Phase control of local and non-local entanglement in a triple spin qubit
Takashi Nakajima, Matthieu R. Delbecq, Tomohiro Otsuka, S. Amaha, Jun Yoneda, Akito Noiri, Kenta Takeda, Giles Allison, Arne Ludwig, Andreas Dirk Wieck, Seigo Tarucha
Entanglement is at the heart of quantum mechanics and an essential ingredient of spin correlations in superconductivity and the Kondo effect and quantum non-locality of noninteracting particles1–4. Non-local entanglement is also key to quantum computation outperforming classical computers5 and has now begun to be applied to various atom and photonbased quantum information technologies6. Despite many technical advances in manipulating electron spins in solids, entanglement control has only been realized for two coupled qubits in quantum dots (QDs), mainly by local SWAP operations7–11. Here we use a triple QD with three addressable spin qubits to demonstrate phase control of locally entangled two or three electron spins and two non-locally entangled uncoupled spins. Singlet-triplet oscillation of two locally entangled spins in adjacent dots is induced by a built-in local Zeeman field, and phase-modulated using the exchange-coupled third spin as a control bit. Non-local entangle-
Entanglement is at the heart of quantum mechanics and an essential ingredient of spin correlations in superconductivity and the Kondo effect and quantum non-locality of noninteracting particles1–4. Non-local entanglement is also key to quantum computation outperforming classical computers5 and has now begun to be applied to various atom and photonbased quantum information technologies6. Despite many technical advances in manipulating electron spins in solids, entanglement control has only been realized for two coupled qubits in quantum dots (QDs), mainly by local SWAP operations7–11. Here we use a triple QD with three addressable spin qubits to demonstrate phase control of locally entangled two or three electron spins and two non-locally entangled uncoupled spins. Singlet-triplet oscillation of two locally entangled spins in adjacent dots is induced by a built-in local Zeeman field, and phase-modulated using the exchange-coupled third spin as a control bit. Non-local entangle-
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.
We study controllable exchange coupling between two singlet-triplet qubits. We start from the original second quantized Hamiltonian of a quadruple quantum dot system and obtain the effective spin-spin interaction between the two qubits using the projection operator method. Under a strong uniform external magnetic field and an inhomogeneous local micromagnetic field, the effective interqubit coupling is of the Ising type, and the coupling strength can be expressed in terms of quantum dot parameters. Finally, we discuss how to generate various two-qubit operations using this controllable coupling, such as entanglement generation, and a controlled-not gate.
Coherent control of solid-state qubits lies at the heart of most quantum information architectures. In quantum dots (QDs), optical fields are an attractive medium for qubit manipulation and readout. The entanglement between a QD spin qubit and an emitted photonic qubit allows for the transport of quantum information between distant quantum memories via decoherence resistant photon channels. I will present recent experimental work showing the entanglement between a single electron spin confined to an InAs QD and its spontaneously emitted photon. This entanglement is significant for the further development of quantum information technologies using QDs and forms the foundation of on-chip technologies using photonic crystal pathways. In addition, I will discuss on-going work on teleportation of information from a single photon generated in a spontaneous parametric down conversion (SPDC) process to a QD spin through intermediate interference between the SPDC photon and the dot’s emitted photon. The ability to integrate two quantum information platforms is not only exciting in its own right, but this technique could allow for an entanglement swapping bridge between other matter-qubit (ions, NV centers, etc.) based quantum memories.
The recent discovery of efficient quantum algorithms for factoring and database search has shown that quantum computing would allow to solve important problems which are intractable with conventional computers. In contrast to the very demanding task of building a large-scale quantum computer, there are quantum communication protocols, e.g. quantum key distribution for cryptography, which—though still difficult—require much less effort and can be implemented with current technology. Apart from the technological motivation, the study of quantum information offers (at least) two additional benefits. First, new insight into fundamental questions on quantum mechanics, e.g. concerning non-locality and entanglement, are gained from an information-theoretical approach. And second, investigating a particular physical implementation of quantum information can give rise to independent physical results. Spintronics, the use of spin as opposed to charge in (classical) electronics is a new field for which some results presented here could be relevant. In this dissertation we investigate several theoretical aspects of the physical implementation of quantum computation and communication in which the fundamental unit of quantum information, the qubit, is represented by the spin of electrons in semiconductor quantum dots. The required coupling between the spins can be obtained by allowing for tunneling of electrons between adjacent dots, leading to a Heisenberg exchange coupling J S1 · S2 between the spins, a scenario which we study for laterally coupled quantum dots in a two-dimensional electron system, and for a three-dimensional setup with vertically coupled quantum dots. Furthermore, an alternative scheme to couple the spins via the interaction with an optical cavity mode is presented. Quantum error correction represents one of the important ingredients for the physical implementation of a quantum computer by protecting \nit from the e�ects of a noisy environment. As a �rst test for errorcorrection \nin a solid-state device using spins, we propose an optimized \nimplementation of the most primitive error correction scheme (the threebit \ncode). In this context, we introduce parallel switching, allowing \nto operate a quantum computer more e�ciently than the usual serial \nswitching. \nCoupling spins with the exchange interaction J S1 �S2 is not su�cient \nfor quantum computation; the spins also have to be addressed individually \nusing controllable local magnetic �elds or g-factors giBi �Si in order \nto allow for single-qubit operations. On the one hand, we discuss several \nschemes for overcoming the di�culty of applying local magnetic �elds \n(requiring large gradients), e.g. g-factor engineering, which allows for \nall-electric operation of the device. On the other hand, we show that \nat the expense of additional devices (spins) and switching operations, \nsingle-spin rotations can be dispensed with completely. \nAddressing the feasibility of quantum communication with entangled \nelectrons in mesoscopic wires, i.e. interacting many-body environments, \nwe propose an interference experiment using a scattering set-up with an \nentangler and a beam splitter. The current noise for electronic singlet \nstates turns out to be enhanced (bunching), while it is reduced for triplets \n(antibunching). Due to interactions, the �delity of the entangled singlet \nand triplet states is reduced by z4F \nin a conductor described by Fermi \nliquid theory, zF being the quasiparticle weight factor. \nFinally, we study the related but more general problem of the noise of \nthe cotunneling current through one or several tunnel-coupled quantum \ndots in the Coulomb blockade regime. The various regimes of weak and \nstrong, elastic and inelastic cotunneling are analyzed for quantum-dot \nsystems (QDS) with few-level, nearly-degenerate, and continuous electronic \nspectra. In contrast to sequential tunneling, the noise in inelastic \ncotunneling can be super-Poissonian. In order to investigate strong cotunneling \nwe develop a microscopic theory of cotunneling based on the \ndensity-operator formalism and using the projection operator technique. \nWe have derived the master equation for the QDS and the current and \nnoise in cotunneling in terms of the stationary state of the QDS. These \nresults are then applied to QDS with a nearly degenerate and continuous \nspectrum.
In this theoretical manuscript I propose a scheme for entangling a single-electron-semiconductor spin qubit with a single-electron-semiconductor charge qubit in a triangular triple-quantum-dot configuration. Two out of three quantum dots are used to define a single-electron-semiconductor charge qubit. Furthermore, the spin qubit is embedded in the Zeeman sublevels of the third quantum dot. Combining single-qubit gates with entangling controlled-not gates allows one to construct a swap gate and therefore to use the semiconductor spin qubit as a long-lived memory for the semiconductor charge qubit.