  Empirical evidence for a gap between the computational powers of classical
and quantum computers has been provided by experiments that sample the output
distributions of two-dimensional quantum circuits. Many attempts to close this
gap have utilized classical simulations based on tensor network techniques, and
their limitations shed light on the improvements to quantum hardware required
to frustrate classical simulability. In particular, quantum computers having in
excess of $\sim 50$ qubits are primarily vulnerable to classical simulation due
to restrictions on their gate fidelity and their connectivity, the latter
determining how many gates are required (and therefore how much infidelity is
suffered) in generating highly-entangled states. Here, we describe recent
hardware upgrades to Quantinuum's H2 quantum computer enabling it to operate on
up to $56$ qubits with arbitrary connectivity and $99.843(5)\%$ two-qubit gate
fidelity. Utilizing the flexible connectivity of H2, we present data from
random circuit sampling in highly connected geometries, doing so at
unprecedented fidelities and a scale that appears to be beyond the capabilities
of state-of-the-art classical algorithms. The considerable difficulty of
classically simulating H2 is likely limited only by qubit number, demonstrating
the promise and scalability of the QCCD architecture as continued progress is
made towards building larger machines.
