A continuous-time quantum walk (CTQW) is a quantum walk on a given (simple) graph that is dictated by a time-varying unitary matrix that relies on the Hamiltonian of the quantum system and the adjacency matrix.
The concept of a CTQW is believed to have been first considered for quantum computation by Edward Farhi and Sam Gutmann;[1] since many classical algorithms are based on (classical) random walks, the concept of CTQWs were originally considered to see if there could be quantum analogues of these algorithms with e.g. better time-complexity than their classical counterparts.
In recent times, problems such as deciding what graphs admit properties such as perfect state transfer with respect to their CTQWs have been of particular interest.
denote the adjacency matrix of
It is also possible to similarly define a continuous-time quantum walk on
relative to its Laplacian matrix; although, unless stated otherwise, a CTQW on a graph will mean a CTQW relative to its adjacency matrix for the remainder of this article.
Mixing matrices are symmetric doubly-stochastic matrices obtained from CTQWs on graphs:
are said to admit perfect state transfer at time
admit perfect state transfer at time t, then
itself is said to admit perfect state transfer (at time t).
of pairs of distinct vertices on
is said to admit perfect state transfer (at time
admits perfect state transfer at time
is said to admit perfect state transfer (at time
admit perfect state transfer at time
such that all of its vertices are periodic at time
A graph is periodic if and only if its (non-zero) eigenvalues are all rational multiples of each other.
admit perfect state transfer at time
admit perfect state transfer at time
admits perfect state transfer at time
admits perfect state transfer at time
admits perfect state transfer at time
If a walk-regular graph admits perfect state transfer, then all of its eigenvalues are integers.
is a graph in a homogeneous coherent algebra that admits perfect state transfer at time
admit perfect state transfer at time
must have a perfect matching that admits perfect state transfer if it admits perfect state transfer between a pair of adjacent vertices and is a graph in a homogeneous coherent algebra.
A regular edge-transitive graph
cannot admit perfect state transfer between a pair of adjacent vertices, unless it is a disjoint union of copies of the complete graph
A strongly regular graph admits perfect state transfer if and only if it is the complement of the disjoint union of an even number of copies of
The only cubic distance-regular graph that admits perfect state transfer is the cubical graph.