  Hrube\v{s} and Wigderson (2015) initiated the complexity-theoretic study of
noncommutative formulas with inverse gates. They introduced the Rational
Identity Testing (RIT) problem which is to decide whether a noncommutative
rational formula computes zero in the free skew field. In the white-box
setting, deterministic polynomial-time algorithms are known for this problem
following the works of Garg, Gurvits, Oliveira, and Wigderson (2016) and
Ivanyos, Qiao, and Subrahmanyam (2018).
  A central open problem in this area is to design efficient deterministic
black-box identity testing algorithm for rational formulas. In this paper, we
solve this problem for the first nested inverse case. More precisely, we obtain
a deterministic quasipolynomial-time black-box RIT algorithm for noncommutative
rational formulas of inversion height two via a hitting set construction.
Several new technical ideas are involved in the hitting set construction,
including key concepts from matrix coefficient realization theory
(Vol\v{c}i\v{c}, 2018) and properties of cyclic division algebra (Lam, 2001).
En route to the proof, an important step is to embed the hitting set of Forbes
and Shpilka for noncommutative formulas (2013) inside a cyclic division algebra
of small index.
