Quantum phase slips (QPS) generate voltage fluctuations in superconducting nanowires. Employing Keldysh technique and making use of the phase-charge duality arguments we develop a theory of QPS-induced voltage noise in such nanowires. We demonstrate that quantum tunneling of the magnetic flux quanta across the wire yields quantum shot noise which obeys Poisson statistics and is characterized by a power law dependence of its spectrum $S_��$ on the external bias. In long wires $S_��$ decreases with increasing frequency $��$ and vanishes beyond a threshold value of $��$ at $T \to 0$. Quantum coherent nature of QPS noise yields non-monotonous dependence of $S_��$ on $T$ at small $��$.

4.5 pages + 2-page supplemental material, 3 figures