  Quantum K-theory is a K-theoretic version of quantum cohomology, which was
recently defined by Y.-P. Lee. Based on a presentation for the quantum K-theory
of the classical flag variety Fl_n, we define and study quantum Grothendieck
polynomials. We conjecture that they represent Schubert classes (i.e., the
natural basis elements) in the quantum K-theory of Fl_n, and present strong
evidence for this conjecture. We describe an efficient algorithm which, if the
conjecture is true, computes the quantum K-invariants of Gromov-Witten type for
Fl_n. Two explicit constructions for quantum Grothendieck polynomials are
presented. The natural generalizations of several properties of Grothendieck
polynomials and of the quantum Schubert polynomials due to Fomin, Gelfand, and
Postnikov are proved for our quantum Grothendieck polynomials. For instance, we
use a quantization map satisfying a factorization property similar to the
cohomology quantization map, and we derive a Monk-type multiplication formula.
We also define quantum double Grothendieck polynomials and derive a Cauchy
identity. Our constructions are considerably more complex than those for
quantum Schubert polynomials. In particular, a crucial ingredient in our work
is the Pieri formula for Grothendieck polynomials due to the first author and
Sottile.
