A discrete-time quantum walk is the evolution of a unit vector of amplitudes by repeated application of a unitary
matrix 
. Its state at integer time
is 
,
and the probability of a coordinate outcome is the
squared absolute value of the corresponding amplitude.
Unlike a random walk, amplitudes can interfere (Guo
and Schmeits 2026, pp. 2-4).
A two-reflection walk has
 |
(1)
|
where 
and 
are projection
matrices and 
is the identity matrix. Choose a matrix 
with orthonormal
columns satisfying 
, and set
 |
(2)
|
where 
denotes the conjugate transpose. Then
 |
(3)
|
where 
and 
is the Chebyshev
polynomial of the first kind (Guo and Schmeits 2026, pp. 6-7).
For distinct states indexed by 
and 
and a real matrix 
, write 
, where
projects onto the eigenspace
of eigenvalue 
. The transition amplitude
satisfies
 |
(4)
|
Guo and Schmeits (2026) call equality in the first bound, with a nonzero right side at a positive
integer time, peak state transfer. More precisely, let 
be nonempty.
Peak state transfer occurs iff every 
can be written as 
with 
and integers 
and 
having greatest
common divisor 1, and there is a common 
such that
 |
(5)
|
where 
is the sign function and 
is the least common
multiple of the positive denominators 
for 
(Guo and Schmeits 2026, pp. 17-19). Transfer
then occurs precisely at the odd multiples
of 
. Perfect state transfer means that 
. When 
for every 
, where 
is the 
th vector of the standard
basis, the second bound is also an equality, and
peak state transfer is perfect state transfer. They apply this criterion to strongly
regular graphs, Levi graphs of block
designs, and walks associated with toroidal grid
graphs.
