Bayesian Modeling of Quantum-Dot-Cellular-Automata CircuitsSanjukta Bhanja and Saket Srivastava

Electrical Engineering, University of South Florida, Tampa, Florida, USA.

Pss is the steady state

polarization.E is the total kink energy of neighboring cells.Ω is the Rabi frequency andΔ is the thermal ratio.

and are the probabilities of observing the system in each of the two states. and

where

&

H is the Hamiltonian of a QCA Cell using Hartree approximation.Ek is the kink energy.γ is the tunneling energy.

fi is the geometric distance factor.

is the weighted sum of neighborhood polarizations.

Quantum-Dot Cellular Automata (QCA) Overview

P=+1 P=-1

In a QCA cell two electrons occupy diagonally opposite dots in the cell due to mutual repulsion of like charges. Hence a QCA cell can be in any one of the two possible states depending on the polarization of charges in the cell.

Electrostatic Interaction between charges in two QCA Cells Is given as:

This interaction is determines the kink energy between two cells.

Ekink = Eopp. polarization – Esame polarization

Purpose:• Fast Bayesian Computing Model• Quick Estimation and Comparison of Quantum Mechanical Quantities in QCA Circuits• Directly models the quantum mechanical steady state probabilities

Computation Model using Hartree Approximation

Bayesian Network based Modeling

Results

• Results of our Bayesian model of a three input clocked majority gate, Inverter and Full Adder. • The Bayesian Network structure shown here is based on the flow of the wave function and information regarding the clock. • We used Hugin software tool for Bayesian inference. • We also show the temperature and input dependence on the probability of the output node in the Graph. • We have also successfully modeled other logic circuits. • The output node probabilities of all the models have similar dependence on temperature and the applied input vector

(a) QCA Clocked Majority Gate (b) Bayesian net Structure (c) Temperature and Input Dependence

(a) QCA Inverter(b) Bayesian model

Bayesian Network is a DAG in which:Nodes: Random variablesLinks: Causal dependencies amongst random variables.

Each Node has a Conditional Probability Table (CPT) that quantifies the effect of parents on that node.

Ne(X) = set of all neighboring cells that can effect a cell X.C(X) = Clocking zone of cell X.T(X) = Time taken by wave function to propagate from the inputs or from the nodes nearest to the previous clock zone.Pa(X) = Parent set of cell X.Ch(X) = Non parent neighbors of X.

Joint Probability Distribution function:

Minimal factored Joint Probability Distribution function:

General representation:

How do we model Causality?• Clocks determine the causal order between cells. Within each clock zone causality is determined by the direction of propagation of wave function.How do we perform ground state calculation?• We choose children states so as to maximize Ω, which would minimize ground state energy over all possible ground states of the cell.

Steady state density matrix diagonal entries are used with child state assignments (ch*(X)) to determine the CPT of all cells in the Bayesian Network.

(a) QCA Majority Gate (b) Bayesian model