  Quantum control in large dimensional Hilbert spaces is essential for
realizing the power of quantum information processing. For closed quantum
systems the relevant input/output maps are unitary transformations, and the
fundamental challenge becomes how to implement these with high fidelity in the
presence of experimental imperfections and decoherence. For two-level systems
(qubits) most aspects of unitary control are well understood, but for systems
with Hilbert space dimension d>2 (qudits), many questions remain regarding the
optimal design of control Hamiltonians and the feasibility of robust
implementation. Here we show that arbitrary, randomly chosen unitary
transformations can be efficiently designed and implemented in a large
dimensional Hilbert space (d=16) associated with the electronic ground state of
atomic 133Cs, achieving fidelities above 0.98 as measured by randomized
benchmarking. Generalizing the concepts of inhomogeneous control and dynamical
decoupling to d>2 systems, we further demonstrate that these qudit unitary maps
can be made robust to both static and dynamic perturbations. Potential
applications include improved fault-tolerance in universal quantum computation,
nonclassical state preparation for high-precision metrology, implementation of
quantum simulations, and the study of fundamental physics related to open
quantum systems and quantum chaos.
