Chapter 5: Architecture - Computer and Information Science - CUNY

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24 CHAPTER 5. ARCHITECTURE also has three inputs and three outputs. |x〉 • |x〉 (5.65) |y〉 × |y ′ 〉 |z〉 × |z ′ 〉 The top |x〉 input is the control input. The output is always the same |x〉. If |x〉 is set to |0〉, then |y ′ 〉 = |y〉 and |z ′ 〉 = |z〉, i.e., the values stay the same. If, on the other hand, the control |x〉 is set to |1〉, then the outputs are reversed: |y ′ 〉 = |z〉 and |z ′ 〉 = |y〉. In short |0, y, z〉 ↦→ |0, y, z〉 and |1, y, z〉 ↦→ |1, z, y〉. Exercise 5.3.4 Show that the Fredkin gate is its own inverse. The matrix that corresponds to the Fredkin gate is 000 001 010 011 100 101 110 111 ⎡ ⎤ 000 1 0 0 0 0 0 0 0 001 0 1 0 0 0 0 0 0 010 0 0 1 0 0 0 0 0 011 0 0 0 1 0 0 0 0 100 0 0 0 0 1 0 0 0 . (5.66) 101 0 0 0 0 0 0 1 0 ⎢ ⎥ 110 ⎣ 0 0 0 0 0 1 0 0 ⎦ 111 0 0 0 0 0 0 0 1 The Fredkin gate is also universal. By setting y to |0〉 we get the AND gate as follows:
5.4. QUANTUM GATES 25 |x〉 • |x〉 (5.67) |0〉 × |x ∧ z〉 |z〉 × |(¬x) ∧ z〉 The NOT gate and the fanout gate can be obtained by setting |y〉 to |1〉 and |z〉 to |0〉. This gives us |x〉 • |x〉 (5.68) |1〉 × |¬x〉 |0〉 × |x〉 So both the Toffoli and the Fredkin gates are universal. Both are not only reversible gates, but a look at their matrices show that they are also unitary. In the next section we shall look at other unitary gates. 5.4 Quantum Gates A quantum gate is simply any unitary matrix that manipulates qubits. We have already worked with some quantum gates such as the Identity matrix, the Hadamard gate, the NOT gate, the controlled-NOT gate, the Toffoli gate and the Fredkin gate. What else is there? Let us first concentrate on quantum gates that manipulate a single qubit.
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 and 10: 5.2. CLASSICAL GATES 9 This matrix
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: 5.3. REVERSIBLE GATES 23 output wil
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

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