  Schubert polynomials give polynomial representatives of Schubert classes in
the cohomology of the complete flag variety and have a combinatorial
formulation in terms of bumpless pipe dreams. Quantum double Schubert
polynomials are polynomial representatives of Schubert classes in the
torus-equivariant quantum cohomology of the complete flag variety, but they
have no analogous combinatorial formulation. We introduce a generalization of
the bumpless pipe dreams called quantum bumpless pipe dreams, giving a novel
combinatorial formula for quantum double Schubert polynomials as a sum of
binomial weights of quantum bumpless pipe dreams. We give a bijective proof for
this formula by showing that the sum of binomial weights satisfies a defining
recursion that is a special case of Monk's rule.
