Quantum information processing in semiconductor quantum dots using single and multi-spin qubitsThe idea of a quantum computer, a device capable of processing information using quantum bits (qubit) instead of classical bits that outperforms classical computer in terms of computation speed, is still science ction. However, recent advances in quantum information processing showed that this ction may come true faster than anticipated. A promising platform for the physical realization of a qubit are spin qubits where the information is encoded in the quantum mechanical spin of an electron con ned in a semiconductor quantum dot. There are a variety of di erent implementations how the logical qubit is physically encoded, the use of a single electron spin or the use of the spins of multi electrons. This thesis focuses on the implementation of the spin- 1/2 qubit using the spin of a single electron in a single quantum dot, the exchangeonly (EO) qubit using three electron spins distributed in three quantum dots, and the novel quadrupolar exchange-only (QUEX) qubit using four electron spins distributed in three quantum dots. In this thesis we investigate several theoretical aspects of the physical implementation of the aforementioned qubit encodings. Since all three qubit encodings are allelectric controlled these implementations su er decoherence, the loss of information, from electric uctuations of the environment commonly called charge noise. Typically, the use of more quantum dots increases the decoherence due to charge noise while it decreases the decoherence due to magnetic noise. The rst main aspect investigated in this thesis is the protection of the quantum information encoded in the di erent qubit implementations against charge noise. Special attention is paid to the multi-electron spin implementations where high-symmetry points of operation are discovered which are less sensitive to the charge noise. Protection against charge noise is even better by increasing the number of electrons while keeping the number of quantum dots making the QUEX qubit superior to the EO qubit. So far the main direction in this thesis is the reduction of the qubit sensitivity to electric uctuations. However, coherent electromagnetic elds in form of microwaves v can be used to interconnect qubits. In this thesis also a high-performance coupling between two spin-1/2 is investigated. The use of a resonator, subsequently, allows for a coupling of qubits over macroscopic distances as well as to extract valuable information about the physical system. Because of the larger susceptibility of multi-spin qubits to the electric eld of the resonator, this thesis focuses on the coupling between an EO or a QUEX qubit and a microwave resonator. Points of operations are discovered for both qubit implementations which o er a strong coupling towards a microwave resonator while maintaining their protection against charge noise. Subsequently, a series of measurements are proposed to extract information about the microscopic structure of the quantum dots. Finally, a promising high-performance protocol for coupling two qubits remotely via the resonator is investigated in detail.
Quantum information processing in semiconductor quantum dots using single and multi-spin qubitsThe idea of a quantum computer, a device capable of processing information using quantum bits (qubit) instead of classical bits that outperforms classical computer in terms of computation speed, is still science ction. However, recent advances in quantum information processing showed that this ction may come true faster than anticipated. A promising platform for the physical realization of a qubit are spin qubits where the information is encoded in the quantum mechanical spin of an electron con ned in a semiconductor quantum dot. There are a variety of di erent implementations how the logical qubit is physically encoded, the use of a single electron spin or the use of the spins of multi electrons. This thesis focuses on the implementation of the spin- 1/2 qubit using the spin of a single electron in a single quantum dot, the exchangeonly (EO) qubit using three electron spins distributed in three quantum dots, and the novel quadrupolar exchange-only (QUEX) qubit using four electron spins distributed in three quantum dots. In this thesis we investigate several theoretical aspects of the physical implementation of the aforementioned qubit encodings. Since all three qubit encodings are allelectric controlled these implementations su er decoherence, the loss of information, from electric uctuations of the environment commonly called charge noise. Typically, the use of more quantum dots increases the decoherence due to charge noise while it decreases the decoherence due to magnetic noise. The rst main aspect investigated in this thesis is the protection of the quantum information encoded in the di erent qubit implementations against charge noise. Special attention is paid to the multi-electron spin implementations where high-symmetry points of operation are discovered which are less sensitive to the charge noise. Protection against charge noise is even better by increasing the number of electrons while keeping the number of quantum dots making the QUEX qubit superior to the EO qubit. So far the main direction in this thesis is the reduction of the qubit sensitivity to electric uctuations. However, coherent electromagnetic elds in form of microwaves v can be used to interconnect qubits. In this thesis also a high-performance coupling between two spin-1/2 is investigated. The use of a resonator, subsequently, allows for a coupling of qubits over macroscopic distances as well as to extract valuable information about the physical system. Because of the larger susceptibility of multi-spin qubits to the electric eld of the resonator, this thesis focuses on the coupling between an EO or a QUEX qubit and a microwave resonator. Points of operations are discovered for both qubit implementations which o er a strong coupling towards a microwave resonator while maintaining their protection against charge noise. Subsequently, a series of measurements are proposed to extract information about the microscopic structure of the quantum dots. Finally, a promising high-performance protocol for coupling two qubits remotely via the resonator is investigated in detail.
A two-level quantum system is the building block of a quantum computer. This pair of quantum states defines the computational unit of a quantum computer, and it is called a quantum bit or qubit in analogy to the bit that is the binary unit of a classical computer. The spin of a single electron naturally defines such a two-level quantum system. A quantum dot can be tuned to the single-electron regime, and an array of singly occupied quantum dots provides a multi-qubit register. This thesis describes the multielectron encoding of a spin qubit using a pair or a trio of quantum dots, and it analyzes the coherence properties and manipulation protocols of this system. The singlet-triplet qubit, coded using the singlet state and the spinless triplet state of a pair of singly occupied quantum dots, can be controlled all electrically when applying voltages to gates close to the quantum dot structure. Rapid, subnanosecond modifications of the double quantum dot's charge configuration have been realized successfully. A small magnetic field gradient across the double quantum dot enables universal qubit control. Two modifications of the normal singlet-triplet encoding are introduced. (1) The six-electron configuration of a double quantum dot encodes a singlet-triplet qubit in the same way as for the two-electron double quantum dot. Two electrons at each quantum dot are irrelevant for the qubit manipulations because they are paired in a singlet state; the remaining two electrons encode the qubit. The qubit's wave function is immune to charge noise at moderate out-of-plane magnetic fields. (2) A singlet-triplet qubit encoded using two quantum dots of different sizes has an orbital state degeneracy of the singlet state and the spinless triplet state at finite out-of-plane magnetic fields. Spin-orbit interactions lift this state degeneracy, while the magnitude of the state coupling is determined by the size difference of the QDs. This setup enables the manipulations of singlet-triplet qubits without the need for magnetic field gradients. Finally, two-qubit gates between singlet-triplet qubits are proposed that use mediated exchange interactions via one quantum state. These operations are well controlled and highly noise insensitive. Orbital interactions alone can control spin qubits coded in a three electron Hilbert space. For example, the exchange-only qubit is encoded using three singly occupied quantum dots. The exchange interactions of two quantum dot pairs need to be modified to manipulate this qubit. The noise sensitivity of the exchange-only qubit is discussed. Alternatively, a three-electron qubit at a double quantum dot can be operated when single electrons are transferred between the quantum dots. Fast, subnanosecond manipulations of the double quantum dot's charge configuration are required to realize single-qubit gates. A novel two-qubit pulse gate for the three-electron double quantum dot qubit is proposed.
There are many successful proposals for using electron spins in quantum dots as qubits. However, a semiconductor quantum dot-based qubit that is scalable, reliable and robust enough for actual quantum information applications has not been realized yet. Common problems in semiconductor quantum dot-based spin qubits are (i) the manipulation and coupling of spin states in a scalable way and (ii) the decoherence of the qubit. While the first problem has been overcome with the so-called exchange-only qubit, the second problem is still hindering the progress towards large-scale quantum computation using quantum dot-based spin qubits.\n \n In this regard, we propose a feasible and scalable exchange-only spin qubit composed of four quantum dots that is, to leading order, intrinsically insensitive to randomly fluctuating magnetic noise, while still offering a full electric control. Motivated by our findings, we then analyze in full detail the main relaxation mechanisms in exchange-only spin qubits composed of three and four dots, and find the regimes where the coherence time of the qubit can be extended by several orders of magnitude. We then explore the possibility of implementing our proposal for an exchange-only qubit in a device composed of three quantum dots—which would offer an unprecedented degree of tunability—and in an already existing device with five quantum dots, proving that a straightforward implementation of our proposal is feasible.\n \n We conclude this project with a thorough analysis of a common electric dipole spin resonance (EDSR) experiment in a system composed of two quantum dots with intrinsic spin-mixing mechanisms. EDSR is a broadly used tool for spectroscopy of quantum dots, yet a detailed theoretical analysis of the physics of an electrically driven system of quantum dots at resonance is still lacking.
The computer industry has seen an immense development in the last decades. \nPersonal computers have become available for everybody living in industrialized \ncountries with rapidly increasing performance in terms of speed and storage capacities. \nHowever, the performance of nowadays' computers is fundamentally limited by the \nlaws of classical physics: a classical bit can only take on either of the two distinct \nvalues `0' or `1'. \nIn contrast, a quantum computer could, in principle, make direct use \nof quantum phenomena, such as state superpositions \n-- a quantum bit can be in both states `0' and `1' simultaneously --, \nto perform complex computational tasks much faster than any classical computer. \n \nThe idea of building computers that work according to the laws of quantum physics \nhas opened various fields of research, one of which is the search for the best \nphysical system to use as a quantum bit (qubit). One important criterion for determining \nthe optimal qubit system is the lifetime of state superpositions. \nTypically, once initialized, such superpositions are destroyed on remarkably short \ntimescales due to interactions with the environment -- a process which is referred to \nas decoherence --, posing the question which physical qubit candidate system might show a high-enough robustness against the \ninfluence of the `outside world' to allow for viable quantum computation. \n \nIn this thesis, we will consider three particular realizations of one specific and \nvery promising type of qubit candidate system: \nan electron (or hole) confined to a quantum dot -- a nanoscale structure \nwithin a (typically semiconducting) material --, where the spin states `down' \nand `up' of our electron (or hole) encode the logical states `0' and `1'. \nOur task will be to study the decay of spin-state superpositions in such quantum-dot \nsystems. The main objective of this thesis is to understand the most important physical \nprocesses that lead to spin decoherence and to show ways to suppress this undesirable effect. \nIt turns out that at low temperatures, the main source of decoherence \nis the coupling of the electron (hole) to the surrounding nuclear spins. \n \nThis thesis is divided into three logical parts, corresponding to the three qubit candidate \nsystems under consideration. First we will study electron-spin qubits in III-V semiconductor \nquantum dots, where the electron spin interacts with the nuclear spins of the semiconducting \nhost material \nvia the isotropic Fermi contact hyperfine interaction. Second we consider quantum-dot-confined \nheavy holes and the decoherence of their (pseudo-)spin states due to anisotropic interactions with the \nnuclear spins. Third and last, we study electron-spin qubits made from carbon-nanotube and \ngraphene quantum dots. Quantum dots made of carbon have the advantage of a low abundance \nof spin-carrying nuclear isotopes, therefore reducing decoherence effects significantly. \n \nFor each of the systems under consideration, we will carry out analytical calculations on \nthe nuclear-spin interactions and the spin dynamics of the qubit. Although one main goal of \nthis thesis is to show ways to extend spin decoherence times, we will also focus on physically more \nfundamental questions. Not only the timescale of the decay is relevant for \nthe system's applicability as a qubit, but also the form of the decay which can vary \nsignificantly from system to system. For example, the decay of spin-state superpositions \ncan follow an exponential, super-exponential or power-law decay, and can even pass through \nvarious stages. This is not only of academic interest, but also important for practical \npurposes, such as the implementation of quantum error-correction schemes in a potential \nquantum computer. \n
Quantum physics describes the universe on a microscopic scale. Even though this scale is not accessible without complex experimental setups, quantum effects are respon- sible for the properties of matter and in some instances we have learned to exploit them technologically. A modern computer constituted of transistors passively utilizes the quantum nature of charge carriers in the band structure of semiconductors to encode and process information. Although such a classical computer is a formidable tool already, what could be achieved with active control over the quantum states in the computer is an intriguing question. This is the idea of the quantum computer, which offers an entirely new paradigm of computing by encoding information in quantum binary digits (qubits). Indeed, for several applications it has been shown that a quantum computer is superior to its classical counterpart, even though its realization remains elusive up to date. One candidate to host qubits are gate-defined semiconductor quantum dots, confin- ing single or few charge carriers with spin. The spin of a single confined electron is a quantum two-level system which can be manipulated and measured, allowing the storage and processing of quantum information. This thesis can be viewed as part of the endeavor to give birth to quantum computers based on semiconductor spin qubits. It primarily focuses on the challenge to enable scalable architectures of well-characterized qubits. In the first part, spin shuttling is investigated. This denotes the coherent transport of a single spin between spatially separate nodes of a quantum processor, driven by a time-dependent electric potential. The object of study is a minimal version of a bucket- brigade-like spin shuttling between two quantum dots. The probability of maintaining the spin state is identified as figure of merit and the effect of spin-orbit interaction and the Zeeman effect in an inhomogeneous magnetic field is explored with analytical and numerical methods. Special attention is paid to the repercussions of near-degenerate valleys. The second part of the thesis addresses the challenge to probe the microscopic properties of a quantum dot array, which is a crucial first step before any quantum operation can be performed. The transmission of a microwave resonator coupled to a triple quantum dot occupied by two electrons is investigated, and a scheme to infer the exchange interaction between electrons in adjacent quantum dots is proposed. By means of analytical derivations it is shown that the transmission profile of the resonator directly reveals the value of the exchange coupling strength between two electrons. This holds even if magnetic gradients or a valley splitting comparable to the inter-dot tunnel coupling are included. Subsequently, quantum transport measurements in the Pauli blockade regime are modelled in the presence of a microwave resonator, bringing together two powerful techniques to probe quantum dots. Based on generalized input-output theory, a theoretical framework is derived to investigate how a double quantum dot in a transport setup interacts with a coupled microwave resonator while the current through the double qauntum dot is rectified by Pauli blockade. It is shown that the output field of the resonator can be used to infer the leakage current and thus obtain insight into the blockade mechanisms and to estimate a large number of parameters of the quantum dots. Furthermore, the back-action of the resonator photons on the steady state leakage current is described and quantified. In the context of Pauli blockade, the spin and valley physics of bilayer graphene quantum dots in the few-electron regime is touched upon, describing a situation where spin blockade is observed despite the presence of the valleys. In the final part of the thesis, the attention is shifted towards the readout of the spin qubits. A new paradigm of dispersive readout is devised, which is performed while the qubit dephases, implementing a measurement in the x-basis. Laplace transforming the time dependent cavity response allows a separation of contributions from multiple qubits coupled to the same resonator. Thus, they can be read out simultaneously in a single shot. Modelling silicon spin qubits, as an example, results in in a readout time that compares favourably to conventional dispersive readout with comparable fidelity, even for a single qubit. With an auxiliary qubit the readout scheme can be extended to a quantum non-demolition measurement.
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.
The growing interest in quantum computers has triggered various approaches for the realization of a large-scale quantum processor that can eventually outperform classical high-performance computers. Quantum computers are attracting increasing attention for their potential to enormously accelerate the simulation of quantum systems and the solution of optimization problems, promising an immense impact, particularly in fields such as chemistry, medicine, and cybersecurity. Although it is still a long way to a practical quantum device, the progress over the last decades has shown encouraging results for a so far limited number of quantum bits, or qubits for short, which represent the computational units of the processor. One promising platform for the implementation of qubits is the spin of charge carriers in semiconductor quantum dots. Especially the use of purified silicon as a host material has significantly improved the performance of quantum operations. Advanced fabrication techniques from the established semiconductor industry in principle allow for the manufacturing of a large number of qubits on a small chip, making them specifically competitive from the perspective of scalability. In this thesis, we investigate challenges for electron spin qubits, particularly in the context of scaling to large qubit arrays for quantum computation. We evaluate the performance of quantum gates, present error mitigation strategies, and propose new quantum operations for two particular spin qubit implementations. In a dense array of Loss-DiVincenzo spin qubits on a chip, quantum information can be electrically controlled by microwave pulses. For neighboring qubits, capacitive couplings of nearby electrodes lead to crosstalk that affects the quantum computation. The first examination in this thesis addresses the effect of crosstalk on idling and operating spin qubits due to an interfering microwave drive. After the analysis and characterization of the performance, we suggest mitigation protocols for different constellations of important quantum gates, which can be easily implemented experimentally. We further show how to reduce crosstalk in a two-qubit gate using an asymmetric driving scheme, resulting in an enormous improvement of the gate performance, and may be used to generally overcome the obstacle of crosstalk. An additional source of errors in spin qubit registers originates from the finite residual exchange interaction between neighboring qubits. It arises due to a finite overlap of the electron wave functions and is caused by the limited speed at which the applied voltages can be controlled. We evaluate the fidelity of doing quantum operations under these circumstances with a varying number of neighboring qubits and present robust mitigation schemes with a decent performance also in the presence of noise. Besides crosstalk, another crucial observation in experiments using microwaves is the heating caused by the delivery of high-power radio frequency pulses to the quantum chip and results in a change of the qubit properties. We therefore investigate the impact of phonons on the energy levels in spin qubits to identify a temperature-dependent correction to the qubit state splitting. As microwave pulses are associated with potential errors, exchange-only qubits represent a promising alternative to achieve universal quantum computation only using baseband control. Therefore, we shift our focus towards exchange-only qubits and outline the main error sources in these systems, including residual exchange interactions between spins, charge noise from the applied voltages, and magnetic fluctuations in the material. A consequential complication of this implementation is that the quantum information can leak out of the qubit states. This effect, however, can be reduced in the presence of a finite magnetic field, as we will demonstrate in this thesis. The gate construction for exchange-only qubits usually requires more pulse steps than in the previous qubit type. Nevertheless, if exchange interactions are switched on simultaneously, this can result in faster quantum gates, as we demonstrate in this thesis. Especially the discovery of a quantum operation that introduces relative phases between computational and leakage states can be of interest for the construction of two-qubit gates and will be discussed in an example for a special operation space.