Quantum-dot cellular automata (QCA) may offer a viable alternative of traditional transistor-based technology at the nanoscale. When modeling a QCA circuit, the number of degrees of freedom necessary to describe the quantum mechanical state increases exponentially making modeling even modest size cell arrays difficult. The intercellular Hartree approximation largely reduces the number of state variables and still gives good results especially when the system remains near ground state. This suggests that a large part of the correlation degrees of freedom are not essential from the point of view of the dynamics. In certain cases, however, such as, for example, the majority gate with unequal input legs, the Hartree approximation gives qualitatively wrong results. An intermediate model is constructed between the Hartree approximation and the exact model, based on the coherence vector formalism. By including correlation effects to a desired degree, it improves the results of the Hartree method and gives the approximate dynamics of the correlation terms. It also models the majority gate correctly. Beside QCA cell arrays, our findings are valid for Ising spin chains in transverse magnetic field, and can be straightforwardly generalized for coupled two-level systems with a more complicated Hamiltonian.

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If the nine elements of the correlation vector are placed in a 3×3 tensor, K̂, the equation has the form ℏ(d/dt)K̂(i,j)=Ω̂jK̂(i,j)−K̂(i,j)Ω̂iij.
13.
The reason for that is the structure of the Eq. (2) Hamiltonian. It contains only double terms which are the multiplication of two Pauli spin matrices, i.e., σ̂z(i)σ̂z(j). If the Hamiltonian contained triple terms of the form σ̂z(i)σ̂z(j)σ̂z(k) then in the differential equations for the nth order correlation vector elements we could even find correlations of the order n+2.
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The correlation proper is often called the connected part of the correlation.
15.
Notice that now two complex, that is, four real variables are used to describe the state of a cell. This number can be reduced to two based on the overall phase arbitrariness and unity norm of the wave function. For details see Ref. 7.
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17.
Although in the example presented in Sec. V A the NNPC method will be used as an approximation, there are cases when it gives the same dynamics as the exact model does. For example, when quantum computing operations are done on coupled two-level systems in such a way that only nearest neighbor entanglement occurs. This highly restricts the possible operations, but makes it possible to handle big arrays and still realize, for example, the controlled NOT or the qubit exchange.
18.
The switching was carried out very slowly, since we are using the example to compare the different dynamical descriptions and would like to obtain smooth curves. For the possible speed of the adiabatic switching see Ref. 4.
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