Systems and methods for coherent states based quantum information processing and quantum computing analogs

Abstract

A universal set of quantum gates classical analogs can be implemented in integrated optics by employing classical polarization states derived from the coherent states. Further, quantum qudit analogs can be implemented based on orbital angular momentum (OAM) states and corresponding qudit gates. A controlled-phase gate analog operation can also be implemented between the classical coherent states, where a CNOT operation can be implemented as a type of controlled-phase gate. These devices can be used to create a decoherence-free optical quantum information processing computing analog which does not rely on fragile quantum states, and instead relies on robust classical states.

Description

Attorney Docket No.: 085067-827975 (UA24-089) SYSTEMS AND METHODS FOR COHERENT STATES BASED QUANTUM INFORMATION PROCESSING AND QUANTUM COMPUTING ANALOGS CROSS REFERENCE TO RELATED APPLICATIONS

[0001] This is a PCT patent application that claims benefit to U.S. Provisional Patent Application Serial No.63 / 611,688 filed on December 18, 2023, which is herein incorporated by reference in its entirety. FIELD

[0002] The present disclosure generally relates to quantum information processing, and in particular, to a system and associated method for implementing classical computing analogs for quantum information processing. BACKGROUND

[0003] Decoherence is a problem for quantum information processing due to the fragility of quantum states. As such, advancements in quantum information processing are still needed for practical implementation, particularly with respect to analogs for classical computing counterparts.

[0004] It is with these observations in mind, among others, that various aspects of the present disclosure were conceived and developed. SUMMARY

[0005] A method of entangling two classical coherent states includes: obtaining a first classical coherent state and a second classical coherent state; providing the first classical coherent state to a first waveguide, wherein the first waveguide generates as output a first idler state and a first photon addition signal state; providing the second classical coherent state to a second waveguide, wherein the second waveguide generates as output a second idler state and a second photon addition signal state; and applying a phase shift to the second idler state at an output of the second waveguide. Following application of the phase shift to the second idler state, the method can further include mixing, using a beam splitter associated with a pair of outputs respectively connected to upper and lower branch single photon detectors (SPDs), an input including: a first idler photon associated100510876.21Attorney Docket No.: 085067-827975 (UA24-089) with the first idler state generated by the first waveguide; and a second idler photon associated with the second idler state generated by the second waveguide and following application of the phase shift to the second idler state.

[0006] An integrated optics apparatus outlined herein includes: a polarization beam splitter (PBS) operable for splitting a polarized beam having an input classical polarization state into a vertical polarization portion and a horizontal polarization portion; an optical device configured to perform a single-qubit quantum analog operation on the input classical polarization state based on applying an operation (described by the scattering matrix) to the vertical polarization portion and the horizontal polarization portion of the polarized beam to thereby yield a modified vertical polarization portion and a modified horizontal polarization portion, wherein the optical device includes two or more phase trimmers each having a selectable phase shift, and wherein different values of the selectable phase shifts correspond to different single-qubit quantum analog operations; and a polarization beam combiner (PBC) operable for combining the modified vertical polarization portion and the modified horizontal polarization portion into an output polarized beam having an output classical polarization state modified from the input classical polarization state according to the scattering matrix.

[0007] The vertical polarization portion of the polarized beam can include vertical polarization state photons; the horizontal polarization portion of the polarized beam can include horizontal polarization state photons; and the input classical polarization state can be a superposition of the vertical and horizontal polarization states.

[0008] In a first embodiment of the integrated optics apparatus, the optical device is a 2x2 optical hybrid coupler including four phase trimmers and operable for: splitting, using a first input Y-junction of the 2x2 optical hybrid coupler, the vertical polarization portion of the polarized beam into a first split vertical component and a second split vertical component based on a power splitting ratio; splitting, using a second input Y-junction of the 2x2 optical hybrid coupler, the horizontal polarization portion of the polarized beam into a first split horizontal component and a second split horizontal component based on the power splitting ratio; and using each respective phase trimmer of the four phase trimmers to apply a corresponding selected phase shift to a particular one of the first or second split100510876.22Attorney Docket No.: 085067-827975 (UA24-089) horizontal or split vertical components, wherein each respective phase trimmer receives a different split component as input.

[0009] Using each respective phase trimmer to apply a corresponding selected phase shift can include: using a first phase trimmer to apply a corresponding first phase shift to the second split vertical component to obtain a phase-adjusted second split vertical component; using a second phase trimmer to apply a corresponding second phase shift to the first split vertical component to obtain a phase-adjusted first split vertical component; using a third phase trimmer to apply a corresponding third phase shift to the first split horizontal component to obtain a phase-adjusted first split horizontal component; using a fourth phase trimmer to apply a corresponding fourth phase shift to the second split horizontal component to obtain a phase-adjusted second split horizontal component; and combining the phase-adjusted second split vertical component and the phase- adjusted first split horizontal component into the modified vertical polarization portion of the output polarized beam and combining the phase-adjusted first split vertical component and the phase-adjusted second split horizontal component into the modified horizontal polarization portion of the output polarized beam; the power splitting ratio, the first phase shift, the second phase shift, the third phase shift, and the fourth phase shift each being governed by elements of the scattering matrix, the elements of the scattering matrix being dependent upon on a target gate implementation of the integrated optics apparatus.

[0010] In a second embodiment of the integrated optics apparatus, the optical device can be a directional coupler-based single-qubit gate analog based on Y-Z decomposition and operable for: using a first phase trimmer to apply a first phase shift to the horizontal polarization portion of the polarized beam, the firstphase shift being governed by ^^ − ^^ / 2 − ^^ / 2; using a second phase trimmer toapply a second phase shift to the vertical polarization portion of the polarized beam,the second phase shift being governed by ^^ + ^^ / 2; using a directional coupler toapply a coupling phase shift to the horizontal polarization portion and the vertical polarization portion of the polarized beam following phase shift, the coupling phaseshift governed by ^^^^ = ^^ / 2; and using a third phase trimmer to apply a third phaseshift to the horizontal polarization portion of the polarized beam following applicationof the coupling phase shift, the third phase shift being governed by −^^ + ^^ / 2.100510876.23Attorney Docket No.: 085067-827975 (UA24-089)

[0011] In a third embodiment of the integrated optics apparatus, the optical device can be a directional coupler-based single-qubit gate analog based on a Barenco theorem and operable for: using a first phase trimmer to apply a first phase shift to the horizontal polarization portion of the polarized beam, the first phase shift being governed by ^^; using a directional coupler to apply a coupling phase shift to the horizontal polarization portion and the vertical polarization portion of the polarized beam following phase shift, the coupling phase shift being governedby ^^^^ = ^^; using a second phase trimmer to apply a second phase shift to thehorizontal polarization portion of the polarized beam at an output of the directionalcoupler, the second phase shift being governed by −(^^ + ^^); and using a thirdphase trimmer to apply a third phase shift at an output of the polarization beam combiner, the third phase shift being governed by ^^.

[0012] In a fourth embodiment of the integrated optics apparatus, the optical device can be a Mach-Zehnder interferometer-based single-qubit gate nalog based on Y-Z decomposition and operable for: using a first phase trimmer to apply a first phase shift to the horizontal polarization portion of the polarized beam, the firstphase shift being governed by −^^ − ^^; using a first 3 dB coupler to combine thehorizontal polarization portion and the vertical polarization portion of the polarized beam at an output of the first phase trimmer; using a second phase trimmer to apply a second phase shift to the vertical polarization portion of the polarized beam at anoutput of the first 3 dB coupler, the second phase shift being governed by ^^ − ^^;using a second 3 dB coupler to combine the horizontal polarization portion and the vertical polarization portion of the polarized beam at an output of the second phase trimmer; using a third phase trimmer to apply a third phase shift to the horizontal polarization portion of the polarized beam at an output of the second 3 dB coupler, the third phase shift being governed by −^^; and using a fourth phase trimmer to apply a fourth phase shift at an output of the polarization beam combiner, the fourth phase shift being governed by ^^.

[0013] In a further example, the integrated optics apparatus applies a controlled-phase operation, in which the first polarization state serves as a control qubit and the second polarization state as the target qubit. The PBS can be a first PBS that receives a first polarized beam having a first input classical polarization state, and the PBC can be a first PBC associated with the first polarized beam. The integrated optics apparatus can further include: a second PBS operable for splitting100510876.24Attorney Docket No.: 085067-827975 (UA24-089) the second polarized beam having a second input classical polarization state into a vertical polarization portion and a horizontal polarization portion; a Kerr nonlinearity device that applies a controlled-phase operation between the vertical polarization portion of the first polarized beam and the vertical polarization portion of the second polarized beam resulting in a phase-flipped vertical polarization portion of the second polarized beam; and a second PBC associated with the second polarized beam that receives the horizontal polarization portion of the second polarized beam and the vertical polarization portion of the second polarized beam at an output of the Kerr nonlinearity device. The roles of control and target qubit can be swapped.

[0014] In yet a further example where the optical device applies a CNOT operation to the first polarized beam and the second polarized beam, the optical device can be a first optical device configured as a Hadamard gate resulting in a first modified vertical polarization portion and a first modified horizontal polarization portion of the first polarized beam, the integrated optics apparatus further comprising: a second optical device configured as a Hadamard gate that receives the horizontal polarization portion of the first polarized beam from an output of the first optical device and receives the vertical polarization portion of the first polarized beam from an output of the Kerr nonlinearity device, the second optical device being applied before an input of the first PBC.

[0015] Further, in another example where the integrated optics apparatus applies a Bell states preparation operation to the first polarized beam and the second polarized beam, the integrated optics apparatus can further include: a third optical device configured as a Hadamard gate that receives the vertical polarization portion and the horizontal polarization portion of the second polarized beam from an output of the second PBC, where the Kerr nonlinearity device receives the vertical polarization portion of the first polarized beam from an output of the first optical device and receives the vertical polarization portion of the second polarized beam from an output of the third optical device.

[0016] In another aspect, a device that applies an arbitrary single-qubit logical operation includes: an orbital angular momentum (OAM) mode demultiplexer that separates OAM basis modes of a multimode beam; a plurality of electro-optical modulators in parallel with one another, each electro-optical modulator of the plurality of electro-optical modulators receiving a respective output of the OAM demultiplexer, the plurality of electro-optical modulators collectively introducing100510876.25Attorney Docket No.: 085067-827975 (UA24-089) complex weights to the OAM basis modes; and an OAM mode multiplexer that recombines the OAM basis modes of the beam.

[0017] In one example, the device can further include: a first Few- Mode-Fiber unit having an output in communication with the OAM mode demultiplexer, the OAM mode demultiplexer including a first taper core; and a second Few-Mode-Fiber unit having an input in communication with the OAM mode multiplexer, the OAM mode multiplexer including a second taper core.

[0018] In another example, the OAM mode demultiplexer can include a power splitter in communication with a first plurality of computer-generated holography (CGH) units, each CGH of the first plurality of CGH units corresponding with a respective electro-optical modulator of the plurality of electro-optical modulators; and the OAM mode multiplexer can include a power combiner in communication with a second plurality of CGH units that each receive an output of a respective electro-optical modulator of the plurality of electro-optical modulators.

[0019] In a further aspect, a device that applies a controlled-phase operation to a first beam and a second beam can include: a first computer-generated holography (CGH) unit that receives a first beam having a first state; a second CGH unit that receives a second beam having a second state; and a Kerr nonlinearity device that receives an output of the first CGH unit and an output of the second CGH unit, the Kerr nonlinearity device being in communication with the first CGH unit and the second CGH unit by a pair of Few-Mode-Fiber linkages.100510876.26Attorney Docket No.: 085067-827975 (UA24-089) BRIEF DESCRIPTION OF THE DRAWINGS

[0020] The present patent or application file contains at least one drawing executed in color. Copies of this patent or patent application publication with color drawing(s) will be provided by the Office upon request and payment of the necessary fee.

[0001] FIGS.1A and 1B are a pair of illustrations showing transformation of a polarization basis, where FIG.1A illustrates rotation of a state vector and FIG.1B illustrates rotation of a coordinate system;

[0002] FIG.2 is a simplified illustration showing a LiNbO3 technology- based circuit to entangle two coherent states (where PDC: parametric down conversion, PPLN: periodically poled lithium niobate crystal (or waveguide), BS: beam splitter, SPD: single-photon detector);

[0003] FIG.3 is a simplified illustration showing a Poincaré sphere representation of a classical polarization state;

[0004] FIG.4 is an illustration showing a 2x2 optical hybrid device suitable for coherent detection and single-qubit gates implementations on an input classical polarization state;

[0005] FIG.5 is a simplified diagram showing a first embodiment of an integrated optics apparatus including an optical device that applies an arbitrary single-qubit logic operation on an input classical polarization state based on the 2x2 optical hybrid device of FIG.4;

[0006] FIG.6 is a simplified diagram showing a second embodiment of an integrated optics apparatus including an optical device that applies an arbitrary single-qubit logic operation on an input classical polarization state based on a directional coupler and a Y-Z decomposition theorem;

[0007] FIG.7 is a simplified diagram showing a third embodiment of an integrated optics apparatus including an optical device that applies an arbitrary single-qubit logic operation on an input classical polarization state based on a directional coupler and a Barenco gate;

[0008] FIG.8 is a simplified diagram showing a fourth embodiment of an integrated optics apparatus including an optical device that applies an arbitrary single-qubit logic operation on an input classical polarization state based on a Mach- Zehnder interferometer and a Y-Z decomposition theorem;100510876.27Attorney Docket No.: 085067-827975 (UA24-089)

[0009] FIG.9 is a simplified diagram showing an integrated optics apparatus that applies a controlled-phase logic operation analog on an input classical polarization state using a Kerr nonlinearity device;

[0010] FIG.10 is a simplified diagram showing an integrated optics apparatus that applies a CNOT logic operation analog on an input classical polarization state using a Kerr nonlinearity device and two Hadamard gates constructed using the optical device of FIG.5, 6, 7, or 8;

[0011] FIG.11 is a simplified diagram showing an integrated optics apparatus that applies a Bell states preparation logic operation analog on an input classical polarization state using a Kerr nonlinearity device and three Hadamard gates constructed using the optical device of FIG.5, 6, 7, or 8;

[0012] FIG.12 is a simplified diagram showing an integrated optics apparatus that implements a quantum relay on an input classical polarization state using a CNOT gate of FIG.10, a controlled-phase gate of FIG.9, two Hadamard gates constructed using the optical device of FIG.5, 6, 7, or 8, a Pauli-X gate constructed using the optical device of FIG.5, 6, 7, or 8, and a Pauli-Z gate constructed using the optical device of FIG.5, 6, 7, or 8;

[0013] FIG.13 is a simplified diagram showing an experimental setup to demonstrate the controlled-phase logic operation on classical qubit analogs, where “PC" refers to “personal computer”;

[0014] FIG.14 is a graphical representation showing bit-error-rate measured on vertical polarizations of target coherent states (at 1551 nm);

[0015] FIG.15 is a graphical representation showing normalized amplitude for a BPSK sequence received on vertical polarization states at 1551nm, where a control qubit analog is operated at 1550nm;

[0016] FIG.16 is a graphical representation showing a portion of a target qubit analog waveform recorded in real-time (corresponding to 1551nm wavelength);

[0017] FIG.17 is a simplified diagram showing a fifth embodiment of an integrated optics apparatus including an Orbital Angular Momentum (OAM)-based optical device that applies an arbitrary single-qubit logic operation on an input classical coherent state, the optical device incorporating Few Mode Fibers (FMF);

[0018] FIG.18 is a simplified diagram showing a sixth embodiment of an integrated optics apparatus including an Orbital Angular Momentum (OAM)-based100510876.28Attorney Docket No.: 085067-827975 (UA24-089) optical device that applies an arbitrary single-qubit logic operation on an input classical coherent state, the optical device incorporating Computer Generated Holography (CGH); and

[0019] FIG.19 is a simplified diagram showing an integrated optics apparatus that applies a controlled-phase logic operation on two input classical coherent states using a Kerr nonlinearity device, CGH units, and FMF.

[0020] Corresponding reference characters indicate corresponding elements among the view of the drawings. The headings used in the figures do not limit the scope of the claims.100510876.29Attorney Docket No.: 085067-827975 (UA24-089) DETAILED DESCRIPTION

[0021] Recent advancements in quantum information processing demonstrate that macroscopic states, such as coherent states, can be entangled by the delocalized photon addition. Further, phase bits gates can be implemented by employing topological acoustics (TA) principles to implement the TA-based quantum computing analog. This motivates implementation of universal quantum gates in integrated optics using optical hybrid, directional coupler, Mach-Zehnder interferometer, and periodically poled lithium niobate (PPLN) waveguides, but in a different context. The present disclosure outlines systems and methods for implementing a universal set of quantum gates classical analogs in integrated optics by employing classical polarization states derived from coherent states. A main problem for integrated optics implementation on a single-photon level was to implement a controlled-phase gate because existing optical nonlinear devices were incapable of introducing a ^^ rad phase shift on a single photon level through the Kerr effect. However, this is not a problem at all when the classical polarization states are used instead. The present disclosure also describes how to implement quantum qudit analogs based on orbital angular momentum (OAM) states and corresponding qudit gates. To highlight the importance of the contributions outlined herein, experimental demonstration is also provided for a controlled-phase gate analog operation between classical coherent states. 1. Introduction

[0022] Quantum information processing (QIP) is a very active research area with large number of applications including quantum computing, quantum communications, quantum networks, and quantum sensing, to mention few. Typically, the following three features of QIP are considered different from classical computing: linear superposition, entanglement, and quantum parallelism. The linear superposition indicates that the qubit represents a linear combination of basis states, while in classical computing only two discrete bits (0 and 1) are used. On the other hand, superposition principle is a basic principle applicable to any linear system, not necessarily quantum. Even though state-of-the-art digital computers indeed use just 0 and 1 bits to store information to compute, the future classical computers do not need to be binary only. The entangled states are particular quantum states that cannot be decomposed into independent quantum states.100510876.210Attorney Docket No.: 085067-827975 (UA24-089)

[0023] However, recent advancements have experimentally demonstrated that macroscopic states, including coherent states, can be entangled by the delocalized photon addition. Quantum parallelism is the capability to perform a large number of operations in parallel. Recently, it has been shown that the quantum algorithms can be run on topological acoustic (TA)-based quantum analogs, which are essentially classical computers and that the nonlinearity is a key enabler for so-called quantum parallelism. These recent findings have motivated revisiting previous QIP proposals for integrated optics implementations, but now in the context of optical quantum computing analogs.

[0024] The present disclosure outlines how to implement optical quantum computing classical analogs in integrated optics by employing coherent states as qubit analogs. The disclosure experimentally demonstrates the controlled- phase operation between classical coherent states, and a also describes qubit classical analogs and corresponding gates based on the orbital angular momentum (OAM) states.

[0025] The organization of this disclosure is summarized here. The classical polarization states and the action on them by the polarizing elements, characterized by using the Jones formalism, is described in Sec. II. Sec. II also explains how to entangle coherent states. Sec. III describes various embodiments of single-qubit analogs’ implementations in integrated optics by employing an optical hybrid, a directional coupler, or a Mach-Zehnder interferometer. Sec. IV describes how to implement two-qubit analogs in integrated optics, namely controlled-phase and CNOT gate analogs. Sec. IV also describes how to implement the generic Bell state preparation circuit and quantum relay analogs. To illustrate the high-potential of the proposed integrated optics-based quantum computing analog concepts, experimental demonstration of the controlled-phase operation is performed. Sec. V describes the quantum qudit equivalent based on OAM and introduce single-qudit and two-qudit gates analogs’ implementations in integrated optics. II. Classical Polarization States and Entanglement of Classical States

[0026] The Jones representation of the polarized light is given by: ^^^^^^0^^ ^^^^^^(1)100510876.211Attorney Docket No.: 085067-827975 (UA24-089) where ^^^^0(^^^^0) is the amplitude in the horizontal (vertical) polarization state and^^^^(^^^^) is the corresponding phase. The intensity ^^ can be determined by:^^ = ^^ϯ^^ = [^^^^^^0^∗∗^0^^^^0 ] [^^^^0] = |^^^^0|2 + |^^^^0|2, (2)when normalizedpolarization |^^^ = [^^^^^^^^] = ^^^^ [1 ^ 0]+ ^^^^ [0 ^1]= ^^^^|^^^ + ^^^^|^^^, (3)wherein ^^ = ^^and polarization states, respectively (that are clearly orthogonal). The condition is obviously satisfied: |2 + |^^ |2 |^^2^^|+ |^^^^|2^^ =^^0|^^0 = 1. (4)^^ In Dirac notation, withvector (“bra”) ^^^| is associated as follows: ^^^| = [^^ ∗^∗^^^^^]. (5)The dot (scalar) product of ket^∅|^^^ = ^^^∗^^^^^ + ^^^∗^^^^^ = ^^^|^^^. (6)Using this notation, therespectively as |^^^ and |^^^, can be represented by: |^^^ =1 1 1 ^^ [] = |^^^ +|^^^, (7)which are clearly

[0027] The action of the polarizing element on a polarization state can be described by the following Jones operator (matrix): ^^^^^^^^^^^^^^^^^^^^^ = ^^ ^^^ = [^^] ^. (8)The linear polarizer, characterized by the absorption coefficients along H- and V-axis, denoted as ^^^^and ^^^^, respectively as follows:100510876.212Attorney Docket No.: 085067-827975 (UA24-089) ^^polarizer = [^^^^0 0^^^^] ; 0 ≤ ^^^^ ≤ 1, ^^ ∈ {^^,^^}. (9)The wave plateselements that along V-axis, wherein the Jones matrix of this action is represented by: ^^^^^^^^ / 20^^^^ / 2wave plate = [ 1 0−^^^^ / ] = ^^ [ −^^^^]. (10)0 ^^ 2 0 ^^Two popularhalf-wave plate (HWP), for which ^^ = ^^, while the corresponding Jones matricesdescribing their actions are given respectively by: ^^ 1 0QWP = [ 0 −^^] , ^^HWP = [1 00 −1]. (11)To illustrate, when the|±45°^ = [1 ±1]^^ / √2, at the output it gets converted to the left- / right-circularpolarization state as follows: ^^QWP| ± 45°^ =1 [1 00 −^^] [1] =1 1 [ ±^^]. (12)

[0028] is illustrated in FIGS.1A and 1B, wherein the polarization state |^^^ is rotated for ^^ in a clockwise direction, which is equivalent to the rotation of coordinate system in the opposite (counterclockwise) direction for the same angle (^^), as shown in FIG.1B. The polarization state |Ψ^ can be represented in original {|^^^,^^^} basis by: |^^^ = |^^^^^^|^^^ + |^^^^^^|^^^. (13)From Eq. (13), notice that the following is valid: |^^^^^^| + |^^^^^^| = ^^2, (14)where ^^ is 2 × 2 identitythe quantum mechanics. However, the polarization states are classical, wherein each polarization state is composed of large number of photons having the same polarization. By multiplying Eq. (13) by ^^^′| and ^^^′ from the left side: ^^′^ =^^^′|^^^= ^^^′|^^^ ^^^′|^^^ ^^ ^′ ′^ |^^(15) ′−cos^100510876.213Attorney Docket No.: 085067-827975 (UA24-089) =^^(^^)|^^^,Where ^^(^^) is the Jones rotation simultaneously the basistransformation matrix.

[0029] By using these three polarizing elements, an arbitrary elliptic polarization state can be obtained starting from the coherent state. Therefore, several concepts relevant in quantum mechanics have already been introduced such as the superposition principle (13), the completeness relationship (14), and the change of basis (15), while manipulating the classical polarization states.

[0030] Now, the present disclosure outlines how to entangle classicalcoherent states. The coherent state |^^^, where ^^ is a complex number (that is ^^ =|^^|exp(^^^^)), can be represented in terms of number (Fock) states |^^^ as follows: +∞ |^^^ = exp[−|^^|2 / 2]∑(^^!)−1 / 2 ^^^^|^^^,^^ = |^^|^^^^^^ (16)One of thevector of the annihilation operator ^̂^ (decreasing the degree of excitation for 1), that is: ^̂^|^^^ = ^^|^^^ (17)The coherent state can also be represented as a displaced vacuum state by: ^^ (^^)|0^ = |^^^,^^ (^^) = exp(^^^̂^† − ^^∗^̂^), (18)where ^^(^^)is thebe entangled with the help of two periodically polled lithium niobate (PPLN) crystals or waveguides serving the role of single-photon addition modules. The corresponding circuit to entangle two coherent states|α^and|β^is provided in FIG. 2, where ^̂^†denotes the creation operator (increasing the degree of excitation for 1). laser is used for both PPLN modules, whose power is split by the power splitter. Each pump photon in either PPLN modules generates signal and idler photons, which are entangled. The idler modes are interacted on the beam splitter. If the input states where CV states, after the beam splitter it will not be possible to identify if the click on single photon detector (SPD1) / SPD2 originating from upper or lower PPLN module, thus making the input CV states entangled: |^^out^ = ^^−1 / 2(^̂^†|^^^1|^^^2 + ^^^^^^|^^^ †1^^ |^^^2), (19)100510876.214Attorney Docket No.: 085067-827975 (UA24-089) where ^^ is the normalization factor. However, this is not always true when the input states are coherent states. By using the symmetry properties of the displacementoperator we arrive to the following relationship ^̂^†^^ (^^) = ^^ (^^)(^̂^† + ^^⋆), whichallows re-writing Eq. (19) as follows: 1 |^^out^ = ^^− 2^^1(^^)^^2(^^)(|1^1|0^2+^^^^^^|0^1|1^2) −1 / 2 ∗ ^^(20) +^^ ^^ 1 + ^^ ^^wherein theseparable on ^^ = ^^rad, the separable term becomes zero, and leading to entanglement of two coherent states: |^^out^^^−0 = ^^−1 / 2^^ 1(^^)^^ 2(^^)(|1^1|0^2 + |0^1|1^2). (21)

[0031] principle and entanglement are also applicable to the classical states and are not exclusive to quantum mechanics, which is contrary to the common belief.

[0032] Before concluding this section, it is convenient to introduce the Poincaré sphere representation of the classical polarization state given by Eq. (3), which is in quantum mechanics known as the Bloch sphere. The coefficients ^^Hand ^^Vare complex numbers, and therefore can be represented as ^^^^=|^^^^|exp(^^^^^^),^^^^ = |^^^^|exp(^^^^^^), so Eq. (3) can be rewritten as follows:^^ =^^^^^^^^ + ^^ ^^(^^^^−^^^^) = ^^^^−^^^^ (22)and by2), |^^^^| = sin(^^ / 2) and ignoring the global phase shift the following representationcan be derived: co^^ ^^ ^^s ()where ^^ ∈,pole correspond to the computational base {|^^^, |^^^}. The diagonal basis is definedby two unit vectors along x-axis {|+^, |−^}, while the circular basis by two unit vectorsalong y-axis {|^^, |^^^}.100510876.215Attorney Docket No.: 085067-827975 (UA24-089) III. Single-Qubit Gates Analogs Implemented in Integrated Optics

[0033] A 2x2 optical hybrid has been previously considered in context of coherent detection to mix the received signal and local oscillator (LO) laser signal as well as in the context of single-qubit gates implementation in integrated optics. Here, the present disclosure outlines a particular version of 2x2 optical hybrid with four phase trimmers, provided in FIG.4, suitable for manipulation of the classical polarization states. This device includes two input ports, two input Y-junctions, two output Y-junctions, two output ports, and four phase trimmers. Based on FIG.4, one can conclude that the output electric fields ^^^^,1and ^^^^,2are related to the input electric fields ^^^^,1and ^^^^,2as follows: ^^ − ^^^^ ^^^^The corresponding ^^ =^^^^,1 1 − ^^ ^^^^1 ^^^^3^^ ^^= √ ^^ √^^^^ ^^,1^^ [ ] [ ^^^^2 ^^^^^^^^4] [ ]= ^^^^^^, (25)where ^^ is1− ^^^^^^^^1 ^ ^^^^3^^ = [√ √ ^^^^^^^ ], (26)^^^^and ^^ is the power

[0034] For instance, by setting ^^1 = ^^2 = ^^3 = 0,^^4 = ^^, ^^ = 1 / 2 the ^^-matrix becomes the Hadamard matrix: ^^ =1 [1 1]. (27) By placing the 2x2 opticalpolarization beam combiner (PBC), the circuit shown in FIG.5 can be constructed, which is suitable to perform arbitrary single-qubit analog operation on a given classical polarization state.

[0035] By parametrizing the power splitting ratio by √^^ = sin(^^ / 2) andsetting the phase shifts as follows: ^^1 = −^^ / 2 − ^^ / 2, ^^2 = ^^ / 2 − ^^ / 2^^3 = −^^ / 2 + ^^ / 2 + ^^,^^4, = ^^ / 2 + ^^ / 2(28)100510876.216Attorney Docket No.: 085067-827975 (UA24-089) the unitary matrix connecting the output in input classical polarization states is given by: ^^ =^^(^^ / 2−^^ / 2) ^^(^^ / 2+^^ / 2)[cos(^^ / 2)^^ −sin(^^ / 2)^^ ^^(], (29)sin(^^ / 2)^^^^ / 2−^^ / 2)cos(^^ / 2)^^^^(^^ / 2+^^ / 2)which is knownrepresentation, can =^^ / 2,^^ = 0, ^^ = ^^ / 2, and ^^ = ^^ the ^^-gate becomes the Hadamard gate as in (27).

[0036] The Pauli X-, Y-, and Z-gates can be implemented as follows.By setting ^^ = ^^ / 2,^^ = −^^, ^^ = ^^, and ^^ = 0 the ^^-gate becomes the Pauli-X gate:^^ = [0 11 0]. (30.1)By setting ^^ = ^^ / 2,^^ = 0, ^^ = ^^,= 0 the ^^-gate becomes the Pauli-Y gate:^^ = [0 −^^^^ 0]. (30.2)By setting ^^ = ^^ / 2,^^ = ^^, ^^ = 2^^, and ^^ = 0 the ^^-gate becomes the Pauli-Z gate:^^ = [1 00 −1]. (30.3)Further, by setting ^^ = ^^ / 8,^^^^ / 8-gate: ^^ = [1 00 ^^^^^^ / 4]. (31)Finally, by setting ^^ =^^ ^^4,^^ =2 gate: ^^ = [1 00 ^^]. (32) The scattering matrix of directional coupler, used in FIG.6, is given by: ^^ = [ cos(^^^^) ^^ sin(^^^^)^^ ^^^^ ^^^^ ] (33)where ^^ is the couplingthe coupling phase shift to be ^^^^ = ^^ / 2, and by selecting the phase shifts on threephase trimers as indicated in FIG.5, the overall scattering matrix is identical to that given by Eq. (29). Therefore, this device represents an alternative single-qubit gate analog implementation. Another version of the single-qubit gate analog is provided in100510876.217Attorney Docket No.: 085067-827975 (UA24-089) FIG.6 and it is derived from the Barenco theorem, claiming the following two-qubit gate is universal: ^^(^^,^^,^^) = [^^2] ^^^^By selecting theon two phase trimers as indicated in FIG.7, the overall scattering matrix is identical to that given by ^^^^-matrix in Eq. (34).

[0037] Finally, by using the Mach-Zehnder interferometer-based circuit provided in FIG.8, and by selecting the phase shifts as specified in FIG.8, the overall scattering matrix given by Eq. (29) can be obtained. IV. Controlled-Phase and CNOT Gates Analogs Implemented In Integrated Optics

[0038] The Barenco gate, introduced by the Eq. (34) represents the universal quantum gate. The three-qubit Deutsch gate also represents the universal quantum gate. Another popular universal set of gates is {^^,^^,^^,^^^^^^^^}. The ^^,^^,and ^^ gates are described in the previous section, what remains is to describe howto implement the CNOT-gate. Given that^^2 − ^^, and that CNOT can be expressed interms of controlled-phase gate ^^(^^) = [^^20 ^^] by: ^^^^^^^^|^^^|^^^ =|^^^^(^^^^^^)^^|^^^^^^^^ ^^the set ofphotonic implementation of the ^^(^^) gate in integrated optics is illustrated in FIG.9, where the Kerr nonlinear effect is employed to introduce the nonlinear phase shift of ^^ rad only between vertically polarized states and the action of the gate can be described by: ^^(^^)|^^^|^^^ = |^^^|^^^,^^(^^)|^^^|^^^ = |^^^|^^^^^(^^)|^^^|^^^ = |^^^|^^^,^^(^^)|^^^|^^^ = |^^^|^^^ (36)Namely, for sufficiently high light intensity the refractive index of nonlinear optical medium, such as optical fiber, is not only a function of frequency ^^, but also afunction of intensity of the light beam propagating ^^ through the nonlinear medium,100510876.218Attorney Docket No.: 085067-827975 (UA24-089)that is, ^^(^^, ^^) = ^^(^^) + ^^2^^, where ^^2 is the Kerr coefficient. The cross-phasemodulation (XPM) will introduce the nonlinear phase shift to the target polarization state, denoted by ΔΦ^^, as follows: 1− ^^−^^^^ΔΦ = 2Γ^2^^^^2^^ ^^^^^, Γ =, ^^^^^^^^^^(37) where Γ is thefiber, ^^ is the attenuation coefficient, and ^^c is the power in the controlpolarization state. By properly adjusting the power ^^c, a ^^ phase-shift can be introduced. Given that the typical values of the nonlinear coefficient for standard SMF is small and range from 0.92.75 W-1km-1, the highly nonlinear fiber (HNLF) can be used for this purpose. Introducing the ^^ rad XPM phase shift is extremely difficult on a single-photon level; however, it straightforward to do on a classical level with the help of HNLFs. Unfortunately, the HNLF is not compatible with integrated optics. Alternatively, the HNLF can be replaced by the type-0 PPLN waveguide, where the nonlinear conversion efficiency is very efficient for the vertical polarization states, and the overall circuit is suitable for implementation in the lithium niobate (LN) technology. Such PPLN waveguides are commercially available, and have been routinely used in recent experiments to generate bright entangled photons for distribution of entanglement over atmospheric turbulence channels and in entanglement assisted communication demonstrations over turbulent free-space optical links. Based on Eq. (35), to implement the CNOT gate analog, two Hadamard gates on target polarization state can be inserted just before and after C(Z)- operation, which is illustrated in FIG.10.

[0039] Now, an alternative way to entangle two classical polarization states can be derived, with the help of the Bell state preparation circuit, shown in FIG.11. After applying the Hadamard gates on both control and target qubits, the CNOT-gate followed by the Hadamard gate can be applied on target qubit so that the output polarization state becomes: ^^^^^^^^ + ^^^^^^^^100510876.219Attorney Docket No.: 085067-827975 (UA24-089) Compared to the implementation from FIG.2, only one PPLN waveguide is needed.To illustrate, by setting ^^^^ = 1, ^^^^ = 0 and ^^^^ = 0, ^^^^ = 1, we obtain the following Bellstate: |^^01^ = 2−1 / 2[^^ ^^ ^^ ^^] = 2−1 / 2(|^^^|^^^ + |^^^|^^^|). (39)By using the relay analogCNOT and C(Z) gates are applied between the highlighted (in green) vertical polarization basis states. Measurements are performed at intermediate nodes with the help of two avalanche photodiodes (APDs) and when the vertical polarization state is detected, Pauli X- and Z-gates can be conditionally executed on the bottom polarization state. Clearly, the complexity of this approach is high. An alternative strategy is to employ entanglement swapping and teleportation concepts by photon addition.

[0040] To illustrate high-potential of the concepts outlined herein, a controlled-phase quantum qubit analog demonstration was performed with an experimental setup shown in FIG.13.

[0041] The laser operated at 1550 nm generates the classical states that serve the role of the control qubit, while the bottom laser generates the states at 1551 nm that serve the role of the target qubit. To demonstrate the controlled-phase gate analog operation from FIG.9, we need to demonstrate that control qubit coherent states can introduce the phase shift of ^^ rad to the vertical polarization states of the target qubit analog. To do so we impose the BPSK modulation at 10 Gb / s on the control coherent states, while the target coherent states are unmodulated. We then combine the control and target coherent states by the beam combiner and conduct the cross-phase modulation (XPM) interaction of control and target coherent states by the PPLN waveguide. At the output of PPLN waveguide we separate horizontal (H) and vertical (V) polarization states by the polarization beam splitter (PBS) and perform the coherent balanced detection on vertical photons related to the target coherent states at 1551 nm. To do so we mix the local oscillator (LO) laser signal at 1551 nm with the vertical photons on optical hybrid (not shown in Figure) followed by the homodyne balanced detector. The detected signal is sampled by the real-time scope and sampled RF signal waveform is transferred to the personal computer (PC), where we re-sample the waveform and calculate bit log-100510876.220Attorney Docket No.: 085067-827975 (UA24-089) likelihood ratios (LLRs), which are used to make decisions and determine the bit- error rates (BERs). The corresponding BER related to the target coherent state are provided in FIG.14. The input power to the power combiner, the control power ^^^^, was varied, while the input power of the target coherent states was used as the parameter. For the control qubit analog powers higher than 1.4 dBm, the measured BER was zero, indicating that the BPSK get transferred completely from the control coherent states to the target coherent states, which demonstrates that the circuit from Fig.9 indeed operates as the controlled-phase analog.

[0042] For illustrative purposes, FIG.15 shows the portion of the BPSK sequence received on the vertical polarization states at 1551 nm. The ^^ rad voltage level is clearly indicated. Therefore, the XPM made it possible to introduce the ^^ rad phase shift on vertical polarization states at 1551 nm, while the control qubit analog was operated at 1550 nm thus demonstrating the controlled-phase operation. Finally, in FIG.16, the portion of the waveform record on the real-time scope corresponding to the 1551 nm target qubit analog is provided, confirming that the sequence transferred from the control to target qubit analog is error free. V. Optical Orbital Angular Momentum Gates Analogs

[0043] Previous work has introduced the photon angular momentum states by combining polarization and orbital angular momentum (OAM) states, and proposed corresponding gates suitable for universal quantum computing and quantum communication applications. In this section, the corresponding classical quantum gates’ analogs are outlined. The OAM modes have been intensively studied for various classical communication applications, including OAM multiplexing and OAM modulations. The OAM degree-of-freedom is associated with the azimuthal phase dependence of the complex electric field. Among various vortex optical beams carrying the OAM, the Laguerre-Gauss (LG) vortex beams / modes are very popular and the electric distribution of an LG beam traveling along the z-axiscan be expressed in cylindrical coordinates (^^,^^, ^^) (^^: the radial distance frompropagation axis, ^^: the azimuthal angle, ^^: the propagation distance) as follows: |^^2^^! 1|2^^2100510876.221Attorney Docket No.: 085067-827975 (UA24-089) where ^^(^^) = ^^0√1 + (^^ / ^^^^)2 (^^0 is the zero order Gaussian radius at the waist),^^^^ = ^^^^2 (with ^^ being the wavelength), ^^ = 2^^ / ^^ is theassociated Laguerre polynomial, with ^^ and ^^ representing the radial and angular mode numbers, respectively. It can be seen from Eqn. (40) that the ^^-th mode of the LG beam has the azimuthal angular dependence given by term exp(−^^^^^^), where ^^ is the azimuthal mode index. In free space, for ^^ = 0, ^^(^^,^^, ^^) becomes a zero-order Gaussian beam known as TEM00 mode. For^^ = 0, ^^^^^^(∙) for all ^^’s, so that the intensity of an LG mode is a ring of radiusproportional to |^^|1 / 2. It can be shown that for fixed ^^, the following principle of orthogonality is satisfied: 〈^^^^,^^|^^^^,^^〉 = ∫^^ ∗^^,^^ (^^,^^, ^^)^^^^,^^(^^,^^, ^^)^^^^^^^^^^

[0044] are mutually orthogonal, and as such they can be used as the basis functions for either OAM modulation or OAM multiplexing. The OAM modes can easily be generated by the computer-generated holograms (CGHs). The Bessel modes also belong to the class of OAM modes, and they are obtained as solutions of the wave equation in a step-index multi-mode fiber (MMF) of core radius ^^, with corresponding electric field z-component in cylindrical coordinates (^^,^^, ^^) being:^^ (^^,^^, ^^) = ^^^^ (^^ ^^)^^−^^^^^^ ^^^^^^^^ ^^ ^^ ^^ , ^^ ≤ ^^ (42)where ^^ = √^^2 2^^^^the free-and ^^ the propagation constant. ^^^^(∙) denotes the Bessel function of the ^^-th order. The ideal Bessel beams satisfy the relationship |^^(^^,^^, ^^)|2 = |^^(^^,^^)|2 and are therefore diffraction free and as such representexcellent candidates for free-space optical (FSO) applications. It the practice, it is not trivial to generate the ideal Bessel beams and instead various approximations need to be used.

[0045] Given that the OAM modes are orthogonal to each other, as shown by Eq. (41), they can be used as the basis functions {|^^^}(^^ =−^^, … , −1,0,1, … , ^^). The arbitrary spatial mode can be decomposed in terms of thebasis OAM modes by: 100510876.222Attorney Docket No.: 085067-827975 (UA24-089) ^^ |^^^ = ∑ ^^1|^^^ , (43)assuming that the number of

[0046] With this space dimensional, the OAM single-qudit and two-qudit analogs are described here. The MMFs can be used for this purpose, given Eq. (41); however, the MMFs support too many modes to be of practical interest. Instead, the few-mode fibers (FMFs), supporting the limited number of modes should be used instead. The single-qudit analog is provided in FIG.17. The output of the FMF will include all basis OAM modes with the same weight. In OAM modes-demultiplexer, the OAM basis modes are separated, the desired complex weights for each OAM basis mode are introduced by the electro-optical (E / O) modulators, and the OAM modes-multiplexer recombines them before coupling into the FMF.

[0047] A computer-generated holography (CGH)-based single-qudit analog is provided in FIG.18. The OAM modes with corresponding complex coefficients are first separated with the help of power splitter and complex-conjugate CGHs. Each coordinate gets modified by the E / O modulators to perform a desiredqudit operation. The CGHs impose the ^^-th OAM mode (^^ = −^^, … ,0, … , ^^)corresponding to the ^^-th coordinate, and such weighted OAM modes are combined to generate the output mode. A generalized controlled-phase gate analog is provided in FIG.19. The CGHs impose desired spatial modes. Highly nonlinear FMF is used to interact the control |^^^ and target |^^^ qudit. Assuming that (2L+1) is a prime number, the generalized Z-gate is defined by: ^^(^^)^^^ = ^^^^2^^2^^+1^^|^^^, (44)and by adjusting the powerintroduced on the target qubit with the help of XPM. Instead of the HN-FMF, the PPLN waveguide / crustal can be used but now supporting a desired number of higher order spatial modes. VI. Concluding Remarks

[0048] Inspired by recent findings that the classical states can be entangled, the CNOT gate can be implemented using the classical acoustic qubit- analog, and that quantum parallelism can be achieved on classical quantum100510876.223Attorney Docket No.: 085067-827975 (UA24-089) computing acoustic analog, the present disclosure outlined how to implement quantum information processing analogs in integrated optics based on classical polarization and OAM states.

[0049] By using the Jones formalism it is shown that arbitrary classical polarization state can be represented as the superposition of horizontal and vertical basis polarization states. The disclosure also shows that the quantum mechanics relevant concepts such as the superposition principle, the completeness relationship, and the change of basis are also applicable to the classical polarization states. To obtain any classical polarization state only three polarizing elements are necessary: polarizer, wave-plate, and rotator.

[0050] The present disclosure has also shown that any two classical coherent states can be entangled with the help of single photon addition module, based on two PPLN waveguides.

[0051] The present disclosure has also further described how to implement an arbitrary single-qubit gate analog in integrated optics using any of the following three devices: 2x2 optical hybrid with four phase trimmers, directional coupler with three phase trimmers, and Mach-Zehnder interferometer with four phase trimmers. The disclosure has also described how to implement controlled- phase and CNOT gates analogs operating on classical polarization states in the lithium niobate technology. This completes the implementation of universal quantum gates analogs for classical polarization states. Further, the present disclosure has also described how to implement Bell states preparation circuit and quantum relay in the same technology. The controlled-phase operation between the classical coherent states has also been experimentally demonstrated.

[0052] The focus has been then moved to the spatial modes, and the present disclosure has shown that an arbitrary classical spatial mode can be decomposed in terms of basis OAM modes. Therefore, the spatial modes can be used as the classical qudit analogs. Further, the disclosure has outlined how to implement arbitrary single-qudit gate analog in either FMF or CGH technologies, and has also described how to implement the generalized control-phase two-qudit classical analog.

[0053] The concepts outlined herein, therefore, represent a step forward in creating a decoherence-free optical quantum information processing and computing analog, which does not rely on the fragile quantum states but rather100510876.224Attorney Docket No.: 085067-827975 (UA24-089) robust classical states. Moreover, the classical states can be measured without causing the state collapse, which is unavoidable for quantum states. Further, these classical quantum computing analogs do not require any error correction, let alone the quantum error correction. Finally, fault tolerance is not needed in classical quantum computing analogs.

[0054] The functions performed in the processes and methods may be implemented in differing order. Furthermore, the outlined steps and operations are provided as examples, and some of the steps and operations may be optional, combined into fewer steps and operations, or expanded into additional steps and operations without detracting from the essence of the disclosed embodiments.

[0055] It should be understood from the foregoing that, while particular embodiments have been illustrated and described, various modifications can be made thereto without departing from the spirit and scope of the invention as will be apparent to those skilled in the art. Such changes and modifications are within the scope and teachings of this invention as defined in the claims appended hereto.100510876.225

Claims

Attorney Docket No.: 085067-827975 (UA24-089) CLAIMS What is claimed is:

1. An integrated optics apparatus, comprising: a polarization beam splitter (PBS) operable for splitting a polarized beam having an input classical polarization state into a vertical polarization portion and a horizontal polarization portion; an optical device configured to perform a single-qubit quantum analog operation on the input classical polarization state based on applying a scattering matrix, representing a single-qubit analog operation, to the vertical polarization portion and the horizontal polarization portion of the polarized beam to thereby yield a modified vertical polarization portion and a modified horizontal polarization portion, wherein the optical device includes two or more phase trimmers each having a selectable phase shift, and wherein different values of the selectable phase shifts correspond to different single-qubit quantum analog operations; and a polarization beam combiner (PBC) operable for combining the modified vertical polarization portion and the modified horizontal polarization portion into an output polarized beam having an output classical polarization state modified from the input classical polarization state according to the scattering matrix.

2. The integrated optics apparatus of claim 1, wherein: the vertical polarization portion of the polarized beam comprises vertical polarization state photons; the horizontal polarization portion of the polarized beam comprises horizontal polarization state photons; and the input classical polarization state is a superposition of the vertical and horizontal polarization states.

3. The integrated optics apparatus of claim 1, the optical device being a 2x2 optical hybrid coupler including four phase trimmers and operable for:100510876.226Attorney Docket No.: 085067-827975 (UA24-089) Splitting the first classical polarization state, using a first input Y-junction of the 2x2 optical hybrid coupler, the vertical polarization portion of the polarized beam into a first split vertical component and a second split vertical component based on a power splitting ratio; Splitting the second polarization state, using a second input Y-junction of the 2x2 optical hybrid coupler, the horizontal polarization portion of the polarized beam into a first split horizontal component and a second split horizontal component based on the power splitting ratio; and using each respective phase trimmer of the four phase trimmers to apply a corresponding selected phase shift to a particular one of the first or second split horizontal or split vertical components, wherein each respective phase trimmer receives a different split component as input.

4. The integrated optics apparatus of claim 3, wherein using each respective phase trimmer to apply a corresponding selected phase shift comprises: using a first phase trimmer to apply a corresponding first phase shift to the second split vertical component to obtain a phase-adjusted second split vertical component; using a second phase trimmer to apply a corresponding second phase shift to the first split vertical component to obtain a phase-adjusted first split vertical component; using a third phase trimmer to apply a corresponding third phase shift to the first split horizontal component to obtain a phase-adjusted first split horizontal component; using a fourth phase trimmer to apply a corresponding fourth phase shift to the second split horizontal component to obtain a phase-adjusted second split horizontal component; and combining the phase-adjusted second split vertical component and the phase-adjusted first split horizontal component into the modified vertical polarization portion of the output polarized beam and combining the phase-adjusted first split vertical component and the phase-adjusted second split horizontal component into the modified horizontal polarization portion of the output polarized beam;100510876.227Attorney Docket No.: 085067-827975 (UA24-089) the power splitting ratio, the first phase shift, the second phase shift, the third phase shift, and the fourth phase shift each being governed by elements of the scattering matrix, the elements of the scattering matrix being dependent upon on a target gate implementation of the integrated optics apparatus.

5. The integrated optics apparatus of claim 1, the optical device being a directional coupler-based single-qubit gate analog based on Y-Z decomposition and operable for: using a first phase trimmer to apply a first phase shift to the horizontal polarization portion of the polarized beam, the first phase shift being governed by ^^ − ^^ / 2 − ^^ / 2;using a second phase trimmer to apply a second phase shift to the vertical polarization portion of the polarized beam, the second phase shift being governed by ^^ + ^^ / 2;using a directional coupler to apply a coupling phase shift to the horizontal polarization portion and the vertical polarization portion of the polarized beam following phase shift, the coupling phase shift governed by ^^^^ =^^ / 2; and using a third phase trimmer to apply a third phase shift to the horizontal polarization portion of the polarized beam following application of the coupling phase shift, the third phase shift being governed by −^^ + ^^ / 2.

6. The integrated optics apparatus of claim 1, the optical device being a directional coupler-based single-qubit gate analog based on a Barenco theorem and operable for: using a first phase trimmer to apply a first phase shift to the horizontal polarization portion of the polarized beam, the first phase shift being governed by ^^; using a directional coupler to apply a coupling phase shift to the horizontal polarization portion and the vertical polarization portion of the polarized beam following phase shift, the coupling phase shift being governed by ^^^^ = ^^;100510876.228Attorney Docket No.: 085067-827975 (UA24-089) using a second phase trimmer to apply a second phase shift to the horizontal polarization portion of the polarized beam at an output of the directional coupler, the second phase shift being governed by −(^^ +^^); and using a third phase trimmer to apply a third phase shift at an output of the polarization beam combiner, the third phase shift being governed by ^^.

7. The integrated optics apparatus of claim 1, the optical device being a Mach- Zehnder interferometer-based single-qubit gate analog based on Y-Z decomposition and operable for: using a first phase trimmer to apply a first phase shift to the horizontal polarization portion of the polarized beam, the first phase shift being governed by −^^ − ^^;using a first 3 dB coupler to combine the horizontal polarization portion and the vertical polarization portion of the polarized beam at an output of the first phase trimmer; using a second phase trimmer to apply a second phase shift to the vertical polarization portion of the polarized beam at an output of the first 3 dB coupler, the second phase shift being governed by ^^ − ^^;using a second 3 dB coupler to combine the horizontal polarization portion and the vertical polarization portion of the polarized beam at an output of the second phase trimmer; using a third phase trimmer to apply a third phase shift to the horizontal polarization portion of the polarized beam at an output of the second 3 dB coupler, the third phase shift being governed by −^^; and using a fourth phase trimmer to apply a fourth phase shift at an output of the polarization beam combiner, the fourth phase shift being governed by ^^.

8. The integrated optics apparatus of claim 1, the PBS being a first PBS that receives a first polarized beam having a first input classical polarization state, serving as a control qubit, and the PBC being a first PBC associated with the first polarized beam, the integrated optics apparatus further comprising:100510876.229Attorney Docket No.: 085067-827975 (UA24-089) a second PBS operable for splitting a second polarized beam having a second input classical polarization state, serving as a target qubit, into a vertical polarization portion and a horizontal polarization portion; a Kerr nonlinearity device that applies a controlled-phase operation to the vertical polarization portion of the second polarized beam and the vertical polarization portion of the first polarized beam serving as the control qubit, resulting in a phase-flipped vertical polarization portion of the first polarized beam or the second polarized beam; and a second PBC associated with the second polarized beam that receives the horizontal polarization portion of the second polarized beam and the vertical polarization portion of the second polarized beam at an output of the Kerr nonlinearity device; wherein the integrated optics apparatus applies a controlled-phase operation between the first and second polarization states, wherein the first polarization state serves as control qubit and the second polarization state as a target qubit.

9. The integrated optics apparatus of claim 8, configured to apply an identity gate applied on the top polarization states, serving the role of the control qubit, and the Hadamard gate on the bottom polarization state, serving the role of the target qubit, the integrated optics apparatus further comprising: the Kerr nonlinearity device, operating on the vertical polarizations of the top and bottom polarization states to apply a controlled-phase operation between the top and bottom polarization states, wherein the top polarization state serves as control qubit and the bottom polarization state as a target qubit; the Hadamard gate being applied on modified bottom polarization state.

10. The integrated optics apparatus of claim 9, wherein the optical device applies a CNOT operation between the first polarized beam / polarization state (serving the role of control qubit) and the second polarized beam / polarization state (serving the role of a target qubit).

11. The integrated optics apparatus of claim 9, further comprising:100510876.230Attorney Docket No.: 085067-827975 (UA24-089) two optical devices configured as Hadamard gates being applied on the top and bottom polarization states; the Kerr nonlinearity device receiving the vertical polarization portion of the top polarized beam / polarization state from an output of the top Hadamard gate and receiving the vertical polarization portion of the bottom polarized beam / polarization state from the vertical output of the bottom Hadamard gate and performing controlled-phase operation between the top and bottom polarization state; another Hadamard gate is applied to the bottom modified polarization state (after the Kerr nonlinearity device); and wherein the integrated optics apparatus applies a Bell states preparation operation to the first polarized beam and the second polarized beam.

12. A device, comprising: an orbital angular momentum (OAM) mode demultiplexer that separates OAM basis modes of a multimode beam; a plurality of electro-optical modulators run in parallel with one another, each electro-optical modulator of the plurality of electro-optical modulators receiving a respective output of the OAM demultiplexer, the plurality of electro-optical modulators collectively introducing complex weights to the OAM basis modes; and an OAM mode multiplexer that recombines the OAM basis modes of the beam.

13. The device of claim 12, further comprising: a first Few-Mode-Fiber unit having an output in communication with the OAM mode demultiplexer, the OAM mode demultiplexer including a first taper core; and a second Few-Mode-Fiber unit having an input in communication with the OAM mode multiplexer, the OAM mode multiplexer including a second taper core; wherein the integrated optics and FMF-based apparatus serves the role of the single-qudit gate.100510876.231Attorney Docket No.: 085067-827975 (UA24-089) 14. The device of claim 12, the OAM mode demultiplexer including a power splitter in communication with a first plurality of computer-generated holography (CGH) units, each CGH of the first plurality of CGH units corresponding with a respective electro-optical modulator of the plurality of electro-optical modulators; and the OAM mode multiplexer including a power combiner in communication with a second plurality of CGH units that each receive an output of a respective electro-optical modulator of the plurality of electro-optical modulators; wherein the integrated optics and CGHs-based apparatus serves the role of the single-qudit gate.

15. A device, comprising: a first computer-generated holography (CGH) unit that receives a first beam having a first state; a second CGH unit that receives a second beam having a second state; and a Kerr nonlinearity device that receives an output of the first CGH unit and an output of the second CGH unit, the Kerr nonlinearity device being in communication with the first CGH unit and the second CGH unit by a pair of Few-Mode-Fiber linkages; wherein the device applies a generalized controlled-phase qudit operation between the first beam servings as a control qudit and the second beam serving a target qudit.

16. A method of entangling two classical coherent states, comprising: obtaining a first classical coherent state and a second classical coherent state; providing the first classical coherent state to a first waveguide, wherein the first waveguide generates as output a first idler state and a first photon addition signal state; providing the second classical coherent state to a second waveguide, wherein the second waveguide generates as output a second idler state and a second photon addition signal state;100510876.232Attorney Docket No.: 085067-827975 (UA24-089) applying a phase shift to the second idler state at an output of the second waveguide; and mixing, using a beam splitter associated with a pair of outputs respectively connected to upper and lower branch single photon detectors (SPDs), an input including: a first idler photon associated with the first idler state generated by the first waveguide; and a second idler photon associated with the second idler state generated by the second waveguide and following application of the phase shift to the second idler state.

17. An integrated optics apparatus configured to implement a quantum relay analogue, the apparatus comprising: a plurality of optical waveguide pairs corresponding to a source node, an intermediate node, and a destination node of the quantum relay analogue, wherein each optical waveguide pair is configured for operating on vertical and horizontal polarizations of an input polarization state to the optical waveguide pair; a plurality of optical devices each operable to apply a controlled-NOT (CNOT) operation between a different pair of vertical polarization basis states of the source node and the intermediate node; an optical device configured to implement a controlled-phase gate operation between the vertical polarization basis states of the source node and the intermediate node, wherein the controlled-phase gate operation is performed after a first CNOT operation and a second CNOT operation; two single-qubit gate analog optical devices configured as Hadamard gates applied on a modified polarization state at the source node and a modified polarization state at the intermediate node, respectively; two avalanche photodiodes (APDs) operable to obtain vertical polarization measurements at the intermediate node; and an additional two single-qubit gate analog optical devices configured to conditionally execute a Pauli X-gate and a Pauli Z-gate, respectively, on the destination node polarization state, in response to detection of the vertical polarization state using the two APDs.100510876.233

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