  Grover's search algorithm is one of the first quantum algorithms to exhibit a
provable quantum advantage. It forms the backbone of numerous quantum
applications and is widely used in benchmarking efforts. Here, we report
better-than-classical success probabilities for a complete Grover search
algorithm on the largest scale demonstrated to date, of up to five qubits,
using two different IBM superconducting transmon qubit platforms. This is
enabled, on the four and five-qubit scale, by error suppression via robust
dynamical decoupling pulse sequences, without which we do not observe
better-than-classical results. Further improvements arise after the use of
measurement error mitigation, but the latter is insufficient by itself for
achieving better-than-classical performance. For two qubits, we demonstrate a
success probability of 99.5% via the use of the [[4,2,2]] quantum
error-detection (QED) code. This constitutes a demonstration of quantum
algorithmic breakeven via QED. Along the way, we introduce algorithmic error
tomography, a method of independent interest that provides a holistic view of
the errors accumulated throughout an entire quantum algorithm, filtered via the
errors detected by the QED code used to encode the circuit. We demonstrate that
algorithmic error tomography provides a stringent test of an error model based
on a combination of amplitude damping, dephasing, and depolarization.
