How can we avoid the dimensionality curse? Many possibilities
During the last two years we have started a large scientific activity on Quantum Computing and Machine Learning at University of Oslo and Michigan State University.
This activity spans from the development of quantum-mechanical many-particle theories for studying systems of interest for making quantum computers, via the development of quantum algorithms for solving quantum mechanical problems to exploring quantum machine learning algorithms.
Electrons on superfluid helium represent a promising platform for investigating strongly-coupled qubits.
Therefore a systematic investigation of the controlled generation of entanglement between two trapped electrons under the influence of coherent microwave driving pulses, taking into account the effects of the Coulomb interaction between electrons, is of significant importance for quantum information processing using trapped electrons.
Entanglement is the fundamental characteristic that distinguishes quantum systems composed of two or more coupled objects from their classical counterparts. The study of entanglement in precisely engineered quantum systems with countably many degrees of freedom is at the forefront of modern physics and is a key resource in quantum information science (QIS). This is particularly true in the development of two-qubit logic for quantum computation.
The generation of two-qubit entanglement has been demonstrated in a wide variety of physical systems used in present-day quantum computing, including superconducting circuits, tapped ions, semiconductor quantum dots, color-center defects in diamond, and neutral atoms in optical latticesjust to name a few.
Generating an entanglement between two quantum systems rely on exploiting interactions in a controllable way. The details in the interaction Hamiltonian between two systems defines the protocol schemes for two-qubit logic.
In superconducting circuits the interaction between qubits may arise from direct capacitive coupling between circuit elements or by indirect coupling of two qubits to a common resonator (virtually populating resonator mode) which results in a non-local Hamiltonian in the form of exchange interaction. This allow to implement various schemes for entanglement, such as \( \sqrt{i\text{SWAP}} \), controlled-phase gate, resonator-induced phase gate, cross-resonance gate.
Entanglement gates in trapped ions are produced by means of the Coulomb interaction, where shared motional modes of two or more ions, entangled to their internal states, used for transferring excitations between ion qubits. This has been experimentally demonstrated.
In photonic quantum computing schemes two-qubit entangling operations are realized by nonlinear interactions between two photons scattering from quantum dots, plasmonic nanowires, diamond vacancy centers and others embedded into waveguides. Two-qubit gates in semiconductor quantum dots are based on spin-spin exchange interactions or generated by coupling to a superconducting resonator via artificial spin-orbit interaction.
Coulomb interaction governed entanglement can be realized in the system of electrons on the surface of superfluid helium, where qubit states are formed by in-plane lateral motional or out-of plane Rydberg states. Trapped near the surface of liquid helium these states have different spatial charge configurations and the wavefunctions of different electrons do not overlap.
This results in a strong exchange free Coulomb interaction which depends on the states of the electrons. The lack of disorder in the systems also leads to slow electron decoherence, which has attracted interest to the system as a candidate for quantum information processing.
The static Coulomb interaction arises from a virtual photon exchange process between two charge particles according to quantum electrodynamics. This results in a correlated motion of two charges generating quantum entanglement.
Surface state electrons (SSE) 'floating' above liquid helium originates from quantization of electron's perpendicular to the surface motion in a trapping potential formed by attractive force from image charge and a large \( \sim \) 1 eV barrier at the liquid-vacuum interface. At low temperatures the SSE are trapped in the lowest Rydberg state for vertical motion some 11 nm above the helium surface, which is perfectly clean and has a permittivity close to that of vacuum.
The weak interaction with enviroment, which is mainly governed by interaction with quantized surface capillary waves (ripplons) and bulk phonons, ensures long coherence times - a vital ingredient for any qubit platform. SSE's in-plane motion can be further localized by using microdevices on the length scales approaching the interelectron separation (at the order of one micron).
As we are only studying a model comprised of two electrons restricted to move in a one-dimensional external potential we have employed the configuration-interaction theory to compute the steady-state properties of the system. We have used a static, one-dimensional, grid-based basis set for the single-particle functions. This allows for flexibility in the choice of the external potential, and fits the interpolated potential particularly well.
The Hamiltonian of \( N \) interacting electrons confined by some potential \( v(r) \) can be written on general form
$$ \begin{equation} \hat{H} = \sum_{i=1}^N \left(-\frac{1}{2}\nabla_i^2 + v(r_i) \right) + \sum_{i < j} \hat{u}(r_i, r_j), \label{_auto1} \end{equation} $$where \( \hat{u}(i,j) \) is the electron-electron (Coulomb) interaction.
We consider a one-dimensional model where the confining potential is parametrized/obtained from finite element calculations.
The bare Coulomb interaction is divergent in 1D (REF) and it is customary to use a smoothed Coulomb interaction given by
$$ \begin{align} u(x_1, x_2) = \frac{\alpha}{\sqrt{(x_1 - x_2)^2 + a^2}}, \label{_auto2} \end{align} $$where \( \alpha \) adjusts the strength of the interaction and \( a \) removes the singularity at \( x_1 = x_2 \).
The single-particle functions are chosen as the eigenfunctions of the single-particle Hamiltonian
$$ \begin{equation} \left( -\frac{d^2}{dx^2}+v(x) \right) \psi_p(x) = \epsilon_p \psi_p(x). \label{_auto3} \end{equation} $$It should be noted that this implies that Slater determinants built from the single-particle are eigenfunctions of the non-interacting many-body Hamiltonian
$$ \begin{equation} \left( \sum_{i=1}^N -\frac{1}{2}\nabla_i^2 + v(r_i) \right) \vert\Phi_p\rangle = E_p \vert\Phi_p\rangle, \label{_auto4} \end{equation} $$where \( \vert\Phi_p\rangle = \vert\psi_{p_1\rangle\cdots \psi_{p_N}} \) and
$$ \begin{equation} E_p = \sum_{i=1}^N \epsilon_{p_i}. \label{_auto5} \end{equation} $$