By Smita Krishnaswamy

Logic circuits have gotten more and more vulnerable to probabilistic habit brought on by exterior radiation and strategy version. additionally, inherently probabilistic quantum- and nano-technologies are at the horizon as we process the boundaries of CMOS scaling. making sure the reliability of such circuits regardless of the probabilistic habit is a key problem in IC design---one that necessitates a basic, probabilistic reformulation of synthesis and trying out ideas. This monograph will current thoughts for studying, designing, and trying out common sense circuits with probabilistic behavior.

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Previously computed partial results of the eliminate_r edundant_ variables operation are stored in a common hash table, which is searched first to avoid traversing common paths or recomputing existing results. 1 Compressing Matrices with Decision Diagrams 43 Fig. 7 Algorithm to compute fidelity In Fig. 6, capitalized variables refer to ADDs, and lower case variables refer to nodes. This algorithm searches the ADD, starting from the root, for the first of two redundant variables v1 , v2 with v1 < v2 in the ADD node ordering.

Inn−1 = i n−1 occur. When an input vector is right-multiplied by the PTM, the result is an output vector of size 1 × 2m . The output vector gives the resulting output distribution. Examples of an ITM and PTM are shown in Fig. 2. 1 Basic Operations PTMs can be defined for all the gates of a logic circuit by taking into account errors affecting the gates. A PTM for the entire circuit can then be derived from the PTMs 24 2 Probabilistic Transfer Matrices of the gates and their interconnections. The basic operations needed to compute the circuit PTM from component PTMs are the matrix and tensor products.

It can also be used to compute the probability of error of individual outputs. • eliminate_redundant_variables. This operation eliminates redundant input variables that result from tensoring matrices of gates that are in different fan-out branches of the same signal We now formally define and describe these operations in more detail. First, we define the element-wise product used in computing fidelity. 4 The element-wise product of two matrices A and B, both of dimension n × m, is denoted A. ∗ B = M and defined by M(i, j) = A(i, j) × B(i, j).