Though we don't come across negative probabilities in a quantum computation problem in the general sense, there is a historic context on the discussion and debate around negative probabilities in quantum mechanics.
In 1942, Paul Dirac wrote a paper "The Physical Interpretation of Quantum Mechanics" where he introduced the concept of negative energies and negative probabilities. The idea of negative probabilities later received increased attention in physics and particularly in quantum mechanics. Richard Feynman introduced ghosts as "negative probability" in perturbative gauge theories. The main purpose of the ghosts is to cancel the contributions from unphysical polarisations of gauge fields in loops.
Another example is known as the Wigner distribution in phase space, introduced by Eugene Wigner in 1932 to study quantum corrections, often leads to negative probabilities. For this reason, it has later been better known as the Wigner quasiprobability distribution. The Wigner distribution function is routinely used in physics nowadays and provides the cornerstone of phase-space quantization. Its negative features are an asset to the formalism and often indicate quantum interference.
However, one never obtains "negative probability" densities when one discusses single observables. One obtains "negative probability" densities only when one discusses joint distributions of incompatible observables.
There are two works of Feynman about negative probabilities.
R. P. Feynman, Negative probability in Quantum implications: Essays in Honor of David Bohm, edited by B. J. Hiley and F. D. Peat (Routledge and Kegan Paul, London, 1987), Chap. 13, pp 235 – 248.
R. P. Feynman, Simulating physics with computers (Chapter 6), Int. J. Theor. Phys., 21, 467 – 488 (1982).