Chapter 5: Architecture - Computer and Information Science - CUNY
Chapter 5: Architecture - Computer and Information Science - CUNY Chapter 5: Architecture - Computer and Information Science - CUNY
2 CHAPTER 5. ARCHITECTURE 5.1 Bits and Qubits What is a bit? A bit is an atom of information that represents one of two disjoint situations. There are many examples of bits: • A bit is electricity traveling through a circuit or not (or high and low.) • A bit is a switch turned on or off. • A bit is a way of denoting “true” or “false.” All these examples are saying the same thing: a bit is a way of describing a system whose set of states is of size two. We usually write these two possible states as 0 and 1, or or F and T, etc. Since we have become adept at matrices, let us use them as a way of representing a bit. We shall represent 0–or better the state |0〉–as a two by one matrix with a 1 in the 0’s row and a 0 in the 1’s row: |0〉 = We shall represent a 1, or state |1〉 as: [ ] 0 1 1 0 (5.1) |1〉 = [ ] 0 0 1 1 (5.2) Since these are two different representations (indeed orthogonal), we have an honest-to-goodness bit. We will explore how to manipulate these bits in section 5.2.
5.1. BITS AND QUBITS 3 A bit is either in state |0〉 or in state |1〉, which was sufficient for the classical world. Either electricity is running through a circuit or it is not. Either a switch is on or it is off. Either a proposition is true or it is false. But either/or is not sufficient in the quantum world. In that world, there are situations where we are in one state and in the other simultaneously. In the realm of the quantum world, there are systems where a switch is both on and off. One quantum system can be in state |0〉 and in state |1〉. Hence we are led to the definition of a qubit: Definition 5.1.1 A quantum bit or a qubit is a way of describing a quantum system of dimension two. We shall represent a qubit as a two by one matrix with complex numbers [ ] 0 c 0 , (5.3) 1 c 1 where |c 0 | 2 + |c 1 | 2 = 1. Notice that a classical bit is a special type of qubit. |c 0 | 2 is to be interpreted as the probability that after measuring the qubit, it will be found in state |0〉. |c 1 | 2 is to be interpreted as the probability that after measuring the qubit it will be found in state |1〉. Whenever we measure a qubit, it automatically becomes a bit. So we shall never “see” a general qubit. Nevertheless, they do exist and are the main characters in our tale. We might visualize this “collapsing” of a qubit to a bit as [1, 0] T [c 0 , c 1 ] T [0, 1]. T (5.4) Following the normalization procedures that we learned in chapter 4 on page ???, any element of C 2 can be converted into a qubit. For example, the vector ⎡ ⎤ V = ⎣ 5 + 3i ⎦ (5.5) 6i
- Page 1: Chapter 5 Architecture Noson S. Yan
- 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
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- Page 17 and 18: 5.3. REVERSIBLE GATES 17 Exercise 5
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- Page 21 and 22: 5.3. REVERSIBLE GATES 21 |x〉 |x
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- Page 25 and 26: 5.4. QUANTUM GATES 25 |x〉 • |x
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- Page 37: Bibliography [1] Charles H. Bennett
2 CHAPTER 5. ARCHITECTURE<br />
5.1 Bits <strong>and</strong> Qubits<br />
What is a bit? A bit is an atom of information that represents one of two<br />
disjoint situations. There are many examples of bits:<br />
• A bit is electricity traveling through a circuit or not (or high <strong>and</strong> low.)<br />
• A bit is a switch turned on or off.<br />
• A bit is a way of denoting “true” or “false.”<br />
All these examples are saying the same thing: a bit is a way of describing a<br />
system whose set of states is of size two. We usually write these two possible<br />
states as 0 <strong>and</strong> 1, or<br />
or F <strong>and</strong> T, etc.<br />
Since we have become adept at matrices, let us use them as a way of representing<br />
a bit. We shall represent 0–or better the state |0〉–as a two by one<br />
matrix with a 1 in the 0’s row <strong>and</strong> a 0 in the 1’s row:<br />
|0〉 =<br />
We shall represent a 1, or state |1〉 as:<br />
[ ]<br />
0 1<br />
1 0<br />
(5.1)<br />
|1〉 =<br />
[ ]<br />
0 0<br />
1 1<br />
(5.2)<br />
Since these are two different representations (indeed orthogonal), we have an<br />
honest-to-goodness bit. We will explore how to manipulate these bits in section<br />
5.2.
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