Modelling Techniques for Simulating Large QCA Circuits - Field-Coupled Nanocomputing: Paradigms, Progress, and Perspectives - page 264

Once the coherence vector has been computed, the polarization of the i th cell
can be found by evaluating, P i

− z ( i ). It is worthwhile to point out that
K ab ( i, j )= K ba ( j, i )( a, b = x, y, z ), and as such, in the above example, it is not
necessary to compute the two-cell coherence vector

=

(2 , 1). If we increase the
size of our system to three cells, i.e., N = 3, then our coherence vector would
expand to include the expectation values for the additional basis operators of
our system, i.e.,

K

⊡ ⊤

λ (1)

⊣ ⊦

λ (2)

λ (3)

λ = K (1 , 2) , (12)

K (1 , 3)

K (2 , 3)

K (1 , 2 , 3)

where (1 , 2 , 3) is the three-cell coherence vector containing the 27 three-cell

correlation terms (i.e., the expectation values of the three-cell operators). Note
that there are now three two-cell coherence vectors corresponding to the three
unique two-cell groupings possible within a three cell circuit. The increasing
number of unique two-cell, three-cell, etc, groupings as the circuit grows is what
leads to the exponential growth in the coherence vector. While this exponen-
tial growth makes solving larger systems intractable, it has been shown in [ 40 ]
that the contribution of the higher-order (i.e., three-cell, four-cell, etc.) correla-
tion terms in the coherence vector are negligible in calculating the dynamics of
QCA systems. Thus, if we remove these higher-order correlation terms, then our
coherence vector grows according to s =4 . 5 N 2

K

1 . 5 N , which offers a much more
manageable growth in state variables, and is far more suitable for solving QCA
systems with a large number of cells. If we limit the scope of our Hamiltonian
to simply nearest-neighbour (NN) coupling, then the number of state variables
in our system can be reduced even further to s =12 N

−

9. Comparatively,
the number of state variables when modelling a QCA system using the ICHA
grows according to s =3 N . Thus, by considering only two-cell correlations and
nearest-neighbour coupling, we can achieve the accuracy of the full-basis two-
state approximation, but with the computational complexity of the ICHA.
A comparison of the number of state variables being solved for in each
approximation is shown in Fig. 4 .

−

2.1 The Liouville Equation

In the Heisenberg picture of quantum mechanics, the state vector, |ˈ≥ ,doesnot

change with time, and an observable, A , satisfies,

H, A ,

d
dt

A ( t )= i

(13)