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# Quantum Computing

Multi-Photon, Multi-Dimensional Hyper-Entanglement Quantum Computing

These patents include Multi-Photon, Multi-Dimensional Hyper-Entanglement using Higher-Order Radix qudits with Applications to Quantum Computing, QKD and Quantum Teleportation. Google recently announced that they had achieved quantum supremacy with 53 qubits (base-2 binaries or radix-2), corresponding to a computational state-space of dimension (about ). In multiple patents, we claim that you can potentially achieve a similar computational dimension as Google with a much lower number of qudits (not qubits) where each qudit is of a higher radix (greater than 2) using a photonics system (i.e. 16 qudits with radix-10). We use multiple Orbital Angular Momentum (OAM) modes as a new degree of freedom for quantum computing and a multi-dimensional QKD. We believe that the convergence of both broadband, secure communications, quantum computing and quantum teleportation is only possible in a photonics realization. Therefore, the use of photonic qudits allows extending the security and capacity of the quantum teleportation beyond what was achieved by the Chinese Micius quantum satellite. Both the defense and commercial computing industries need such quantum computing systems. Qubits are state-2 binary systems constructed from superposition of two orthogonal binary basis. We can create multi-dimensional qudits (not qubits). In quantum algorithms, we need maximal superposition of states. The maximal superposition of 2-state qubit basis can be constructed using Hadamard operator. However, we use the generalized Chrestenson operator that would extend to multi-dimensional qudits.

Higher Radix Quantum Computing

The bandwidth and security qualities of Free-Space Optics (FSO) make it an attractive technology for military and commercial communications and computing. Photons have a very weak interaction with the environment, and that makes them perfectly suitable for encoding and transmitting quantum information. Alternatively, photon-matter interaction can also be used to directly create deterministic gates. We can also use graphene that can provide a strong nonlinearity without the technical drawbacks of atomic systems. In conjunction with high capacity communications using SAM and OAM, one can use these two properties to entangle them creating a hyper-entangled photonic state that can be used for multi-dimensional QKD as well as quantum computing.

One way to achieve spin-orbital entanglement is by using metamaterial. The metamaterial could be a dielectric metasurface. SAM-OAM entanglement can also be achieved using q-plates based on liquid crystals. Such an approach can one day produce integrated photonics chips that perform quantum processing. We also use a Geometric Phase Metasurface (GPM) which could be a Si-based component. GPM adds or subtracts one quanta of OAM depending on sign of spin. Therefore, we can construct multi-dimensional Bell states or GHZ states using qudits. We can also extend to hyper-entangled states with qudits and in fact construct C-Not gate with Polarization as a control bit and helicity as target bit (we can also do that in reverse) or even beam path as the target bit and combine the methods.

For photons, if we only use polarization, we will only present a qubit (radix-2). However, if we leverage OAM, we can present a qudit (higher order radix). Therefore, for a pump beam of zero-angular momentum we can have multi-dimensional entangled state for 2-photons. Entanglement happens because of indistinguishability of HH and VV pairs prior to polarization measurements where H is the Horizontal polarization and V is vertical polarization. We can use any two-state system as a qubit and use them to create hyper-entanglement of qudits. A two-state qubit system could be polarization (HV, DD, RL) horizontal/ vertical, Diagonal (45 degree/135 degree), or Right- and Left-hand circular polarizations. A two-state qubit system could be two different spatial modes or paths (say up and down paths).

Higher Radix Quantum Computing to Achieve Quantum Supremacy

A two-state qubit system could be two different path lengths (longer path and shorter path) or earlier photon and a later photon using prisms. We can have n-state behavior to hyper-entangle with a two-state system. For example, resolvable wavelength or frequency can be used and those states could have more than two states. We can have n-state behavior to hyper-entangle with a two-state system. For example, resolvable OAM states can be used and those states could have more than two states.

We can also combine all two-state systems with all the n-state systems into a multi-dimensional hyper-entangled state to create qudits. It is possible to achieve quantum supremacy by using 10 photons with radix-40 qudits which is possible to achieve as ten-photon entanglement has already been achieved. We can achieve a convergence of an advanced FSO system for communications, QKD, quantum computing and quantum teleportation with qudits that can perform beyond what was achieved with the Chinese Micius quantum satellite.

We can build all major quantum gates using photonics (X, Z, C, F, C-NOT). We can use the quantum gates and new degrees of freedom to create hyper-entangled states and perform quantum computation, multi-dimensional QKD as well as quantum teleportation using qudits with fewer number of entangled qudits compared to qubits for the same computational state-space dimension.

Photon–Matter Interaction Hamiltonian & Quantum Computing

We also have multiple patents on photon-matter interactions and their application in quantum computing. When a group of photons are in an entangled state, the measurement of one or more of these photons' states causes the state of the remaining photons to collapse. This collapse can be used to get photons to interact and entangle. The quantum states of a photon (spectrum, pulse shape, time of arrival, transverse wave vector, and position) must be controlled to perform entanglement. Since photons do not interact with one another well, we make photons interact with quantum matter and control the quantum transitions in the semiconductor material for quantum computing.

To go beyond 2- states qubits, we have to use OAM of twisted light in order to create higher-radix systems that could represent a qudits (not qubits). A framework for transitions between quantized states of a material (i.e. optical transitions with light frequencies above the bandgap) is built so that free carriers rather than excitons are produced. There is a transfer of angular momentum between the light and the photoexcited electrons so that a net electric current is created. Magnetic field can also be induced by these photocurrents. The framework describes the inter-band transitions in a 2-band model of a bulk semiconductor induced by twisted light. First a semiclassical Hamiltonian is constructed where we use quantum mechanical operators for electrons and classical variables for the light field with two terms, the bare electron energy and the interaction Hamiltonian. The transfer OAM to the photo-excited electrons causes an electric current and a magnetic field. These can be used in opto-electronic applications. The total current/magnetic field produced by all electrons can be found by multiplying the total number of photo-excited electrons excited by the field of the current/magnetic field of a single electron. We believe that the excitation process generates a superposition of conduction-band states with a well-defined angular momentum. The framework should include an OAM- generalized semiconductor Bloch equations and coherences of the photo-excited electrons. The Coulomb interaction does not play a major role in the basic physics of band-to-band optical transitions (except in the case of exitons).

Photon–Matter Interaction & Application to Solid States

This framework can be applied to 2-dimensional systems excited at normal incidence which would be the basis of the experimental work. For bulk systems, we can consider a solid cylinder and quantize the electron states in cylindrical coordinates, and finally take the limit of large system. The bulk properties are independent of the geometry of the solid. We use cylindrical rather than cartesian coordinates to decouple the Heisenberg equations of motion based on the electron's angular momentum. Using these generalized Block states, we can predict the kinetics of electrons, look for the electric currents and demonstrate the transfer of OAM from the light beam to the electrons.

This framework is applicable to many systems, from semiconductors to insulators having larger band gaps if the frequency of the twisted field is tuned above the energy bandgap. Therefore, twisted fields in the near infrared to UV spectrum can be used from this 2-level framework, and we can easily generalize to the case of more than one valence band. The excitation of solids by twisted light beams also produces a space-dependent carrier kinetics which requires local variables. We can potentially visualize the pattern of motion of the photo-excited electrons from the spatially inhomogeneous electric current density. The inter-band coherence contributions have fast oscillations similar to the inter-band polarization. However, intra-band coherence contributions are slower processes where we have a transfer of momentum from light to matter. The first quantization framework considers observables as operators and state as a wave function. In second quantization, we consider states as operators as well. Second quantization allows us to do multi-body dynamics. The second quantization allows the use of “big” Hilbert space with vectors that represent states of large numbers of particles because second quantization is a link between the quantum mechanics of many-particle systems and the relativistic quantum field theory. In structured beams, photons with OAM and SAM travel at a speed slower than velocity of light. This is due to a projection effect of the effective motion of the photon on the axis of propagation of the diverging beam and depends on the geometrical properties of the beam.

Multi-Dimensional Quantum Key Distribution & Quantum Computing

Multi-dimensional Quantum Key Distribution (QKD) for cybersecurity using OAM states in photonics and RF. The current state of technology uses two-states of polarization for QKD. However, we have introduced new OAM states of photons as the basis for QKD. We have extended Bennett-Brassard (BB84) beyond polarization (2 state system) using LG modes or HG modes (multi-state system) for qubits. We also have extended Ekert 91 (E91) scheme with entangled pairs of photons using LG modes or HG modes (multi-state system) for qubits. We explore new techniques to combat denial of service attacks by routing communication via alternate links in case of disruption. We have included new techniques to combat Trojan horse attacks which does not require physical access to the endpoints as well as new techniques to combat faked-state attacks, phase remapping attacks and time-shift attacks. We introduced cloud-based Quantum Key Distribution for a multiuser network in a HetNet configuration.

Structured Photons to Improve the Efficiency of Solar Cells

In this patent class we introduce an artificial photo conversion system using an amplitude or a phase mask which can enable the suppression of electron-hole recombination, thereby increasing or enhancing the efficiency of any PN junction or solar cell or any solid state used for energy harvesting or display. When a hologram is used in the path of a photon the orbital angular momentum generated by the photon can be transferred to the electron and a new quantum state can be created, and suppression of electron-hole recombination is supported. This suppression is due to the change in total angular momentum of the electron (spin + orbital) using a variety of methods to twist the photon field (passively using a hologram or actively using other methods).

In photovoltaic diodes, recombination of photogenerated electrons and holes is a major loss process. Biological light harvesting complexes (LHCs) prevent recombination via the use of cascade structures, which lead to spatial separation of charge carriers. Therefore, the twisted photons can be used in conjunction with organic photovoltaic cells (OPVs) to suppress recombination of electron-holes. The suppression of recombination can be engineered through the interplay between spin, OAM, and delocalization of electronic excitations in organic semiconductors. OPVs would have poor quantum efficiencies if every encounter leads to recombination, but state-of-the-art twisted photons with OPVs can demonstrate better quantum efficiencies.

Time-resolve spectroscopy can be used to study different engineered models with high efficiency systems in which the lowest lying molecular triplet exciton (T1) lies below the intermolecular charge transfer state (CT). Encounters of spin-uncorrelated electrons and holes generate CT states with both spin singlet (1CT) and spin triplet (3CT). Triplet exciton formation can be a major loss mechanism in OPVs. However, even when energetically favored, the relaxation of 3CT spin triplet to T1 can be strongly suppressed via, control over wave function delocalization allowing for disassociation of CT back to free charges (FC) thereby reducing recombination and enhancing performance. Extracted kinetics can demonstrate that triplets may grow as charges decay. One can consider that the primary decay channel for triplets is triplet-charge annihilation, due to the high charge density present, and model the time evolution with Langevin equation. The CT energy lies below T1 making relaxation from 3CT to T1 energetically favored, however, the more efficient 1:3 blend no triplet formation is possible at room temperature. But at low temperatures bi-molecular triplet formation can be observed in this blend. This suggests that there is a thermally activated process that competes with relaxation to T1. This process is disassociation of 3CT back to free charges. At high temperatures the disassociations of 3CT back to free charges with an associated time scale τ3, outcompetes relaxation of 3CT to T1. At lower temperatures this disassociation process is suppressed leading to a buildup of triplet excitons.

Structured Photons with Majorana Quasiparticle States in Quantum Computing

The structured beams carrying OAM have a group velocity similar to Majoranas for bosonic and fermionic particles. In addition, structured light beams and other photonic systems can behave like Majorana particles. In a plasma, photons carrying OAM acquire an effective Proca mass through the Anderson-Higgs mechanism and obey a mass/OAM relationship that resembles a Majorana structure. With coherent superposition of OAM states, we can get Fermionic and anionic-like behaviors with non-abelian gauge structure. The slowing down of structured light beam in a vacuum is related to geometrical divergence of the beam. Phase velocity is also different for different OAM modes.

The time delays or different group velocities can be interpreted as the effect of a fictitious mass of a quasiparticle state that characterizes the dynamics of the beam and is described by a Schrodinger-like equation at low velocities propagating with the group velocity. This fictitious mass term leads to a Majorana mass. Any of these quasiparticle states have a precise relationship between their angular momentum value and their fictitious mass in vacuum that resembles the behavior of Proca photons with OAM in a plasma. Therefore, photons propagating in vacuum belonging to this class of structured beams obey the mathematical rules of Majorana model of quasiparticle states. The dynamics and structure of the beam is uniquely characterized in space and time through the rules of the Poincare´ group that build up the Majorana model.

Through the projection effect along the axis of propagation, one can resize and shape the beam at will to obtain different apparent sub-luminal velocities for different values of OAM and spatially buffer information in time. Moving Majorana quasiparticles around each other and exchanging their positions is a non-Abelian unitary transformation that depend only on the topology of the trajectories. The information can then be encoded in the twist of the state function (wave function). Unitary transforms are robust against decoherence and dephasing due to local environments which would make this approach interesting for topological quantum computing. There is a Clifford algebra generalization of the quaternions and its relationship with braid group representations related to Majorana fermions. Majorana operators represent a braid group that can represent exchanges of Majoranas in space.

Topological Quantum Computing using Electromagnetic Knots

Braiding of Majorana fermions are natural Clifford representations and with the quaternions representations as SU (2) to the braid group. This could also explain the vortices of the quantum Hall effect. Braids can be multiplied by attaching the bottom row of one braid to the top row of the other braid. This multiplication of braids forms a group. The braids are critical to the wave function or electromagnetic knots. We can perform braiding of Majorana fermions with Majorana photons which are the vortex vector beams that carry OAM. We can perform sophisticated braiding using superconducting Y-Junctions in a cavity with OAM beam acting on this Y-junction. Hamiltonian constructed from a rotating wave approximation couples the states with photons in the cavity at cavity’s frequency. Quaternions are intimately connected to topology and braiding is fundamental to Majoranas and their application to quantum computing. Quaternions can also be used to create electromagnetic knots. With temperature control, one can go from one symmetry space to another and the quaternionic algebra used for braiding wave functions can be used to describe these braiding states.

Graphene behaves like a semi-metallic material, and its low-energy excitations behave as massless Dirac fermions. Graphene absorbs white light, and its transparency is governed by quantum electrodynamics rather than condensed matter physics. There is a transfer of angular momentum from OAM of the photons to the electrons in graphene. There is another angular momentum (pseudospin) associated with the honeycomb lattice of graphene, and the total angular momentum (orbital plus pseudospin) is conserved. The interaction Hamiltonian between OAM and matter developed can be used to study the interaction of graphene with twisted light and calculate relevant physical observables such as the photo-induced electric currents and the transfer of angular momentum. The low-energy states of graphene are 2-component spinors. These spinors are not the spin states of the electron, but they are related to the physical lattice. Each component is associated with the relative amplitude of the Bloch function in each sublattice of the honeycomb lattice. They have a SU(2) algebra. We can also use OAM to produce the “guide sequence” for gene editing called "OAM-Assisted CRISPR".

OAM-Assisted CRISPR can impact both precision genetics as well as precision pharma industries.

Electromagnetic Knots and Applications

Some of the highlights include electromagnetic knots. Natural processes such as reflection, diffraction and scattering degrade the information signals in wireless communications. However, they cannot open an electromagnetic knot and such EM knots are resistant to channel impairments. Natural processes also destroy coherence in quantum computing superposition of states for qubits. However, they can not open an electromagnetic knot and such topological knots are resistant to de-coherence. If electromagnetic waves can be knotted using sophisticated antenna structures (ring, toroidal and 3-dimensional patch antennas), then information could be encoded into the electromagnetic knots using a new modulation using knots as the states (modulated knots). It is possible to create paraxial solutions using EM knots that produce knotted OAM states. These states can be muxed to achieve improvements in wireless, security and QKD and quantum computing.

We introduce a new way of combating the degradations due to fading in wireless communications as well as quantum de-coherence in quantum computing using electromagnetic knots. In general, natural processes can degrade fabrics and signals, but generally they are not able to undue a knotted fabric or a knotted electromagnetic wave. Maxwell equations have an underlying topological structure given by a scalar field which represents a map that determines the electromagnetic field through a certain transformation (from 3-sphere to 2-sphere). Maxwell equations in a vacuum also have topological solutions, characterized by a Hopf index equal to the linking number of any pair of magnetic lines. This allows the classification of the electromagnetic fields into homotopy classes, labeled by the value of the helicity. This helicity is proportional to an integer action constant. Different methods of generation will be required depending upon the form of disturbance required and the frequencies required (ring, toroidal and 3D- patch antennas). We have included cylindrically polarized vector beams to generate an electric ring. We can also use toroidal antennas to create electromagnetic knots. We can wrap a solenoid on the ring element and control the electromagnetic knots based on homotopy classes. We can also have more sophisticated antennas such as knotted toroids. Multiple patch antennas can be configured in circular, elliptical or mixed configuration to generate electromagnetic knots. Each component of patch antennas would have a different phase to produce knotted eigen-channels. Multiple layers of such antennas can be overlaid to naturally mux the independent modes using the EM knots. Design and simulations of the EM knots can be performed using HFSS with microstrip feed structure to prepare for manufacturing. Cleanroom and lithography process are used to build the 3D structure of patch antennas that produce EM knots. We can generate qubits using the EM knots in photonics using polarization states. We can also generate knotted OAM states in both photonics as well as EM waves.

We can then multiplex the muxed OAM knotted beams with a specific radio frequency or transmit the muxed EM knotted states and multiplex them with different radio channels. We can also de-Multiplex the Muxed knotted OAM beams. We can transmit multiple knotted beams each carrying different message signals and can do a Knot Division Multiplexing (KDM) with one frequency. This KDM can have improvements in Backhaul and Fronthaul scenarios. It can also be configured in MIMO setting for a compactified massive MIMO with patch antennas.