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).
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).
For distinct states indexed by 
and real 
, 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 
having greatest
common divisor 1, and there is a common 
such that
 |
(5)
|
where 
is the least common multiple of the positive
denominators 
for 
(Guo and Schmeits 2026,
Theorem 3.5). Transfer then occurs precisely at the odd multiples of 
. Perfect state transfer means that 
. When 
for every 
, the second bound is also an equality, and peak state
transfer is perfect state transfer. They apply this criterion to strongly
regular graphs, incidence graphs of block
designs, and walks associated with toroidal grid graphs.
