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The magnetic field dependence of energy levels in gapped single- and bilayergraphene quantum dots (QDs) defined by electrostatic gates is studiedanalytically in terms of the Dirac equation. Due to the absence of sharp edgesin these types of QDs, the valley degree of freedom is a good quantum number.We show that its degeneracy is efficiently and controllably broken by amagnetic field applied perpendicular to the graphene plane. This opens up afeasible route to create well-defined and well controlled spin- andvalley-qubits in graphene QDs. We also point out the similarities anddifferences in the spectrum between single- and bilayer graphene quantum dots.Striking in the case of bilayer graphene is the anomalous bulk Landau level(LL) that crosses the gap which results in crossings of QD states with thisbulk LL at large magnetic fields in stark contrast to the single-layer casewhere this LL is absent. The tunability of the gap in the bilayer case allowsus to observe different regimes of level spacings directly related to theformation of a pronounced ``Mexican hat'' in the bulk bandstructure. We discussthe applicability of such QDs to control and measure the valley isospin andtheir potential use for hosting and controlling spin qubits.

Bound states and magnetic field-induced valley splitting in gate-tunable graphene quantum dots