  Khrushchev's formula is the cornerstone of the so called Khrushchev theory, a
body of results which has revolutionized the theory of orthogonal polynomials
on the unit circle. This formula can be understood as a factorization of the
Schur function for an orthogonal polynomial modification of a measure on the
unit circle. No such formula is known in the case of matrix-valued measures.
This constitutes the main obstacle to generalize Khrushchev theory to the
matrix-valued setting which we overcome in this paper.
  It was recently discovered that orthogonal polynomials on the unit circle and
their matrix-valued versions play a significant role in the study of quantum
walks, the quantum mechanical analogue of random walks. In particular, Schur
functions turn out to be the mathematical tool which best codify the return
properties of a discrete time quantum system, a topic in which Khrushchev's
formula has profound and surprising implications. We will show that this
connection between Schur functions and quantum walks is behind a simple proof
of Khrushchev's formula via `quantum' diagrammatic techniques for CMV matrices.
This does not merely give a quantum meaning to a known mathematical result,
since the diagrammatic proof also works for matrix-valued measures. Actually,
this path counting approach is so fruitful that it provides different matrix
generalizations of Khrushchev's formula, some of them new even in the case of
scalar measures.
  Furthermore, the path counting approach allows us to identify the properties
of CMV matrices which are responsible for Khrushchev's formula. On the one
hand, this helps to formalize and unify the diagrammatic proofs using simple
operator theory tools. On the other hand, this is the origin of our main result
which extends Khrushchev's formula beyond the CMV case, as a factorization rule
for Schur functions related to general unitary operators.
