1 Entanglement and quantum computing with ballistic electrons

Today only a few quantum computers have been build and these have only been able to operate on a few qubits. However, there have not been any experimental reports of many qubit gates, but controlled entanglement of two qubits have been described. Here implementations based on electron charges in coupled quantum wires is described, and simulations are performed for a model system. The simulaions are performed as an initial step towards a physical implementation of a new kind of quantum computing devices.

An other reason of performing the simulations, is that practical implementations of deviced for quantum computers based on many qubit gates seems to be most easily fullfilled by solid-state approaches. Solid-state qubits are interesting because it might be possible to build them into devises based on the conventional methods of nanotecnology. However, a major draw-back of solidstate implementation is decoherence, that is interaction with the envoronment, as described in section ??. Thus for the simulations, in order to have a negilible number of electrons in the conduction band of the structure and to minimize the decoherence caused by electron interaction with the lattice vibrations, the device were assumed to work at very low temperatures.

A quantum computer device has to contain a well-characterized qubit by means of a two-state quantum system and a sugestion of transformations on single qubits as well as on two-qubit states. Furthermore, to perform quantum computing is has to be possible to preperate the input states, control the quantum evolution, the decorence state and the readout of the results.

1.1 Simulations of ballistic electrons in coupled quantum wires

Electrons are spin-\frac{1}{2} particles, and thus the state of a qubit corresponds to a two-component vector. A set of single-qubit logic transformations on such states cab be constituted by a whole class of transformations perfomed by rotations.

The basis device used in the model consists of two parallel quantum wires, or two qubits, which were realized by means of a GaAs/alGaAs or a Si/SiO_{2} structure. A qubit was defined by the x-component, transversal to the wires, of the wavefunction of an electron propagating along the wires. The two wires are coupled by use of a coupling region between these. Transformations of single qubits between the two wires could then be performed through the coupling region, because the potential barrier at this point was lowered so that an energy split, ΔE, was produced between the two lower eigenstates of the wires.

The wavefunction of a propagating electron, when it is crossing the coupling region, will oscillate between the two wires with a periode τ = \frac{ΔE}{h}. When the electron reaches the end of the coupling region, the oscillation process terminates and the wavefunction will sometimes be diveded into two parts, with one part running along each wire. This transformation can be represented by a 2 × 2 rotation matrix.

A second kind of single-qubit transformation were also obtained by introduction of a potential barriere on one of the two wires. At the barrier the propagation of the wavefunction would be delayed, because the barrier was low enough to advoid reflections, and thus a phase shift between the two components of the qubit was introduced. This transformation can be represented by another rotation matrix.

In order to implement a two-qubit gate, two couples of wires were modelled such that the wire state |1⟩ for the first qubit and the wire state |0⟩ for the second qubit got closer to each other as shown in figure 1 so that the the two coubits would bocome entagled by introducion of a phase factor to the |01⟩ state of the system. The reason for this is, that for the qubits only the state correspondning to that the two electrons are in the inner wires, that is the |01⟩ component of the two-qubit state, would acquires a phase factor.

The geometry of this system creates a so-called Coulomb coupler that gives rise to a Coulomb interaction between the electrons propagating along the two inner wires. Because of the Coulomb coupler the phase of the |01⟩ state of the two-particle system were able to shift the angle, γ. This phase shift would be corresponding to a transformation, S(γ), which were dependent on the distance between the wires and the length of the Coulomb coupler. In order to check for

Figur 1: Shematic potential profile of a two-qubit S(γ) gate.

the feasibility of the gates under investigations, a realistic profile of a conducting band inside a doped two-dimensional electron gas were simulated, where the two coupled wires were formed inside a GaAs/AlGaAs structure as shown i figure 2. Then, to inject single electrons into the system, the region containing the electron gas was connected to the wire region.

1.2 Measurements of entanglement

The dynamics of two electrons running along two couples of wires was described by the time-dependent Schrödinger equation for the two-particle wavefunction in order to describe the obtained entangled, or non-separable, qubit states. Af-

Figur 2: o two-qubit gate

terwards the potential was calculated for the two wires taking the geometry of the system into acount.

As prevented, for the simlations it was found that the Coulomb coupler was able the entangle the two qubits by introduction of a phase factor to the |01⟩ state.

On the other hand, another major effect was otained. Because of the Coulomb coupler, the two electrons were also allowed to interact electrostatically and thus these got entangeled i their y-position, because the broadening along the wire direction af one of the electrons was dependent of the possition of the other electronen. Thus it was found that because of the electrostatic interaction between the two electrons, these were entangled in their y-position variables.

To study why the two electrons were entangled in their y-position, simulations where performed for two electrons propagation either in the same direction or in opposite directions in two wires realized by Si/SiO_{2}. For this model the entanglement were calculated at different times as a von Neumann entropy of the reduced density matrix calculated fr the system.

Nummerical results obtained for simulations of two electrons propagating in the same direction along two parallel wires is shown in figure 3. Because of the Coulomb interaction the entanglement increases very fast an reaches a constant value, corresponding to the maximum entanglement, as shown to the left in the figure, where the entanglement is plottet as function of the time. The the maximum entanglement is plottet as function of the distance between the wires to the right in the figure, from which it is seen that at any given time the mutual maximal entanglement is deceasing when the distance between the wires is lowered. In constrast, if two electrons are moving in opposite directions along the two parallel wires, the obtained results are very different from the privious results as shown in figure 4. As shown to the left in the figure the entanglement