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this extended syndrome measurement, we need to consider two aspects: (i) the entanglement between the additional ancilla and codewords should not be disturbed during the syndrome measurement, since after the measurement, we need to disentangle the extra ancilla from the codeword and keep the codeword intact; and (ii) the new stabilizers (or generators) should also stabilize the codeword and satisfy all the stabilizers’ requirements. In our approach, the additional ancilla qubit A is a copy of the control qubit in Z basis. To keep the entanglement, we copy all X operations on the control qubit to qubit A. Similarly, the additional ancilla qubit B is a copy of the control qubit in X basis. We replicate all Z operations performed on the control qubit to this qubit. Since it is in X basis, the Z operations will change into X operations. Consider the example of the Steane code and the time-correlated error on qubit 3, Figure 6.2(b). In this case we replicate all X operations performed on qubit 3 to ancilla qubit A, and replicate all Z operations to additional ancilla qubit B as X operations. These operations will keep the entanglement intact during the syndrome measurement process. Also, it is easy to check that the new stabilizers (i) commute with each other, and (ii) stabilize the codewords. 3. Disentangle and measure the two additional ancilla qubits. After the syndrome measurement, we use a circuit similar to the one in Figure 6.2(a) to disentangle the two additional ancilla qubits from the codeword. The gates in this circuit are in reverse order as in Figure 6.2(a). Since the entanglement is not affected during the extended syndrome measurement, this process will disentangle the extra ancilla from the codewords and keep 93
... ... ... Extra XXX Codeword ancilla X H H |0> Ancilla H 1324567 Z X XX HH (a) (b) X ZZZ Figure 6.2: (a) Duplicate the qubit affected error during the last cycle in both X and Z basis; (b) Extended syndrome measurement on both the codeword and the additional ancilla qubits. AB the codeword in its original state. If qubit 3 is affected by the time-correlated error, bit-flip, phase-flip or both, the disentanglement circuit will propagate these errors to the extra ancilla qubits: a CNOT gate will propagate a bit-flip of the control qubit to the target, and an HCNOT gate will propagate a phase-flip of the control qubit to the target qubit as a bit-flip, Figure 6.3. Therefore, measuring the additional ancilla qubits after the disentanglement will give us the information regarding the type of error qubit 3 has relapsed to. 4. Correct the errors according to the output of the extended syndrome measurement and of the measurement of the two additional ancilla qubits. Consider the following scenarios: 94
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STUDIES OF A QUANTUM SCHEDULING ALG
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ABSTRACT Quantum computation has be
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ACKNOWLEDGMENTS The first person I
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3.2 Amplitude Amplification . . . .
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LIST OF FIGURES 2.1 (a) The measure
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LIST OF TABLES 4.1 An example of 8
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lots of scientists to study on diff
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CHAPTER 2 BASIC CONCEPTS AND RELATE
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All these achievements continuously
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Different methods of implementing q
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| 11〉. For example, | 0〉⊗ | 1
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Suppose the state we want to copy i
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2.3 Quantum Circuits In the classic
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Operation U1 followed by U2 is equi
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c t c t ⊕ c c c Figure 2.3: Circu
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2.4 Physical Implementations of Qua
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an arbitrary number of qubits. A qu
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Up to date, there have been several
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iophysics and materials technology.
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In a key development for Topologica
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calculations many times. The advant
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very limited. To see the spectacula
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Example. Consider a 3-dimensional s
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CHAPTER 3 GROVER’S ALGORITHM 3.1
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lack box performs the following tra
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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
Page 77 and 78: Quantum error correction is used in
Page 79 and 80: discovered independently by Shor [S
Page 81 and 82: Since the first qubit was flipped,
Page 83 and 84: Let us take the C1 [7,4,3] Hamming
Page 85 and 86: For example, consider the case n =
Page 87 and 88: 2. The identity matrix multiplied b
Page 89 and 90: which means that the stabilizer M c
Page 91 and 92: example shows that the code cannot
Page 93 and 94: CHAPTER 6 QUANTUM ERROR-CORRECTION
Page 95 and 96: j Qubit flips i and j Qubit i is re
Page 97 and 98: of the model can be written as: H =
Page 99 and 100: We also assume that: • There are
Page 101 and 102: the original codeword. Now we need
Page 103: We use the Steane code to illustrat
Page 107 and 108: flip, 11 - bit-and-phase-flip, see
Page 109 and 110: Table 6.2: The syndromes for each o
Page 111 and 112: correcting code capable of correcti
Page 113 and 114: # Sample code for testing the algor
Page 115 and 116: # "Disentangle" the extra ancilla g
Page 117 and 118: 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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