A QCA cell holds two mobile electrons among four quantum dots at the corners of a square. Coulomb repulsion forces the pair onto one of two diagonals, giving a bistable polarization
P = (ρ13 − ρ24) / (ρ13 + ρ24) ∈ [−1, +1]
where ρ13, ρ24 are the electron occupation probabilities of the two diagonals — P = +1 encodes bit 1, P = −1 encodes bit 0. A neighboring cell's field pushes an undecided cell toward the opposite-sign kink-energy minimum, so polarization propagates cell-to-cell purely through electrostatics, with no current flow.
Real QCA wires also need a clock: an inter-dot tunneling barrier that is raised and lowered on each cell. This simulator models the standard 4-phase scheme — the wire is split into repeating zones, each driven by a clock signal 90° out of phase with its neighbor:
c(t) = ½(1 + cos(2π(t/T − k/4))) barrier height, zone k = 0..3
dP/dt = R·(1−c)·(tanh(β·P_left) − P) relaxation while barrier is low
- Switch — barrier rising; the cell adiabatically locks onto its neighbor's field.
- Hold — barrier near maximum; polarization is frozen and can itself drive the next zone.
- Release — barrier falling; the cell lets go of its state.
- Relax — barrier near zero; the cell sits unpolarized, ready for a new input.
Because each zone is 90° ahead of the next, data moves exactly one zone per quarter clock cycle — a genuine clocked pipeline, not a single instantaneous relaxation. This is what lets real QCA circuits gain power (each cell only ever switches against a locally low field, dissipating far less energy than a CMOS gate) and be timed like a shift register, at the cost of introducing latency. Landauer/Lent's group at Notre Dame built and measured working four-phase QCA wires and majority gates this way in the 1990s–2000s.