How Many Shots Does a Quantum Reservoir Need? Measurement Cost and Its Scaling in Buffered Quantum Reservoir Computing

Abstract

Quantum reservoir computing (QRC) inherits from classical reservoir computing the property that only a linear readout is trained, but unlike a classical reservoir its state is never directly observable: every feature is an expectation value estimated from a finite number of measurement shots. The shot budget is therefore a first-class hyperparameter with no classical analogue. Its effect on QRC has recently begun to be studied; what remains unsettled is how a required budget should be defined, and whether competing definitions agree. This paper argues that they do not, and that the disagreement is systematic rather than incidental. This paper distinguishes a \emph{precision budget} $N_\varepsilon$, the shots needed to estimate the reservoir features to a fixed relative accuracy, from a \emph{convergence budget} $N_\star$, the shots needed for the trained readout to approach its own noise-free ceiling. Proposition~\ref{prop:confound} shows that $N_\star$ depends on the noise-free error floor by construction, so that a configuration with a worse floor appears cheaper, and that configurations should therefore be compared at a fixed measurement budget rather than by the budget each needs to reach its own ceiling.