Chapter 5: Architecture - Computer and Information Science - CUNY

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26 CHAPTER 5. ARCHITECTURE The following three matrices are called Pauli matrices and are very important. ⎡ X = ⎣ 0 ⎤ 1 ⎦ , ⎡ Y = ⎣ 0 ⎤ −i ⎦ , ⎡ Z = ⎣ 1 0 ⎤ ⎦ . (5.69) 1 0 i 0 0 −1 ⎤ They occur everywhere in quantum mechanics and quantum computing. 3 Notice that the X matrix is nothing more than our NOT matrix. Other important matrices that will be used are ⎡ ⎤ ⎡ 0 i 0 e iπ/4 S = ⎣ 1 0 ⎦ , T = ⎣ 1 0 ⎦ . (5.70) There are other quantum gates on single qubits. However these will be enough to tell our tale. Exercise 5.4.1 Show that each of these matrices are unitary. Exercise ⎡ ⎤ 5.4.2 Show the action of each of these matrices on an arbitrary qubit ⎣ c 0 c 1 ⎦. Exercise 5.4.3 These operations are intimately related to each other. Prove the following relationships between the operations: 1. H = 1 √ 2 (X + Z) 2. X = HZH 3. Z = HXH 4. −1Y = HY H 5. S = T 2 6. −1Y = XY X There is a beautiful geometric way of representing one-qubit operations. Remember from chapter 1, page CITE PAGE, that for a given complex number c = x + yi whose modulus is 1, there is a nice way of visualizing c as an arrow of length 1 from the origin to the circle of radius 1. |c| 2 = c × c = (x + yi) × (x − yi) = x 2 + y 2 = 1 = r (5.71) 3 Sometimes the notation σ x, σ y and σ z is used for these matrices.
5.4. QUANTUM GATES 27 In other words, every complex number of radius 1 can be identified by the angle φ that the vector makes with the x axis. There is an analogous representation of a qubit as an arrow in three dimensions: let us see how it works. A generic qubit is of the form |ϕ〉 = c 0 |0〉 + c 1 |1〉 (5.72) where |c 0 | 2 +|c 1 | 2 = 1 Although at first sight there are four real numbers involved in the definition above, it turns out that there are only two actual degrees of freedom. Let us see why. To begin with, let us rewrite our qubit in polar form: and and so c 0 = r 0 e iφ0 (5.73) c 1 = r 1 e iφ1 (5.74) |ϕ〉 = r 0 e iφ0 |0〉 + r 1 e iφ1 |1〉 (5.75) still four real parameters: r 0 , r 1 , φ 0 , φ 1 . However, a quantum physical state does not change if we multiply its corresponding vector by an arbitrary complex number of norm 1 (see chapter 4, page ???), thus we can obtain an equivalent expression for our qubit, where the amplitude for |0〉 is real, by ”killing” its phase: e −iφ0 |ϕ〉 = e −iφ0 (r 0 e iφ0 |0〉 + r 1 e iφ1 |1〉 = (5.76) r 0 |0〉 + r 1 e i(φ1−φ0) |1〉 (5.77) now we only have three real parameters, namely r 0 , r 1 and φ = φ 1 − φ 0 . But we can do better: using the constraint above, we have so we can rename them as and r 2 0 + r 2 1 = 1 (5.78) r 0 = cos(θ) (5.79) r 1 = sin(θ) (5.80) summing up, our qubit is now in the canonical representation (♥) |ϕ〉 = cos(θ)|0〉 + sin(θ)e iφ |1〉 (5.81) with only two real parameters left! What is the range of the two angles θ and φ? We invite you to show that 0 ≤ φ ≤ 2π and 0 ≤ θ ≤ π 2 are enough to cover all possible qubits: Exercise 5.4.4 Prove that every qubit in the canonical representation (♥) with θ > π 2 is equivalent to another one where θ lies in the first quadrant of the plane. Hint: use a bit of trigonometry and change φ according to your needs.
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 and 24: 5.3. REVERSIBLE GATES 23 output wil
Page 25: 5.4. QUANTUM GATES 25 |x〉 • |x
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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