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Quantum Computation with Spins in Quantum Dots

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Quantum Computation with Spins in Quantum Dots Quantum Computation With Spins In Quantum Dots Fikeraddis Damtie June 10, 2013 1 Introduction Physical implementation of quantum computating has been an active area of research since the last couple of decades. In classical computing, the transmission and manipulation of classical information is carried out by physical machines (computer hardwares, etc.). In theses machines the manipulation and transmission of information can be described using the laws of classical physics. Since Newtonian mechanics is a special limit of quantum mechanics, computers making use of the laws of quantum mechanics have greater computational power than classical computers. This need to create a powerful computing machine is the driving motor for research in the field of quantum computing. Until today, there are a few different schemes for implementing a quantum computer based on the David Divincenzo chriterias. Among these are: Spectral hole burning, Trapped ion, e-Helium, Gated qubits, Nuclear Magnetic Resonance, Optics, Quantum dots, Neutral atom, superconductors and doped silicon. In this project only the quantum dot scheme will be discussed. In the year 1997 Daniel Loss and David P.DiVincenzo proposed a spin-qubit quantum computer also called The Loss-DiVincenzo quantum computer. This proposal is now considered to be one of the most promising candidates for quantum computation in the solid state. The main idea of the proposal was to use the intrinsic spin-1=2 degrees of freedom of individual elecrons confined in semiconductor quantum dots. The proposal was made in a way to satisfy the five requirements for quantum computing by David diVincenzo which will be described in sec. 2 in the report. Namely 1. A scalable physical system with well characterized quibits 2. The ability to initialize the state of the qubits to a simple fiducial state such as j000:::i 3. Long relevant decoherence times, much longer than the gate operation time 4. A "universal" set of quantum gates 5. A qubit-specific measurement capability A good candidate for such quantum computer is single and double lateral quantum dot systems. This report is organized as follows. In section 2 general requirements for the physical implementa- tion of quantum computation the so called diVincenzo criteria is discussed briefly. Section 3 will introduce the basics of quantum dot physics. In this section the properties of single and double quantum dot will be discussed. In section 4 a description of the Loss-diVincenzo proposal for quan- tum computation based on single electron spin in semiconductor quantum dot will be given. In section 5, analysis of the theoretical and experimental development in the last decade will be given. The last section will be devoted to discussion and assesment of main qualitative and quantitative 1 obstacles for an ultimate realization of the spin based quantum computer. For the report, I mainly based on the papers [9] and [5] 2 DiVincenzo requirements for the physical implementation of quantum computation In his paper "The Physical Implementation of Quantum Computation"[3], David P. DiVincenzo and his co-workers described Five (plus two) requirements for the implementation of quantum computations. Below I will try to briefly mention the requirements. 1. A scalable physical system with well characterized quibits: The requirement here is that a physical system containing a collection of qubits is needed at the beginning. Here a qubit being "well characterized" can mean different things. It can for example mean that its physical parameters should be accurately known including the internal Hamiltonian of the qubit, the presence of and couplings to other states of the qubit, the interaction with other qubits and the couplings to external fields that might be used to manipulate the state of the qubit. 2. The ability to initialize the state of the qubits to a simple fiducial state such as j000:::i: This is analogues to say that registers should be initialized to a known value before start of computation, which is a straight forward computing requirement for classical computation. Another reason for this is initialization requirement is from the point of quantum error correction which requires a continous, fresh supply of qubits in a low entropy state. (link the j0i) state. 3. Long relevant decoherence times, much longer than the gate operation time: An overly sim- plified definition for a coherence time can be described as the time for a generic qubit state j i = aj0i + bj1i to be transformed in to the mixture ρ = jaj2j0ii0j + jbj2j1ii1j. This time helps characterize the dynamics of a qubit in contact with its enviroment. Decoherance is an important concept in quantum mechanics. It is ientified as the principal mechanism for the emergence of classical behaviour. For quantum computating, decherance can be very can be very dangerous. If the qubit system has a fast decoherence time, the capability of the quantum computer will not be very different from that of the classical ones.Hence in quan- tum computing, it is desirable to have long enough decoherence time such that the uniquely quantum features of quantum computating can have a chance to come in to play. The answer to "How long is long enough?" is determined by quantum error correction which will not be 2 discussed in this report see for example [7]. 4. A "universal" set of quantum gates:This fourth requirement can be considered as the most important for quantum computing. One and two qubit gates are needed. 5. A qubit-specific measurement capability:This is also called a readout. The result of a compu- tation must be readout. This requires the ability to measure specific qubits. For computation alone, the above five requirements are enough. But the advantages of quantum information processing are not manifest solely for straight forward computation only. There are different kinds of information- processing tasks that involve more that just computation and for which quantum tools provide a unique advantage. One of such tasks is quantum communication: the transmission of intact qubits from place to place. If one consider additional information processing tasks than just computation only, two more re- qurements are needed to be fulfilled. 6. The ability to interconvert stationary and flying qubit: 7. The ability faithfully to transmit flying qubits between specified locations: [3] Before discussing the Loss-diVincenzo proposal for quantum computer based on spin in quantum dots, it might be a good idea to spend some time discussing the basics of quantum dot physics briefly. 3 Semiconductor Quantum Dots Quantum dot’s are artificial sub-micron structures in a solid, typically consisting of 103 −109 atoms and comparable number of electrons. [6] As they are confined in all three dimensions, the resulting electronic states exhibit discreteness. 3 Figure 1: Top: Schematic representation of three, two, one, and zero-dimensional nanostructures made using semiconductor hetrostructures. Bottom: The corresponding densities of electronic states. [1] In a three dimensional bulk semiconductor electrons are free in all three dimensions. In 2DEG (Two dimensional electron gas) electrons are free to move in a plane in two dimensions. In one Dimensional quantum wires, electrons are allowed only to move along the direction of length of the wire. In zero dimensional quantum dots, electronic motion is restricted in all three dimensions and the density of state is discrete. By using the constant interaction model, it is possible to describe for example the current voltage characterstics of a quantum dot system. In the constant interaction model, Coulomb interaction between electrons in the dot and electrons in the surrounding enviroment is replaced by self- capacitances. 3.1 Resonant Tunneling Through Quantum Dots Due to the discreteness of the electronic states in quantum dots electron transport is restricted to resonant conditions. For a quantum dot coupled to a source and drain contact, resonant tunneling occurs when an electronic state which can either be occupied or empty in one of the contacts align with any of the available states in the dot. Schematically this situation is shown as in the picture below 4 Figure 2: Schematic diagram of resonant condition through a single quantum dot showing the electrochemical potential level of a dot coupled to the source and drain reservoirs. [4] As one can see from the above schematic, resonant tunneling occurs when there is available discrete level of the dot in between the source and drain chemical potentials. As a result, electrons flow through the dot sequentially and are detected as a change in source-drain current. [4] Practically, in experiments, the chemical potentials of the source and drain contact can be varied by varying the source drain bias as µs − µd = eVsd. Similarly, the electrochemical potential in the dot can be adjusted by varying the gate voltage Vg. By plotting the gate voltage versus the source drain bias on a two-dimensional map, which is called the charge-stability diagram, one can get information about the sequential tunneling and total number of particles involved as there are diamond-shaped regions with well-defined charge numbers in the dot. These diamond-shaped regions are called Coulomb diamonds. [4] The typical charge stability diagram of a single quantum dot coupled to source and drain contacts is shown in the figure below. 5 Figure 3: A typical charge stability diagram of a single quantum dot coupled to the source and drain contact which shows Coulomb diamond due to a transport through the ground state of the quantum dot and additional transport lines running parallel to the diamond edges at an energy ∆E. These lines can be attributed to transport due either to other single particle levels or inelastic processes such as emission or absorption of phonon or phonon with energy ∆E. [4] 3.2 Resonant Tunneling Through Double Quantum Dots Double quantum dots are sometimes called artificial molecules as they are made by coupling two quantum dots either vertically or in parallel.
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