Can You Trust Your Shot Budget? Benchmarking Conformal Risk Control for Variational Quantum Algorithms

How many times must you run a variational quantum circuit before you can trust its output? Run too few shots and the energy estimate is just noise; run too many and you waste scarce, expensive quantum hardware time. In practice the number is chosen by folklore, with no guarantee attached. We ask whether that choice can carry a formal risk guarantee, and we insist on charging each method for the shots its own guarantee consumes. Benchmarking five ways of choosing the budget, from distribution-free risk control to classical statistical intervals, on molecular ground-state problems and a real 156-qubit IBM processor, we find no winner among the standard methods: the safe ones never save a shot, and the thrifty ones break their promised error rate in up to three of every five runs when budgets are tight. The tradeoff, though, is not fundamental. A smarter distribution-free method that recycles the same measurements to calibrate many budgets at once keeps both promises once the budget is large enough, cutting the shot count by more than half; and we show a hard floor below which no method can certify any saving at all. On real hardware the certificate does what a trustworthy tool should: it refuses to approve a budget when device noise, not shot noise, dominates the error. The gap between certificates one can afford and certificates one can trust is real, but it closes above a floor set by the quantum resources themselves.