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arXiv:1004.1344 · Physical Review B

Quantum circuits based on coded qubits encoded in chirality of electron spin complexes in triple quantum dots

Chang‐Yu Hsieh, Paweł Hawrylak

Published Nov 9, 2010• 24 citations
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Abstract

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.

DOI
10.1103/physrevb.82.205311

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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.
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.
We present a theoretical proposal for the Herzberg circuit and controlled accumulation of Berry's phase in a chirality-based coded qubit in a triangular triple quantum dot molecule with one electron each. The qubit is encoded in the two degenerate states of a three-spin complex with total spin S=1/2S=1/2S=1/2. Using a Hubbard and Heisenberg model the Herzberg circuit encircling the degeneracy point is realized by adiabatically tuning the successive on-site energies of quantum dots and tunnel couplings across a pair of neighboring dots. It is explicitly shown that encircling the degeneracy point leads to the accumulation of the geometrical Berry's phase. We show that only the triangular, not the linear, quantum dot molecule allows for the generation of Berry's phase and we discuss a protocol to detect this geometrical phase.
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 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 present a method of changing the effective spin-spin interactions of electrons localized in quantum dot molecules from antiferromagnetic to ferromagnetic using electrical means. We show that starting with antiferromagnetic interaction of two electrons localized in two quantum dots, the effective spin-spin interaction can change to ferromagnetic by allowing interaction with a third dot containing two spin singlet electrons. The total spin of this four-electron complex can be tuned by changing the third dot potential from singlet to triplet. This is demonstrated using the linear combination of harmonic orbitals combined with the configuration-interaction method mapped onto the Hubbard and an effective $t\text{\ensuremath{-}}J$ model for the triple dot system. All three approaches predict singlet-triplet spin transition, i.e., a mechanism for changing the magnitude and sign of the effective Heisenberg interaction between two localized spins.
In the last decade, a lot of effort have gone to find the most appropriate physical implementation of quantum computer. Quantum dot based system is one of the most promising system owing to its integrability with classical computation hardware and its versatility in realizing qubits and quantum gates. In this work, we propose a model for quantum dot based realization of qubits and quantum gates. The proposed system makes use of semiconductor hetrostructure based quantum dots and gate voltages to confine electrons and thus defining the qubits. Electron spin resonance is used to manipulate the spin state of the electrons. We implement single, double and triple qubit systems and two most widely used universal sets of quantum gates. We show full control of Bloch sphere in single qubit system, CNOT gate implementation in double qubit system and a novel way to implement Toffoli gate in triple qubit system.