Bayesian Macromodeling for Circuit Level QCA Design Saket Srivastava and Sanjukta Bhanja Department of Electrical Engineering University of South Florida, Tampa

IEEE-Nano2006 Table of Contents  Purpose of this work Purpose of this work  Overview of Quantum-Dot Cellular Automata Overview of Quantum-Dot Cellular Automata  Overview of Bayesian Modeling Overview of Bayesian Modeling  Layout Level Bayesian Modeling Layout Level Bayesian Modeling  Bayesian Macro modeling Bayesian Macro modeling  Thermal Studies with Macro models Thermal Studies with Macro models  Circuit Level Modeling Circuit Level Modeling  Experimental Results Experimental Results  Conclusion Conclusion

IEEE-Nano2006 Purpose of this work Fast Bayesian Computing Model. Abstracts the behavior of circuit components in QCA design using Probabilistic Macromodeling. Quick Estimation and Comparison of Quantum Mechanical Quantities in QCA architecture at Layout level and Circuit level. Directly models the quantum mechanical steady state probabilities at a hierarchical level. Can be used to identify weak spots in the design in the early design process, at the circuit level itself.

IEEE-Nano2006 Prior Work  S. Henderson, E. Johnson, J Janulis, and P. Tougaw, “incorporating standard cmos design process methodologies into the QCA logic design process”, IEEE Transactions on Nanotechnology, vol.3, pp. 2-9, March 2004.

IEEE-Nano2006 Quick Overview - Quantum-Dot Cellular Automata  In a QCA cell two electrons occupy diagonally opposite dots in the cell due to mutual repulsion of like charges.  A QCA cell can be in any one of the two possible states depending on the polarization of charges in the cell. P = +1 P = - 1

IEEE-Nano2006 Quick Overview - Quantum-Dot Cellular Automata  Electrostatic Interaction between charges in two QCA Cells is given as: P = +1 P = - 1 E kink = E opp. polarization – E same polarization  This interaction is determines the kink energy between two cells.  Kink energy is the is the Energy cost of two neighboring cells having opposite polarization. P = +1

IEEE-Nano2006 Quick Overview - Bayesian Modeling  Bayesian Network is a DAG in which:  Nodes: Random variables  Links: Causal dependencies amongst random variables. General representation: Minimal factored Joint Probability Distribution function: Joint Probability Distribution function: (a) QCA Majority Gate  Each Node has a Conditional Probability Table (CPT) that quantifies the effect of parents on that node. (b) Bayesian model

IEEE-Nano2006 Quick Overview - Bayesian Modeling  The steady state polarization of a QCA cell is obtained from the Hamiltonian matrix using Hartree approximation and is given by: E k is the kink energy. γ is the tunneling energy. f i is the geometric distance factor. is the weighted sum of neighborhood polarizations. ρ ss is the steady state polarization.  The probabilities of observing the system in each of the two states is given as:

IEEE-Nano2006 Layout Level Bayesian Model of Cell Arrangements  We then determine the parent and child nodes of each QCA cell.  Each QCA cell is represented as a random variable (node) taking on two possible values.  Conditional Probabilities for each cell (node) is then given by: where: and also known as Rabbi frequency.

IEEE-Nano2006 Bayesian Macro modeling  A macromodel is a functional block containing a small number of cells.  The macromodels of different circuit elements are the conditional probability of output cells given the values of the input cells.  It can be obtained by marginalizing the joint probability distribution of those cells over all the remaining cells in a layout.  For example if a macromodel block contains three cells (x i,x j and x k ) out of n cells in a layout, then its joint probability is obtained by: is the joint probability distribution over all n cells in a layout where:  To compute the marginal probabilities for each macromodel, we first transform the DAG into a junction tree of cliques and then the marginal probabilities are calculated using local message passing between cliques.

IEEE-Nano2006 Bayesian Macro modeling Figure: Validation of the Bayesian network modeling of QCA circuits with Hartree­Fock approximation based coherence vector­based quantum mechanical simulation of same circuit. Probabilities of correct output are compared for basic circuit elements. SymbolMacromodel MAJSimple Majority CMClocked Majority INVInverter LINELine ICInverter Chain COCorner ETEven Tap OTOdd Tap CBCrossbar ANDAnd Gate OROr Gate Table 1: Abbreviations for Macromodel blocks used in the circuit design of Adder-1 and Adder-2.

IEEE-Nano2006 Thermal Studies with Macromodels MacromodelQCA LayoutBayesian Model Block Diagram Thermal Behavior (a) Majority Gate 1 clock zone 3 inputs 1 output (a) Clocked Majority Gate 2 clock zones 3 inputs 1 output (a) Inverter 1 clock zone 1 input 1 output

IEEE-Nano2006 Thermal Studies with Macromodels  As the Temperature increases, the polarization probability at the output node decreases.  Thermal behavior is also dependent on the input vector set.  A clocked majority gate consists of two clocking zones as it has been seen that circuit reliability increases when majority gates are clocked separately from the outputs.  This is done in order to synchronize the input signals reaching the majority gate irrespective of the path length they have traversed.  Larger number of cells in clocked majority lead to overall increased uncertainty that accounts for a larger drop in polarization at the output node at higher temperature.

IEEE-Nano2006 Thermal Studies with Macromodels MacromodelQCA LayoutBayesian Model Block Diagram Thermal Behavior (a) Majority Gate 1 clock zone 3 inputs 1 output (a) Clocked Majority Gate 2 clock zones 3 inputs 1 output (a) Inverter 1 clock zone 1 input 1 output

IEEE-Nano2006 Circuit Level Modeling – A NAND Gate Example AND INVLINE A B Out A B  The QCA layout of a NAND gate consists of a Majority gate with one fixed cell, an Inverter and a Line and in three clock zones.  The Macromodel circuit of a NAND gate is modeled using the macromodel blocks of an AND gate, an Inverter and a Line.  A Bayesian network of the macromodel circuit is then formed. A B Out  A QCA layout Bayesian model is also developed.  The two models are then studied and a compared for output node polarization at different temperatures. Out

IEEE-Nano2006 Circuit Level Modeling – Full Adder-2 QCA layout of Adder-2 Bayesian network for of Adder-2 layout

IEEE-Nano2006 Circuit Level Modeling – Full Adder-2 Macromodel block design of Adder-2 Bayesian model for macromodel circuit design of Adder-2

IEEE-Nano2006 Circuit Level Modeling – Full Adder-2 Probability of correct output for sum and carry of Adder­2 based on the layout­level Bayesian net model and the circuit level macromodel, at different temperatures, for different inputs (a) (0,0,0) (b) (0,0,1) (c) (0,1,0) (d) (0,1,1). (a) (d)(c) (b)

IEEE-Nano2006 Circuit Level Modeling – 2x2 Multiplier Macromodel block design of 2x2 Multiplier QCA layout of 2x2 Multiplier

IEEE-Nano2006 Circuit Level Modeling – 2x2 Multiplier Bayesian model for macromodel circuit design of 2x2 Multiplier

IEEE-Nano2006 Circuit Level Modeling – 2x2 Multiplier Probability of correct output at the four output nodes of 2x2 Multiplier circuit based on the layout-level Bayesian net model and the circuit level macromodel, at different temperatures, for different inputs (a)(1,0),(0,1) (b) (1,0),(1,1) (c) (1,1),(0,1) (d) (1,1),(1,1).

IEEE-Nano2006 Experimental Results Table 3: Comparison between simulation timing of a Full Adder circuits and 2x2 Multiplier circuit in QCADesigner(QD) and Genie Bayesian Network(BN) Tool for Full Layout and Macromodel Layout.

IEEE-Nano2006 Experimental Results – Error Modes We compute the near-ground state configurations that results in error in the output carry bit C out of a QCA full using both the layout and circuit level models. We show the case for input vector set (1,0,0). The other input vector sets have similar results. We use red marker to point to the components that are weak (high error probabilities) in both the layout and circuit level. If a node (a macromodel block) in macromodel circuit layout is highly error prone for a given input set, then some or all the QCA cells forming that macromodel block are highly prone to error. This indicates that weak spot in the design can be identified early in the design process, at the circuit level itself.

IEEE-Nano2006 Conclusion  We proposed an efficient Bayesian Network based probabilistic macro modeling strategy for QCA circuit.  This model can estimate cell polarizations, ground state probability, and lowest- energy error state probability, without the need for computationally expensive quantum-mechanical computations.  We showed that the polarization estimates at layout and circuit levels are in good agreement.  We illustrated a full adder design and a 2-bit multiplier design.  We showed that the weak spots at the layout level can be effectively identified at the circuit level using this model.  One possible future direction of this work involves the extension of the BN model to handle sequential logic.

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