  We propose a set of Bell-type nonlocal games that can be used to prove an
unconditional quantum advantage in an objective and hardware-agnostic manner.
In these games, the circuit depth needed to prepare a cyclic cluster state and
measure a subset of its Pauli stabilizers on a quantum computer is compared to
that of classical Boolean circuits with the same, nearest-neighboring gate
connectivity. Using a circuit-based trapped-ion quantum computer, we prepare
and measure a six-qubit cyclic cluster state with an overall fidelity of 60.6%
and 66.4%, before and after correcting for measurement-readout errors,
respectively. Our experimental results indicate that while this fidelity
readily passes conventional (or depth-0) Bell bounds for local hidden-variable
models, it is on the cusp of demonstrating a higher probability of success than
what is possible by depth-1 classical circuits. Our games offer a practical and
scalable set of quantitative benchmarks for quantum computers in the
pre-fault-tolerant regime as the number of qubits available increases.
