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in states corresponding to a blend or superposition of these states. In other words, a qubit can exist as a zero, a one, or simultaneously as both 0 and 1, with a coefficient representing the probability amplitude for each state. Actually, a qubit is a 2-dimensional, normalized vector in a Hilbert space with base vectors | 0〉 and | 1〉. An arbitrary single-qubit state can be represented as: | ψ〉 = α | 0〉 + β | 1〉 This is a normalized state where the probability amplitudes α and β are complex numbers, with |α| 2 + |β| 2 = 1. A classical computer with n bits has 2 n possible states and this is an n-dimensional vector space over Z2. The inner product of two state vectors represents the generalized ”angle” between the states and gives an estimate of the overlap between the states | ψa〉 and | ψb〉. The inner product 〈ψa | ψb〉 = 0 defines orthogonal states; the implication of 〈ψa | ψb〉 = 1 is that | ψa〉 and | ψb〉 are the same state. A quantum computer with n qubits is a state in a 2 n -dimensional complex vector space. It can be represented as the tensor product ”⊗” of the respective state spaces of all the individual qubits. We abbreviate: | 0〉⊗ | 1〉 =| 01〉, etc. The tensor product of two qubits over the bases {| 0〉, | 1〉} can be expressed with the four basic vectors | 00〉, | 01〉, | 10〉, 9
| 11〉. For example, | 0〉⊗ | 1〉 = ⎛ ⎜ ⎝ ⎛ ⎞ ⎞ 1 ⎟ ⎠ 0 ⊗ ⎛ ⎞ ⎜ 0 ⎟ ⎜ ⎟ ⎝ ⎠ 1 = ⎜ 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 ⎟ ⎜ ⎟ ⎜ ⎟ = 0 | 00〉 + 1 | 01〉 + 0 | 10〉 + 0 | 11〉 ⎜ 0 ⎟ ⎜ ⎟ ⎝ ⎠ 0 A quantum register of n qubits has the associated Hilbert space H2 ⊗ H2 ⊗ · · · ⊗ H2 = H2 n and is of the form with � 2 n −1 i=0 |αi| 2 = 1 2n �−1 i=0 αi | i〉 Here, in Hilbert space H2 n of dimension 2n , the orthogonal basis {| i〉} ⎛ ⎞ ⎜ 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ . ⎟ ⎜ ⎟ ⎝ ⎠ 0 is defined as the computational basis. state: ⎛ ⎞ ⎜ 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ . ⎟ ⎜ ⎟ ⎝ ⎠ 0 · · · ⎛ ⎞ ⎜ 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ . ⎟ ⎜ ⎟ ⎝ ⎠ 1 The Pauli matrices: G = {σI, σx, σy, σz} describe transformations of interest of one qubit σI = ⎛ ⎜ ⎝ 1 0 0 1 ⎞ ⎟ ⎠ σx = ⎛ ⎜ ⎝ 0 1 1 0 ⎞ ⎟ ⎠ 10 σy = ⎛ ⎜ ⎝ 0 -i i 0 ⎞ ⎟ ⎠ σz = ⎛ ⎜ ⎝ 1 0 0 -1 ⎞ ⎟ ⎠
Page 1 and 2: STUDIES OF A QUANTUM SCHEDULING ALG
Page 3 and 4: ABSTRACT Quantum computation has be
Page 5 and 6: ACKNOWLEDGMENTS The first person I
Page 7 and 8: 3.2 Amplitude Amplification . . . .
Page 9 and 10: LIST OF FIGURES 2.1 (a) The measure
Page 11 and 12: LIST OF TABLES 4.1 An example of 8
Page 13 and 14: lots of scientists to study on diff
Page 15 and 16: CHAPTER 2 BASIC CONCEPTS AND RELATE
Page 17 and 18: All these achievements continuously
Page 19: Different methods of implementing q
Page 23 and 24: Suppose the state we want to copy i
Page 25 and 26: 2.3 Quantum Circuits In the classic
Page 27 and 28: Operation U1 followed by U2 is equi
Page 29 and 30: c t c t ⊕ c c c Figure 2.3: Circu
Page 31 and 32: 2.4 Physical Implementations of Qua
Page 33 and 34: an arbitrary number of qubits. A qu
Page 35 and 36: Up to date, there have been several
Page 37 and 38: iophysics and materials technology.
Page 39 and 40: In a key development for Topologica
Page 41 and 42: calculations many times. The advant
Page 43 and 44: very limited. To see the spectacula
Page 45 and 46: Example. Consider a 3-dimensional s
Page 47 and 48: CHAPTER 3 GROVER’S ALGORITHM 3.1
Page 49 and 50: lack box performs the following tra
Page 51 and 52: amplitude amplification is an opera
Page 53 and 54: Each iteration Q will rotate the sy
Page 55 and 56: CHAPTER 4 GROVER-TYPE QUANTUM SCHED
Page 57 and 58: 4.2 Introduction to Scheduling Prob
Page 59 and 60: equires that the number of differen
Page 61 and 62: schedules: | � �� ST Cmax
Page 63 and 64: m index qubits to control the job-m
Page 65 and 66: The makespan of schedule S1 is equa
Page 67 and 68: J1 J2 ... ... JN |0> |0> |0> ... ..
Page 69 and 70: Max(| 5〉, | 7〉, | 4〉) =| 7〉
Page 71 and 72:
the machine and remaining q qubits
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1. Devise an encoding scheme for th
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state, a well-defined desired state
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Quantum error correction is used in
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discovered independently by Shor [S
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Since the first qubit was flipped,
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Let us take the C1 [7,4,3] Hamming
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For example, consider the case n =
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2. The identity matrix multiplied b
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which means that the stabilizer M c
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example shows that the code cannot
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CHAPTER 6 QUANTUM ERROR-CORRECTION
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j Qubit flips i and j Qubit i is re
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of the model can be written as: H =
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We also assume that: • There are
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the original codeword. Now we need
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We use the Steane code to illustrat
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... ... ... Extra XXX Codeword anci
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flip, 11 - bit-and-phase-flip, see
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Table 6.2: The syndromes for each o
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correcting code capable of correcti
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# Sample code for testing the algor
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# "Disentangle" the extra ancilla g
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LIST OF REFERENCES [AAK01] D. Aharo
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[Got97] D. Gottesman. “Stabilizer
Page 121 and 122:
[RG05] B. W. Reichardt and L. K. Gr
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