  Mesoscopic integrated circuits achieving high-fidelity control of elementary
quantum systems require new methodology for benchmarking. We offer
circuit-level statistical description of rare-error accumulation in terms of a
universal random-walk model for on-demand electron transfer. For a
high-fidelity single-electron circuit, realized in the experiment as a chain of
quantum dots in a GaAs/AlGaAs heterostructure, the error of the transfer
operation is probed by charge counting. Error rates for extra ($P_+$) or
missing ($P_-$) electrons of the electron shuttle are measured to $P_{-}=(6.92
\pm 0.14) \times 10^{-5}$ and $P_{+}=(2.13 \pm 0.08)\times 10^{-5}$ with
uncertainty due to correlated noise in the environment. Furthermore, precise
control over the timing of the random walk allows to explore the role of memory
as the clock frequency is increased.
