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DESCRIPTION

The present invention concerns a quantum processor, quantum logic gate, method of modulating a unit of quantum information (qudit) in a quantum processor, a qudit and register of a quantum processor.

The quantum processor, the method and the unit have been developed with a focus on the realisation of a quantum apparatus, e.g. a quantum computer, however, other applications are not excluded, such as the realisation of a classical computer, where they still allow for lower power consumption and a higher computing speed than today's bit-based computers.

DEFINITIONS

"Quantum": In physics, the quantum (from the Latin quantum meaning quantity) is the discrete and indivisible elementary quantity of a certain magnitude. By extension, the term is also used here as a synonym for an elementary particle associated with a force field (wikipedia). The preferred example of the quantum in the present invention is the photon.

A "quantum apparatus" comprises at least one "quantum circuit", e.g. part of a quantum processor, configured to perform a quantum calculation in order to generate output data.

The output data may be quantum (qubits), classical (bits), or their extensions such as qudit or combinations thereof, usually in string form.

The quantum circuit also has input data, which preferably include classical data such as bits, objects, events, symbols or signals also in quantum regime, usually in string form.

Quantum computing is used to build output data on the basis of input data

Quantum apparatus may comprise a quantum computer, a network of them, a quantum data transmission device, sensors, classics and combinations of them, or a network of them.

The definition of a quantum apparatus also includes quantum cryptographic systems, possibly coupled to other "classical" devices, capable of generating cryptographic keys or sequences of objects, events, or symbols.

The concept of an "event" is understood as a generic physical manifestation occurring within the Universe or any Multiverse or even more abstractly in Metaverse.

It is therefore understandable how, for example, the apparatus can be a quantum sensor, whereby quantum computing is in this case a measurement.

Each of these quantum apparatuses are realised on relative pieces of hardware, here referred to as supporting hardware apparatuses.

A quantum apparatus may contain a single quantum circuit or a collection of them. The preferred example of a quantum apparatus of the present invention is a photonic quantum apparatus, i.e., one operating by means of photons.

A quantum processor is a processor dominated or regulated by the laws we associate with quantum mechanics, by which we mean any processor capable of performing at least one "quantum computation" . The quantum processor may comprise one or more quantum circuits. In the present invention, preferred examples of quantum processors are photonic quantum processors, i.e., configured to perform calculations using photons.

Quantum computing is defined as "any process performed by a quantum circuit" such as, for example, any of the following operations based on the laws of quantum mechanics: measuring, generating, manipulating quantum states, sequences of symbols derived from them, et similia, without further restrictions.

Quantum computing is traditionally based on a basic unit called qubit, i.e. the unit of quantum information described by an overlapping of two states. Quantum computing can also be based on extensions of qubits. For example, the qudit is known, i.e. the unit of quantum information described by an overlapping of a plurality of states, where the number of states is an integer greater than two (wikipedia).

Also known is the qutrit; that is, the unit of quantum information described by an overlapping of three states; it is thus a three-state qudit.

Qubits are, for example, states of subatomic particles such as photons or electrons, where since each particle, due to the overlapping principle, can be in several different states at the same time and with different probabilities, it is possible to "overcome" the dualism of the classical binary 0/1 codes and convey much more information, thus being able to perform several operations simultaneously.

TE: For TE waves we mean electromagnetic waves that have only electric field components perpendicular to the direction of propagation of the electromagnetic wave (i.e., components in an x and y plane). TE modes are often referred to as TE _mn , where m and n are the mode index representing the number of nodes along the x and y directions respectively.

A=TE ₁₀ = electrical fundamental mode B=TE ₀₁ = electrical second-order mode C= (QAM 1=+1) = positive orbital angular momentum of value +1 D= (OAM 1=-1) negative angular momentum of value -1

PRIOR ART

It is well known that quantum computing systems are undergoing ever-increasing development, offering new horizons of hitherto unimaginable computing power and services that will affect and involve an ever-increasing number of users and activities in the future.

It is also well known that the growing demand for greater computational capacity has taken the quest for the classical computer based on standard bits practically to its physical limit in terms of materials and circuitry, despite the fact that modern electronic technologies allow the use of ever faster computers, beyond the petabit, capable of reaching very high computational efficiencies (expressed in terms of bits per second).

In this scenario, it is particularly important to develop new methods together with new technological solutions that make it possible to make deeper use of knowledge of the laws of nature to solve the increasingly complex computational problems as it is now required.

Also in this context, of particular importance is the process of computation performed by means of quantum logic, the so-called quantum computation, which has recently proven to be much more powerful for solving certain classes of problems than classical computation. This superiority derives from the ability of the quantum analogue of the bit, the qubit, to maintain a stable coherence between different classical states, being expressed as a coherent overlapping of states 0 and 1 representing the knowledge associated with the classical bit. This property enables the quantum computer to perform calculations on many classical input states simultaneously, allowing a quantum computer to have, at least in theory, an exponential increase in computational speed.

Well-known examples of this are apparatuses for detecting quantum effects, from astronomy to other experimental sciences, communication, quantum and classical cryptography and their combination, to quantum computing.

In order to carry out a quantum calculation, it is necessary to manipulate the quantum state associated with quanta in a controlled manner, i.e. in practice to change the quantum state of particles, e.g. photons, that have an associated wavelength or polarization or other quantum-mechanical property.

This is traditionally achieved by means of a waveguide, e.g. an optical fibre, which establishes the path of the photons, where components that change the quantum state of the photons, called "quantum modulators", are inserted along this path. Examples of such components are polarizers that change the polarization of photons in a controlled manner, phase modifiers, etc. These insertions, however, interrupt the fibre and therefore represent "break points" in the path of the photons. No matter how carefully the interruption and insertion of devices are carried out, such as anti- reflection treatments, etc., they will still "lose photons". Since there will be many of these interruptions and insertions along the fibre of the final circuit, which will be used in the computation, the defects introduced will add up, and thus there will be significant losses of photons and more generally of the information associated with them. All this is inevitably reflected in important errors in quantum computation.

Finally, it must be recognised that in order to have high computing power at the quantum level, a very large number of qubits must be used to cover the entire state space we wish to represent . In particular, to date we need at least n ₁ =log ₂ N qubits to represent an N- dimensional system in terms of qubits. This has an impact on the complexity of the circuits, which grows "exponentially" .

A known solution for obtaining qudits with only two photons is described in D1 - US2022171133, which makes use of a plurality of waveguides (202, 204) and couplers between them (200, 618 etc.) to "couple" the modes of the two photons. D1's couplers are either beam splitters 200 or optical couplers 618 etc.

This, however, does not solve the problem of photon losses, as photons are forced to "jump" from one waveguide to the other. Despite this, the prior art confirms a unanimous orientation towards the two-photon coupling solution, also discussed in the scientific literature in:

• D2 - ASEEMA MOHANTY ET AL. - QUANTUM INTERFERENCE BETWEEN TRANSVERSE SPATIAL WAVEGUIDE MODES- NATURE COMMUNICATIONS, VOL. 8, 20 JANUARY 2017, PAGE 14010, XP55562943.

• D3 - RAKESH RANJAN KUMAR ET AL - QUANTUM STATES OF HIGER-ORDER WHISPERING GALLERY MODES IN A SILICON MCRO DISK RESONATOR, ARXIV.ORG, CORNELL UNIVERSITY LIBRARY,201 OLIN LIBRARY CORNELL UNIVERSITY ITHACA, NY14853, 15 MARCH 2020, XP081621671.

In particular, these two articles always refer to two-photon coupling where the couplers are respectively unidirectional couplers used to realise a Hong Hou Mandel interferometer in D2, and a coupler with a disk resonator in D3.

The purpose of the present invention is to overcome all or part of the problems of the prior art.

A further purpose of the present invention is to obtain a qudit from a single photon or generally from a single quantum.

A further purpose of the present invention is to manipulate a qudit corresponding to a single photon or generally a single quantum.

A further purpose of the present invention is to provide a circuit for quantum computing with minimal loss of quantum information.

A further purpose of the present invention is to reduce the number of computing units required to cover the state space we wish to represent and thus simplify quantum circuits.

Another further purpose of the present invention is to provide an easily and inexpensively realised quantum circuit.

Another further scope of the present invention is to provide easy modulation of quantum states in a quantum circuit.

GENERAL INTRODUCTION

According to its first general aspect the present invention concerns a quantum processor comprising:

- at least one input register (5) of information units corresponding to at least one quantum placed as input to at least one waveguide (10) defining a path of said at least one quantum and having a longitudinal development in a predetermined direction L, the waveguide individually considered being configured to support "d" propagation modes of said at least one quantum, and defining a continuous optical path between an input and an output comprising at least one modulating geometric modification (24) comprising at least one variation of a orthogonal section to the longitudinal direction L, and placed in optical continuity with the remaining part of said waveguide, wherein the modulating geometric modification is configured to manipulate said propagation modes of said at least one quantum, the processor comprising at least one output register (15) placed at the output to said waveguide (10) and configured to decode the information corresponding to the manipulated modes.

Conveniently, the present invention is able to generate a qudit even with a single photon and manipulate its states. The photon does not make driving jumps for manipulation and therefore there are no losses. This is in contrast to the diversion of the prior art.

However, the processor can conveniently also operate with many indistinguishable or even entangled photons with multiple states associated with each individual photon, increasing the power of quantum computing.

Preferably, said modulating geometric modification comprises a reduction in section of said waveguide.

According to certain preferred implementation forms, said geometrical modification comprises a concavity located at an indentation of said section, wherein preferably said concavity has a section comprising two branches projecting in different directions. Said branches, for example, are such that said waveguide at said concavity has a substantially V- or L-shaped section.

According to some preferred implementation forms of the invention, this waveguide is a single optical fibre.

In that case, preferably said optical fibre is of substantially rectangular section or has a core of substantially rectangular section, and wherein said geometric modification is an indentation at an edge of said rectangular section.

Preferably, said manipulation comprises a multimodal conversion defined as the switching from one to the other of at least some of the "d" modes of at least one quantum.

According to some preferred implementation forms, the input register is a qudit register with "d" states or a qubit register with "d" levels, corresponding to said "d" propagation modes of at least one quantum, and the waveguide (10) has a geometry supporting exclusively "d" propagation modes and their overlapping.

According to a particularly preferred example, the modulating geometric modification (24) is configured to excite, starting from one of the input propagation modes, two linearly polarised LP modes of the at least one quantum, referred to hereafter as LP modes, with different propagation constants (β1 and β2), wherein at the output of the geometric modification (24) the two LP modes interfere by generating at least one of the "d" quanta propagation modes. According to some particularly preferred implementation forms, d=4 and the waveguide (10) has a rectangular section (12) configured to exclusively support the following four mutually orthogonal quantum propagation modes:

A=TE ₁₀ = electrical fundamental mode

B=TE ₀₁ = electrical second-order mode

C= (0AM 1=+1) = positive orbital angular momentum of value +1

D= (0AM 1=-1) = negative angular momentum of value -1 and their overlapping.

In that case, preferably, the waveguide has main development dimensions X and Y orthogonal to each other and to the direction Z=L of quantum propagation, wherein said cavity comprises a dimension "a" along X and a dimension "b" along y such that a≥b and a≤2b, and preferably said modulating geometric modification (24) comprises a waveguide tract of L or V section in a plane transverse to the direction of propagation Z.

Preferably the modulating geometric modification extends for a length L in the propagation direction z, wherein L is equal to: its multiples where β1 and β2 are the propagation constants of the two LP modes

According to some preferred implementation forms, the input register comprises at least one 4-state qudit A, B, C, D together with at least two polarization states, preferably the polarization states are a horizontal state |H> along x, and a vertical state |v> along y of the corresponding quantum, preferably used as control bits (or qubits) to run a desired circuit, or alternatively used to double the computational capacity by making a qudit for each polarization, alternatively to the polarization it is possible to use a pair of the OAM states (or the two OAM states) as a control register and use the polarization as part of the qudit.

Preferably, the input register has at least 8, preferably at least 16, of said 4-state qudits, encoded in a single quantum or photon and allows to do even with only a single quantum (e.g. a photon) all operations that in prior art require at least two photons. However, the processor of the present invention can conveniently also operate with many indistinguishable or even entangled photons with multiple states associated with each individual quantum (for example a photon), increasing the power of quantum computing.

According to some preferred implementation forms, the processor is configured to generate at least one CO- NOT gate comprising at least said waveguide 10 with said at least one modulating geometric modification, wherein said C-NOT gate is defined by the following truth table: and where it is configured in such a way that said modulating geometric modification (24) changes the phase of the qudit by cyclically shifting the qudit state.

Preferably the qudit has 4 states A, B, C, D and the value 1 stands for a phase change of π/2, i.e., the cyclic evolution of one state to the next of A B C D.

Preferably the processor is characterised by the fact that it is configured to generate at least one Toffoli gate CCNOT configured to generate said C-NOT gate wherein the Toffoli gate is characterised by the following truth table for each photon:

where the C-NOT polarization imposes an additional phase shift on the "d" propagation modes that build the states of said input qudit.

Preferably the processor is configured to generate at least one Hadamard gate comprising said C-NOT gate wherein the Hadamard gate is characterised by the fact that for each state "d" of the qudit indicated by the symbol |Sd(n)> i.e. S=state of d of the qudit, we have a "+" input which overlaps our starting state with the next one |Sd(n+1)> and the input which is obtained by running backwards the basic CNOT circuit which overlaps |Sd(n)> with the previous one |Sd(n-1)>.

According to some preferred implementation forms, said input register is placed at the input of a plurality of waveguides each comprising at least one of said modulating geometric modifications (24), configured to generate parallel paths and which combine by interacting to overlap, modulate and modify qudit states, where the output register is in common with said plurality of waveguides.

According to some preferred implementation forms, at least some of said modulating geometric modifications (24) are different from each other, for example they have at least different extensions in the direction of propagation.

According to its second aspect, the invention concerns a qudit with 4 states A, B, C and D corresponding to the following modes of propagation of a quantum :

A=TE ₁₀ B=TE ₀₁ C= (OAM 1=+1) D= (OAM 1=-1) According to its third aspect, the invention concerns an input register of a quantum processor that comprises at least a 4-state qudit according to the type mentioned above and at least two polarization states e.g. projected onto two punctually orthogonal bases, e.g. the horizontal polarization states |H> along x, and vertical |V> along y of the corresponding quantum, preferably used as control bits (or qubits) to make it run a desired circuit, or alternatively used to double the computational capacity by making a qudit for each polarization.

According to its fourth aspect, the invention concerns a method of modulating a qudit in a quantum processor, characterised by providing a quantum processor (1) of the type indicated above, placing to the input of the waveguide (10) at least one quantum characterised by at least one state corresponding to a propagation mode supported by said waveguide, and manipulating said propagation mode by means of at least one of said modulating geometric modifications in such a way that in output said mode, and the corresponding state, corresponds to one of said "d" propagation modes even if different from said input mode.

Preferably the method is characterised by using a qudit characterised by "d" states of propagation of said quantum and a control register comprising the polarization state of said quantum.

According to some preferred implementation forms, there are four "d" states, indicated by A, B, C, D, corresponding to the following propagation modes supported by the waveguide (10):

A=TE ₁₀

B=TE ₀₁

C= (0AM 1=+1)

D= (0AM 1=-1) wherein said manipulation comprises the phase of generating from at least one of said propagation modes or their overlapping, by beating on a modulating geometrical modification (24) of the waveguide (10), two LP linear polarization modes with different propagation constants (β1 and β2), wherein at the output of the geometrical modification (24) the two LP linear polarization modes interfere generating one of the "d" quantum propagation modes according to their phases.

DETAILED DESCRIPTION Further features and advantages of the present invention will best be seen from the following detailed description of preferred implementations of the invention, made with reference to the accompanying drawings, exemplifying but not limitative. In said drawings:

- figure 1 shows schematically a quantum processor where a quantum circuit is represented comprising a waveguide having structural continuity, said waveguide being for example an optical fibre;

- figure 2 shows schematically a mode selection section of the waveguide of figure 1 having a rectangular section;

- figure 3 shows schematically a mode modulation tract of the waveguide of figure 1, having a modulating geometrical modification with respect to the selection tract and structurally continuous with respect to it.

- figure 4 shows the fibre states formed by coupling the linearly polarised LP modes (also called LP- like modes) of the modulation tract of figure 1 according to the Phase-shift they have with respect to each other at the exit of that tract.

- figure 5 shows the electric field and magnetic field in the waveguide of figure 1 in a section with a rectangular cross-sectional cavity prior to the modulating geometric modification;

- figure 6 shows the first 12 modes of a waveguide with a known generic rectangular cavity;

- figures 7 and 8 show the equations demonstrating the selection effect, generated by the cross-sectional dimensions a and b, of the four waveguide modes TE ₁₀ , TE ₀₁ , (OAM) 1=+1, (0AM) 1=-1, known as A, B, C, D;

- figure 9 shows the change from a TE ₁₀ state to an OAM state as a result of a modulating geometric modification in figure 1;

- figure 10 shows a quantum C-NOT gate comprising at least one circuit according to the present invention;

- figure 11 shows a quantum C-NOT Toffoli circuit comprising at least one circuit according to the present invention;

- figure 12 shows a quantum Hadamard gate comprising at least one circuit according to the present invention;

- figure 13 shows a quantum processor according to the present invention wherein the input and output registers are in common with a plurality of parallel quantum paths.

GENERAL PRINCIPLE

The general principle underlying the present invention is to generate a quantum circuit comprising at least one waveguide (e.g. optical fibre) comprising at least one local geometry modification, called a modulating geometric modification, acting on the quantum state of each quantum (e.g. a photon, even alone) travelling through the waveguide. Modulating geometric modifications have the function of manipulating the quantum state (and its associated quantum information), and said manipulation (also called modulation) is part of the quantum computation process. This reduces the loss of quanta, e.g. photons, and consequently calculation errors, compared to the known case of inserting extraneous devices into the waveguide resulting in interruptions to the quantum path.

The quantum processor may comprise a plurality of said modulating geometric modifications, distributed over one or more waveguides.

In general, the waveguide can be solid, or tubular, in both cases it is often referred to as a cavity in the literature, so this term is understood as a general synonym indicating the material and relative geometry where the radiation propagates.

Hereafter we use as an example an apparatus based on photonic quantum computation, without excluding apparatuses based on similar procedures that make use of the quantum-mechanical wave properties of any other quantum such as atoms, ions, elementary and non- elementary particles and their assemblies, etc. Nor do we exclude hybrid circuits that make use of two or more types of quantum.

Quantum computation preferably takes place through the injection (input) of one, or several, quanta (photons in our example) single or overlapping states, or more generally entangled or correlated states, generated by means of a single quantum or multi quantum states.

Each of these quanta possesses a predetermined quantum state with which information is associated. Quantum information is expressed in terms of qubits, by an overlapping of two states orthogonal to each other, or in terms of qudits, by an overlapping of "d" states orthogonal to each other. To a qudit is associated a Hilbert space of dimension d, to a qubit a Hilbert space of dimension 2. Projective techniques for the geometry of Hilbert spaces can map Hilbert spaces of different dimensions into subspaces or product spaces, i.e. it is possible to express a qudit by means of an appropriate set of qubits.

To make a quantum calculation, the quanta (photons in our example) are placed in a circuit (optical in our example) along which one or more local modifications of the geometry of the waveguide cavity are arranged, configured to generate a modulation path for the quantum states of the quanta (photons). Single quantum (photon) (photo)detectors or equivalent sensors are placed at the end of the circuit, in order to reveal the presence or absence of quanta (photons) at a certain instant of time, thus obtaining the result sought by quantum computation.

The path of each quantum participating in the computation, whether by quanta of light or otherwise, is obtained by confining it in a set of waveguides.

(which in our example is one or more cavities even simultaneously) in which the information carried by it, is preserved and is linked to its quantum state, i.e. for example the polarization of the photon or a field propagation mode such as orbital angular momentum (OAM), cavity modes such as Hermite-Gauss modes or more general properties that can be associated with a quantum state forming a set of preferably orthogonal states.

A photon used as a quantum in the computation process, for example, can be found simultaneously in all parallel paths within a quantum circuit or have all polarization or OAM states overlapped in a quantum mixture. As the calculation goes on, new paths and thus new quantum states overlapped with the previous, co- existing ones can be generated. The final quantum calculation is to tame all the overlapping (e.g. for a module of 16^16) and have them projected into a final bit string of, for example, 64 bits) with the final collapse of the wave function, which is the result of the calculation. It is to all intents and purposes a real physical process and not simulated using quantum logic as in a classical calculation. The language of nature is used.

In general, the invention comprises a processor comprising at least one input register 5 of information units, in the form of qubits or preferably qudits, in which certain information is encoded and placed as input to at least one waveguide 10 defining the path of the corresponding quanta. The waveguide is configured to support a plurality "d" of propagation modes of the quanta, at the input and preferably also at the output, and comprises local geometric modifications configured to manipulate them by transforming at least one propagation mode into one of the other modes of the qubit or qudit. The local geometric modifications hereafter are also called modulating geometric modifications. At the output of the waveguide, the result is read by decoding the information in an output register 15.

The modulating geometric modifications break the original rotation symmetry and split the mode degeneracy into two linearly polarised LP propagation modes (also called LP-like modes and hereafter for brevity LP modes)). Upon exiting the modulating geometric modifications, the two LP modes interfere and generate one of the supported "d" modes, depending on the phase shift, whereby modulating geometric modifications make it possible to switch from one mode to another.

In this way, each qubit or qudit has "d" states manipulated by waveguide geometry. In a specific preferred example, the waveguide geometry supports 4 propagation modes so each qudit used for calculation has 4 states, preferably orthogonal to each other, preferably TE ₀₁ , TE ₁₀ , OAM+1 OAM-1 . Each waveguide modulating geometric modification is configured to transform at least one of the four propagation modes, and thus the four states of the qudit, into one of the others. The transformation occurs because the modulating geometric modifications are configured to support for each mode two orthogonal LP linear polarization modes, which have two different propagation speeds. In the case of a waveguide with a rectangular section transversal to the propagation direction z and with sides parallel to the directions X and Y, for example, the two LP modes have optical axes rotated by approximately 45° with respect to X and Y. When the two LP modes interfere, they generate one of the modes supported by the rectangular waveguide at the output of the modulating geometric modification, depending on their phases. When the two LP modes are in phase, they form the TE ₁₀ mode.

Preferably, the modulating geometric modification is interposed between an input waveguide section and an output section that share a common geometry.

Figure 4 shows the transformation table according to the phase differences (called shift or phase-shift) of the LP linearly polarised modes at the output of modulation trait 24.

Since the four modes are mutually orthogonal and independent, they can be used both for channel multiplexing and to set four independent modes for quantum computing.

Based on these basic-principle premises, figure 1 represents a quantum processor 1 comprising at least one quantum circuit 2. Quantum circuit 2 comprises: a waveguide 10, a source of quanta 5 (e.g. photons) part of an input register to waveguide 10 and a detector device of quanta 15 (e.g. photons) part of an output register from waveguide 10.

Waveguide 10 generally defines an electromagnetic quantum propagation path in a propagation direction Z, corresponding to its main direction of development.

The waveguide 10 in this example comprises the cavity 12 of a tubular body, which extends along its entire length z. Said cavity 12 may be empty or full, which does not exclude it being filled, e.g. with dielectric material.

The waveguide, for example, is an optical fibre with a rectangular section for the transport of photons.

The waveguide 10 has at least one propagation mode selection section 19 having a first section 20 transversal to the propagation direction Z. In the example of figure 1, and as better seen in figure 2, the section is characterised by a rectangular cavity 12, i.e. it has two main development directions X and Y, where the dimensions of cavity 12 in said directions are "a" and "b" respectively, and where preferably the ratio between a and b is such as to define a selection of supported propagation modes, in our example the cavity of the first section 20 is configured to maintain TE ₁₀ as the fundamental mode and TE ₀₁ as the only second-order TE mode, suppressing the other higher modes.

More precisely, in the example we use rectangular fibres with dimension (x,y) = (a,b) dimensioned to have transversal electric modes (TE) limited to the fundamental and second-order modes of TE ₁₀ and TE ₀₁ , respectively, and two orbital angular momentum modes (0AM) 1 = +/-1. The z-axis, as mentioned, is that of propagation.

These four are independent, mutually orthogonal propagation modes denoted A, B, C and D, and are the four modes that define the preferred qudit states according to the present invention.

We remind you that by convention we define propagation modes according to the wave-field approach based on the well-known Maxwell equations.

According to this approach, waveguide propagation modes are divided into transverse-electric (TE) or transverse-magnetic (TM) modes, depending on whether the electric or magnetic field is purely transversal to the direction of propagation.

Section 20 above is obtained when a≥b since the low-frequency cutoff for the TE01 mode is when fc, ₀₁ =1/(2b√ (με)) is active for a≤2b.

By definition we have f _cl =l/(2a√ (με)), where μ and s are the dielectric and magnetic permittivity of the fibre and c=1/√ (με) is the propagation speed of light in the medium.

Figure 5 shows the electric field and magnetic field in the rectangular-section cavity 12 of guide 10 when a≥b, in particular showing the fundamental mode TE ₁₀ .

Figure 6 shows in general the first 12 modes of a waveguide with a rectangular cavity. Arrows indicating the direction and polarization of the E-fields are drawn.

The three "rules" for the E and H (M) fields within the waveguide, according to Maxwell's equations are:

• Electromagnetic waves do not pass through or cross conductors, they are always reflected by the conductors. • Electric field lines touching a conductor must be perpendicular to it.

• Magnetic field lines close to a conductor must be parallel to it.

Based on these rules, it is possible to formulate the equations in figures 7 and 8, which show how the choice of a and b is able to carry out the selection of supported propagation modes mentioned above.

The waveguide 10 comprises at least one modulation section 24 having a second section 25, at the end of the selection section 19, with respect to the propagation direction z. The modulation section 24 defines a modulating geometric modification according to the present invention.

Considering for example the second-order mode (TE ₀₁ ) supported by the selection tract 19, its passage in the second tract 24 excites two orthogonal LP modes with different propagation constants (β1 and β2). While propagating in the modulation tract 24, they have a phase shift between them induced by the modified cavity geometry 12a. Modulation section 24 is configured to generate phase shifts of π/2 or multiples thereof. This results in an output of one of four independent modes forming the qudit according to the present invention. Figure 9 illustrates an example of modulation to transform a TE ₁₀ input mode into an 0AM output mode. Turning to the preferred geometric examples of section 24, the second section 25 defines a local modification of the geometry of the cavity 12, best seen in figure 3, in particular the modified cavity is indicated as 12a and comprises two main development directions orthogonal to the propagation direction Z. In the example, the two directions are the X- and Y- direction. In order to achieve this in the example, the section 25 comprises with respect to the section 20 an indentation 26, preferably in such a way as to form an L. As visible in figure 1, assuming that the outer contour of the waveguide traces the cavity, except the thickness of the walls, the indentation 26 forms a trench along the second section 24 which interrupts one of the edges of the rectangular section of the selection section 19.

In general, the geometry of modulation trait 24 has the task of generating LP modes (linear polarization) and keeping them distinct.

At the exit of the modulation section 24 there is an interference section 30 of the LP modes where they rejoin and form one of the four modes A, B, C, D depending on their phase. The interference tract 30 in fact has a geometry that supports said four modes, and preferably has the same geometry as the selection tract 19.

The phase shifts between the output LP modes depend on the length of the modulating geometric modification in the propagation direction z.

This length is indicated by L in Figure 1 and to obtain phase shifts of π/2 (as in the case of generating OAM = 1=+1 or 1+1 from TE10) is given by: while to achieve phase shifts of 3/2 π(as in the case of generating OAM = 1=-1 or 1-1 from TE10) where β1 and β2 are the propagation constants of the two LP modes (Linear Polarization / LP-like modes)

In general L is equal to or its multiples.

With regard to the dimensions w and h of trench 26 forming the L-shape, they are generally not the result of a formula calculation, but are more conveniently optimised on the basis of backwards calculation techniques starting from an expected result, in particular they are optimised backwards to adjust, depending on the materials chosen, the desired effect of modifying electromagnetic fields and minimising losses.

This effect is described by means of refractive index perturbations on a multi-mode waveguide (as is the case in the present invention), the energy can be coupled from one waveguide mode to other modes (which are A B C D in our example), and the amplitude of each mode along the propagation direction z can be determined by a set of differential equations of this type, for each mode p of the cavities of our interest (in our example, there are two differential equations for the modes TE01 and TE10 and two others to describe the phase shift of +/- π/2 of the two overlapping modes TE01 and TE10, which are the modes OAM+1 and OAM-1), where Ap and Aq are the amplitudes of p and q modes of the waveguide, and pp and pq are respectively their propagation constants. The index m indicates the highest order mode in the multi-mode waveguide. K _pq represents the modal coupling coefficient between the p and q waveguide modes, which can be defined as where the integration region S is the waveguide cross-section of the material (which may be, for example, silicon), Ep (x, y) and Eq (x, y) are the electric field profiles of the waveguide modes p and q in the cross- section, respectively, and Δε (x, y, z) represents the perturbation of the refractive index on the waveguide given by the shape of trench 26. According to equation 2 above, K _pq is proportional to the spatial integral of the perturbation Δε (x, y, z) with Ep (x, y) and Eq (x,y), which we call the electric field overlapping.

There are two requirements for multi-mode conversion (i.e., switching between modes A, B, C, D) based on the Equation (1). The first is to obtain large values of the modal coupling coefficients between the incident modes (e.g. starting mode A) and the target modes (where it transforms as e.g. mode B or the following). This concept also applies to overlapping of modes and thus to a generic qudit made by modes A, B, C, D.

The second requirement is to fulfil the condition of phase matching along the direction of propagation, which compensates for the discontinuity of the propagation constant Δβ (i.e. β _P - β _q ) of the oscillating exponential term in Equation 1. The phase matching condition for modal coupling between p and q modes can be deduced from the following relationship where δ _pq represents the cyclic period of perturbation of the refractive index. The discontinuity given by Δβ which is also found in Equation (3) between different pairs of cavity modes (of modes A, B, C, D) has different values in the periods where this perturbation takes place which transforms one mode into another and also therefore leads to solutions with different coupling lengths.

Therefore, by using LP modes to switch from one state to another or transform one qudit into another, we have different L lengths of the cavity with trench 26 to transform with high precision and distinctively one cavity mode into another.

The simplest example for realising a quantum apparatus according to the present invention is to adopt a circuit architecture that is based on a set of rectangular waveguide fibres (e.g. shown in figure 13) that are easy to construct, for example, using Si02 or other materials such as polyethylene or silicon nitride, which can be printable, for example in a silicon-on- insulator (SOI) chip as a support. Trench 26 can be generated, for example, by locally cutting the optical fibre or by growing the cavity 12a in a suitable manner.

The first advantage of selecting these TE modes is that they are stable with respect to propagation in circuits (rectangular fibres) and are not suppressed. By selecting transversal fibre dimension a and b, all other higher-order modes are suppressed, preventing spurious mode mixing.

The second key advantage is that with the modulating geometric modification one of the states can be transformed (or rather modulated) into another, e.g. by means of the trench modulating geometric modification 26 it is possible to transform one of the four states of the rectangular waveguide 12 mentioned above into another (quantum state shift). In this way we can operate on the four basic states of each qudit by crossing sections of single-trench waveguides that are easier and cheaper to build instead of phase masks, plasmonic devices and so on, reducing reflection losses and discontinuities that will still be present in smaller numbers in the switches introduced in the processor in order to build a computation path for the photon within the processor. By manipulating the quantum states of the photon in this way, part or all of the result of the quantum computation is obtained.

The other advantage is that in this way the circuit, based on rectangular waveguide geometry, supports four natural fibre modes, i.e. two transversal electric states and two OAM orbital angular momentum states (TE ₁₀ , TE ₀₁ , 1= -1, 1=+1).

Formally, to each of these orthogonal modes we associate each eigenvector with which to build a qudit with d=4. The direct advantage is that with the use of a qudit of order d, when compared to the qubit, fewer qudits are required to cover the state space of a given problem, such as a quantum register of cardinal order N, since nl= log2 N qubits can represent an N-dimensional system, whereas only n2 = loga N qudits are required. In our case 112 = log ₄ N.

An equivalent (d=2) binary of their construction requires a number of qubit gates in the scale of O(n ² iN ² ). By analogy, the scale of qudit gates required using the same construction is O(n ² 2N ² ). Thus, the qudit method has a (log ₂ d) ² scaling advantage over the simple qubit case. The scaling advantage of qudits is (log ₂ d) ² and in our case is a factor of 4.

A 16-qubit port with d=4 becomes equivalent to a 32-qubit or 64-qubit port when considering repetition for each polarization.

The dimension of the Hilbert space of n qudits is dn, where d is the dimension of the qudit, in our case with n=16 we get 4 ¹⁶ = 2 ³² .

The circuit can also work with qubits by factoring the size of the qudit with respect to the qubit, gaining an advantage in circuit complexity.

We observe that each photon has two independent states of polarization, so it is possible for each of them to obtain the 4 independent states A, B, C, D to build the 4-state qudit, potentially resulting in an 8- independent-state qudit.

Without excluding this possibility, it is however preferable, in order to simplify the circuitry and reduce the calculation error, instead of also using the polarization state for the calculation, to configure the processor to use it for controlling the quantum calculation, i.e. the 4 states of the qudit.

Therefore, the preferred configuration of the processor according to the present invention is one with the 4-state qudit mentioned above for quantum computing, and quantum polarization as the control register.

Input/output: Input register 5 is made e.g. with NI = N/4 indistinguishable photons, entangled or by a single photon distributed along different simultaneous paths. The input register is preferably made with 16 qudits with the 4 states A, B, C, D together with the horizontal polarization states |H> along x, and vertical polarization states |V> along y, used as control bits (or qubits) to make it travel along the desired circuit.

An example could be input photons generated with a photon gun machine and timing + mode coding (time delay input to equalise photon timing), as known from the literature.

The input into the waveguide can conveniently be generated via a fixed phase device, comprising e.g. phase masks or 0AM sources, so as to directly impose a state of those supported by cavity 12 of waveguide 10, e.g. the input generates photons with orbital angular momentum (0AM) 1 = -1.

This simplifies input circuitry that would require a complicated device in the case of adopting multiple inputs, as desirable in the present invention, e.g. 16 qubits. At output register 15, a separation of the qudits into bits and reading of N events takes place by means of a demultiplexer that can have various shapes and configurations. Preferably photons detectors are integrated in the circuit at the end of the processor, reducing the losses. They are preferably SPAD, single photon avalanche diode.

Optionally, it is possible to provide a read wait state: to possibly reduce the number of SPADs at the end of the read, we can consider dividing the read output quantum states instead of 64 simultaneous states into 16 or 32 states divided by time slots and send them instead of 64 different SPADs, to 16 or 32 SPADs in order to reduce costs.

We prefer to stop at the waveguide configuration that supports the four independent modes in order to have fewer errors in the quantum calculation procedure. In fact, if we do not limit ourselves to the four selected modes (A, B, C, D) and want to push the limits of our configuration by adopting higher fibre modes and more than two LP modes, errors may be introduced due to mode mixing. This is because we have observed that when studying the evolution of the electromagnetic field profile of LP modes with different propagation constants, it must be considered that a simple mode demultiplexer cannot accurately discriminate LP modes at the output end if there are more than two LP modes and mode mixing, and thus additional errors are introduced.

Circuit 2 described so far is a basic circuit, which can be used to build any kind of complex logic- quantum circuit within a quantum processor 1, in particular the logic gates, also known as gates.

One of the fundamental logic gates that we can build using the basic circuit 2 is, for example, the controlled NOT known as the C-NOT gate. The Toffoli gate, which is built from a series of the C-NOT controlled-controlled NOT (CCNOT) gate, and the Hadamard gate, which is also related to the entanglement of quantum states, are also given as examples.

A quantum computing processor can comprise a series of interconnected C-NOT gates. In addition, in order to have a connection to classical computing and algorithms, following the correspondence principle of quantum mechanics, we can generate the Toffoli circuit or gate by means of a series of C-NOTs resulting in a controlled-controlled NOT, CCNOT.

Polarization-controlled C-NOT gate for each 4- state qudit with polarization as control register of the present invention (Figure 10)

The classical C-NOT gate operates on a quantum register consisting of two qubits, a control qubit and target qubit. The C-NOT circuit flips the second qubit (the target qubit) if and only if the first qubit (the control qubit) is |1).

The C-NOT circuit can be represented by the following matrix.

In our case, using circuit 2 and the 4-state qudit, the truth table is the same where the value 1 stands for a phase change of π/2, i.e., the cyclic evolution of one state to the next A B C D of the 4-state qudit we are using.

Figure 10 shows an example where PSB stands for polarising beamsplitter. The trench 26 waveguide changes the phase of the qudit by cyclically shifting the qudit state.

CCNOT Toffoli gate (controlled controlled c-not)

Introduction: The Toffoli gate, also called CCNOT gate or Deutsch gate, D(π/2), is a 3-bit gate that is universal for classical but not for quantum computing. Toffoli's quantum gate here is defined for 3 qubits. If we only accept input qubits that are |0) and |1), then if the first two bits are in the |1) state it applies a Pauli-X (or NOT) on the third bit, otherwise it does nothing. This is an example of a CC-U (controlled- controlled Unitary) gate. Since it is the quantum analogue of a classical gate, it is completely specified by its truth table:

The following table shows the implementation of the Toffoli gate using Hadamard, phase, controlled-NOT and π/8 gates.

Polarization-controlled Toffoli gate for each 4- state qudit with polarization as control register of the present invention (Figure 11).

The Toffoli gate is a universal reversible logic gate. A classic reversible circuit can be broken down into a series of Toffoli gates.

By definition, a logic gate n fed into input L with input x, and output y, , is reversible if the following conditions are met: bijective i.e., for each output y, there is a unique input x and the gate L is reversible if there exists an inverse function n or gate L '(y) = x such that which maps from y to x.

An example of a reversible gate is the NOT gate.

Reversible gates can describe logical operations that can be performed in quantum computers, as they behave as unitary transformations and thus evolve reversibly.

In fact, the evolution of quantum states is described either by the Schrödinger equation in terms of unitary transformations (i.e., preserving the inner product of the eigenstates, which before the transformation remain the same as after the transformation) or by the collapse of the wave function, which in this case corresponds to a projection onto an eigenstate .

A unitary transformation U is an isomorphism between two Hilbert spaces Hl and H2, which means that U is a bijective function,

The Toffoli gate is a universal reversible C-CNOT gate. By definition, following the pigeonhole principle, the number of input bits is equal to the number of output bits. This gate has a 3-bit input and 3 outputs. Identity occurs for any input, but when the first two bits are both set to 1 with the effect of inverting the third bit.

The Toffoli gate is the mapping of the three input bits to the output

The universality of the Toffoli gate means that one can represent any operation described by a given Boolean function f (x1, x2, .. xm) of the variables x1, x2, .. xm, by means of a circuit constructed with Toffoli's gate. The circuit maps the input variables x1, x2, ..., xm and some extra bits set to 0 or 1 to the output made with f (x1, x2, ..., xm), possibly together with the initial variables x1, x2, ..., xm and some extra bits. With a Toffoli gate, a quantum computer can implement all possible classical calculations, although it cannot be used for universal quantum calculation.

Referring to Figure 11, the qudit-based Toffoli gate according to the present invention is generated in the following way: the first bit is given by the polarization state, the second bit is a phase shift of 0 or π/2 of modes A, B, C, D and the second bit represents a further phase shift 0 or it of the mode A, B, C, D encoded on the qudit thus selected to have a 1-1 mapping of the polarization and fibre states avoiding overlapping of states from the Toffoli truth table.

The quantum Toffoli gate according to the present invention comprises a circuit 2 and is controlled by the C-NOT polarization which imposes an additional phase shift π/2 to the A B C D modes which build the qudit of the present invention. For example, if we select the vertical polarization V to represent the first bit with value 1, the horizontal polarization H will represent the state 0.

Hadamard gate for each 4-state qudit with polarization as control register of the present invention (Figure 12)

In general, the Hadamard gate makes an action on a single qubit or qudit states expressed in qubits as follows:

Thus, a measurement of the output state from the Hadamard gate will produce an entangled state or, rather, overlapped and shared state in the various qubits with the same probability of giving an output 1 or 0. It is preferably used in path entanglement with single quanta.

In this case, two states of the qudit overlap, where + denotes a positive state shift and - a negative one.

The advantage of using "+ " and state shifts is that this language simplifies the circuitry effectively. We start from one of the states A B C D and input + with polarization equivalently distributed along the states |H> and |V> (meaning similarly along the x- or y-axis respectively) obtained by passing the photon through a quarter- or three-quarter-wave foil that makes any polarization into a balanced overlapping of polarization states

For each quantum-mechanical state of the photon associated with one of the orthogonal propagation modes A, B, C, D, denoted by the symbol |S4(n)> i.e. S= state of 4 of the qudit, I have a "+" input which overlaps our starting state onto the next one |S4(n+1)> and the input which is obtained by running backwards through the basic CNOT circuit which overlaps |S4(n)> onto the previous one |S4(n-1)>.

For example, in the figure we have |S4(1)> =C, A and |S4(4)> B without loss of generality (we can rename them as we wish) and with right- or left-turn cyclicity as indicated by figure 12.

Finally, with reference to figure 13, a processor 101 according to the present invention is shown which differs from the processor 1 of figure 1 in that the input and output registers 5 and 15 are in common with a plurality of waveguides 2 defining corresponding parallel quantum paths.

Waveguides 2 can have different structural geometric modifications 24a, 24b, 24c.

In this way, it is also possible for a quantum, e.g. a photon, to travel through all the waveguides simultaneously, new paths can be created in the processor, and new overlapping of states can be created with respect to the starting state or a previous calculation . In general, in the present invention, it is preferable to use photons as quanta,

GENERAL MEANING OF TERMS

In understanding the purpose of the present invention, the term "comprising" and its derivatives, as used herein, are intended as open-ended terms specifying the presence of the declared characteristics, elements, components, groups, integers and/or phases, but not excluding the presence of other undeclared characteristics, elements, components, groups, integers and/or phases. The above also applies to words with similar meanings such as the terms "including", "having" and their derivatives. Furthermore, the terms "part", "section", "portion", "member" or "element" when used in the singular may have the dual meaning of a single part or a plurality of parts. As used herein to describe the form/forms of implementation above, the following directional terms "forward", "backward", "above", "under", "vertical", "horizontal", "below" and "transversal" as well as any other similar directional terms refer to the form of implementation described in operative position. Finally, grade terms such as "substantially", "about" and "approximately" as used herein mean a reasonable amount of deviation of the modified term such that the end result is not significantly changed.

While only selected implementations have been chosen to illustrate the present invention, it will be clear to those expert in the field from this description that various modifications and variations may be made without departing from the purpose of the invention as defined in the attached claims. For example, the size, shape, position or orientation of the various components may be modified as needed and/or desired. Components shown directly connected or in contact with each other may have intermediate structures interposed between them. The functions of one element can be performed by two and vice versa. The structures and functions of one form of implementation can be adopted in another one. It is not necessary that all advantages are present in a particular form of implementation at the same time. Each characteristic that is original compared to the prior art, alone or in combination with other characteristics, should also be considered a separate description of further inventions by the applicant, including structural and/or functional concepts incorporated by those characteristics. Therefore, the previous descriptions of implementation forms according to the present invention are provided for illustrative purposes only and not for the purpose of limiting the invention as defined by the attached claims and their equivalents.