Chapter 5: Architecture - Computer and Information Science - CUNY

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10 CHAPTER 5. ARCHITECTURE Exercise 5.2.2 Show that this matrix performs the OR operation. The NAND gate is of special importance because every logical gate can be made out of NAND gates. Let us try to work out which matrix would correspond to NAND. One way is to sit down and consider for which of the four possible input states of two bits (00,01,10,11) does NAND output a 1 (answer: 00,01,10) and in which states does NAND output a 0 (answer: 11). From this we realize that NAND can be written as 00 01 10 11 [ 0 0 0 0 1 NAND = 1 1 1 1 0 ] . (5.35) Notice that the column names correspond to the inputs and the row names correspond to the outputs. 1 in the jth column and ith row means that on entry j the matrix/gate will output i. There is, however, another way in which one can determine the NAND gate. The NAND gate is really the AND gate followed by the NOT gate. In other words, we can perform the NAND operation by first performing the AND operation and then the NOT operation. In terms of matrices we can write this as ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ NOT ⋆ AND = ⎣ 0 1 ⎦ ⋆ ⎣ 1 1 1 0 ⎦ = ⎣ 0 0 0 1 ⎦ = NAND. (5.36) 1 0 0 0 0 1 1 1 1 0 Exercise 5.2.3 Find a matrix that corresponds to NOR. This way of thinking of NAND rings to light a general situation. When we perform a computation, often we have to carry out one operation followed by another.
5.2. CLASSICAL GATES 11 A B (5.37) We call this performing sequential operations. If matrix A corresponds to performing an operation and matrix B corresponds to performing another operation, then the multiplication matrix B ⋆ A corresponds to performing the operation sequentially. Notice that B ⋆ A looks like the reverse of our picture which has, from left to right, A and then B. Do not be alarmed by this. The reason for this is because we read from left to right and hence we depict processes as going from left to right. We could have easily drawn the above figure as B A (5.38) with no confusion. 1 We shall follow the convention that computation flows from left to right and leave out the arrows. And so a computation of A followed by B shall be denoted A B (5.39) 1 If this text were written in Arabic or Hebrew, this problem would not even arise.
Page 1 and 2: Chapter 5 Architecture Noson S. Yan
Page 3 and 4: 5.1. BITS AND QUBITS 3 A bit is eit
Page 5 and 6: 5.1. BITS AND QUBITS 5 It is import
Page 7 and 8: 5.1. BITS AND QUBITS 7 In the class
Page 9: 5.2. CLASSICAL GATES 9 This matrix
Page 13 and 14: 5.2. CLASSICAL GATES 13 A (5.42) B
Page 15 and 16: 5.2. CLASSICAL GATES 15 A is a 2 m
Page 17 and 18: 5.3. REVERSIBLE GATES 17 Exercise 5
Page 19 and 20: 5.3. REVERSIBLE GATES 19 Figure 5.5
Page 21 and 22: 5.3. REVERSIBLE GATES 21 |x〉 |x
Page 23 and 24: 5.3. REVERSIBLE GATES 23 output wil
Page 25 and 26: 5.4. QUANTUM GATES 25 |x〉 • |x
Page 27 and 28: 5.4. QUANTUM GATES 27 In other word
Page 29 and 30: 5.4. QUANTUM GATES 29 Let us spend
Page 31 and 32: 5.4. QUANTUM GATES 31 will work. R
Page 33 and 34: 5.4. QUANTUM GATES 33 Throughout th
Page 35 and 36: 5.4. QUANTUM GATES 35 Multiplying t
Page 37: Bibliography [1] Charles H. Bennett

Chapter 5: Architecture - Computer and Information Science - CUNY

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