  We study the class of non-commutative Unambiguous circuits or
Unique-Parse-Tree (UPT) circuits, and a related model of Few-Parse-Trees
(FewPT) circuits (which were recently introduced by Lagarde, Malod and Perifel
[LMP16] and Lagarde, Limaye and Srinivasan [LLS17]) and give the following
constructions:
  (1) An explicit hitting set of quasipolynomial size for UPT circuits,
  (2) An explicit hitting set of quasipolynomial size for FewPT circuits
(circuits with constantly many parse tree shapes),
  (3) An explicit hitting set of polynomial size for UPT circuits (of known
parse tree shape), when a parameter of preimage-width is bounded by a constant.
  The above three results are extensions of the results of [AGKS15], [GKST15]
and [GKS16] to the setting of UPT circuits, and hence also generalize their
results in the commutative world from read-once oblivious algebraic branching
programs (ROABPs) to UPT-set-multilinear circuits. The main idea is to study
shufflings of non-commutative polynomials, which can then be used to prove
suitable depth reduction results for UPT circuits and thereby allow a careful
translation of the ideas in [AGKS15], [GKST15] and [GKS16].
