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protect such a stream only by cutting it into successive K-qubit blocks. Some introduction and details can be found in [Cha98, Cha99, OT03, OT04]. Quantum Codes for Burst Error: Most quantum error correction codes assume the ”independent qubit decoherence”(IQD) model in which the decoherence of each qubit is uncorrelated to the decoherence of any other qubit as they all interact with separate reservoirs. If this is not true, the errors are space-correlated, typically shown as a burst of errors. Many approaches are introduced for burst error correction, such as Quantum cyclic code [VRA99], Quantum interleaver [Kaw00]. Quantum Reed-Solomon Code [GGB99]: a classical [N; K; d] Reed-Solomon code over GF (2 k ) with length N = 2 k − 1, distance d, and dimension K = N − d + 1, is a cyclic code with the generator polynomial: g(x) = (x − β j )(x − β j+1 · · · (x − β j+d−2 )) with β a primitive element of GF (2 k ) of order N. The quantum Reed-Solomon [N; K; d] code encodes k(2N-K) qubits in states | ϕ1〉, | ϕ2〉 · · · | ϕk(2N−K)〉into kN qubits. It can be constructed as a family of Calderbank-Shor-Steane (CSS) codes. 81
CHAPTER 6 QUANTUM ERROR-CORRECTION FOR TIME-CORRELATED ERRORS 6.1 Time-Correlated Errors Different physical implementations reveal that the interactions of the qubits with the en- vironment are more complex and force us to consider spatially- as well as, time-correlated errors. If the qubits on an n-qubit register are confined to a 3D structure, an error affecting one qubit will propagate to the qubits in a volume centered around the qubit in error. Spatially-correlated errors and means to deal with the spatial noise are analyzed in [AHH02,CSG04,KF05]. An error affecting qubit i of an n-qubit register may affect other qubits of the register. An error affecting qubit i at time t and corrected at time t + ∆ may have further effect either on qubit i or on other qubits of the register. The error model con- sidered in this thesis is based upon a recent study which addresses the problem of reliable quantum computing using solid state systems [NB06]. There are two obvious approaches to deal with quantum correlated errors : • (i) design a code capable to correct these two or more errors, or • (ii) use the classical information regarding past errors and quantum error correcting codes capable to correct a single error. 82
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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
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
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: example shows that the code cannot
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 and 104: We use the Steane code to illustrat
Page 105 and 106: ... ... ... Extra XXX Codeword anci
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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