As the complexity of mesoscopic quantum devices increases, simulations are becoming an invaluable tool for understanding their behavior. This is especially true for the superconductor-semiconductor heterostructures used to build Majorana-based topological qubits, where quantitatively understanding the interplay of topological superconductivity, disorder, semiconductor quantum dots, Coulomb blockade and noise has been essential for progress on device design and interpretation of measurements. In this paper, we describe a general framework to simulate the low-energy quantum dynamics of such complex systems. We illustrate our approach by computing the dispersive gate sensing (DGS) response of quantum dots coupled to topological superconductors. We start by formulating the DGS response as an open-system quantum dynamics problem, which allows a consistent treatment of drive backaction as well as quantum and classical noise. For microscopic quantum problems subject to Coulomb-blockade, where a direct solution in the exponentially large many-body Hilbert space would be prohibitive, we introduce a series of controlled approximations that incorporate ideas from tensor network theory and quantum chemistry to reduce this Hilbert space to a few low-energy degrees of freedom that accurately capture the low-energy quantum dynamics. We demonstrate the methods introduced in this paper on the example of a single quantum dot coupled to a topological superconductor and a microscopic realization of the fermion parity readout setup of Aghaee et al. arXiv:2401.09549 (2024).
As the complexity of mesoscopic quantum devices increases, simulations are becoming an invaluable tool for understanding their behavior. This is especially true for the superconductor-semiconductor heterostructures used to build Majorana-based topological qubits, where quantitatively understanding the interplay of topological superconductivity, disorder, semiconductor quantum dots, Coulomb blockade and noise has been essential for progress on device design and interpretation of measurements. In this paper, we describe a general framework to simulate the low-energy quantum dynamics of such complex systems. We illustrate our approach by computing the dispersive gate sensing (DGS) response of quantum dots coupled to topological superconductors. We start by formulating the DGS response as an open-system quantum dynamics problem, which allows a consistent treatment of drive backaction as well as quantum and classical noise. For microscopic quantum problems subject to Coulomb-blockade, where a direct solution in the exponentially large many-body Hilbert space would be prohibitive, we introduce a series of controlled approximations that incorporate ideas from tensor network theory and quantum chemistry to reduce this Hilbert space to a few low-energy degrees of freedom that accurately capture the low-energy quantum dynamics. We demonstrate the methods introduced in this paper on the example of a single quantum dot coupled to a topological superconductor and a microscopic realization of the fermion parity readout setup of Aghaee et al. arXiv:2401.09549 (2024).
Spin and topological-qubit based quantum computers require an easily scalable and highly sensitive method for readout and manipulation, for which Dispersive Gate Sensing (DGS) is a promising candidate [1, 2]. DGS enables sensing of single electron tunneling events in a mesoscopic system by measuring a reflected Radio Frequency (RF) signal from a resonator capacitively coupled to one of the system’s gate electrodes. From the resonator’s perspective, the system is modeled as an effective ‘parametric capacitance’ [3]. Motivated by its viability for qubit readout and control, we measure transport characteristics of a semiconducting-superconducting hybrid charge island system using this method. Doing so, we observe a spin degeneracy modulation of interdot tunnel couplings resulting from hybridization of a dot orbital with a subgap state of the superconductor, measured entirely within Coulomb blockade. Without DC measurement, we also track a subgap state’s energy via voltage intervals between island charge states as a function of field. Subsequently, we attempt to correlate changes in the resonator’s internal dissipation with coherence of tunneling into the superconducting island. In particular, we collapse a superconducting island’s lowest energy state to the Fermi level by applying a magnetic field. Contrary to expectations, internal dissipation in the resonator did not change dramatically with field, in fact decreasing at charge degeneracy points at all field strengths. We corroborate measurements with existing theory, augmented by simple analytical and numerical models demonstrating that coherence factors of superconducting quasiparticle states and the degeneracy of dot orbitals modulate parametric capacitance. A master equation model of finite temperature interdot tunneling is solved, agreeing with experimental evidence that Sisyphus dissipation is negligible when a dot orbital is hybridized with multiple independent quasiparticle states. To be employed in quantum computers, a full understanding of all phenomena contributing to a DGS signal in these hybrid systems is critical. Hence, these results mark a step towards unambiguous readout of Majorana parity in topological qubits [4].
Superconductivity in its various manifestations has been stimulating both experimental and theoretical progress in condensed-matter physics for more than a hundred years. The remarkable property of electrons to pair up and form quasi-particles gives rise to a plethora of phenomena featuring important practical applications not only in science, but, for instance, also in medicine and metrology. Recently, new directions in investigating this fascinating subject emerged, such as superconductivity out-of equilibrium and topological superconductors. Providing experimental evidence for enhanced superconducting correlations in optically pumped copper oxides at temperatures far above the equilibrium transition temperature, the first issue caused considerable excitement. On the other hand, topological superconductors are believed to provide realizations of highly fault-tolerant qubits by means of hosting non-Abelian quasi-particles, which can be the building blocks of scalable quantum computers. Experimentally verifying the emergence of these Majorana edge modes, exotic quasi-particles in heterostructures consisting of a conventional superconductor and semiconductors or topological insulators, is one of the most urgent questions to be answered right now. Both subjects cannot be accounted for with analytically solvable approximations only, and also provide very challenging numerical problems. We implemented a matrix-product state (MPS) based toolkit exploiting U(1) symmetries, providing a flexible and efficient platform to study these complex systems. In order to efficiently simulate out-of equilibrium setups we studied, compared, and developed time-evolution algorithms for MPS enabling us to choose the most suitable method for a given task. We also developed a new framework to represent operators in an enlarged Hilbert space so that benefits from conserving U(1) symmetries can also be exploited in systems that originally break such symmetries (projected purification). Using this method we could efficiently model mesoscopic phenomena such as a charging energy controlled by a gate electrode without further approximations. Equipped with this techniques we studied out-of equilibrium spectral functions to explore how to identify superconducting correlations more reliably on ultra-short timescales. We found conclusive evidence that in particular two-particle spectral functions yield excellent probes for the formation of a (quasi-)condensate out-of equilibrium. Furthermore, we also investigated the question whether in a particular model system there is the possibility of true long-range order out-of equilibrium by studying correlation matrices and the scaling of their eigenvalues. Here, we observe a change in the algebraic decay of the correlations, even though the extrapolated order parameter is still zero within the error bounds. Furthermore, we also investigated the effects of coupling a superconductor-semiconductor heterostructure, which is subject to an in-plane magnetic field and a charging energy controlled by a gate voltage, to normal leads. In the context of experimentally verifying the existence of Majorana edge modes, such systems are believed to be the most promising and recent studies seem to underline this expectation. However, in order to consistently analyze the experimental data, the effects of quantum fluctuations caused by hybridization of the heterostructure with the leads have to be understood. Here, only perturbative limits are available so far, i.e., the weak and strong tunneling limit, while the experimentally relevant regime is expected to be somewhere inbetween. We aimed to fill this gap using the projected purification method to calculate the ground state phase diagram over a wide parameter regime. Our results indicate that the experimental situation is much more involved than what is predicted from perturbative analysis.
Quantum computers can solve some problems exponentially faster than classical computers. Unfortunately, the computational power of quantum computers is currently limited by the number of working qubits. It is difficult to scale up these systems, because qubits are easily affected by noise in their environment. This noise leads to decoherence: loss of the qubit’s encoded information. A possible solution to diminish decoherence is using Majorana box qubits, as these qubits are predicted to be insensitive to local noise. However, this promising type of qubit does not exist yet. With the research described in this thesis, we aim to contribute to the development of Majorana box qubits (MBQs). In these qubits, Majorana zero modes, the basic elements of MBQs, are contained within a superconducting island to suppress Majorana parity fluctuations caused by quasiparticle poisoning. To enable parity readout of the MBQ, these modes are coupled to quantum dots within a nanowire network. To help realize MBQs, we need a better understanding of quasiparticles in superconducting islands, parity-readout techniques, and ways to fabricate nanowire networks. These three aspects are the focus of the experiments presented in this thesis. To study superconducting islands and readout techniques, we used InAs semiconductor nanowires with an epitaxially grown Al shell. Majorana signatures have already been observed in such nanowires. We addressed quasiparticle dynamics in superconducting islands by measuring the gate-charge modulation of the switching current. We found a consistent 2e-periodic modulation at zero magnetic field, and an exponential decrease of parity lifetime with increasing magnetic field. We explored MBQ readout, using a quantum dot level as a proxy for a Majorana zero mode, and measured its charge hybridization with another dot using gate-based readout. We showed that we can rapidly discriminate between two settings with different tunnel couplings, demonstrating the potential of gate-based readout to measure MBQs. And, using gate-based readout, we could study charge-transfer processes occurring in hybrid structures of superconducting islands coupled to quantum dots. Finally, to find a good material platform for nanowire networks, we characterized two two-dimensional systems. We realized quantum point contacts in InSb, which we used to measure the g-factor anisotropy, and effective electron mass in this system. And, we studied the spin-orbit interaction in InAs/GaSb by extracting the difference in density between electrons with different spin orientations. This thesis finishes with a proposal for a series experiments to realize MBQs. These experiments make use of superconducting islands and the reflectometry setup we developed for gate-based readout.
Superconducting devices subject to strong charging energy interactions and Coulomb blockade are one of the key elements for the development of nanoelectronics and constitute common building blocks of quantum computation platforms and topological superconducting setups. The study of their transport properties is nontrivial and some of their nonperturbative aspects are hard to capture with the most ordinary techniques. Here we present a matrix product state approach to simulate the real-time dynamics of these systems. We propose a study of their transport based on the analysis of the currents after quantum quenches connecting such devices with external leads. Our method is based on the combination of a Wilson chain construction for the leads and a mean-field BCS description for the superconducting scatterers. In particular, we employ a quasiparticle energy eigenbasis which greatly reduces their entanglement growth and we introduce an auxiliary degree of freedom to encode the device's total charge. This approach allows us to treat nonperturbatively both their charging energy and coupling with external electrodes. We show that our construction is able to describe the Coulomb diamond structure of a superconducting dot with subgap states, including its sequential tunneling and cotunneling features. We also study the conductance zero-bias peaks caused by Majorana modes in a blockaded Kitaev chain and compare our results with common Breit-Wigner predictions.
Hybrid semiconductor-superconductor qubits have recently emerged as a promising alternative to traditional platforms, combining material advantages with device-level tunability. A defining feature is their gate-tunable Josephson coupling, enabling superconducting qubit architectures with full electric-field control and offering a path toward scalable, low-crosstalk quantum processors. This approach seeks to merge benefits of superconducting and semiconductor qubits, for instance by encoding quantum information in the spin of a quasiparticle occupying an Andreev bound state, thus combining long coherence times with fast, flexible control. Progress has accelerated through bottom-up engineering of Andreev states in coupled quantum dot arrays, leading to architectures such as minimal Kitaev chains hosting Majorana zero modes. In parallel, Hamiltonian-protected designs aim to enhance resilience against local noise and decoherence by exploiting superconducting phase dynamics and discrete charge or flux degrees of freedom. This article reviews recent theoretical and experimental advances in hybrid qubits, providing an overview of physical mechanisms, device implementations, and emerging architectures, with emphasis on their potential for (topologically) protected quantum information processing. While many designs remain at proof-of-concept stage, rapid progress suggests practical demonstrations may soon be achievable.
Topology-related ideas might lead to noise-resilient quantum computing. For example, it is expected that the slow spatial exchange (``braiding'') of Majorana zero modes in superconductors yields quantum gates that are robust against disorder. Here, we report our numerical experiments, which describe the dynamics of a Majorana qubit built from quantum dots controlled by time-dependent gate voltages. Our protocol incorporates nonprotected control, braiding-based protected control, and readout of the Majorana qubit. We use the Kitaev chain model for the simulations, and we focus on the case when the main source of errors is quasistatic charge noise affecting the hybridization energy splitting of the Majorana modes. We provide quantitative guidelines to suppress both diabatic errors and disorder-induced qubit dephasing, such that a fidelity plateau is observed as the hallmark of the topological quantum gate. Our simulations predict realistic features that are expected to be seen in future braiding experiments with Majorana zero modes and other topological qubit architectures.