We report on our recent progress in applying semiconductor quantum dots for spin-based quantum computation, as proposed by Loss and DiVincenzo (1998 Phys. Rev. A 57 120). For the purpose of single-electron spin resonance, we study different types of single quantum dot devices that are designed for the generation of a local ac magnetic field in the vicinity of the dot. We observe photon-assisted tunnelling as well as pumping due to the ac voltage induced by the ac current driven through a wire in the vicinity of the dot, but no evidence for ESR so far. Analogue concepts for a double quantum dot and the hydrogen molecule are discussed in detail. Our experimental results in laterally coupled vertical double quantum dot device show that the Heitler–London model forms a good approximation of the two-electron wavefunction. The exchange coupling constant J is estimated. The relevance of this system for two-qubit gates, in particular the SWAP operation, is discussed. Density functional calculations reveal the importance of the gate electrode geometry in lateral quantum dots for the tunability of J in realistic two-qubit gates.
We report on our recent progress in applying semiconductor quantum dots for spin-based quantum computation, as proposed by Loss and DiVincenzo (1998 Phys. Rev. A 57 120). For the purpose of single-electron spin resonance, we study different types of single quantum dot devices that are designed for the generation of a local ac magnetic field in the vicinity of the dot. We observe photon-assisted tunnelling as well as pumping due to the ac voltage induced by the ac current driven through a wire in the vicinity of the dot, but no evidence for ESR so far. Analogue concepts for a double quantum dot and the hydrogen molecule are discussed in detail. Our experimental results in laterally coupled vertical double quantum dot device show that the Heitler–London model forms a good approximation of the two-electron wavefunction. The exchange coupling constant J is estimated. The relevance of this system for two-qubit gates, in particular the SWAP operation, is discussed. Density functional calculations reveal the importance of the gate electrode geometry in lateral quantum dots for the tunability of J in realistic two-qubit gates.
This thesis addresses the concept of quantum computing with semiconductor quantum dots. The basic unit of a quantum computer is a quantum mechanical two-level system, the so-called quantum bit (qubit). The qubit can be defined as the spin of an electron confined in a quantum dot or as a two-dimensional subspace of the Hilbert space for several spins. Some semiconductors have several minima in their conduction band, socalled valleys. A two-dimensional valley degree of freedom can also be considered as a qubit. In this thesis, quantum registers storing spin as well as valley qubits are described theoretically. Virtual hopping yields an exchange interaction between neighboring quantum dots, which reduces the energy of antisymmetric spin-valley states according to the Pauli principle. Considering the spin and the valley degrees of freedom of two electrons in neighboring quantum dots as qubits, the exchange interaction is a four-qubit interaction. In this thesis, it will be shown that it is, nevertheless, possible to generate a universal two-qubit gate for spin qubits or for valley qubits by combining the exchange interaction and individual single-qubit gates. The exchange interaction in this two-electron double quantum dot directly provides a universal two-qubit gate for one spin and one valley qubit as well. For this, the qubits are defined as suitable subspaces, spanned by the singlet and one triplet state of the spin and the valley states, respectively, Using those gates, arbitrary quantum operations can be performed for single-spin and single-valley qubits in the same quantum register, where in one double quantum dot the quantum states are restricted to the singlet-triplet subspace. While in this register single-spin and single-valley rotations are required, these gates are dispensable in a quantum register storing spin and valley singlet-triplet qubits. The single-qubit operations, in this case, can be provided by the exchange interaction and a gradient in the energy splitting of spin and valley states when the spin qubit is stored in a double quantum dot with spin degrees of freedom only and when the spin states in the double quantum dot which stores the valley qubit are polarized. For providing the two-qubit gate, the spin and the valley qubits need to be stored in the same double quantum dot. The crucial task for this register is to interchange the spin of a spin-only and a spin-valley quantum dot in order to switch between the single-qubit and the two-qubit operation mode. This interchanging of the spin can be achieved by a valley-depending virtual hopping between the quantum dots. Furthermore, echo sequences for a so-called exchange-only qubit are considered. An exchange-only qubit is defined as a subspace in a system of three electron spins in a triple quantum dot. An inhomogeneous magnetic field, which may be caused by nuclear spins in the semiconductor, can lead to decoherence and leakage, i.e., the state might leave the qubit subspace. Suppressing this effect by an echo sequence which consists of pulses interchanging neighboring spin states, so-called SWAP gates, was proposed. Controllability over these quantum operations is provided via the exchange interaction. In this thesis, the performance of different such sequences is investigated. The analysis shows that optimization strategies developed for single spin echoes can be adapted for the three spin system. The echo sequences can be different in their order of applied SWAP gates interchanging the spins in the first and second dots or in the second and third dots. The noise suppression performance depends slightly on this order of SWAP gates. For two spins in a double quantum dot a scheme for quantum state tomography is presented. The corresponding measurement operators are constructed from a set of mutually unbiased bases of the four-dimensional two-qubit Hilbert space. These measurements are represented by short sequences of no more than three elementary quantum gates and projective measurements by the well-established spin-to-charge conversion.
In this thesis, I present a microscopic theory of quantum circuits based on interacting electron spins in quantum dot molecules. We use the Linear Combination of Harmonic Orbitals-Configuration Interaction (LCHO-CI) formalism for microscopic calculations. We then derive effective Hubbard, t-J, and Heisenberg models. These models are used to predict the electronic, spin and transport properties of a triple quantum dot molecule (TQDM) as a function of topology, gate configuration, bias and magnetic field. With these theoretical tools and fully characterized TQDMs, we propose the following applications: 1. Voltage tunable qubit encoded in the chiral states of a half-filled TQDM. We show how to perform single qubit operations by pulsing voltages. We propose the "chirality-to-charge" conversion as the measurement scheme and demonstrate the robustness of the chirality-encoded qubit due to charge fluctuations. We derive an effective qubit-qubit Hamiltonian and demonstrate the two-qubit gate. This provides all the necessary operations for a quantum computer built with chirality-encoded qubits. 2. Berry's phase. We explore the prospect of geometric quantum computing with chirality-encoded qubit. We construct a Herzberg circuit in the voltage space and show the accumulation of Berry's phase. 3. Macroscopic quantum states on a semiconductor chip. We consider a linear chain of TQDMs, each with 4 electrons, obtained by nanostructuring a metallic gate in a field effect transistor. We theoretically show that the low energy spectrum of the chain maps onto that of a spin-1 chain. Hence, we show that macroscopic quantum states, protected by a Haldane gap from the continuum, emerge. In order to minimize decoherence of electron spin qubits, we consider using electron spins in the p orbitals of the valence band (valence holes) as qubits. We develop a theory of valence hole qubit within the 4-band k.p model. We show that static magnetic fields can be used to perform single qubit operations. We also show that the qubit-qubit interactions are sensitive to the geometry of a quantum dot network. For vertical qubit arrays, we predict that there exists an optimal qubit separation suitable for the voltage control of qubit-qubit interactions.
Micromagnet-free operation of electron spin qubits in Si/Si1−xGex vertical double quantum dots
We study a vertical double quantum dot (DQD) in a Si/Si1−xGex/Si double-well heterostructure for full electrical control of electron Loss-DiVincenzo (LD) spin qubits, using realistic device modeling and numerical simulations. Due to the emerging spin-orbit interaction in the DQD, as well as strain from the gate electrodes, small (percentage range) but finite g tensor variations emerge. In addition, we find a large valley splitting, on the order of Ev∼250μeV. As a result, multiple avenues for fast electrical single qubit rotations emerge. An ac electric field gives rise to electric dipole spin resonance (EDSR), while electron spin resonance (ESR) in the presence of an ac magnetic field can be electrically controlled by local gates due to varying g factors in DQDs. We also show that shuttling between neighboring dots, in vertical and horizontal direction, results in ultrafast single qubit gates of less than a nanosecond. Remarkably, this DQD architecture completely eliminates the need for micromagnets, significantly facilitating the scalability of LD spin qubits in semiconductor foundries.
The spin of an electron confined in semiconductor quantum dots is currently a promising candidate for quantum bit (qubit) implementations. Taking advantage of the existing CMOS integration technologies, such devices can offer a platform for large scale quantum computation. However, a quantum mechanical framework bridging a device's physical design and operational parameters to the qubit's energy space is lacking. Furthermore, the spin to charge coupling introduced by intrinsic or induced spin-orbit interaction (SOI) exposes the qubits to charge noise compromising their coherence properties and inducing quantum gate errors. We present here a comodeling framework for double quantum dot (DQD) devices and their charge noise environments. We use a combination of an electrostatic potential solver, full configuration interaction quantum mechanical methods, and two-level-fluctuator models to study the quantum gate performance in realistic device designs and operation conditions. We utilize the developed models together alongside the single electron solutions of the quantum dots to simulate one- and two-qubit gates in the presence of charge noise. We find an inverse correlation between quantum gate errors and quantum dot confinement frequencies. We calculate X-gate fidelities >97 in the simulated Si-MOS devices at a typical TLF density of ${10}^{11}\phantom{\rule{4pt}{0ex}}{\mathrm{cm}}^{\ensuremath{-}2}$. We also find that exchange driven two-qubit SWAP gates show higher sensitivity to charge noise with fidelities down to 91 in the presence of the same density of TLFs. We further investigate the one- and two-qubit gate fidelities at different TLF densities. We find that given the small size of the quantum dots, sensitivity of a quantum gate to the distance between the noise sources and the quantum dot creates a strong variability in the quantum gate fidelities, which can compromise the device yields in scaled qubit technologies.
Recent achievements in the field of gate-defined semiconductor quantum dots reinforce the concept of a spin-based quantum computer consisting of nodes of locally connected qubits which communicate with each other via superconducting circuit resonator photons. In this paper, we theoretically demonstrate a versatile set of quantum gates between adjacent spin qubits defined in semiconductor quantum dots situated within the same node of such a spin-based quantum computer. The electric dipole acquired by the spin of an electron that moves across a double quantum dot potential in a magnetic field gradient has enabled strong coupling to resonator photons and low-power spin control. Here we show that this flopping-mode spin qubit also provides the tunability to program multiple two-qubit gates. Since the capacitive coupling between these qubits brings about additional dephasing, we calculate the estimated infidelity of different two-qubit gates in the most immediate possible experimental realizations.
The goal of this article is to review the progress of three-electron spin qubits from their inception to the state of the art. We direct the main focus towards the exchange-only qubit (Bacon et al 2000 Phys. Rev. Lett. 85 1758-61, DiVincenzo et al 2000 Nature 408 339) and its derived versions, e.g. the resonant exchange (RX) qubit, but we also discuss other qubit implementations using three electron spins. For each three-spin qubit we describe the qubit model, the envisioned physical realization, the implementations of single-qubit operations, as well as the read-out and initialization schemes. Two-qubit gates and decoherence properties are discussed for the RX qubit and the exchange-only qubit, thereby completing the list of requirements for quantum computation for a viable candidate qubit implementation. We start by describing the full system of three electrons in a triple quantum dot, then discuss the charge-stability diagram, restricting ourselves to the relevant subsystem, introduce the qubit states, and discuss important transitions to other charge states (Russ et al 2016 Phys. Rev. B 94 165411). Introducing the various qubit implementations, we begin with the exchange-only qubit (DiVincenzo et al 2000 Nature 408 339, Laird et al 2010 Phys. Rev. B 82 075403), followed by the RX qubit (Medford et al 2013 Phys. Rev. Lett. 111 050501, Taylor et al 2013 Phys. Rev. Lett. 111 050502), the spin-charge qubit (Kyriakidis and Burkard 2007 Phys. Rev. B 75 115324), and the hybrid qubit (Shi et al 2012 Phys. Rev. Lett. 108 140503, Koh et al 2012 Phys. Rev. Lett. 109 250503, Cao et al 2016 Phys. Rev. Lett. 116 086801, Thorgrimsson et al 2016 arXiv:1611.04945). The main focus will be on the exchange-only qubit and its modification, the RX qubit, whose single-qubit operations are realized by driving the qubit at its resonant frequency in the microwave range similar to electron spin resonance. Two different types of two-qubit operations are presented for the exchange-only qubits which can be divided into short-ranged and long-ranged interactions. Both of these interaction types are expected to be necessary in a large-scale quantum computer. The short-ranged interactions use the exchange coupling by placing qubits next to each other and applying exchange-pulses (DiVincenzo et al 2000 Nature 408 339, Fong and Wandzura 2011 Quantum Inf. Comput. 11 1003, Setiawan et al 2014 Phys. Rev. B 89 085314, Zeuch et al 2014 Phys. Rev. B 90 045306, Doherty and Wardrop 2013 Phys. Rev. Lett. 111 050503, Shim and Tahan 2016 Phys. Rev. B 93 121410), while the long-ranged interactions use the photons of a superconducting microwave cavity as a mediator in order to couple two qubits over long distances (Russ and Burkard 2015 Phys. Rev. B 92 205412, Srinivasa et al 2016 Phys. Rev. B 94 205421). The nature of the three-electron qubit states each having the same total spin and total spin in z-direction (same Zeeman energy) provides a natural protection against several sources of noise (DiVincenzo et al 2000 Nature 408 339, Taylor et al 2013 Phys. Rev. Lett. 111 050502, Kempe et al 2001 Phys. Rev. A 63 042307, Russ and Burkard 2015 Phys. Rev. B 91 235411). The price to pay for this advantage is an increase in gate complexity. We also take into account the decoherence of the qubit through the influence of magnetic noise (Ladd 2012 Phys. Rev. B 86 125408, Mehl and DiVincenzo 2013 Phys. Rev. B 87 195309, Hung et al 2014 Phys. Rev. B 90 045308), in particular dephasing due to the presence of nuclear spins, as well as dephasing due to charge noise (Medford et al 2013 Phys. Rev. Lett. 111 050501, Taylor et al 2013 Phys. Rev. Lett. 111 050502, Shim and Tahan 2016 Phys. Rev. B 93 121410, Russ and Burkard 2015 Phys. Rev. B 91 235411, Fei et al 2015 Phys. Rev. B 91 205434), fluctuations of the energy levels on each dot due to noisy gate voltages or the environment. Several techniques are discussed which partly decouple the qubit from magnetic noise (Setiawan et al 2014 Phys. Rev. B 89 085314, West and Fong 2012 New J. Phys. 14 083002, Rohling and Burkard 2016 Phys. Rev. B 93 205434) while for charge noise it is shown that it is favorable to operate the qubit on the so-called '(double) sweet spots' (Taylor et al 2013 Phys. Rev. Lett. 111 050502, Shim and Tahan 2016 Phys. Rev. B 93 121410, Russ and Burkard 2015 Phys. Rev. B 91 235411, Fei et al 2015 Phys. Rev. B 91 205434, Malinowski et al 2017 arXiv: 1704.01298), which are least susceptible to noise, thus providing a longer lifetime of the qubit.