  Imaginary-time response functions of finite-temperature quantum systems are
often obtained with methods that exhibit stochastic or systematic errors.
Reducing these errors comes at a large computational cost -- in quantum Monte
Carlo simulations, the reduction of noise by a factor of two incurs a
simulation cost of a factor of four. In this paper, we relate certain
imaginary-time response functions to an inner product on the space of linear
operators on Fock space. We then show that data with noise typically does not
respect the positive definiteness of its associated Gramian. The Gramian has
the structure of a Hankel matrix. As a method for denoising noisy data, we
introduce an alternating projection algorithm that finds the closest positive
definite Hankel matrix consistent with noisy data. We test our methodology at
the example of fermion Green's functions for continuous-time quantum Monte
Carlo data and show remarkable improvements of the error, reducing noise by a
factor of up to 20 in practical examples. We argue that Hankel projections
should be used whenever finite-temperature imaginary-time data of response
functions with errors is analyzed, be it in the context of quantum Monte Carlo,
quantum computing, or in approximate semianalytic methodologies.
