Robert Joynt

Practical design and simulation of silicon-based quantum-dot qubits

Spins based in silicon provide one of the most promising architectures for quantum computing. Quantum dots are an inherently scalable technology. Here, we combine these two concepts into a workable design for a silicon-germanium quantum bit. The novel structure incorporates vertical and lateral tunneling, provides controlled coupling between dots, and enables single electron occupation of each dot. Precise modeling of the design elucidates its potential for scalable quantum computing. For the first time it is possible to translate the requirements of faulttolerant error correction into specific requirements for gate voltage control electronics in quantum dots. We demonstrate that these requirements are met by existing pulse generators in the kHzMHz range, but GHz operation is not yet achievable. Our calculations further pinpoint device features that enhance operation speed and robustness against leakage errors. We find that the component technologies for silicon quantum dot quantum computers are already in hand. Quantum computing offers the prospect of breaking out of the classical von Neumann paradigm that dominates present-day computation. It would enable huge speedups of certain very hard problems, notably factorization. Constructing a quantum computer (QC) presents many challenges, however. Chief among these is scalability: the 10 qubits needed for simple applications far exceed the potential of existing implementations. This requirement points strongly in the direction of Si-based electronics for QC. Silicon devices offer the advantage of long spin coherence times, fast operation, and a proven record of scalable integration. Specific Si-based qubit proposals utilize donor-bound nuclear or electronic spins as qubits. However, quantum dots can also be used to house electron spins, and they have the advantage that the electrostatic gates controlling qubit operations are naturally aligned to each qubit. These proposals describe an intriguing possibility. Our aim here is to describe a new SiGe qubit design, and, just as importantly, to carry out detailed modeling of a specific design for the first time. Modeling provides a proof of principle, pinpoints problem areas, and suggests new directions. The fundamental goal of our design is the ability to reduce the electron occupation of an individual dot precisely to one, as in vertically coupled structures. It may be possible to use the spin of multi-electron quantum dots as qubits, but single occupation is clearly desirable. The spin state “up” = 0 or “down” = 1 , stores the quantum bit of information. At the same time, it is necessary to have tunable coupling between neighboring dots. This is achieved by controlled movement of electrons along the quantum well that contains two dots. The solution is to draw on two distinct quantum dot technologies: lateral and vertical tunneling quantum dots. The design, shown in Fig. 1, incorporates a back-gate that serves as an electron reservoir, a quantum well that confines electrons vertically, and split top gates that provide lateral confinement by electrostatic repulsion. All semiconductor layers are formed of strainrelaxed x xGe Si 1 except the quantum well, which is pure, strained Si. Relaxation is achieved by step-graded compositional growth on a Si wafer. Here, we consider the composition 077 . 0 = x , consistent with a quantum well band offset meV 84 ≅ ∆ c E , with respect to theSpins based in silicon provide one of the most promising architectures for quantum computing. A scalable design for silicon-germanium quantum-dot qubits is presented. The design incorporates vertical and lateral tunneling. Simulations of a four-qubit array suggest that the design will enable single electron occupation of each dot of a many-dot array. Performing two-qubit operations has negligible effect on other qubits in the array. Simulation results are used to translate error correction requirements into specifications for gate-voltage control electronics. This translation is a necessary link between error correction theory and device physics.

Controllable valley splitting in silicon quantum devices

Silicon has many attractive properties for quantum computing, and the quantum-dot architecture is appealing because of its controllability and scalability. However, the multiple valleys in the silicon conduction band are potentially a serious source of decoherence for spin-based quantum-dot qubits. Only when a large energy splits these valleys do we obtain well-defined and long-lived spin states appropriate for quantum computing. Here, we show that the small valley splittings observed in previous experiments on Si–SiGe heterostructures result from atomic steps at the quantum-well interface. Lateral confinement in a quantum point contact limits the electron wavefunctions to several steps, and enhances the valley splitting substantially, up to 1.5 meV. The combination of electrostatic and magnetic confinement produces a valley splitting larger than the spin splitting, which is controllable over a wide range. These results improve the outlook for realizing spin qubits with long coherence times in silicon-based devices.

Tunable Spin Loading and T-1 of a Silicon Spin Qubit Measured by Single-Shot Readout

The remarkable properties of silicon have made it the central material for the fabrication of current microelectronic devices. Silicon’s fundamental properties also make it an attractive option for the development of devices for spintronics [1] and quantum information processing [2–5]. The ability to manipulate and measure spins of single electrons is crucial for these applications. Here we report the manipulation and measurement of a single spin in a quantum dot fabricated in a silicon/silicon-germanium heterostructure. We demonstrate that the rate of loading of electrons into the device can be tuned over an order of magnitude using a gate voltage, that the spin state of the loaded electron depends systematically on the loading voltage level, and that this tunability arises because electron spins can be loaded through excited orbital states of the quantum dot. The longitudinal spin relaxation time T1 is measured using single-shot pulsed techniques [6] and found to be ∼ 3 seconds at a field of 1.85 Tesla. The demonstration of single spin measurement as well as a long spin relaxation time and tunability of the loading are all favorable properties for spintronics and quantum information processing applications. Silicon is a material in which spin qubits are expected to have long coherence times, thanks to the predominance of a spin-zero nuclear isotope and relatively weak spin-orbit coupling. However, silicon quantum dots have yet to demonstrate the reproducibility and controllability achieved in gallium arsenide devices [7–10]. Here, we demonstrate the control and manipulation of spin states of single electrons in a silicon/silicon-germanium (Si/SiGe) quantum dot and report the first single-shot measurements of the longitudinal spin relaxation time T1 in such devices. We also show that the presence of a relatively low-lying spin-split orbital excited state in the dot can be exploited to increase the speed and tunability of the loading of spins into the dot. Our results demonstrate that Si/SiGe quantum dots can be fabricated that are sufficiently tunable to enable single-electron manipulation and measurement, and that long spin relaxation times are consistent with the orbital and/or valley excitation energies in these systems. The measurements we report were performed on a gate-defined quantum dot with the gate configuration shown in Fig. 1a, tuned to be in the single-dot regime. The dot is measured at low temperature and in a parallel magnetic field. As shown in Fig. 1b, an electron can be loaded into one of four energy eigenstates; we denote 2.0 1.0 0.0 B (T) -0.158 -0.152

Phase diagram of d-wave superconductors in a magnetic field

A theoretical analysis of the upper critical field of d-wave superconductors with hexagonal or tetragonal crystal symmetry is carried out. This generalises the earlier work of Volovik (1988) and shows that a magnetic field can induce many different kinds of transitions between distinct d-wave superconducting phases. This occurs by changing the field strength at fixed temperature. Ultrasound absorption measurements in UPt3 are analysed using this physical picture, and the high- and low-field superconducting phases are determined. Specific heat measurements on YBa2Cu3O7 may indicate that this is a d-wave superconductor as well. It has a slight orthorhombic distortion which can split the superconducting transition into two, which also occurs in UPt3. The structure of the vortex lattice and the associated form factor for neutron scattering from this lattice changes during the transition, and criteria for distinguishing the various possibilities by this scattering are given.

Experiments concerning the connective nature of superconductivity in YBa2Cu3O7

Samples of YBa2Cu3O7 have been prepared with rather sharp inductive transitions having in the best cases breadths of 7 K and midpoint Tc values of 88 K. The resistive Tc midpoints are 92–95 K with transition widths of ±1–2 K. Flux shielding at 4.2 K is normally 100% and flux expulsion at 4.2 K reaches 95%. However, even small fields of order 1 mT decouple some 15%–20% of the volume, allowing flux to enter the samples. Resistive Hc2 measurements suggest that Hc2(0) varies from 300 T, depending on the criterion chosen. ac susceptibility measurements suggest that Hc2(0) is ∼60 T. Magnetization current densities are relatively high (150–200 A/mm2 at 1–10 T at 4.2 K) but measured transport current densities are small (≤1 A/mm2). Magnetization current densities at 77 K are about two orders of magnitude lower. The samples were seen to be heavily twinned by light microscopy (scale of 1–5 μm) and by transmission electron microscopy (scale of ∼250 nm). It is concluded that these measurements are consistent w...

One-dimensional quantum walks with absorbing boundaries

In this paper we analyze the behavior of quantum random walks. In particular, we present several new results for the absorption probabilities in systems with both one and two absorbing walls for the one-dimensional case. We compute these probabilities both by employing generating functions and by use of an eigenfunction approach. The generating function method is used to determine some simple properties of the walks we consider, but appears to have limitations. The eigenfunction approach works by relating the problem of absorption to a unitary problem that has identical dynamics inside a certain domain, and can be used to compute several additional interesting properties, such as the time dependence of absorption. The eigenfunction method has the distinct advantage that it can be extended to arbitrary dimensionality. We outline the solution of the absorption probability problem of a (D-1)-dimensional wall in a D-dimensional space.

Two-particle quantum walks applied to the graph isomorphism problem

We show that the quantum dynamics of interacting and noninteracting quantum particles are fundamentally different in the context of solving a particular computational problem. Specifically, we consider the graph isomorphism problem, in which one wishes to determine whether two graphs are isomorphic (related to each other by a relabeling of the graph vertices), and focus on a class of graphs with particularly high symmetry called strongly regular graphs (SRGs). We study the Green’s functions that characterize the dynamical evolution single-particle and two-particle quantum walks on pairs of nonisomorphic SRGs and show that interacting particles can distinguish nonisomorphic graphs that noninteracting particles cannot. We obtain the following specific results. (1) We prove that quantum walks of two noninteracting particles, fermions or bosons, cannot distinguish certain pairs of nonisomorphic SRGs. (2) We demonstrate numerically that two interacting bosons are more powerful than single particles and two noninteracting particles, in that quantum walks of interacting bosons distinguish all nonisomorphic pairs of SRGs that we examined. By utilizing high-throughput computing to perform over 500 million direct comparisons between evolution operators, we checked all tabulated pairs of nonisomorphic SRGs, including graphs with up to 64 vertices. (3) By performing a short-time expansion of the evolution operator, we derive distinguishing operators that provide analytic insight into the power of the interacting two-particle quantum walk. DOI: 10.1103/PhysRevA.81.052313

Decoherence of electron spin qubits in Si-based quantum computers

Direct phonon spin-lattice relaxation of an electron qubit bound by a donor impurity or quantum dot in SiGe heterostructures is investigated. The aim is to evaluate the importance of decoherence from this mechanism in several important solid-state quantum computer designs operating at low temperatures. We calculate the relaxation rate 1/T 1 as a function of [100] uniaxial strain, temperature, magnetic field, and silicon/germanium content for Si:P bound electrons and quantum dots. The quantum dot potential is much smoother, leading to smaller splittings of the valley degeneracies. We have estimated these splittings in order to obtain upper bounds for the relaxation rate. In general, we find that the relaxation rate is strongly decreased by uniaxial compressive strain in a SiGe-Si-SiGe quantum well, making this strain an important positive design feature. Ge in high concentrations (particularly over 85%) increases the rate, making Si-rich materials preferable. We conclude that SiGe bound electron qubits must meet certain conditions to minimize decoherence but that spin-phonon relaxation does not rule out the solid-state implementation of error-tolerant quantum computing.

Charge Sensing and Controllable Tunnel Coupling in a Si/SiGe Double Quantum Dot

We report integrated charge sensing measurements on a Si/SiGe double quantum dot. The quantum dot is shown to be tunable from a single, large dot to a well-isolated double dot. Charge sensing measurements enable the extraction of the tunnel coupling t between the quantum dots as a function of the voltage on the top gates defining the device. Control of the voltage on a single such gate tunes the barrier separating the two dots. The measured tunnel coupling is an exponential function of the gate voltage. The ability to control t is an important step toward controlling spin qubits in silicon quantum dots.

Superconducting Instability in the Large-U Limit of the Two-Dimensional Hubbard Model

We have investigated numerically the pairing instabilities of Gutzwiller wavefunctions. These are equivalent to a certain form of the resonant valence bond wavefunction. The case considered is a nearly half-filled two dimensional band with interactions given by a Hubbard model with large on-site Coulomb interactions. We find that the paramagnetic normal state is unstable tod-wave pairing but stable againsts-wave pairing. The antiferromagnetic state is marginally stable against both types of pairing. These results can be explained as an interference effect resulting in enhanced antiferromagnetic spin correlation in the paired state.