Probabilistic Modeling of QCA Circuits Using Bayesian Networks

To push the frontiers of quantum-dot cellar automata (QCA) based circuit design, it is necessary to have design and analysis tools at multiple levels of abstractions. To characterize the performance of QCA circuits it is not sufficient to specify just the binary discrete states (0 or 1) of the individual cells, but also the probabilities of observing these states. We present an efficient method based on graphical probabilistic models, called Bayesian networks (BNs), to model these steady-state cell state probabilities, given input states. The nodes of the BN are random variables, representing individual cells, and the links between them capture the dependencies among them. BNs are minimal, factored, representation of the overall joint probability of the cell states. The method is fast and its complexity is shown to be linear in terms of the number of cells. This BN model allows us to analyze clocked QCA circuits in terms of quantum- mechanical quantities, such as steady-state polarization and thermal ratios for each cell, without the need for full quantum-mechanical simulation, which is known to be very slow and is best postponed to the final stages of the design process. We can also estimate the most likely (or ground) state configuration for all the cells and the lowest energy configuration that results in output errors. We validate the model with steady-state probabilities computed by the Hartree-Fock self-consistent approximation (HT-SCA). Using full adder designs, we demonstrate the ability to compare and contrast QCA circuit designs with respect to the variation of the output state probabilities with temperature and input. We also show how weak spots in clocked QCA circuit designs can be found using our model by comparing the (most likely) ground-state configuration with the next most likely energy state configuration that results in output error