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arXiv:0908.2105 · Physical Review B

Valence holes as Luttinger spinor based qubits in quantum dots

Chang‐Yu Hsieh, Ross Cheriton, Marek Korkusiński, Paweł Hawrylak

Published Dec 16, 2009• 19 citations
arXivHTMLPDF

Abstract

We present a theory of valence holes as Luttinger spinor based qubits in $p$-doped self-assembled quantum dots within the four-band $k\ensuremath{\cdot}p$ formalism. The two-qubit levels are identified with the two chiralities of the doubly degenerate ground state. We show that single-qubit operations can be implemented with static magnetic field applied along the $z$ axis (growth direction) for ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\sigma}}}_{z}$ operation and with magnetic field in the quantum dot plane, $x$ direction, for ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\sigma}}}_{x}$ operation. The coupling of two dots and hence the double-qubit operations are shown to be sensitive to the orientation of the two quantum dots. For vertical qubit arrays, there exists an optimal qubit separation suitable for the voltage control of qubit-qubit interactions.

DOI
10.1103/physrevb.80.235320

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We present a theory of valence holes as Luttinger spinor based qubits in ppp-doped self-assembled quantum dots within the four-band k\ensuremath⋅pk\ensuremath{\cdot}pk\ensuremath⋅p formalism. The two-qubit levels are identified with the two chiralities of the doubly degenerate ground state. We show that single-qubit operations can be implemented with static magnetic field applied along the zzz axis (growth direction) for \ensuremathσ\ifmmode^\elseˆ\fiz{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\sigma}}}_{z}\ensuremathσ\ifmmode^\elseˆ\fi​z​ operation and with magnetic field in the quantum dot plane, xxx direction, for \ensuremathσ\ifmmode^\elseˆ\fix{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\sigma}}}_{x}\ensuremathσ\ifmmode^\elseˆ\fi​x​ operation. The coupling of two dots and hence the double-qubit operations are shown to be sensitive to the orientation of the two quantum dots. For vertical qubit arrays, there exists an optimal qubit separation suitable for the voltage control of qubit-qubit interactions.
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.
A pair of self-assembled InGaAs quantum dots filled with two electrons can act as a singlet-triplet spin qubit that is robust against nuclear spin fluctuations as well as charge noise. This results in a T2* coherence time two orders of magnitude longer than that of a single electron, provided the qubit is operated at a particular "sweet spot" in gate voltage. However, at this fixed operating point the ground-state splitting can no longer be tuned into resonance with e.g. another qubit, limiting the options for coupling multiple qubits. Here, we propose using a four-electron coupled quantum dot to implement a singlet-triplet qubit that features a magnetically tunable level splitting. As a first step towards full experimental realization of this qubit design, we use optical spectroscopy to demonstrate the tunability of the four-electron singlet-triplet splitting in a moderate magnetic field.
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We study the two-qubit controlled-NOT gate operating on qubits encoded in the spin state of a pair of electrons in a double quantum dot. We assume that the electrons can tunnel between the two quantum dots encoding a single qubit, while tunneling between the quantum dots that belong to different qubits is forbidden. Therefore, the two qubits interact exclusively through the direct Coulomb repulsion of the electrons. We find that entangling two-qubit gates can be performed by the electrical biasing of quantum dots and/or tuning of the tunneling matrix elements between the quantum dots within the qubits. The entangling interaction can be controlled by tuning the bias through the resonance between the singly occupied and doubly occupied singlet ground states of a double quantum dot.
We present a theory of quantum circuits based on coded qubits encoded in chirality of electron spin complexes in lateral gated semiconductor triple quantum dot molecules with one-electron spin in each dot. Using microscopic Hamiltonian we show how to initialize, coherently control, and measure the quantum state of a chirality-based coded qubit using static magnetic field and voltage tuning of individual dots. The microscopic model of two interacting coded qubits is established and mapped to an Ising Hamiltonian, resulting in nonlocal double-qubit gates.
As a candidate scheme for controllably coupled qubits, we consider two quantum dots, each doped with a single electron. The spin of the electron defines our qubit basis and trion states can be created by using polarized light; we show that the form of the excited trion depends on the state of the qubit. By using the Luttinger-Kohn Hamiltonian we calculate the form of these trion states in the presence of light-heavy hole mixing, and show that they can interact through both the F\"orster transfer and static dipole-dipole interactions. Finally, we demonstrate that by using chirped laser pulses, it is possible to perform a two-qubit gate in this system by adiabatically following the eigenstates as a function of laser detuning. These gates are robust in that they operate with any realistic degree of hole mixing, and for either type of trion-trion coupling.