Chapter 5: Architecture - Computer and Information Science - CUNY
Chapter 5: Architecture - Computer and Information Science - CUNY Chapter 5: Architecture - Computer and Information Science - CUNY
32 CHAPTER 5. ARCHITECTURE One of the central features of computer science is an operation that is done only under certain conditions and not under others. This is equivalent to an IF- THEN statement. If a certain (qu)bit is true, then a particular operation should be performed, otherwise the operation is not performed. For every n-qubit unitary operation U, we can create a unitary n+1-qubit operation Controlled- U or C U. |x〉 • |x〉 (5.93) / n U / n This operation will perform the U operation if the top |x〉 input is a |1〉 and will simply perform the identity operation if |x〉 is |0〉. For the simple case of ⎡ ⎤ U = ⎣ a b ⎦ (5.94) c d the controlled-U operation can be seen to be ⎡ ⎤ 1 0 0 0 0 1 0 0 C U = . (5.95) ⎢ ⎣ 0 0 a b⎥ ⎦ 0 0 c d This same construction works for matrices larger than 2 by 2. Exercise 5.4.9 Show that the constructed C U works as it should when the top qubit is set to |0〉 or set to |1〉. Exercise 5.4.10 Show that if U is unitary, then so is C U. Exercise 5.4.11 Show that the Toffoli gate is nothing more than C ( C NOT)
5.4. QUANTUM GATES 33 Throughout the rest of this text we shall demonstrate the many operations that can be performed with quantum gates. However, there are limitations to what can be done with a quantum gate. For one thing, every operation must be reversible. Another limitation is a consequence of the The No-Cloning Theorem. This theorem says that it is impossible to clone an exact quantum state. In other words, it is impossible to make a copy of an arbitrary quantum state without first destroying the original. In “computerese,” this says that we can “cut” and “paste” a quantum state but we cannot “copy” and “paste” a quantum state. “Move” is possible; “Copy” is imposable. Why can’t we? What would such a cloning operation look like? There would be two different places having the same vector space that describes a quantum system, say V. Although these two systems would be apart, we are interested in looking at the two systems as one, i.e., we are interested in V ⊗ V. A potential cloning operation would be a linear map C : V ⊗ V −→ V ⊗ V (5.96) that should take an arbitrary state |x〉 in the first system and, perhaps, nothing in the second system and clone |x〉, i.e., C(|x〉 ⊗ 0) = (|x〉 ⊗ |x〉). (5.97) This seems like a harmless enough operation, and there is no problem cloning for an arbitrary classical state |x〉. In fact, every Xerox machine clones and every time we copy a file, classical states of information are cloned. However, this information is in a collapsed classical state. It is not in a superposition of states. The problem is cloning a superposition of states, that is, an arbitrary quantum state. Suppose we have a superposition of states √ . Cloning such a 2 |x〉 + |y〉 state would mean that ( ) ( ) |x〉 + |y〉 |x〉 + |y〉 |x〉 + |y〉 C √ ⊗ 0 = √ ⊗ √ . (5.98) 2 2 2 However if we insist that C is a quantum operation, then C must be linear and hence must respect the scalar multiplication and the addition in V ⊗ V. If C was linear, then ( ) ( ) |x〉 + |y〉 1 C √ ⊗ 0 = C √2 (|x〉 + |y〉) ⊗ 0 = √ 1 C((|x〉 + |y〉) ⊗ 0) (5.99) 2 2 = 1 √ 2 (C(|x〉⊗0+|y〉⊗0)) = 1 √ 2 (C(|x〉⊗0)+C(|y〉⊗0)) = 1 √ 2 ((|x〉⊗|x〉)+(|y〉⊗|y〉)) (5.100) (|x〉 ⊗ |x〉) + (|y〉 ⊗ |y〉) = √ . (5.101) 2
- 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 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: 5.4. QUANTUM GATES 31 will work. R
- Page 35 and 36: 5.4. QUANTUM GATES 35 Multiplying t
- Page 37: Bibliography [1] Charles H. Bennett
32 CHAPTER 5. ARCHITECTURE<br />
One of the central features of computer science is an operation that is done<br />
only under certain conditions <strong>and</strong> not under others. This is equivalent to an IF-<br />
THEN statement. If a certain (qu)bit is true, then a particular operation should<br />
be performed, otherwise the operation is not performed. For every n-qubit<br />
unitary operation U, we can create a unitary n+1-qubit operation Controlled-<br />
U or C U.<br />
|x〉<br />
•<br />
|x〉<br />
(5.93)<br />
/ n U / n<br />
This operation will perform the U operation if the top |x〉 input is a |1〉 <strong>and</strong><br />
will simply perform the identity operation if |x〉 is |0〉.<br />
For the simple case of<br />
⎡ ⎤<br />
U = ⎣ a b ⎦ (5.94)<br />
c d<br />
the controlled-U operation can be seen to be<br />
⎡<br />
⎤<br />
1 0 0 0<br />
0 1 0 0<br />
C U =<br />
. (5.95)<br />
⎢<br />
⎣<br />
0 0 a b⎥<br />
⎦<br />
0 0 c d<br />
This same construction works for matrices larger than 2 by 2.<br />
Exercise 5.4.9 Show that the constructed C U works as it should when the top<br />
qubit is set to |0〉 or set to |1〉.<br />
Exercise 5.4.10 Show that if U is unitary, then so is C U.<br />
Exercise 5.4.11 Show that the Toffoli gate is nothing more than C ( C NOT)
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