  We consider steady state heat conduction across a quantum harmonic chain
connected to reservoirs modelled by infinite collection of oscillators. The
heat, $Q$, flowing across the oscillator in a time interval $\tau$ is a
stochastic variable and we study the probability distribution function $P(Q)$.
In the large $\tau$ limit we use the formalism of full counting statistics
(FCS) to compute the generating function of $P(Q)$ exactly. We show that $P(Q)$
satisfies the steady state fluctuation theorem (SSFT) regardless of the
specifics of system, and it is nongaussian with clear exponential tails. The
effect of finite $\tau$ and nonlinearity is considered in the classical limit
through Langevin simulations. We also obtain predictions of universal heat
current fluctuations at low temperatures in clean wires.
