Chapter 5: Architecture - Computer and Information Science - CUNY

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8 CHAPTER 5. ARCHITECTURE 5.2 Classical Gates Classical logical gates are ways of manipulating bits. Bits enter and exit logical gates. We shall need ways of manipulating qubits and will therefore study classical gates from the point of view of matrices. As stated in section 5.1, we represent n input bits as a 2 n by 1 matrix and m output bits as a 2 m by 1 matrix. How should we represent our logical gates? When one multiplies a 2 m by 2 n matrix with a 2 n by 1 matrix, the result is a 2 m by 1 matrix. In symbols: (2 m by 2 n ) ⋆ (2 n by 1) = (2 m by 1). (5.25) So bits will be represented by column vectors and logic gates by matrices. Let us try a simple example. Consider the NOT gate NOT takes as input one bit, or a 2 by 1 matrix, and outputs one bit, or a 2 by 1 matrix. NOT of |0〉 equals |1〉 and NOT of |1〉 equals |0〉. Consider the matrix ⎡ ⎤ NOT = ⎣ 0 1 ⎦ . (5.26) 1 0 This matrix satisfies ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎣ 0 1 ⎦ ⎣ 1 ⎦ = ⎣ 0 ⎦ 1 0 0 1 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎣ 0 1 ⎦ ⎣ 0 ⎦ = ⎣ 1 ⎦ , (5.27) 1 0 1 0 which is exactly what we want. What about the other gates? Consider the AND gate. The AND gate is different from the NOT gate because AND accepts two bits and outputs one bit. Since there are two inputs and one output, we will need a 2 1 by 2 2 matrix. Consider the matrix ⎡ ⎤ AND = ⎣ 1 1 1 0 ⎦ . (5.28) 0 0 0 1
5.2. CLASSICAL GATES 9 This matrix satisfies We can write this as ⎡ ⎤ 0 ⎡ ⎤ ⎡ ⎤ ⎣ 1 1 1 0 0 ⎦ = ⎣ 0 ⎦ . (5.29) 0 0 0 1 ⎢ ⎣ 0⎥ ⎦ 1 1 AND|11〉 = |1〉. (5.30) In contrast, consider another 4 by 1 matrix: ⎡ ⎤ 0 ⎡ ⎤ ⎡ ⎤ ⎣ 1 1 1 0 1 ⎦ = ⎣ 1 ⎦ . (5.31) 0 0 0 1 ⎢ ⎣ 0⎥ ⎦ 0 0 We can write this as Exercise 5.2.1 Calculate AND|10〉. AND|01〉 = |0〉. (5.32) What would happen if we put an arbitrary 4 by 1 matrix to the right of AND? ⎡ ⎤ 3.5 ⎡ ⎤ ⎡ ⎤ ⎣ 1 1 1 0 2 ⎦ = ⎣ 5.5 ⎦ (5.33) 0 0 0 1 ⎢ ⎣ 0 ⎥ ⎦ −4.1 −4.1 This is clearly nonsense. We are only permitted to multiply these classical gates with vectors that represent classical states, i.e., column matrices with a single 1 entry and all the other entries must be 0. In the classical world, the bits are only one state and are described by such vectors. Only later, when we delve into quantum gates will we have more room (and more fun). The OR gate can be represented by the matrix ⎡ OR = ⎣ 1 0 0 ⎤ 0 ⎦ . (5.34) 0 1 1 1
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: 5.1. BITS AND QUBITS 7 In the class
Page 11 and 12: 5.2. CLASSICAL GATES 11 A B (5.3
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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