The field of quantum simulations promises to tackle formidable problems which are inaccessible to classical computers due to their complexity. These questions range from the simulation of many-body effects in materials and fundamental questions of physics, to the exact calculation of chemical and biological processes. A possible way to tackle these problems is through the use of analog quantum simulators, a class of controllable quantum systems in which the underlying Hamiltonian is directly mappable to the problem of interest. Combined with a high degree of control and tunability over individual parameters, analog quantum simulators promise to reach practicality already at intermediate device scales, without the need for scaling up to the hundreds of thousands of sites necessary for their digital counterparts.

In this thesis,we study gate-defined semiconductor quantum dot arrays as a possible candidate for analog quantum simulators. Semiconductor quantum dots have become an attractive system for quantum technologies given their small footprint, fast operation, long coherence times and individual and in-situ tunability of couplings and on-site energies. Additionally, the system is natively described by the Fermi-Hubbard Hamiltonian, a complex many-body Hamiltonian which is predicted to give rise to a variety of interesting phases of matter. This makes quantum dot arrays a promising platform for many-body quantum simulations. However, scaling beyond a few quantum dots while maintaining a high degree of controllability has remained a long-standing challenge for the field.

In this thesis, we study a 2×4 quantum dot array of Ge/SiGe quantum dots as an analog quantum simulator, exploring the emergence of correlated spin and charge phenomena. With the quantum dots arranged in a ladder geometry, this device constitutes one of the first two-dimensional realizations of quantum dot arrays. After introducing the relevant theoretical and experimental concepts, we showcase the experimental simulation of two different types of quasiparticles and their dynamics, each emerging in a different parameter regime of the Fermi-Hubbard model. First, we look at the creation and transport of electron-hole pairs or excitons. Here, we crucially exploit the existence of strong Coulomb repulsion between both legs of the quantum dot ladder. Subsequently, we explore the dynamics of single-spin and two-spin excitations as they evolve through the array. These so-called magnon and triplon excitations propagate thanks to nearest-neighbor exchange interactions which we can locally and independently tune over a large range of parameters. This is achieved thanks to a novel scheme of crosstalk compensation which we develop and showcase in this thesis. In the last chapter, we summarize our findings, discuss challenges going forward and provide a variety of possible experiments which could be attainable with current state-of-the-art quantum dot quantum simulators.

Coupled spins in semiconductor quantum dots are a versatile platform for quantum computing and simulations of complex many-body phenomena. However, on the path of scale-up, crosstalk from densely packed electrodes poses a severe challenge. While crosstalk onto the quantum dot potentials is nowadays routinely compensated for, crosstalk on the exchange interaction is much more difficult to tackle because it is not always directly measurable. Here we propose and implement a way of characterizing and compensating crosstalk on adjacent exchange interactions by following the singlet-triplet avoided crossing in Ge. We show that we can easily identify the barrier-to-barrier crosstalk element without knowledge of the particular exchange value in a 2×4 quantum dot array. We uncover striking differences among these crosstalk elements that can be linked to the geometry of the device and the barrier gate fan-out. We validate the method by tuning up four-spin Heisenberg chains. The same method should be applicable to longer chains of spins and to other semiconductor platforms in which mixing of the singlet and the lowest-energy triplet is present or can be engineered. Additionally, this procedure is well-suited for automated tuning routines as we obtain a standout feature that can be easily tracked and directly returns the magnitude of the crosstalk.

Quantum systems with engineered Hamiltonians can be used to study many-body physics problems to provide insights beyond the capabilities of classical computers. Semiconductor gate-defined quantum dot arrays have emerged as a versatile platform for realizing generalized Fermi-Hubbard physics, one of the richest playgrounds in condensed matter physics. In this work, we employ a germanium 4×2 quantum dot array and show that the naturally occurring long-range Coulomb interaction can lead to exciton formation and transport. We tune the quantum dot ladder into two capacitively coupled channels and exploit Coulomb drag to probe the binding of electrons and holes. Specifically, we shuttle an electron through one leg of the ladder and observe that a hole is dragged along in the second leg under the right conditions. This corresponds to a transition from single-electron transport in one leg to exciton transport along the ladder. Our work paves the way for the study of excitonic states of matter in quantum dot arrays.

The coherent control of interacting spins in semiconductor quantum dots is of strong interest for quantum information processing and for studying quantum magnetism from the bottom up. Here we present a 2 × 4 germanium quantum dot array with full and controllable interactions between nearest-neighbour spins. As a demonstration of the level of control, we define four singlet–triplet qubits in this system and show two-axis single-qubit control of each qubit and SWAP-style two-qubit gates between all neighbouring qubit pairs, yielding average single-qubit gate fidelities of 99.49(8)–99.84(1)% and Bell state fidelities of 73(1)–90(1)%. Combining these operations, we experimentally implement a circuit designed to generate and distribute entanglement across the array. A remote Bell state with a fidelity of 75(2)% and concurrence of 22(4)% is achieved. These results highlight the potential of singlet–triplet qubits as a competing platform for quantum computing and indicate that scaling up the control of quantum dot spins in extended bilinear arrays can be feasible.