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arXiv:cond-mat/0401106 · Physical Review A

Charge qubits and limitations of electrostatic quantum gates

Andreas Weichselbaum, Sergio E. Ulloa

Published Sep 29, 2004• 11 citations
arXivHTMLPDF

Abstract

We investigate the characteristics of purely electrostatic interactions with external gates in constructing full single-qubit manipulations. The quantum bit is naturally encoded in the spatial wave function of the electron system. Single-electron transistor arrays based on quantum dots or insulating interfaces typically allow for electrostatic controls where the interisland tunneling is considered constant, e.g., determined by the thickness of an insulating layer. A representative array of $3\ifmmode\times\else\texttimes\fi{}3$ quantum dots with two mobile electrons is analyzed using a Hubbard Hamiltonian and a capacitance matrix formalism. Our study shows that it is easy to realize the first quantum gate for single-qubit operations, but that a second quantum gate comes only at the cost of compromising the low-energy two-level system that encodes the qubit. We use perturbative arguments and the Feshbach formalism to show that this compromising of the two-level system is a rather general feature for electrostatically interacting qubits and is not just related to the specific details of the system chosen. We show further that full implementation requires tunable tunneling or external magnetic fields.

DOI
10.1103/physreva.70.032328

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We investigate the characteristics of purely electrostatic interactions with external gates in constructing full single-qubit manipulations. The quantum bit is naturally encoded in the spatial wave function of the electron system. Single-electron transistor arrays based on quantum dots or insulating interfaces typically allow for electrostatic controls where the interisland tunneling is considered constant, e.g., determined by the thickness of an insulating layer. A representative array of 3\ifmmode×\else\texttimes\fi33\ifmmode\times\else\texttimes\fi{}33\ifmmode×\else\texttimes\fi3 quantum dots with two mobile electrons is analyzed using a Hubbard Hamiltonian and a capacitance matrix formalism. Our study shows that it is easy to realize the first quantum gate for single-qubit operations, but that a second quantum gate comes only at the cost of compromising the low-energy two-level system that encodes the qubit. We use perturbative arguments and the Feshbach formalism to show that this compromising of the two-level system is a rather general feature for electrostatically interacting qubits and is not just related to the specific details of the system chosen. We show further that full implementation requires tunable tunneling or external magnetic fields.
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We analyze the dynamics of a ground state two-level system built on the spatial distribution of the system wave function and designed to be well-separated from the remainder of the state space. With a total of two operative electrons on a planar array of quantum dots coupled capacitively to a set of external voltage gates, the system is modeled using an extended Hubbard Hamiltonian. The voltage dependence of the low-energy singlet and triplet states is analyzed, respectively, using the Feshbach formalism. Coherent operation of the array is studied with respect to single quantum bit operations. One quantum gate is implemented via voltage controls, while for the necessary second quantum gate, a uniform external magnetic field is introduced. The Aharonov-Bohm phases on the closed loop tunnel connections in the array are used to effectively suppress the tunneling, despite a constant tunneling amplitude in the structure. This allows one to completely stall the qubit in any arbitrary quantum superposition, providing full control of this interesting quantum system.
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 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.
A GaAs∕AlGaAs\mathrm{Ga}\mathrm{As}∕\mathrm{Al}\mathrm{Ga}\mathrm{As}GaAs∕AlGaAs based two-qubit quantum device that allows the controlled generation and straightforward detection of entanglement by measuring a stationary current-voltage characteristics is proposed. We have developed a two-particle Green's function method of open systems and calculate the properties of three-dimensional interacting entangled systems nonperturbatively. We present concrete device designs and detailed charge-self-consistent predictions. One of the qubits is an all-electric Mach-Zehnder interferometer that consists of two electrostatically defined quantum wires with coupling windows, whereas the second qubit is an electrostatically defined double quantum dot located in a second two-dimensional electron gas beneath the quantum wires. We find that the entanglement of the device can be controlled externally by tuning the tunneling coupling between the two quantum dots.
We consider one- and two-qubit operations on charge qubits that represent double quantum dots with one electron in each of them. The dots are formed inside a high-Q semiconductor cavity (disk, toroidal, or spherical) in the antinodes of one of its optical eigenmodes; the frequencies of the transitions between the ground (logic) and excited (auxiliary) states of the discrete electron spectrum in quantum dots are close to the frequency of this mode. The precise tuning of the transition frequency is performed by applying an electric potential on the qubit control gate. Within the model of qubits coherently interacting with a cavity quantum field, a few methods for controlling their states are developed. In particular, we propose different variants for implementing two-qubit CNOT and CZ gates and generating qubit entangled states. By the example of a micro-disk cavity, we calculate the operating characteristics that ensure a high rate of quantum gate implementation.