In this paper we investigate the implementation of single-qubit operations in the Kane solid-state quantum computer by Rabi flipping the nuclear qubits via the application of resonant radio-frequency fields. Specifically, we calculate the change in the resonance frequency of the nuclear qubits with the application of electric potentials to metallic control gates designed to address individual qubits, as well as the effects these fields have on neighbouring qubits. We find that although the resonance frequencies of the neighbouring qubits are shifted, it is still theoretically possible to implement single-qubit operations with the fidelity required for large-scale quantum computation using this technique.
In this paper we investigate the implementation of single-qubit operations in the Kane solid-state quantum computer by Rabi flipping the nuclear qubits via the application of resonant radio-frequency fields. Specifically, we calculate the change in the resonance frequency of the nuclear qubits with the application of electric potentials to metallic control gates designed to address individual qubits, as well as the effects these fields have on neighbouring qubits. We find that although the resonance frequencies of the neighbouring qubits are shifted, it is still theoretically possible to implement single-qubit operations with the fidelity required for large-scale quantum computation using this technique.
The nuclear spin quantum computer proposed by Kane1 exploits as a qubit array 31P dopants embedded within a silicon matrix. Single-qubit operations are controlled by the application of electrostatic potentials via a set of metallic A gates, situated above the donors, on the silicon surface, that tune the resonance frequency of individual nuclear spins, and a globally applied RF magnetic field that flips spins at resonance. Coupling between qubits is controlled by the application of potentials via a set of J gates, between the donors, that induce an electron-mediated coupling between nuclear spins. We report the results of a study of the electric field and potential profiles arising within the Kane device from typical gate operations. The extent to which a single nuclear spin can be tuned independently of its neighbours, by operation of an associated A-gate, is examined and key design parameters in the Kane architecture are addressed. Implications for current fabrication strategies involving the implantation of 31P atoms are discussed. Solution of the Poisson equation has been carried out by simulation using a TCAD modeling package (Integrated Systems Engineering AG).
The nuclear magnetic resonance (NMR) frequency of a single qubit structure of Kane’s solid-state quantum computer is investigated by using the perturbation theory. With higher-order excited states (up to 3d modes) included in our calculation, the perturbation frequencies and energies are obtained numerically. To compute for arbitrary A gate geometries, the perturbation potential inside the qubit structure is determined through an electromagnetic simulation method. Calculations show that the potential distributions for realistic A gate geometries are far from linear ones. Our results show that the A gate voltage has a much more effective control over the NMR frequency of the phosphorus nucleus than that previously shown. Using our method, arbitrary A gate structures of any shapes or geometries can be engineered for the realization of a solid-state scalable quantum computer. We also investigate an alternative A gate structure using SiGe as the insulation barrier. Our study shows that this A gate structure offers a much more efficient utilization of the control voltage than the original A gate structure using SiO2 as the insulation barrier.
Large-scale quantum computers must be built upon quantum bits that are both highly coherent and locally controllable. We demonstrate the quantum control of the electron and the nuclear spin of a single (31)P atom in silicon, using a continuous microwave magnetic field together with nanoscale electrostatic gates. The qubits are tuned into resonance with the microwave field by a local change in electric field, which induces a Stark shift of the qubit energies. This method, known as A-gate control, preserves the excellent coherence times and gate fidelities of isolated spins, and can be extended to arbitrarily many qubits without requiring multiple microwave sources.
A numerical method is introduced to determine the nuclear magnetic resonance frequency of a donor (31P) doped inside a silicon substrate under the influence of an applied electric field. This phosphorus donor has been suggested for operation as a qubit for the realization of a solid-state scalable quantum computer. The operation of the qubit is achieved by a combination of the rotation of the phosphorus nuclear spin through a globally applied magnetic field and the selection of the phosphorus nucleus through a locally applied electric field. To realize the selection function, it is required to know the relationship between the applied electric field and the change of the nuclear magnetic resonance frequency of phosphorus. In this study, based on the wave functions obtained by the effective-mass theory, we introduce an empirical correction factor to the wave functions at the donor nucleus. Using the corrected wave functions, we formulate a first-order perturbation theory for the perturbed system under the influence of an electric field. In order to calculate the potential distributions inside the silicon and the silicon dioxide layers due to the applied electric field, we use the multilayered Green's functions and solve an integral equation by the moment method. This enables us to consider more realistic, arbitrary shape, and three-dimensional qubit structures. With the calculation of the potential distributions, we have investigated the effects of the thicknesses of silicon and silicon dioxide layers, the relative position of the donor, and the applied electric field on the nuclear magnetic resonance frequency of the donor.
In the Kane silicon-based electron-mediated nuclear spin quantum computer architecture, phosphorous is doped at precise positions in a silicon lattice, and the P donor nuclear spins act as qubits. Logical operations on the nuclear spins are performed using externally applied magnetic and electric fields. There are two important interactions: the hyperfine and exchange interactions, crucial for logical qubit operations. Single qubit operations are performed by applying radio frequency magnetic fields resonant with targeted nuclear spin transition frequencies, tuned by the gate-controlled hyperfine interaction. Two qubit operations are mediated through the exchange interaction between adjacent donor electrons. It is important to examine how these two interactions vary as functions of experimental parameters. Here we provide such an investigation. First, we examine the effects of varying several experimental parameters: gate voltage, inter donor separation, donor depth below the silicon oxide interface and back gate depth, to explore how these variables affect the donor electro density. Second, we calculate the hyperfine interaction and the exchange coupling as a function of these parameters. These calculations were performed using an anisotropic effective mass Hamiltonian. The electric field potential was obtained using Technology Computer Aided Design software, and the interfaces were modelled as a barrier using a step function. We aim to provide relevant information for the experimental design of these devices and highlight the significance of environmental factors other than gate potential that affect the donor electron.
Standard CMOS circuits are expected to be no longer able to satisfy nowadays demand for higher performances and lower power consumption. This commits to look for new solutions. Quantum computation, whose fundamental quantity of information is the qubit, is a thoroughly new way of thinking to electronics. There are at least two main reasons for which quantum computing is expected to broaden computation horizons. First, specific algorithms can be engineered, manipulating the probability amplitudes of different states, to exploit the quantum superposition principle and achieve high parallelism. Second, it is expected to be spontaneously suitable to simulate complex quantum physical systems, as large molecules. Quantum computing requires the cooperation of several scientific fields. The leading idea has been to highlight the operations and algorithms a quantum computer must be able to perform and, then, to propose a hardware solution, providing the rigorous physical treatment mandatory for a deep understanding and for an appropriate modelling, something which is hard to find in nowadays literature. Thus, the first chapters address the quantum algorithms, focusing on Grover's search algorithm and deriving, from fundamentals, the corresponding circuit. The latter is then described in Quantum Assembly (QASM), which can be seen as the quantum equivalent of VHDL, and simulated on IBM Q Experience. The following chapters deal with a possible physical implementation of quantum processors: the encoding of qubits on nuclear spins in diamagnetic molecules. When a static magnetic field is applied to a spin-1/2 nucleus, two well-defined energy eigenstates arise, as a result of the Zeeman effect. Each of them can be associated with a qubit eigenstate. Thanks to nuclear magnetic resonance principles, the probability with which one of the two eigenstates is measured can be handled by the superposition of a radio frequency field at the resonance frequency of the nucleus. This can be interpreted as a rotation of the spin, and so of the qubit, about the RF field and enables the execution of single qubit quantum gates, since they are simple rotations of the state vector. The two-qubit CNOT gate is obtained through the interaction of spins via J-coupling. Considering that these gates constitute a universal set, every quantum algorithm can theoretically be run on an NMR processor. Different nuclei have different resonant frequencies, but the addressing of a specific spin can be achieved also in homonuclear molecules, since the resonant frequency depends also on the nuclear environment (nuclear shielding). While unwanted J-couplings can be removed thanks to refocusing, spin decoherence and relaxation phenomena force an upper limit to the allowed timescale. A main advantage of NMR is that this timescale is long, even at room temperature. Conversely, scalability is still an issue. A simple heteronuclear MATLAB model, based on the relations derived from first principles in the thesis, is proposed to prove the feasibility of NMR quantum processors. The input technological parameters, as chemical shielding and J-coupling, can be computed resorting on ORCA, or obtained from experimental data. The model, able to run a universal set of quantum gates and simple algorithms, can be useful to find an optimal operating point as a compromise between molecule physical properties and the quantities which can be controlled by nowadays NMR instrumentation.