A/Prof Lloyd C.L. Hollenberg - University of Melbourne

Device Modelling Researchers
Dr Cameron Wellard - University of Melbourne
Dr Andrew Greentree

Device Modelling Students
Mr Joo Chew Ang (PhD)
Mr Vincent Conrad (PhD)
Mr Jared Cole(PhD)
Mr Simon Devitt (PhD)
Mr Chris Escott (PhD)
Dr Austin Fowler (PhD completed)
Mr Charles Hill (PhD)
Mr Gajendran Kandasamy (PhD)
Mr Tim Starling (PhD)
Mr Ashley Stephens(Hons)
Mr Matthew Testolin (PhD)

Collaborating Centre Researchers
A/Prof David Jamieson - University of Melbourne
Dr Chris Pakes - University of Melbourne
Prof Steven Prawer - University of Melbourne
Prof Robert Clark - University of New South Wales
A/Prof Andrew Dzurak - University of New South Wales
A/Prof Alex Hamilton - University of New South Wales
Prof Michelle Simmons - University of New South Wales
Prof Gerard Milburn - University of Queensland
Dr Wayne Hutchison- ADFA

Other Collaborating Researchers
Prof Sankar Das Sarma - University of Maryland
Dr Charles Tahan - University of Wisconsin Madison
Prof Gerhard Klimeck - Purdue University
Dr Fedor Jelezko - University of Stuttgart
Prof Jörg Wrachtrup - University of Stuttgart
Prof Frank Wilhelm - University of Munich (LMU)/Waterloo
Dr Stefan Ludwig - University of Munich (LMU)
Dr Daniel Oi - University of Cambridge
Dr Sonia Schirmer - University of Cambridge
Dr Tom Stace - University of Cambridge
Dr Adrian Flitney - University of Melbourne
Dr Harry Quiney - University of Melbourne
A/Prof Salvy Russo - RMIT University
Mr Wayne Haig - Department of Defense
Dr Sean Barrett - Hewlett Packard Laboratories, Bristol
Dr Bill Munro - Hewlett Packard Laboratories, Bristol
Dr Tim Spiller - Hewlett Packard Laboratories, Bristol
Dr Ray Beausoleil - Hewlett Packard Laboratories, Palo Alto
Prof Joseph Salzmann - Technion
Prof Eduardo Muciolo - University of Central Florida
Dr Jason Ralph - University of Liverpool

Program Description
The main objectives of the program are to develop realistic and comprehensive theoretical descriptions of the Si:P buried dopant charge and spin devices, and all facets of their operation. Theoretical modelling relevant to the short, medium and long-term goals provides crucial information on device design parameters to the experimental programs. To this end the device modelling program has been active in a number of projects including; fundamental solid-state physics of the Si:P donor system, single and coupled qubit operations,the effects of decoherence, large scalesimulations of the actual implementation of quantum error correction and algorithms on arrays of qubits, and investigations of new paradigms of quantum computing.

1. Over View
In 2005 the researchers of the Device Modelling Program completed a number of investigations on a broad range of topics from solid-state modelling of the silicon devices to implementation issues for fault-tolerant quantum computing. Research highlights include a new proposal for an information-free quantum bus for non-local qubit coupling, entanglement generation, and single-shot syndrome measurement, characterisation methods for generalised two qubit Heisenberg systems, a composite pulse scheme correcting for exchange coupling variations, and a quantitative comparison of direct and fault-tolerant implementations of logical qubit state preparation in both state vector and density matrix simulators. Below we summarise the output of the program for 2005.

2. Full solid-state quantum & nanoelectronic device modeling
The quantum treatment of microwave driven transitions in the P-P+ charge qubit system was completed within the effective mass formalism, and the change in sideband appearance as a function of donor placement (atomic level variations from ideal), field strength and decoherence rates was determined. Work is in progress on the complete integration with nano-electronic modelling (TCAD) of actual devices in order to determine the sideband measurement characteristics with applied gate voltage, and link directly to measurement data on these devices.

Figure 1: TCAD nano-electronic simulations are combined with a full quantum treatment of the P-P+ system in the effective mass approximation.

Figure 2: Calculations determine the response to a driving microwave field, sensitivity to atomic level variations in the placement of the donors, and dephasing time scales. Theory predicts that the sidebands diminish away from the zero field line.

3. Gaussian techniques for the description of buried molecular Si:P structures
The many new possibilities for devices based on single atom level fabrication capability bring some special theoretical problems associated with buried P-donor structures. In the quantum computing context one is interested in controlled exchange coupling between pairs of Pdonors, transport mechanisms along chains of ionised P-donors and so forth. Because of the high level of complexity of the donormolecular problem in the solid-state, in the first level of approximation multi-donor electron wave functions are constructed from symmetry considerations alone. In reality, however, the donor electron wave functions for such structures must be constructed at a molecular level. We have adapted powerful Gaussian techniques from quantum chemistry to the Si:P context allowing for converged full configuration interaction (FCI) multi donor electron molecular wave functions to be constructed.
In Figure 3 the FCI calculation is compared with the Heitler-London method for a range of donor separations, showing agreement of about 10% at the separations of interest.

Figure 3: Percentage difference in the exchange coupling in the P-P system of the Heitler-London approximation compared with the full configuration calculation in the Gaussian basis.

4. NEMO collaboration
The NEMO team lead by Prof Gehard Klimeck (Purdue University) has continued to develop the numerical tight-binding and visualization technology to tackle P impurity states in Si as part of the ARO QCCM funded collaboration with the CQCT. One key achievement of NEMO3D has been the volume rendering visualization of a wave function in a system containing 21 million atoms described by 20 complex numbers per atom. However there was no interactive method to explore the data containing 0.420 trillion numbers. A tool entitled VolQD was developed to visualise such massive amounts of data using a single PC with a graphics processing unit. VolQD enables the dissection of the data by different orbital contributions in a volume rendered, and 2-D slice form. The capability of this tool is demonstrated for the simulation of an InAs quantum dot wave function (Figure 4).

Figure 4: Three-dimensional volume rendered image using VolQD (Klimeck et al) of the 2nd excited electron state in a dome shaped InAs quantum dot – composite wave function.

5. Single and two qubit Hamiltonian characterisation methods
All qubits, but especially solid-state qubits will inevitably suffer from some form of fabrication variability. In the case of Si:P, placement imprecision at the atomic level introduces exchange oscillations – relatively large variations of the exchange coupling strength for only atomic level variations from ideal placement – and/or gate control variations. Even before tomographic or active compensation methods can be applied, some knowledge of the Hamiltonian is required, and hence the need for characterisation of the single and two qubit open systems at some level is inevitable.
In 2005 work was completed on characterising a single qubit Hamiltonian in a decohering environment, as well as an important two qubit case – the generalised Heisenberg interaction. A measurement regime and analysis in Fourier space was found which allows for the precise determination of the two qubit Hamiltonian – the final accuracy is dependent on number of measurements alone.

Figure 5: Simulated time series data for the z-component measurement of a single qubit in the presence of a decohering environment, and the subsequent Fourierbased characterisation analysis which re-constructs the Hamiltonian parameters.

Figure 6: The Weyl chamber of the two-qubit Hamiltonian forming the basic description of two-qubit gates and their degree of entanglement generation.

6. Composite pulse for large exchange coupling variations
Quantum control theory is beginning to be applied to the quantum computing context with interesting results. Complimentary to characterisation, composite pulsing techniques allows robust control of qubit systems containing some unknown residual systematic uncertainty. In other words, once a qubit system (including control) is characterised to a certain level, precise gates can be constructed and implemented using robust pulse control. In 2005 a new composite pulse scheme was developed which compensates for relatively large variations in exchange coupling strength.
In Figure 7 the systematic error of a CNOT operation based on the donor exchange interaction with fractional error on the J parameter is plotted for first and second levels of composite pulse application and compared with the uncorrected case. For up to 50% variation on the value of J, the second level pulse scheme can provide protection down to the 10-4 level. Application of the scheme to the donor exchange variations is shown in Figure 8, showing that characterisation at some level is required before composite pulsing can efficiently compensate for fabrication uncertainties at the atomic level. Work is in progress to determine the time-scaling of such corrections with respect to the variations
in J expected from atomic level donor placement variations..

Figure 7: Fidelity of the composite pulse correction of a CNOT gate based on an exchange interaction with fractional error Δ in the interaction strength. Concatenation (iteration) of the composite pulse provides significant improvement in fidelity over a wide range of the (in-principle) unknown fractional error in the exchange coupling strength.

Figure 8: Application of the composite pulse scheme to the exchange coupling variations due to donor placement variations from the ideal sites (in the 100 direction). The third level composite scheme obtains high-fidelity CNOT gates with placement errors of about 3 lattice sites.

7. Q-switching coupled cavities
Photonic bandgap cavities are prime solidstate systems to investigate light-matter interactions in the strong coupling regime and candidates for transform limited single-photon sources.
However, as the cavity is defined by the geometry of the periodic dielectric pattern, cavity control in a monolithic structure coherence is limited by the read-out channel, or in a high Q cavity, it is nearly decoupled from the external world, making measurement of the state extremely challenging. We have developed a method to ameliorate these difficulties by using a coupled cavity arrangement, where one cavity acts as a switch for the other cavity, tuned by control of the atomic transition (Figure 9).

Figure 9: Coupled cavity system: open cylinders show the photonic Bragg lattice, and missing holes (lattice defects) constitute the cavities and waveguide. The filled spheres are atoms where the transition frequency is controlled via top gates. The left defect is the cavity, the right the gate, and out-coupling is via the waveguide mode on the right.

Figure 10: Schematic of a protocol to deterministically entangle two atom-cavity qubits in a positive or negative Bell state.

9. Optical based donor spin read-out and control
In previous work of the program, it was shown that the doubly occupied D_ which is the final state of the spin-tocharge conversion read-out scheme is dangerously shallow and may not survive the fields required to generate the transition adiabatically. On the other hand, resonant transfer to the D_ state requires far weaker fields. We have completed work on opticalelectric methods involving a resonant far infrared laser (FIR) for use in both electron spin readout and initialisation.

Figure 11: Charge transfer probability in the P-P system as a function of FIR laser detuning for a range of intensities.

Figure 12 Cross-talk and compensation in donor qubit gate control. Potential profiles obtained from Poisson solver: a) potential spreads out significantly as one gate is biased (centre) while the rest are grounded b) after compensation, the bias effect at the target (below centre gate) is more localised.

Figure 13 Comparison of state vector and density matrix simulations of Shor’s quantum factoring algorithm in the presence of gate errors for a 5 bit number. As the cut-off is increased in the density matrix simulator the results agree with the stochastic state vector method within statistical errors.

Figure 14 The crossover error rate pco = 5.3×10-5 at which fault-tolerant and direct encoding become effectively equivalent. Only for physical error rates below this value will the fault-tolerant circuit achieve logical state preparation with a greater reliability than the directcircuit implementation.