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.
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.
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.
In recent years, as datasets have grown larger, quantum computing has shown potential in addressing the rapidly increasing costs of data processing due to its superposition and entanglement capabilities. Limited by quantum hardware and the number of controllable qubits, variational quantum algorithms (VQAs) are algorithms suitable for NISQ devices, capable of computation on medium-scale noisy devices. VQAs are hybrid classical-quantum algorithms that use parameterized quantum circuits to prepare parameterized quantum states and employ classical optimizers to minimize objective functions, enabling efficient solutions for complex problems. Classical optimizers reduce the stringent requirements on quantum hardware and can achieve quantum advantage on current devices. This paper systematically elaborates on the core foundational theories of quantum computing and neural networks, with a focus on reviewing the research and application status of VQAs in three main scenarios: machine learning (supervised, unsupervised, and reinforcement learning), solving systems of equations (linear and nonlinear), and network attack detection. It analyzes the current technical bottlenecks in this field and provides prospects for future development, offering a reference for further research on variational quantum algorithms.
Variational quantum algorithms (VQAs) have emerged as a promising candidate for realizing quantum advantage in machine learning and optimization tasks on noisy intermediate-scale quantum (NISQ) devices. Despite significant theoretical promise, the practical impact of VQAs is constrained by the exponential growth in computational resources required for classical simulation, the onset of barren plateaus in parameterized landscapes, and the inherent difficulties of scaling classical optimization routines to circuits with many qubits and layers. Addressing these intertwined challenges is essential for both advancing quantum algorithm development and extracting actionable physical insights from modern quantum hardware.This dissertation develops, analyzes, and validates an efficient framework based on approximate tensor ring decompositions for the classical simulation, training, and benchmarking of variational quantum circuits and eigensolvers. The first component of this work formalizes the representation of quantum states and gates within a cyclic tensor ring structure, enabling the compression of parameterized quantum circuits to linear scaling in both qubit number and circuit depth, while preserving essential quantum correlations and entanglement structure. Gate operations are efficiently approximated via truncated singular value decomposition, and novel algorithms for forward propagation and gradient-based training in the tensor ring domain are developed. These methodological advances are systematically benchmarked on standard supervised learning tasks, including Iris and MNIST datasets, and compared to conventional statevector and matrix product state baselines, resulting in competitive accuracy and orders-of-magnitude reductions in simulation cost. Extending this framework, the dissertation also introduces a scalable method for gradient evaluation in variational quantum eigensolvers (VQE) using the approximate tensor ring representation. This approach enables the simulation and training of large circuits for combinatorial optimization problems, supporting practical qubit counts and circuit depths inaccessible to traditional simulation methods. Detailed experiments on the Maximum-Cut optimization problem demonstrate both the computational efficiency and robustness of the tensor ring framework in nontrivial optimization landscapes.Collectively, the results establish approximate tensor ring decomposition as a versatile and powerful framework for advancing quantum machine learning and discrete optimization, supporting both pre-deployment benchmarking on classical hardware and algorithm design for next-generation quantum devices. This work provides a foundation for the further exploration of scalable quantum-classical hybrid algorithms, with implications for quantum simulation, optimization, and beyond.
Parameterized quantum circuits (PQCs) play an essential role in the application of variational quantum algorithms (VQAs) in noisy intermediate-scale quantum (NISQ) devices. The PQCs are a leading candidate to achieve a quantum advantage in NISQ devices and have already been applied in various domains such as quantum chemistry, quantum machine learning, combinatorial optimization, and many others. There is no single definitive way to optimize PQCs. The most commonly used methods are based on computing the gradient via the parameter-shift rule to use classical gradient descent (GD) optimizers like Adam, stochastic GD, and others. In addition, sequential single-qubit optimizers have been proposed, such as Rotosolve, Free-Axis Selection (Fraxis), Free-Quaternion Selection (FQS), and hybrid algorithms from the aforementioned optimizers. We further develop hybrid algorithms than those represented in the previous work by drawing inspiration from the early stopping method used in classical machine learning. The switch between the optimizers depends on the previous cost function values compared to the previous ones. We introduce two new hybrid algorithms that are more robust and scalable, and they outperform previous hybrid methods in terms of convergence towards the global minima across various cost functions. In addition, we find that they are feasible for NISQ devices with different noise profiles.
Differential equations underpin a vast range of scientific and engineering domains, including structural mechanics, fluid dynamics, and financial modelling. Their intricate nature often renders classical numerical methods computationally demanding, requiring fine discretization or large basis expansions, which can lead to prohibitive costs. Quantum computing offers a fundamentally different paradigm, leveraging superposition and entanglement with the promise of computational speedup. However, in the current Noisy Intermediate-Scale Quantum (NISQ) era, fully fault-tolerant quantum computers remain decades away, motivating the development of hybrid strategies such as Variational Quantum Algorithms (VQAs), which train parametrised quantum circuits using classical optimizers. This work investigates spectral VQAs for solving differential equations, extending the Chebyshev-encoded VQA of Kyriienko et al. and introducing a novel quantum spectral approach: the Lagrange-encoded VQA. This approach shifts the paradigm by incorporating entanglement directly within the encoding block while maintaining a shallow variational ansatz. Inspired by the Hadamard test structure, the design enables an efficient representation of the complete set of Lagrange polynomial basis functions using a limited number of gates and facilitates the use of the Hadamard test differentiation method, thereby reducing the computational load. VQA performance is analysed through the accuracy of the result in combination with the gate count and circuit count metrics, which serve as proxies for runtime and noise accumulation on NISQ devices. The proposed VQA is benchmarked on multiple problems: (i) a first-order linear ODEs and (ii) the second-order ODEs damped mass-spring system, both comparing Lagrange- and Chebyshev-encoded VQAs, (iii) the Poisson equation, comparing the Lagrange-encoded VQA with the discretized approach of Sato et al., and (iv) a first-order ODE with nonlinear coefficients addressed with the Lagrange-encoded VQA. Across these cases, the Lagrange-encoded VQA consistently achieves comparable or improved accuracy while significantly reducing circuit depth and computational overload, making it a promising candidate for NISQ devices.
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.