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Semiconductor Science and Technology

Quantum computing with quantum-Hall edge state interferometry

Paolo Bordone, Laura Bellentani, Andrea Bertoni

Published Aug 19, 2019• 12 citations
Publisher

Abstract

Abstract Electron interferometers based on Hall edge states (ESs) proved to be robust demonstrators of the coherent quantum dynamics of carriers. Several proposals to expose their capability to build and control quantum entanglement and to exploit them as building block for quantum computing devices has been presented. Here, we review the time-dependent numerical modeling of Hall interferometers operating at the single-carrier level at integer filling factor (FF). By defining the qubit state either as the spatial localization (at FF 1) or the Landau index (at FF 2) of a single carrier propagating in the ES, we show how a generic one-qubit rotation can be realized. By a proper design of the two-dimensional electron gas potential landscape, an entangling two-qubit gate can be implemented by exploiting Coulomb interaction, thus realizing a universal set of quantum gates. We also assess how the shape of the edge confining potential affects the visibility of the quantum transformations.

DOI
10.1088/1361-6641/ab3be6

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•Jan 1, 2013•0 citations
Abstract Electron interferometers based on Hall edge states (ESs) proved to be robust demonstrators of the coherent quantum dynamics of carriers. Several proposals to expose their capability to build and control quantum entanglement and to exploit them as building block for quantum computing devices has been presented. Here, we review the time-dependent numerical modeling of Hall interferometers operating at the single-carrier level at integer filling factor (FF). By defining the qubit state either as the spatial localization (at FF 1) or the Landau index (at FF 2) of a single carrier propagating in the ES, we show how a generic one-qubit rotation can be realized. By a proper design of the two-dimensional electron gas potential landscape, an entangling two-qubit gate can be implemented by exploiting Coulomb interaction, thus realizing a universal set of quantum gates. We also assess how the shape of the edge confining potential affects the visibility of the quantum transformations.
A solid-state implementation of a universal set of gates for quantum computation is proposed and analysed using a time-dependent 2D Schrödinger solver. The qubit is defined as the state of an electron propagating along a couple of quantum wires. The wires are suitably coupled through a potential barrier with variable height and/or width. It is shown how a proper design of the system allows the implementation of any one-qubit transformation. The two-qubit gate is realized through a Coulomb coupler able to entangle the quantum states of two electrons running in two wires of two different qubits. The simulated devices are GaAs—AlGaAs heterostructures that should be on the borderline of present semiconductor technology. An estimate of decoherence effects due to phonon scattering is also presented.
The implementation of a functional quantum computer involves entangling and coherent manipulation of a large number of qubits. For qubits based on electron spins confined in quantum dots, which are among the most investigated solid-state qubits at present, architectural challenges are often encountered in the design of quantum circuits attempting to assemble the qubits within the very limited space available. Here, we provide a solution to such challenges based on an approach to realizing entanglement of spin qubits over long distances. We show that long-range Ruderman-Kittel-Kasuya-Yosida interaction of confined electron spins can be established by quantum Hall edge states, leading to an exchange coupling of spin qubits. The coupling is anisotropic and can be either Ising type or XY type, depending on the spin polarization of the edge state. Such a property, combined with the dependence of the electron spin susceptibility on the chirality of the edge state, can be utilized to gain valuable insights into the topological nature of various quantum Hall states.
With the help of the spin-orbit interaction, we propose a scheme to perform holonomic single-qubit gates on the electron spin confined to a quantum dot. The manipulation is done in the absence (or presence) of an applied magnetic field. By adiabatic changing the position of the confinement potential, one can rotate the spin state of the electron around the Bloch sphere in semiconductor heterostructures. The dynamics of the system is equivalent to employing an effective non-Abelian gauge potential whose structure depends on the type of the spin-orbit interaction. As an example, we find an analytic expression for the electron spin dynamics when the dot is moved around a circular path (with radius RRR) on the two dimensional electron gas (2DEG) and show that all single-qubit gates can be realized by tuning the radius and orientation of the circular paths. Moreover, using the Heisenberg exchange interaction, we demonstrate how one can generate two-qubit gates by bringing two quantum dots near each other, yielding a scalable scheme to perform quantum computing on arbitrary NNN qubits. This proposal shows a way of realizing holonomic quantum computers in solid-state systems.
Quantum computers are in hot-spot with the potential to handle more complex problems than classical computers can. Realizing the quantum computation requires the universal quantum gate set {T, H, CNOT} so as to perform any unitary transformation with arbitrary accuracy. Here we first briefly review the Majorana fermions and then propose the realization of arbitrary two-qubit quantum gates based on chiral Majorana fermions. Elementary cells consist of a quantum anomalous Hall insulator surrounded by a topological superconductor with electric gates and quantum-dot structures, which enable the braiding operation and the partial exchange operation. After defining a qubit by four chiral Majorana fermions, the single-qubit T and H quantum gates are realized via one partial exchange operation and three braiding operations, respectively. The entangled CNOT quantum gate is performed by braiding six chiral Majorana fermions. Besides, we design a powerful device with which arbitrary two-qubit quantum gates can be realized and take the quantum Fourier transform as an example to show that several quantum operations can be performed with this space-limited device. Thus, our proposal could inspire further utilization of mobile chiral Majorana edge states for faster quantum computation.
In the present paper, the first in a series of two, we propose a model of universal quantum computation using a fermionic or bosonic multiparticle continuous-time quantum walk with two internal states (e.g., the spin-up and -down states of an electron). A dual-rail encoding is adopted to convert information: A single qubit is represented by the presence of a single quantum walker in either of the two parallel paths. We develop a roundabout gate that moves a walker from one path to the next, either clockwise or counterclockwise, depending on its internal state. It can be realized by a single-particle scattering on a directed weighted graph with the edge weights 1 and $\ifmmode\pm\else\textpm\fi{}i$. The roundabout gate also allows the spatial information of the quantum walker to be temporarily encoded in its internal states. The universal gates are constructed by appropriately combining several roundabout gates, some unitary gates that act on the internal states, and two-particle scatterings on straight paths. No ancilla qubits are required in our model. The computation is done by just passing quantum walkers through properly designed paths. Namely, there is no need for any time-dependent control. A physical implementation of quantum random access memory compatible with the present model will be considered in the second paper [R. Asaka et al., following paper, Phys. Rev. A 107, 022416 (2023)].
Many approaches to quantum computing use spatially confined qubits in the presence of dynamic fields to perform computation. These approaches are contrasted with proposals using mobile qubits in the presence of static fields. In this thesis, steady state quantum computing using mobile electrons is explored using numerical modeling. Firstly, a foundational introduction to the case of spatially confined qubits embodied via quantum dots is provided. A collection of universal gates implemented with dynamic fields is described using simulations. These gates are combined to implement a five-qubit Grover search to provide further insight on the time-dependent field approach. Secondly, the quantum dot description is contrasted with quantum computing using steady state spin currents. Leveraging the Non-Equilibrium Greens Function formalism to perform numerical simulations, the quantum aspects of steady state spin currents are explored by revisiting the Stern-Gerlach experiment using spin-polarized contacts on a one-dimensional channel. After demonstrating the quantum nature of mobile electrons at steady state, arbitrary single qubit operations using static fields are explored. The model is further extended to incorporate two-qubit interactions to realize the square root of SWAP gate. The two-qubit CNOT gate is used to prepare a Bell state, which is read via quantum state tomography. Finally, Grover's search is revisited to explore the performance benefits of steady state quantum computing. The described multi-particle model is applicable to mobile qubit quantum computing proposals leveraging synchronized electron transport in static fields to perform quantum computing.