One of the most ambitious technological goals is the development of devices working under the laws of quantum mechanics. Among others, an important challenge to be resolved on the way to such breakthrough technology concerns the scalability of the available Hilbert space. Recently, proof-of-principle experiments were reported, in which the implementation of quantum algorithms (the Grover’s search algorithm, iSWAP-gate, etc.) in a single-molecule nuclear spin qudit (with d = 4) known as ¹⁵⁹TbPc₂ was described, where the nuclear spins of lanthanides are used as a quantum register to execute simple quantum algorithms. In this progress report, the goal of linear and exponential up-scalability of the available Hilbert space expressed by the quditdimension “d” is addressed by synthesizing lanthanide metal complexes as quantum computing hardware. The synthesis of multinuclear large-Hilbertspace complexes has to be carried out under strict control of the nuclear spin degree of freedom leading to isotopologues, whereby electronic coupling between several nuclear spin units will exponentially extend the Hilbert space available for quantum inf ... mehrormation processing. Thus, improved multilevel spin qudits can be achieved that exhibit an exponentially scalable Hilbert space to enable high-performance quantum computing and information storage.