The study of quantum many-body systems is crucial to our understanding of nature at the microscopic scale. As analytical solutions quickly become intractable, the development of efficient computational simulation techniques is indispensable to predict the behaviour of quantum systems. This task is particularly challenging as the dimension of the underlying Hilbert space grows exponentially with the system size. While classical computational methods have achieved remarkable success in specific regimes, their applicability is fundamentally limited by this exponential scaling. Quantum computers offer a natural platform for representing and simulating quantum systems, but current devices are constrained by noise and limited qubit counts, drastically restricting the algorithms that can be run. In this context, hybrid quantum-classical algorithms provide a promising approach to combine the strengths of both paradigms. This thesis develops hybrid quantum-classical algorithms for the simulation of quantum many-body systems. The key idea is to perform classically difficult tasks on near-term quantum devices, while leveraging classical computational power to control, correct, and extend the obtained results. Within this framework, we introduce two complementary approaches for the simulation of quantum dynamics. The first method employs circuit knitting techniques to distribute variational quantum simulations across multiple quantum devices. The entanglement between weakly coupled clusters of highly correlated spins is recovered in classical post-processing, where the overhead is controlled through a constrained optimisation problem. The second method combines a simplified, Trotterised quantum dynamics simulation with a parametrised classical ansatz trained to correct for the approximations made on the quantum device. We show that this method can be employed to correct Trotter errors, implement long-range gates with hardware-efficient circuits, or simulate larger systems by incorporating additional degrees of freedom purely classically. In addition, we present a hybrid approach to computing thermal properties of quantum many-body systems via the Quantum Finite Temperature Lanczos Method. A Krylov subspace is generated on a quantum computer by combining the efficient preparation of typical states with real-time evolution. A classical computer then diagonalises the system in the low-dimensional subspace and estimates the finite-temperature expectation values through trace estimation. The results presented in this thesis demonstrate that hybrid quantum-classical algorithms can expand the range of systems that can be studied on near-term quantum devices. Conversely, we show that incorporating quantum subroutines into classical simulation techniques can improve their efficiency. These results suggest that, rather than treating classical and quantum simulation as competing paradigms, their careful integration can provide a more effective route toward practical quantum many-body simulation.
The study of quantum many-body systems is crucial to our understanding of nature at the microscopic scale. As analytical solutions quickly become intractable, the development of efficient computational simulation techniques is indispensable to predict the behaviour of quantum systems. This task is particularly challenging as the dimension of the underlying Hilbert space grows exponentially with the system size. While classical computational methods have achieved remarkable success in specific regimes, their applicability is fundamentally limited by this exponential scaling. Quantum computers offer a natural platform for representing and simulating quantum systems, but current devices are constrained by noise and limited qubit counts, drastically restricting the algorithms that can be run. In this context, hybrid quantum-classical algorithms provide a promising approach to combine the strengths of both paradigms. This thesis develops hybrid quantum-classical algorithms for the simulation of quantum many-body systems. The key idea is to perform classically difficult tasks on near-term quantum devices, while leveraging classical computational power to control, correct, and extend the obtained results. Within this framework, we introduce two complementary approaches for the simulation of quantum dynamics. The first method employs circuit knitting techniques to distribute variational quantum simulations across multiple quantum devices. The entanglement between weakly coupled clusters of highly correlated spins is recovered in classical post-processing, where the overhead is controlled through a constrained optimisation problem. The second method combines a simplified, Trotterised quantum dynamics simulation with a parametrised classical ansatz trained to correct for the approximations made on the quantum device. We show that this method can be employed to correct Trotter errors, implement long-range gates with hardware-efficient circuits, or simulate larger systems by incorporating additional degrees of freedom purely classically. In addition, we present a hybrid approach to computing thermal properties of quantum many-body systems via the Quantum Finite Temperature Lanczos Method. A Krylov subspace is generated on a quantum computer by combining the efficient preparation of typical states with real-time evolution. A classical computer then diagonalises the system in the low-dimensional subspace and estimates the finite-temperature expectation values through trace estimation. The results presented in this thesis demonstrate that hybrid quantum-classical algorithms can expand the range of systems that can be studied on near-term quantum devices. Conversely, we show that incorporating quantum subroutines into classical simulation techniques can improve their efficiency. These results suggest that, rather than treating classical and quantum simulation as competing paradigms, their careful integration can provide a more effective route toward practical quantum many-body simulation.
Computational methods are central to modern physics, yet many problems, especially those involving Quantum Many-Body Systems (QMBSs), remain challenging at scale. The core difficulty of simulating QMBSs is the exponential growth of the solution space with system size. This renders exact treatments intractable and pushes approximate methods to their limits when tackling two- and higher-dimensional systems and long-time dynamics. Quantum computers offer a quantum-native platform for representing and manipulating such systems. Recent algorithmic and technological advancements have enabled experiments on quantum processors in the quantum utility regime, a scale at which exact classical reproduction is no longer feasible. This represents the first step towards realizing quantum advantage, where quantum computers solve specific, practically relevant problems more efficiently than any classical approach. While advancing towards the goal of quantum advantage, useful computations on quantum processors are expected to be realised through hybrid quantum-classical algorithms, which combine quantum experiments and classical computational resources. A prominent example are variational quantum algorithms, where a Parametrized Quantum Circuit (PQC) prepares a trial quantum state whose parameters are iteratively updated by a classical optimisation routine to minimize a target cost function (e.g., the energy of a target Hamiltonian). However, the optimisation of such PQCs is affected by trainability issues, such as barren plateaus in the parameter landscape, which can render the optimisation increasingly difficult for larger system sizes. This motivates the development of novel hybrid approaches overcoming the limits of PQCs. In this thesis, we introduce a new class of approaches based on hybrid Tensor Networks (TNs). A hybrid TN is obtained by integrating quantum states prepared on a quantum processor within TNs -- one of the state-of-the-art approaches for simulating quantum systems. We first show how hybrid TNs can be used for ground-state searches within a variational setting. Then, we further extend the framework to non-variational hybrid TN ansätze for which the quantum tensors are prepared via quantum annealing and subsequently post-processed by classical TNs. In the context of quantum machine learning, which also relies on PQCs, we further investigate heuristic best practices that mitigate the same exponential concentration effects in kernel-based models for anomaly detection. Another class of hybrid algorithms are quantum error-mitigation workflows, where classical post-processing is used to reconstruct expectation values from noise-affected circuit executions. Such methods generally feature a trade-off between the bias and the variance: the price to pay for mitigating noise in a bias-free fashion is that the number of measurements usually grows exponentially with the depth of the circuit in order to achieve a fixed accuracy. In this thesis, we develop a low-resource error-mitigation strategy that reduces the measurement overhead in practical digital Hamiltonian-simulation workflows. We apply this method to Trotterized dynamics in a lattice-gauge-theory model, enabling the extraction of expectation values with improved accuracy at time-scales that allow the observation of post-collision scattering dynamics.
The accurate and efficient simulation of many body systems has been a long-standing challenge for quantum chemists and physicists. At the heart of this challenge lies the correlation between quantum particles, which is crucial in understanding various physical phenomena, but is numerically hard to calculate. Over the decades, there has been remarkable progress in simplifying this problem aided by mathematical and chemical approximations and the availability of powerful computers. However, an exact solution to the problem still remains intractable as it requires exponential re- sources with respect to the system size. Recent advances in the field of quantum computing have shown promise in Feynman’s idea of simulating quantum mechanics on quantum computers. It has been proposed that an ideal fault-tolerant quantum computer can reduce the scaling of these sim- ulations from exponential to polynomial. However, initial quantum algorithms have shown that there is a large gap between the capabilities of current hardware and the resources required for simulating quantum systems of interest. The work of this thesis focuses on reducing this gap by introducing two novel algorithms for quantum xiv chemistry simulations, which are suitable for near-term hardware. First, we intro- duce a hybrid algorithm to decrease the total number of operations required for the quantum simulations. We use a classical computer to generate an effective Hamilto- nian containing O(n2) terms compared to the O(n4) terms of the full Hamiltonian, where n is the number of orbitals. This sparse Hamiltonian can then be used with a high-level method to recover the ground state energies. We demonstrate that this sparsification of the Hamiltonian reduces the number of quantum operations required for the simulation by an order of magnitude, thus making it accessible for the near- term Hardware. Our second algorithm aims to make the calculation of dynamic correlation functions (Green’s functions) more feasible on these quantum machines. This algorithm is designed with the motive of avoiding the use of time-evolution and reduced use of two-qubit gates. We demonstrate that we can reproduce the Green’s function within reasonable error limits using this approach. We have also discussed the use of re-sampling techniques for proper error-propagation of the stochastic data obtained from the quantum machines. The general structure of this thesis is as follows. We begin by motivating this work in Chapter 1. Chapters 2 and 3 provide a general introduction of quantum computing and Green’s function based methods, respectively. In chapter 4, the development of effective Hamiltonian is discussed in detail. Chapter 5 demonstrates the use of these effective Hamiltonians on the quantum machine for time evolution. In chapter 6, we have discussed the algorithm to calculate Green’s functions on a quantum machine. This is followed by conclusions and future directions in Chapters 7 and 8, respectively.
Computer simulation is central to understanding the behaviour of matter. Yet, more than a century after the development of quantum theory, first-principles simulation of chemistry remains a daunting challenge, owing to the exponential growth in memory required to represent interacting quantum particles such as electrons in molecules. Quantum computing offers, in principle, a fundamentally more efficient paradigm for simulating quantum systems, with quantum chemistry often cited as a flagship application. However, many quantum algorithms face intrinsic limitations, and any genuine quantum advantage must surpass the strong baseline set by decades of progress in classical algorithms and hardware. While classical methods often suffice for simulating weakly correlated systems, systems with many strongly correlated electrons remain a challenge for both classical and quantum approaches. In this thesis, we develop quantum algorithms for solving the time-independent Schrödinger equation, with a focus on molecular systems with strongly correlated electrons. We introduce a framework for identifying a compact set of states that parametrise the many-body wavefunction by capturing the correlation due to spin coupling between localised electrons. This representation requires very few variational parameters but incurs a very high memory cost on classical computers. To overcome this, we design quantum circuits that efficiently prepare such states by exploiting their symmetry, encoding the dominant entanglement structure of electronic wavefunctions into the initial qubit states. Through theoretical analysis and classical simulations of quantum computers, we show that this substantially reduces the quantum resources required to accurately compute ground- and excitedstate properties. Finally, we develop an approach for approximating the optimal ansatz within the variational quantum eigensolver, an algorithm designed for near-term quantum devices. Our method constructs wavefunctions from products of symmetrypreserving unitary operators and couples discrete and continuous optimisation to yield highly compact representations of strongly correlated electrons using shallow quantum circuits. Together, these results establish complementary directions for embedding problemspecific structure into quantum simulation algorithms, advancing the practical simulation of strongly correlated electrons.
Quantum computing not only holds the potential to solve long-standing problems in quantum physics, but also to offer speed-ups across a broad spectrum of other fields. Access to a computational space that incorporates quantum effects, such as superposition and entanglement, enables the derivation of promising quantum algorithms for important tasks, including preparing the ground state of a quantum system or predicting its evolution over time. Successfully tackling these tasks promises insights into significant theoretical and technological questions, such as superconductivity and the design of new materials. The aim of quantum algorithms is to use a series of quantum operations in a quantum circuit to solve a problem beyond the reach of classical computers. However, the noise and limited scale of current quantum computers restricts these circuits to moderate sizes and depths. As a result, many prominent algorithms are currently infeasible to run for problem sizes of practical interest. In response, recent research focused on variational quantum algorithms, which allow the selection of circuits that act within a quantum device's capabilities. Yet, these algorithms can require the execution of a large number of circuits, leading to prohibitively long computation times. This doctoral thesis develops two main techniques to reduce these quantum computational resource requirements, with the goal of scaling up application sizes on current quantum processors. The first approach is based on stochastic approximations of computationally costly quantities, such as quantum circuit gradients or the quantum geometric tensor (QGT). The second method takes a different perspective on the QGT, leading to a potentially more efficient description of time evolution on current quantum computers. Both techniques rely on maintaining available information and only computing necessary corrections, instead of re-computing possibly redundant data. The main focus of application for our algorithms is the simulation of quantum systems, broadly defined as including the preparation of ground and thermal states, and the real- and imaginary-time propagation of a system. The developed subroutines, however, can further be utilized in the fields of optimization or machine learning. Our algorithms are benchmarked on a range of representative models, such as Ising or Heisenberg spin models, both in numerical simulations and experiments on the hardware. In combination with error mitigation techniques, the latter is scaled up to 27 qubits; into a regime that variational quantum algorithms are challenging to scale to on noisy quantum computers without our algorithms.
A complex but important challenge in understanding quantum mechanical phenomena is the simulation of quantum many-body dynamics. Although quantum computers offer significant potential to accelerate these simulations, their practical application is currently limited by noise and restricted scalability. In this work, we address these problems by introducing a hybrid ansatz combining the strengths of quantum and classical computational methods. Using Trotterization, we propose to evolve an initial state on the quantum computer according to a simplified Hamiltonian, focusing on terms that are difficult to simulate classically. A classical model then corrects the simulation by including the terms omitted in the quantum circuit. While the classical ansatz is optimized during the time evolution, the quantum circuit has no variational parameters. Derivatives can thus be calculated purely classically, avoiding challenges arising in the optimization of parametrized quantum circuits. We demonstrate three applications of this hybrid method. First, our approach allows us to avoid gates in the quantum circuit by restricting the quantum part of the ansatz to hardware-efficient terms of the Hamiltonian. Second, we can mitigate errors arising from the Trotterization of the time-evolution unitary. Finally, we can extend the system size while keeping the number of qubits in the quantum circuit constant by including additional degrees of freedom in the classical ansatz.
Simulating quantum many-body systems is essential for understanding complex phenomena in condensed matter, chemistry, and materials science. Quantum computing offers a way to overcome the limitations of classical methods by operating directly in exponentially large Hilbert spaces. In particular, classical methods often fail to capture the dynamics of strongly correlated systems, which makes quantum algorithms a promising alternative for exploring these challenging regimes. This thesis uses quantum dynamics simulations to probe the behaviour of many-body systems. We first introduce a spectroscopy-based protocol to extract excitation spectra from the time evolution of observables and analyse its performance on near-term quantum devices. We show that the protocol requires only short evolution times and is resilient to errors, facilitating its implementation on real hardware. Building on this, we simulate fermionic excitation spectra using current quantum devices, introducing resource optimisations - including an approximate free-fermion initial state, a unitary quench without ancillae, and reduced Trotter steps - that significantly lower circuit depth. These improvements enable simulations at system sizes competitive with the state of the art, yielding results that match classical benchmarks. While most studies focus on unitary dynamics of isolated systems, open quantum dynamics is increasingly relevant for describing non-equilibrium processes across biology, materials science, and quantum control. We extend our scope to systems coupled to an environment and governed by Lindbladian evolution. By studying simulation complexity through physically meaningful quantities - such as correlation length and mixing time - we explore whether open-system dynamics can be as tractable as their unitary counterparts. These findings provide insights into the computational complexity landscape of open-system simulation and highlight regimes where quantum methods offer a clear advantage.