The silicon-based quantum computer proposal has been one of the actively pursued ideas during the past three years. Here we calculate the donor electron exchange in silicon and germanium, and demonstrate an atomic-scale challenge for quantum computing in Si (and Ge), as the six (four) conduction-band minima in Si (Ge) lead to intervalley electronic interference, generating strong oscillations in the exchange splitting of two-donor two-electron states. Donor positioning with atomic-scale precision within the unit cell thus becomes a decisive factor in determining the strength of the exchange coupling-a fundamental ingredient for two-qubit operations in a silicon-based quantum computer.
The silicon-based quantum computer proposal has been one of the actively pursued ideas during the past three years. Here we calculate the donor electron exchange in silicon and germanium, and demonstrate an atomic-scale challenge for quantum computing in Si (and Ge), as the six (four) conduction-band minima in Si (Ge) lead to intervalley electronic interference, generating strong oscillations in the exchange splitting of two-donor two-electron states. Donor positioning with atomic-scale precision within the unit cell thus becomes a decisive factor in determining the strength of the exchange coupling-a fundamental ingredient for two-qubit operations in a silicon-based quantum computer.
The discovery of efficient quantum algorithms a decade ago has shown that a quantum computer encoding and processing information quantum mechanically, can solve important problems intractable with conventional computers. The invention of quantum error correction principle, makes quantum computation possibly fault-tolerant against the decoherence of information carriers. Thereafter, intensive research activities have been made toward the implementation of quantum computation with various realistic quantum systems. Among them, the most attractive implementation proposals are using silicon-based materials, which have the advantage of borrowing the existing ingenuity and resources accumulated during the development of modern microelectronics. In this dissertation we investigate several theoretical aspects of a silicon- based quantum computer in which qubits are represented by the spins of electrons bound to phosphorous donors in silicon. Encoding each qubit in terms of three neighboring donor electron spins, we can realize universal quantum gate operations with only the Heisenberg exchange coupling J S1 •S2 between neighboring donors. Therefore, studying the exchange coupling for a phosphorous donor pair in silicon is of central importance for providing the experimentalists with qualitative insights and quantitative guidance for building such a silicon quantum computer. After giving some general considerations on the quantum computer architecture, we develop the necessary theoretical tools. A multi-valley effective mass equation is derived and discussed, to handle impurities in a multi-valley semiconductor. Then we apply it to solve a single Si:P donor embedded in our quantum computer architecture. We show that the width of the silicon quantum well can significantly influence the energy splitting and charge distributions of the ground state. Oscillation of level splitting is observed as the quantum well width or donor position is varied at atomic scale. Elementary gate operations for 3-donor-spin qubits involve a neighboring pair of donors at each step. The exchange coupling in Heisenberg model hamiltonian, defined as the energy splitting of the lowest singlet and triplet states, has to be calculated from the realistic two-donor hamiltonian. We discuss several popular methods to solve the two-donor problem and develop an appropriate extended Hartree-Fock method that can give reliable results for a well-separated donor pair subject to tunable coupling. This method is first applied to a Si:P donor pair in the framework of hydrogenic effective mass theory, to study how to tune the exchange coupling with simple gate potentials. A followed study shows that under a parallel electric field the singlet and triplet states exhibit very different polarization behaviors. This difference can be exploited to measure the state of an electron spin. We also show that, a perpendicular electric field cannot tune the exchange coupling efficiently. Then we apply the realistic multi-valley effective mass equation, coupled with a realistic modeling of the potential generated by gate electrodes, to a pair of phosphorous donors in silicon quantum well. By varying the gate electrode voltages, the exchange coupling can be tuned with exponential efficiency in a wide range. We find that, for best performance, we need to set the quantum well width to be around 10 nm, and the donor separation to be around 10 a∗ b---- 24nm in the doping plane. We analyze in detail an adiabatic half-swap operation between neighboring donor spins. The gate operation time is estimated to be 0.2 ns, which satisfies the constraints put by the donor spin decoherence time and by the validity of adiabatic approximation. The interference between different valley components could lead to oscillations of exchange coupling as donor positions are shifted by occasion. We find the exchange oscillation persists even with only two relevant valleys for donors in a strongly-strained silicon quantum well. It is also shown that the oscillation induced by changing donor separation at atomic scale is strongly suppressed as donors approach each other. Finally, we study the entanglement issue in quantum computing context. We analyze the low-energy Hilbert space for a pair of qubits encoded by localized electron spins and suggest a suitable measure to describe the entanglement between qubits involving indistinguishable electrons and in the presence of leakage errors. The dynamics of inter-qubit entanglement during a gate operation is also studied.
We consider charge qubits based on shallow donor electron states in silicon and coupled quantum dots in GaAs. Specifically, we study the feasibility of P2+ charge qubits in Si, focusing on single qubit properties in terms of tunnel coupling between the two phosphorus donors and qubit decoherence caused by electron-phonon interaction. By taking into consideration the multivalley structure of the Si conduction band, we show that intervalley quantum interference has important consequences for single-qubit operations of P2+ charge qubits. In particular, the valley interference leads to a tunnel-coupling strength distribution centered around zero. On the other hand, we find that the Si band structure does not dramatically affect the electron-phonon coupling and consequently, qubit coherence. We also critically compare charge qubit properties for Si:P2+ and GaAs double quantum dot quantum computer architectures.
Recent experimental efforts have led to considerable interest in donor-based localized electron spins in Si as viable qubits for a scalable silicon quantum computer. With the use of isotopically purified 28Si and the realization of extremely long spin coherence time in single-donor electrons, the recent experimental focus is on two-coupled donors with the eventual goal of a scaled-up quantum circuit. Motivated by this development, we simulate the statistical distribution of the exchange coupling J between a pair of donors under realistic donor placement straggles, and quantify the errors relative to the intended J value. With J values in a broad range of donor-pair separation (5<|R|<60 nm), we work out various cases systematically, for a target donor separation R0 along the [001], [110] and [111] Si crystallographic directions, with |R0|=10,20 or 30 nm and standard deviation σR=1,2,5 or 10 nm. Our extensive theoretical results demonstrate the great challenge for a prescribed J gate even with just a donor pair, a first step for any scalable Si-donor-based quantum computer.
It has been over ten years since Kane's influential proposal for a silicon-based nuclear spin quantum computer using phosphorous donors. Since then, silicon-based architectures have been refined as the experimental challenges associated with the original proposal have become better understood, while simultaneously a number of powerful and generic models for quantum computation have emerged. Here, I discuss how the cluster state or "one-way" model for quantum computing might be advantageously applied to donors in silicon, with the potential to substantially reduce the practical requirements of a successful implementation. The essence of the scheme is to use the electron spin associated with a donor to weave an entangled network between 31P donor nuclear spins. This resource has been shown to have exceptional coherence times and supports universal quantum computation through local measurements on the nuclear spins. Some of the key ingredients, such as global spin manipulation, have been robustly established, while others, such as single spin measurement, have seen much progress in recent years. A key challenge will be the demonstration of electron transfer between donors that preserves spin coherence.
An architecture for a quantum computer is presented in which spins associated with donors in silicon function as qubits. Quantum operations on the spins are performed using a combination of voltages applied to gates adjacent to the spins and radio frequency magnetic fields applied resonant with spin transitions. Initialization and measurement of electron spins is made by electrostatic probing of a two electron system, whose orbital configuration must depend on the spin states of the electrons because of the Pauli Principle. Specific devices will be discussed which perform all the necessary operations for quantum computing, with an emphasis placed on a qualitative presentation of the principles underlying their operation. The likely impediments to achieving large-scale quantum computation using this architecture will be addressed: the computer must operate at extremely low temperature, must be fabricated from devices built with near atomic precision, and will require extremely accurate gating operations in order to perform quantum logic. Refinements to the computer architecture will be presented which could remedy each of these deficiencies. I will conclude by discussing a possible specific realization of the computer using Si/SixGe1–x heterostructures into which donors are deposited using a low energy focused ion beam.
We present a systematic and realistic simulation for single and double phosphorous donors in a silicon-based quantum computer design. A two-valley equation is developed to describe the ground state of phosphorous donors in a strained silicon quantum well (QW), with the central cell effect treated by a model impurity potential. The valley splitting of the donor ground state as a function of QW width or donor position is calculated and a comparison with the valley splitting of the lowest QW states is presented. Oscillation of the valley splitting is observed as the QW width or donor position is varied at atomic scale. We find that the increase of quantum well confinement leads to shrinking charge distribution in all three dimensions. Using an unrestricted Hartree-Fock method with generalized valence bond (GVB) single-particle wave functions, we are able to solve the two-electron Schr\"odinger equation with quantum well confinement and realistic gate potentials. The lowest singlet and triplet energies and their charge distributions for a neighboring donor pair in the quantum computer (QC) architecture are obtained at different gate voltages. The effects of QW width, gate voltages, donor separation, and donor position shift are calculated and analyzed. The gate tunability and gate fidelity are defined and evaluated, for a typical QC design. Estimates are obtained for the duration of spin-half-swap gate operation and the required accuracy in voltage control. A strong exchange oscillation is observed as both donors are shifted along [001] axis but with their separation unchanged. Applying a gate potential tends to suppress the oscillation. The exchange oscillation as a function of donor separation along [100] axis is found to be completely suppressed as the donor separation is decreased. The simulation presented in this paper is of importance to the practical design of an exchange-based silicon quantum computer.