Abstract.The electric resistance between two arbitrary nodes on any infinite lattice structure of resistors that is a periodic tiling of space is obtained. Our general approach is based on the lattice Green's function of the Laplacian matrix associated with the network. We present several non-trivial examples to show how efficient our method is. Deriving explicit resistance formulas it is shown that the Kagomé, the diced and the decorated lattice can be mapped to the triangular and square lattice of resistors. Our work can be extended to the random walk problem or to electron dynamics in condensed matter physics.

We theoretically study a single-electron spin-valley qubit in an electrostatically defined quantum dot in a transition metal dichalcogenide monolayer, focusing on the example of MoS2. Coupling of the qubit basis states for coherent control is challenging, as it requires a simultaneous flip of spin and valley. Here, we show that a tilted magnetic field together with a short-range impurity, such as a vacancy, a substitutional defect, or an adatom, can give rise to a coupling between the qubit basis states. This mechanism renders the in-plane g-factor nonzero, and allows to control the qubit with an in-plane ac electric field, akin to electrically driven spin resonance. We evaluate the dependence of the in-plane g-factor and the electrically induced qubit Rabi frequency on the type and position of the impurity. We reveal highly unconventional features of the coupling mechanism, arising from symmetry-forbidden intervalley scattering, in the case when the impurity is located at a S site. Our results provide design guidelines for electrically controllable qubits in two-dimensional semiconductors.

Recent experiments on silicon nanostructures have seen breakthroughs toward scalable, long-lived quantum information processing. The valley degree of freedom plays a fundamental role in these devices, and the two lowest-energy electronic states of a silicon quantum dot can form a valley qubit. In this work, we show that a single-atom high step at the silicon/barrier interface induces a strong interaction of the qubit and in-plane electric fields, and analyze the consequences of this enhanced interaction on the dynamics of the qubit. The charge densities of the qubit states are deformed differently by the interface step, allowing non-demolition qubit readout via valley-tocharge conversion. A gate-induced in-plane electric field together with the interface step enables fast control of the valley qubit via electrically driven valley resonance. We calculate single-and two-qubit gate times, as well as relaxation and dephasing times, and present predictions for the parameter range where the gate times can be much shorter than the relaxation time and dephasing is reduced.

We present a theoretical study of the spin dynamics of a single electron confined in a quantum dot. Spin dynamics is induced by the interplay of electrical driving and the presence of a spatially disordered magnetic field, the latter being transverse to a homogeneous magnetic field. We focus on the case of strong driving, i.e., when the oscillation amplitude A of the electron's wave packet is comparable to the quantum dot length L. We show that electrically driven spin resonance can be induced in this system by subharmonic driving, i.e., if the excitation frequency is an integer fraction ( 1 2 , 1 3 , etc.) of the Larmor frequency. At strong driving we find that (i) the Rabi frequencies at the subharmonic resonances are comparable to the Rabi frequency at the fundamental resonance, and (ii) at each subharmonic resonance, the Rabi frequency can be maximized by setting the drive strength to an optimal, finite value. In the context of practical quantum information processing, these findings highlight the availability of subharmonic resonances for qubit control with effectivity close to that of the fundamental resonance, and the possibility that increasing the drive strength might lead to a decreasing qubit-flip speed. Our simple model is applied to describe electrical control of a spin-valley qubit in a weakly disordered carbon nanotube.

Topological properties of quantum systems could provide protection of information against environmental noise, and thereby drastically advance their potential in quantum information processing. Most proposals for topologically protected quantum gates are based on many-body systems, e.g., fractional quantum Hall states, exotic superconductors, or ensembles of interacting spins, bearing an inherent conceptual complexity. Here, we propose and study a topologically protected quantum gate, based on a one-dimensional single-particle tight-binding model, known as the Su-Schrieffer-Heeger chain. The proposed Y gate acts in the two-dimensional zero-energy subspace of a Y junction assembled from three chains, and is based on the spatial exchange of the defects supporting the zeroenergy modes. With numerical simulations, we demonstrate that the gate is robust against hopping disorder but is corrupted by disorder in the on-site energy. Then we show that this robustness is topological protection, and that it arises as a joint consequence of chiral symmetry, time-reversal symmetry and the spatial separation of the zero-energy modes bound to the defects. This setup will most likely not lead to a practical quantum computer, nevertheless it does provide valuable insight to aspects of topological quantum computing as an elementary minimal model. Since this model is non-interacting and non-superconducting, its dynamics can be studied experimentally, e.g., using coupled optical waveguides. arXiv:1902.01358v2 [cond-mat.mes-hall]