  In this paper, we explore the impact of noise on quantum computing,
particularly focusing on the challenges when sampling bit strings from noisy
quantum computers as well as the implications for optimization and machine
learning applications. We formally quantify the sampling overhead to extract
good samples from noisy quantum computers and relate it to the layer fidelity,
a metric to determine the performance of noisy quantum processors. Further, we
show how this allows us to use the Conditional Value at Risk of noisy samples
to determine provable bounds on noise-free expectation values. We discuss how
to leverage these bounds for different algorithms and demonstrate our findings
through experiments on a real quantum computer involving up to 127 qubits. The
results show a strong alignment with theoretical predictions.
