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Ordinarily, the logic gates in a classical computer, other than the NOT gate are not reversible. For example, for an AND gate one cannot generally recover the two input bits from the output bit. If a logic gate is irreversible, then some of the input information is irretrievably erased after the execution of the gate. Landauer’s principle provides the connection between energy consumption and irreversibility [PV01]. It states that the erasure of information is inevitably accompanied by the generation of heat. This becomes a big obstacle for the further improvement of classical computers since the amount of energy dissipated keeps increasing as the computer power increases. However, as a first step in describing a quantum circuit, it is instructive to observe that reversible gates are theoretically possible; moreover, these are actually of practical interest, since they do not increase the entropy. A quantum gate executes a quantum operation on quantum information. A quantum operation can also be defined as a rotation of the state | ψ〉 in the N-dimensional Hilbert space. It is also convenient to represent a quantum gate (operation) in matrix form in Hilbert space. Any quantum operation is reversible, linear and preserves the norm. In fact a linear transformation is a valid quantum operation if and only if it is unitary. An operator U is unitary if U † U = UU † = I with U † the adjacent of U 15
Operation U1 followed by U2 is equivalent to U3 = U2U1 If U1 acts on Hilbert space HA and U2 acts on Hilbert space HB, then U3 = U2 ⊗ U1 acts on the composite Hilbert space HAB = HA ⊗ HB Some elemental quantum gates are introduced as follows: • NOT gate |0>+¡|1> X |1>+¡|0> |0>+¡|1> Z |0> -¡|1> |0>+¡|1> H 0 + 1 0 − 1 α + β 2 2 Figure 2.2: Circuit representations of single qubit logic gates The quantum NOT gate acts linearly. It takes the state α | 0〉 + β | 1〉 to the corre- sponding state α | 1〉 + β | 0〉 in which the vectors of | 0〉 and | 1〉 have been flipped. The quantum NOT gate can be represented as: ⎛ X = ⎜ ⎝ 16 0 1 1 0 ⎞ ⎟ ⎠
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 and 20: Different methods of implementing q
Page 21 and 22: | 11〉. For example, | 0〉⊗ | 1
Page 23 and 24: Suppose the state we want to copy i
Page 25: 2.3 Quantum Circuits In the classic
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
Page 73 and 74: 1. Devise an encoding scheme for th
Page 75 and 76: 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
Page 119 and 120:
[Got97] D. Gottesman. “Stabilizer
Page 121 and 122:
[RG05] B. W. Reichardt and L. K. Gr
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