We present results from a programmable quantum computer comprised of a
chain of individually trapped 171Yb+ ions. It features individual laser be
am addressing and individual readout\, and can be configured to run any se
quence of single- and two-qubit gates [1]. We combine this setup with diff
erent classical optimization routines to implement a so-called hybrid syst
em. Quantum-classical hybrid protocols offer a path towards the applicatio
n of near-term quantum computers for different optimization tasks. They ar
e attractive since part of the effort is outsourced to a classical machine
resulting in shallower and narrower quantum circuits\, which can be execu
ted with lower error rates.

\nWe have realized several experimental d
emonstrations relating to this approach\, such as the training of shallow
circuits for Generative Modeling using a Bayesian optimization routine [2]
\, tackling the Max-Cut problem using the Quantum Approximate Optimization
Algorithm (QAOA) [3]\, and the preparation of thermal quantum states [4].

\nRecent results\, limitations of the above methods\, and ideas for
boosting these concepts for scaling up the quantum-classical hybrid archit
ecture will be discussed.

\n[1] S. Debnath et al.\, Nature 563:63 (20
16)\; [2] D. Zhu et al.\, Science Advances 5\, 10 (2019)\; [3] O. Shehab e
t al.\, arXiv:1906.00476 (2019)\; [4] D. Zhu et al.\, arXiv:1906.02699 (20
19)