Heiss W.D. (ed.) Quantum dots.. a doorway to - tiera.ru

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28 J.M. Elzerman et al. takes one input value, 0 or 1, and computes the corresponding output value, f(0) or f(1). A quantum computer with one quantum bit (or “qubit”) could take as an input value a superposition of |0〉 and |1〉, and due to the linearity of quantum mechanics the output would be a superposition of F |0〉 and F |1〉. So, in a sense it has performed two calculations in a single step. For a twoqubit system, the gain becomes even more significant: now the input can be a superposition of four states, so the quantum computer can perform four calculations in one step. In fact, it can be proved [9] that the computing power of a quantum computer scales exponentially with the number of qubits, whereas this scaling is only linear for a classical computer. Therefore, a large enough quantum computer can outperform any classical computer. It might appear that a fundamental problem has been overlooked: according to quantum mechanics, a superposition of possible measurement outcomes can only exist before it is measured, and the measurement gives only one actual outcome. The exponential computing power thus appears inaccessible. However, by using carefully tailored quantum algorithms, an exponential speed-up can be achieved for some problems such as factoring integers [10] or simulating a quantum system [11]. For other tasks, such as searching a database, a quadratic speed-up is possible [12]. Using such quantum algorithms, a quantum computer can indeed be faster than a classical one. Another fundamental problem is the interaction of the quantum system with the (uncontrolled) environment, which inevitably disturbs the desired quantum evolution. This process, known as “decoherence”, results in errors in the computation. Additional errors are introduced by imperfections in the quantum operations that are applied. All these errors propagate, and after some time the state of the computer will be significantly different from what it should be. It would seem that this prohibits any long computations, making it impossible for a quantum computer to use its exponential power for a non-trivial task. Fortunately, it has been shown that methods to detect and correct any errors exist [13, 14], keeping the computation on track. Of course, such methods only help if the error rate is small enough, since otherwise the correction operations create more errors than they remove. This sets a socalled “accuracy threshold” [15, 16], which is currently believed to be around 10 −4 . If the error per quantum operation is smaller than this threshold, any errors can be corrected and an arbitrarily long computation is possible. Due to the development of quantum algorithms and error correction, quantum computation is feasible from a theoretical point of view. The challenge is building an actual quantum computer with a sufficiently large number of coupled qubits. Probably, more than a hundred qubits will be required for useful computations, but a system of about thirty qubits might already be able to perform valuable simulations of quantum systems.
1.2 Implementations Semiconductor Few-Electron Quantum Dots as Spin Qubits 29 A number of features are required for building an actual quantum computer [17]: 1. A scalable physical system with well-characterized qubits 2. A “universal” set of quantum gates to implement any algorithm 3. The ability to initialize the qubits to a known pure state 4. A qubit-specific measurement capability 5. Decoherence times much longer than the gate operation time Many systems can be found which satisfy some of these criteria, but it is very hard to find a system that satisfies all of them. Essentially, we have to reconcile the conflicting demands of good access to the quantum system (in order to perform fast and reliable operations or measurements) with sufficient isolation from the environment (for long coherence times). Current state-ofthe-art is a seven-bit quantum computer that has factored the number 15 into its prime factors 3 and 5, in fewer steps than is possible classically [18]. This was done using an ensemble of molecules in liquid solution, with seven nuclear spins in each molecule acting as the seven qubits. These could be controlled and read out using nuclear magnetic resonance (NMR) techniques. Although this experiment constitutes an important proof-of-principle for quantum computing, practical limitations do not allow the NMR approach to be scaled up to more than about ten qubits. Therefore, many other implementations are currently being studied [19]. For instance, trapped ions have been used to demonstrate a universal set of one- and two-qubit operations, an elementary quantum algorithm, as well as entanglement of up to three qubits and quantum teleportation [19]. Typically, microscopic systems such as atoms or ions have excellent coherence properties, but are not easily accessible or scalable – on the other hand, larger systems such as solid-state devices, which can be accessed and scaled more easily, usually lack long decoherence times. A solid-state device with a long decoherence time would represent the best of both worlds. Such a system could be provided by the spin of an electron trapped in a quantum dot: a spin qubit. 1.3 The Spin Qubit Our programme to build a solid-state qubit follows the proposal by Loss and DiVincenzo [2]. This describes a quantum two-level system defined by the spin orientation of a single electron trapped in a semiconductor quantum dot. The electron spin can point “up” or “down” with respect to an external magnetic field. These eigenstates, |↑〉and |↓〉, correspond to the two basis states of the qubit. The quantum dot that holds the electron spin is defined by applying negative voltages to metal surface electrodes (“gates”) on top of a semiconductor (GaAs/AlGaAs) heterostructure (see Fig. 2). Such gated quantum dots are
Page 1 and 2: Lecture Notes in Physics Editorial
Page 3 and 4: W. Dieter Heiss (Ed.) Quantum Dots:
Page 5: Preface Nanoscale physics, nowadays
Page 9 and 10: Contents A Guide for the Reader ...
Page 11 and 12: A Guide for the Reader Quantum dots
Page 13 and 14: The Renormalization Group Approach
Page 15 and 16: RG for Interacting Fermions 5 where
Page 17 and 18: where � is a shorthand: � ≡ 3
Page 19 and 20: RG for Interacting Fermions 9 These
Page 21 and 22: RG for Interacting Fermions 11 is i
Page 23 and 24: RG for Interacting Fermions 13 the
Page 25 and 26: RG for Interacting Fermions 15 this
Page 27 and 28: RG for Interacting Fermions 17 equa
Page 29 and 30: � p � dθp = g 2π RG for Inter
Page 31 and 32: RG for Interacting Fermions 21 the
Page 33 and 34: References RG for Interacting Fermi
Page 35 and 36: Semiconductor Few-Electron Quantum
Page 37: Semiconductor Few-Electron Quantum
Page 43 and 44: GaAs n-AlGaAs AlGaAs GaAs Semicondu
Page 49 and 50: ε1 ε0 e Semiconductor Few-Electro
Page 53 and 54: 250 Ω twisted pair 20 MΩ powder
Page 61 and 62: 10 5 0 -5 Semiconductor Few-Electro
Page 63 and 64: G ( e / h) 2 2 1 0 Semiconductor Fe
Page 67 and 68: V R (V) V R (V) Semiconductor Few-E
Page 69 and 70: V L (V) -1.084 -1.082 -1.080 -1.078
Page 71 and 72: B // Semiconductor Few-Electron Qua
Page 77 and 78: a RESERVOIR 200 nm Semiconductor Fe
Page 81 and 82: 4.5 QPC Versus SET Semiconductor Fe
Page 83 and 84: a reservoir dot Semiconductor Few-E
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Semiconductor Few-Electron Quantum
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Spin-down counts 200 100 a Semicond
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6.7 Unresolved Issues Semiconductor
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98 M. Pustilnik and L.I. Glazman Co
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132 C.W.J. Beenakker e e N I N S Fi
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134 C.W.J. Beenakker 2 Andreev Refl
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136 C.W.J. Beenakker F E x 8d/hv ga
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138 C.W.J. Beenakker and we have de
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140 C.W.J. Beenakker A stroboscopic
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142 C.W.J. Beenakker largely indepe
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144 C.W.J. Beenakker The effective
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146 C.W.J. Beenakker Fig. 6. Ensemb
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148 C.W.J. Beenakker 1.4 1.2 1.0 0.
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150 C.W.J. Beenakker 〈ρ(E)〉 =
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152 C.W.J. Beenakker P 0.5 0.4 0.3
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154 C.W.J. Beenakker P(x) 0.4 0.3 0
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156 C.W.J. Beenakker tunnel/ Egap 4
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158 C.W.J. Beenakker its momentum).
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160 C.W.J. Beenakker E00 = π� =
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162 C.W.J. Beenakker 〈ρ(E)〉 δ
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164 C.W.J. Beenakker As described i
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166 C.W.J. Beenakker 0.30 � Egap
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168 C.W.J. Beenakker The τE-depend
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172 C.W.J. Beenakker (τ−,τ+), w
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174 C.W.J. Beenakker 61. P. G. Silv
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