quantum optimization algorithms and make the QNDM approach a valuable alternative to implement VQA on near-term quantum computers.
Abstract We present a detailed numerical study of an alternative approach, named quantum non-demolition measurement (QNDM) (Solinas et al 2023 Eur. Phys. J. D 77 76), to efficiently estimate the gradients or the Hessians of a quantum observable. This is a key step and a resource-demanding task when we want to minimize the cost function associated with a quantum observable. In our detailed analysis, we account for all the resources needed to implement the QNDM approach with a fixed accuracy and compare them to the current state-of-the-art method (Mari et al 2021 Phys. Rev. A 103 012405; Schuld et al 2019 Phys. Rev. A 99 032331; Cerezo et al 2021 Nat. Rev. Phys. 3 625). We find that the QNDM approach is more efficient, i.e. it needs fewer resources, in evaluating the derivatives of a cost function. These advantages are already clear in small dimensional systems and are likely to increase for practical implementations and more realistic situations. A significant outcome of our study is the implementation of the QNDM method in Python, provided in the supplementary material (Caletti and Minuto 2024 https://github.com/ simonecaletti/qndm-gradient ). Given that most variational quantum algorithms (VQA) can be formulated within this framework, our results can have significant implications in quantum optimization algorithms and make the QNDM approach a valuable alternative to implement VQA on near-term quantum computers.
We present a detailed numerical study of an alternative approach, named Quantum Non-Demolition Measurement (QNDM), to efficiently estimate the gradients or the Hessians of a quantum observable. This is a key step and a resource-demanding task when we want to minimize the cost function associated with a quantum observable. In our detailed analysis, we account for all the resources needed to implement the QNDM approach with a fixed accuracy and compare them to the current state-of-the-art method. We find that the QNDM approach is more efficient, i.e. it needs fewer resources, in evaluating the derivatives of a cost function. These advantages are already clear in small dimensional systems and are likely to increase for practical implementations and more realistic situations. A significant outcome of our study is the implementation of the QNDM method in Python, provided in the supplementary material. Given that most Variational Quantum Algorithms can be formulated within this framework, our results can have significant implications in quantum optimization algorithms and make the QNDM approach a valuable alternative to implement Variational Quantum Algorithms on near-term quantum computers.
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.
Abstract We present a comparative study of two implementations of a variational quantum algorithm aimed at minimizing the energy of a complex quantum system. In one implementation, we extract the information of the energy gradient by projective measurements. In the second implementation, called the non-demolition approach, the gradient information is stored in a quantum detector, which is eventually measured. As prototypical examples, we study the energy minimization of the lithium-based molecules and then extend the analysis systems with increased complexity. We find that while both approaches are able to identify the energy minimum, the non-demolition approach has a clear advantage in terms of the overall computational resources needed. This advantage increases linearly with the complexity of the quantum system, making the non-demolition approach the ideal candidate to implement such variational quantum algorithms. Graphical abstract Schematic representation of the energy minimization process for the H2 and LiH molecules using direct measurement (DM) and quantum non-demolition measurement (QNDM) approaches. Both methods converge to the minimum energy with similar accuracy, but the QNDM approach requires significantly fewer quantum resources (up to 79% reduction in gate count for LiH), highlighting its efficiency for variational quantum algorithms
Quantum computing has evolved into a promising computational paradigm capable of addressing problems beyond the reach of classical methods. Among current approaches, hybrid quantum–classical algorithms—and in particular Variational Quantum Algorithms (VQAs)—play a central role in the Noisy Intermediate-Scale Quantum (NISQ) era. This thesis analyzes the research conducted during the PhD within the variational framework along two main directions. First, it introduces the Quantum Non-Demolition Measurement (QNDM) method, a novel strategy for efficient gradient estimation in VQAs. By coupling an auxiliary detector qubit through non-demolition operators, QNDM enables derivative extraction from a single measurement, reducing quantum resource requirements by up to an order of magnitude compared to standard direct-measurement methods. The second part explores the application of quantum machine learning to the solution of differential equations, extending the Differential Quantum Circuit (DQC) framework to stochastic systems. This work was carried out during a one-year research internship at the French scale-up Pasqal.
Drawing inspiration from gradient-descent methods developed for data processing in quantum state tomography [\href{https://iopscience.iop.org/article/10.1088/2058-9565/ae0baa}{Quantum Sci.~Technol.~\textbf{10} 045055 (2025)}] and quantum process tomography [\href{https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.130.150402}{Phys.~Rev.~Lett.~\textbf{130}, 150402 (2023)}], we introduce stochastic gradient descent (SGD) algorithms for fast quantum measurement tomography (QMT), applicable to both discrete- and continuous-variable quantum systems -- thus completing the tomography trio. A measurement device or detector in a quantum experiment is characterized by a set of positive operator-valued measure (POVM) elements; the goal of QMT is to estimate these operators from experimental data. To ensure physically valid (positive and complete) POVM reconstructions, we propose two distinct parameterization schemes within the SGD framework: one leveraging optimization on a Stiefel manifold and one based on Hermitian operator normalization via eigenvalue scaling. Within the SGD-QMT framework, we further investigate two loss functions: mean squared error, equivalent to L2 or Euclidean norm, and average negative log-likelihood, inspired by maximum likelihood estimation. We benchmark performance against state-of-the-art constrained convex optimization methods. Numerical simulations demonstrate that, compared to standard methods, our SGD-QMT algorithms offer significantly lower computational cost, superior reconstruction fidelity, and enhanced robustness to noise. We make a Python implementation of the SGD-QMT algorithms publicly available at \href{https://github.com/agtomo/SGD-QMT}{github.com/agtomo/SGD-QMT}.
Sampling noisy intermediate-scale quantum devices is a fundamental step that converts coherent quantum-circuit outputs to measurement data for running variational quantum algorithms that utilize gradient and Hessian methods in cost-function optimization tasks. This step, however, introduces estimation errors in the resulting gradient or Hessian computations. To minimize these errors, we discuss tunable numerical estimators, which are the finite difference (including their generalized versions) and scaled parameter-shift estimators [introduced in Phys. Rev. A 103, 012405 (2021)], and propose operational circuit-averaged methods to optimize them. We show that these optimized numerical estimators offer estimation errors that drop exponentially with the number of circuit qubits for a given sampling-copy number, revealing a direct compatibility with the barren-plateau phenomenon. In particular, there exists a critical sampling-copy number below which an optimized difference estimator gives a smaller average estimation error in contrast to the standard (analytical) parameter-shift estimator, which exactly computes gradient and Hessian components. Moreover, this critical number grows exponentially with the circuit-qubit number. Finally, by forsaking analyticity, we demonstrate that the scaled parameter-shift estimators beat the standard unscaled ones in estimation accuracy under any situation, with comparable performances to those of the difference estimators within significant copy-number ranges, and are the best ones if larger copy numbers are affordable.