Optical quantum gate, series of optical quantum gates, optical quantum computer, and use

Abstract

The invention relates to an optical quantum gate, a series of optical quantum gates, a quantum computer, and a use. The optical quantum gate according to the invention is based on a non-adiabatic and non-Abelian holonomy for photons of a predefined wavelength and polarisation, and comprises a medium having a predefined refractive index and an arrangement having three waveguide structures, of which a central third waveguide structure (22.3) is positioned between a first and a second of the waveguide structures. The first waveguide structure (22.1) and the second waveguide structure (22.2) are spaced apart from one another in such a way that probabilities for a photon of the predefined wavelength to transition directly from the first waveguide structure (22.1) to the second waveguide structure (22.2) and vice versa are negligible. The paths of the first waveguide structure (22.1) on the one hand and the second waveguide structure (22.2) on the other hand in relation to the central third waveguide structure (22.3) are each selected such that the conditions for a non-adiabatic and non-Abelian holonomy are met for photons of the predefined wavelength and polarisation, resulting in transition probabilities from the first waveguide structure (22.1) to the second waveguide structure (22.2) and vice versa.

Description

[0001] Optical quantum gate, array of optical quantum gates, optical quantum computer, and use Description The invention relates to an optical quantum gate based on a non-adiabatic holonomy for photons of a predetermined wavelength, an array of optical quantum gates, an optical quantum computer, and a use. Quantum computing is one of the most promising new research areas in modern physics. At its core, it exploits non-classical effects in certain quantum systems to drastically accelerate computations and information processing and intrinsically enables increased security of communication channels. The experimental and technological challenges associated with the reliable construction of suitable quantum systems have led to extensive research in the field of quantum error correction and fault-tolerant operations.The basis of quantum computing is the processing of qubits, i.e., quantum systems with two superimposable states that are described quantum mechanically and assume only two states that can be reliably distinguished by measurements. The processing of the qubits, i.e., the implementation of the quantum mechanical equivalent of computational operations, takes place in quantum gates that influence the probabilities for the occurrence of the distinguishable states. In the case of optical quantum computing, qubits can, for example, be realized by photons that are present or absent in certain waveguides. The waveguides can be optical fibers or waveguides embedded in a substrate with a lower refractive index, whose refractive index or refractive index profile is higher than that of the substrate.In its simplest form, a quantum gate can be a directional coupler consisting of two waveguides embedded in a common substrate or medium. These waveguides have a higher refractive index than the medium and are aligned so closely parallel over a certain distance that their evanescent fields overlap. This coupling, along the length of the parallel path, creates a certain probability that a photon will tunnel from one waveguide into the other, and vice versa. The strength of this coupling depends on the wavelength and polarization of the photons, the optical properties in the waveguides and substrate, and the distance between the waveguides. It can be calculated if the material properties are known, but can also be determined experimentally through calibration tests with different distances.These waveguides are conventionally called the left and right waveguides, regardless of their actual spatial relationship. Put simply, the quantum mechanical state upon entry into the quantum gate is that a photon enters either the left or right waveguide and then exits the right or left waveguide, depending on the case. The photon has either remained in the same waveguide or changed waveguides. In principle, input states can also be superpositions of "photon left" and "photon right." Multiple coupling back and forth within the gate is also possible. Likewise, the output states are generally superpositions that collapse during measurement. In a quantum computer, measurements only take place after several gates. The states of the qubit logic can thus be defined as |0^^^^^^ = |1. ^ 0 ^ ^ |1^^^^^^ = |0^ 1 ^^. However, the construction of quantum gates with waveguide-based directional couplers leads to limited reproducibility of the functionalities, since these depend strongly on the experimental conditions via the cumulative coupling and can fluctuate considerably both between different measurements on the same gate and between nominally identical realizations of a particular gate. The present invention is therefore based on the object of providing optical quantum gates that offer greater stability of their respective functionality in application than is possible with conventional waveguide-based directional couplers, and additionally enable significant miniaturization. This object is achieved by an optical quantum gate based on a non-adiabatic and non-Abelian holonomy for photons of a predetermined wavelength and polarization.comprising a medium with a predetermined refractive index and an arrangement with a first waveguide structure, a second waveguide structure, and a central third waveguide structure, wherein the central third waveguide structure is arranged between the first waveguide and the second waveguide, each of which runs between a respective input and a respective output, wherein the first waveguide structure and the second waveguide structure are spaced from one another such that transition probabilities for a photon of the predetermined wavelength directly from the first waveguide structure to the second waveguide structure and vice versa are negligible, wherein the courses of the first waveguide structure on the one hand and the second waveguide structure on the other hand are each selected with respect to the central third waveguide structure,that for photons of the predetermined wavelength and polarization the conditions of a non-adiabatic and non-Abelian holonomy ^, ^are fulfilled, resulting in transition probabilities from the first waveguide structure to the second waveguide structure and vice versa. By introducing the central third waveguide structure between the first and second waveguide structures and designing the configuration or relative arrangement of the three waveguide structures as a non-adiabatic holonomy, the stability of the quantum gate's results is increased, and its non-adiabaticity makes it possible to achieve greater miniaturization than was possible with adiabatic components. The invention is based on the concept of holonomic quantum computation, which was described, for example, by P. Zanardi, M. Rasetti, Holonomic quantum computation. Phys. Lett. A. 264, 94–99 (1999).This concept is implemented according to the invention in a technical implementation of quantum gates as non-Abelian holonomies, a class of topologically protected unitary operators. This topological protection can be used to design intrinsically stable quantum computers. Based on the fact that in quantum mechanics, a pure state of a quantum mechanical system (quantum system) can be described by a vector in the Hilbert space ℋ, the optical quantum gate according to the invention mathematically maps a path of a quantum system through the Hilbert space ℋ. During propagation or movement through the Hilbert space ℋ, the quantum system accumulates a geometric phase that depends exclusively on its path. In contrast to the dynamic phases that record the passage of time, these geometric phases cannot be made to vanish by a gauge transformation.In general, geometric phases can exhibit arbitrary dimensionality. Wilczek and Zee introduced the idea of ​​multidimensional, non-abelian geometric phases, so-called holonomies (F. Wilczek, A. Zee, Appearance of gauge structure in simple dynamical systems. Phys. Rev. Lett. 52, 2111–2114 (1984)). The purely geometric nature of these entities is ensured by restricting the propagation to a ^-fold degenerate subspace of ℋ, thus generating a geometric phase belonging to the group U(^). When using holonomies as quantum gates, the main source of error is the requirement of adiabaticity. The simplest method to keep such deviations in check is to slowly vary the system along the closed path through a parameter manifold. However, this severely limits the possible degree of miniaturization of such devices.However, purely geometric phases can also be realized with a less stringent version of this requirement. Non-adiabatic holonomies are based on a subspace ℋgeo of the Hilbert space spanned by a set of states {|Φ^^}^ that satisfy the condition. = 0, where ^ ^is the Hamiltonian of the system. Restricting the propagation to ℋgeo ensures parallel transport and thus a purely geometric phase. The states {|Φ^^}^ are no longer instantaneous eigenstates of the Hamiltonian, as described in J. Anandan, "Non-adiabatic non-abelian geometric phase," Phys. Lett. A.133, 171-185 (1988). Quantum optics represents a particularly versatile platform for quantum information processing and is therefore suitable for the construction of non-adiabatic holonomic quantum computers. Not only does it easily permit the desired integration and miniaturization, the bosonic nature of photons also conveniently allows multiple excitations of the same bosonic mode. This allows ℋgeo to be extended so that holonomies from higher symmetry groups U(^) can be created in the form of larger and more powerful computing units.The quantum gate presented here comprises three waveguide structures. According to the nomenclature of the present application, these are a first waveguide structure, a second waveguide structure, and a central third waveguide structure. However, these can also be easily named, for example, left waveguide structure L, right waveguide structure R, and central waveguide structure C. In the simplest case, the waveguides can be designed in the same way, for example as single-mode waveguides with the same refractive index or as groups of evanescently coupled waveguides that together form a supermode. However, the invention is neither limited to single-mode waveguides, waveguide groups, nor to all three waveguides being designed with the same refractive index. In a planar geometry, for example, the direct coupling between the outer channels orWaveguide structures L and R are effectively suppressed, while each of them can easily interact with the central waveguide structure C. In the tight-binding approximation used, for example, this leads to the coupling equation. in which ^(^) ^ are the bosonic annihilation (respectively creation) operators of the ^-th waveguide and ^ ^ , ^ ^represent the couplings to L and R. Technically, for example, two indistinguishable photons can be generated from a Type I SPDC source (spontaneous parametric down-conversion, in German parametric fluorescence, via a CW laser with a BiB3O6 crystal), which allows the measurement of a heralded coincidence using photodetectors, for example, a set of avalanche photodiodes (APDs). Excitation of the three-waveguide array with a single photon generates a two-dimensional geometric subspace ℋgeo. For a cyclic motion of geometric subspace ℋgeo, a non-adiabatic holonomy ^ ^ of the group U(2). In one embodiment of the optical quantum gate, the non-adiabatic holonomy ^ ^ the shape where θ and ^ are constant, real factors, with a coupling function ^ ^(^) between the first waveguide structure and the central third waveguide structure as well as a coupling function ^ ^ (^) between the second waveguide structure and the central third waveguide structure as and ^ ^ ( ^ ) = Ω ( ^ ) cos^^ ^ 2 ^ (2) are parameterized, where Ω(^) is an enveloping function and the coupling functions ^ ^ (^), ^ ^ (^) depend on the refractive index of the medium, the respective distance of the first or second waveguide structure from the central third waveguide structure at the location z within the distance Z of the course of the waveguide structures as well as the predetermined wavelength and polarization, with an additional cyclicity condition is satisfied. In the paraxial approximation, which can be used as an example, the three waveguide structures follow a common straight principal direction, which is denoted here by z. Here, ^ ^ and ^ ^ for the spatial coordinates of the input and output of the gate, i.e. for the beginning (“initial”) and the end (“final”) of the segment Z. This resulting holonomy belongs to the symmetry group U(2) and behaves like a single-qubit gate for the dual-rail-encoded qubit A dense subset of the symmetry group can be generated by two consecutive non-commuting holonomies. Interestingly, and in contrast to the adiabatic approach, real-valued couplings (^ = 0) are sufficient to realize non-commuting holonomies. Non-adiabatic holonomies are nevertheless topologically protected. In particular, errors in the coupling are quadratically suppressed. The design of the waveguide structures for generating the quantum gate in the form of a holonomy depends on the coupling between the waveguide structures. The coupling ^ ^^ between two adjacent (single-mode) waveguide structures j and k can be represented in the tight-binding approximation as follows: where ^ and P are the wavelength and polarization of the photons used, ∆^ ^ (^, ^; ^, ^) and ^ ^(^, ^; ^, ^) are the profile of the refractive index or the transverse mode field of the kth waveguide or the kth waveguide structure, respectively. Here, x and y are transverse spatial coordinates. If these parameters are known, the coupling strength can be calculated directly. Alternatively, the coupling can be determined via calibration measurements. For example, if a specific polarization P is chosen and the waveguides are chosen to be identical with regard to the shape of their mode fields and refractive index profiles, the only remaining free parameter is the distance, which is simplified as Δ ^^ = − ^ ^ ^ can be expressed as long as the waveguides are offset from each other only in the x-direction or the direction of the offset is defined as the x-direction. For geometries other than planar, the distance must be defined completely accordingly. In any case, the coupling depends exponentially on the distance Δ ^^ away: The parameters a and b have as unit the inverse of a length, for example mm -1, and can be determined experimentally, for example, using appropriately manufactured arrays of directional couplers, each with two differently spaced waveguides of the same type as intended for a quantum gate, and whose length and spacing are known. Using the coupling parameters known from calculations or calibration measurements, the quantum gates according to the invention can then be adjusted by selecting the spacing and path of the three waveguides. Further developments of the quantum gate are described below.In one embodiment, the first, the second and the third waveguide structure run in a common main direction z over a distance Z between the respective input and the respective output, wherein in particular a length of the distance Z as well as the distances and distance profiles of the first and second wave structures with respect to the central third waveguide structure are each selected such that for photons of the predetermined wavelength and polarization the conditions of the non-adiabatic and non-Abelian holonomy ^. ^are met. However, it is not absolutely necessary for the three waveguide structures to follow a common, straight main direction z. It is also possible, for example, for the three waveguide structures to jointly describe a curved curve, which in extreme cases can also represent a ring. It is only necessary that the distance between the waveguide structures assumes the prescribed value at every point on the curve. In this case, the parameterization of the position z on the path Z follows the course of the curved curve. The straight course is a particularly simple special case. In embodiments, the first waveguide structure, the second waveguide structure and the central third waveguide structure are each designed either as a single waveguide or as a group of two or more evanescently coupled waveguides.This means that, within the scope of the present disclosure, each of the first, second, and third waveguide structures can be configured independently of the other two, either as a single waveguide or as a group of two or more waveguides. Within such a group, the individual waveguides forming the group are arranged so close together that they couple evanescently with one another. For example, such a group could be configured as a series of coupling waveguides running perpendicular to the plane in which the three waveguide structures are located (y-direction). In this case, the series of coupling waveguides would carry a supermode, rather than a single waveguide carrying one or more modes. Mathematically speaking, as well as within the scope of the present disclosure, such a group of waveguides is then considered to be one of the three waveguide structures.Other geometric arrangements in which a supermode is guided by multiple waveguides are also possible. In one embodiment, the waveguides of the waveguide structures are designed as optical fibers or as waveguides embedded in a medium whose predetermined refractive index is lower than a refractive index in the center of the waveguide. Various substrates, including fused silica and silicon, and the appropriate manufacturing methods are suitable for this. Implementation with air as the external medium is also possible. In embodiments, the waveguide structures are arranged in a cross-sectional plane perpendicular to the z-direction in a planar or triangular configuration with the central third waveguide structure between the first waveguide structure and the second waveguide structure.Such a configuration helps to effectively decouple the first and second waveguide structures from one another. The planar arrangement means that in the cross-sectional plane, all three waveguide structures are arranged in a line. Overall, this results in a configuration in which the three waveguides are arranged in a single plane that encompasses the z-direction and is particularly easy to manufacture. In non-planar embodiments with a triangular configuration of the waveguides, care must also be taken to ensure that the first waveguide is separated from the second waveguide in such a way that direct coupling between the two waveguides is suppressed. This can be achieved geometrically and / or with an additional separation structure.In the first case, connecting lines from the first waveguide to the central waveguide on the one hand and from the second waveguide to the central waveguide on the other hand can form an angle in the cross-sectional plane perpendicular to the z-direction that deviates from the planar case of 180°. If the first and second waveguide structures are spaced further apart from each other than from the central third waveguide structure, then a direct coupling between the first and second waveguide structures is suppressed compared to the respective coupling to the central third waveguide structure. In the second case, a light-absorbing or reflecting structure can be arranged in a direct connecting line between the first and second waveguide structures, which suppresses a coupling between the first and second waveguide structures.There is no structure that would suppress coupling between the central third waveguide structure and the first and second waveguide structures. Such a structure can, for example, be formed as a defect structure in the medium. The two alternatives of geometric and structural coupling suppression can also be used together. In embodiments, at least one of the first waveguide structure, the second waveguide structure, and the central third waveguide structure has a straight path parallel to the z-direction. Such straight, i.e., uncurved, structures are particularly easy to manufacture and allow simple calculation for setting a holonomy. In embodiments, two or all three of the waveguide structures also run straight.A curvature-free path also reduces losses in the total number of photons passing through the waveguide structures. In embodiments, at least one of the first waveguide structure, the second waveguide structure, and the central third waveguide structure has a curved path, in particular a cosine curve path, which at least partially approximates it to an immediately adjacent waveguide structure. The choice of cosine curve, in turn, reduces radiation losses of photons in the waveguides. In these embodiments, for example, one, two, or all three waveguides can have a curved path. Curved and straight waveguide structures can also be combined. For example, the two outer waveguides can be curved and the central waveguide straight. This variant offers high fidelity.Alternatively, the central waveguide can be curved and, for example, one of the two outer waveguides straight, or all three can be curved. A structure with two straight and one curved waveguide structure is a special case with which some, in particular those with θ = 0, π, 2π, ..., but not all, arbitrary holonomies can be created. In embodiments, at the inlet and / or outlet of the path Z and in the further course in the negative or positive z-direction, the first waveguide structure and the second waveguide structure are each spaced apart from one another in a curved shape, in particular a cosine shape, at a distance that allows optical fibers to be connected to the first waveguide structure and / or the second waveguide structure.In this context, the positive z-direction is associated with the direction of propagation of the photons from the input to the output, while the negative z-direction represents the opposite direction. In embodiments, the central third waveguide structure has a connection for an optical fiber at the output, and in particular also at the input. This makes it possible to measure photons in the central waveguide with sufficient sensitivity via a connection of an optical fiber and a photodetector. On the one hand, photons in the central waveguide represent losses with regard to the functionality of the gate, which can be determined in this way, and on the other hand, this value can be used as an inverse measure of the quality of the gate.The object underlying the invention is also achieved by a series of optical quantum gates in which a plurality of optical quantum gates are coupled in series, wherein for each pair of successive optical quantum gates, a first waveguide structure of the leading optical quantum gate of the pair is connected to a first waveguide structure of the subsequent optical quantum gate of the pair and a second waveguide structure of the leading optical quantum gate of the pair is connected to a second waveguide structure of the subsequent optical quantum gate of the pair, wherein at least one of the optical quantum gates of the series is designed as described above and has a non-adiabatic holonomy. realized. The series of optical quantum gates thus fulfills the same features, advantages, and properties as the previously described individual quantum gates, with the further advantage that any phases can be set and a dense subset of a symmetry group can be generated as a sequence of two or more non-commutating holonomies. In embodiments, the central third waveguide structures of the individual quantum gates in the series are also connected to one another. This simplifies manufacturing and, under certain circumstances, allows photon losses that remain in the central third waveguide structure despite the inventive design of the series of quantum gates to be quantified through appropriate measurements. In alternative embodiments, the central third waveguide structures of the individual quantum gates in the series are at least partially not connected to one another.This opens up the possibility of implementing the quantum gates using a sequence of exclusively rectilinear waveguides, of which the two outer waveguide structures are designed to be rectilinear and continuous, with a constant spacing between them, and the third central waveguide structure has individually defined spacings from the two outer waveguide structures for each of the series of quantum gates, which can vary from quantum gate to quantum gate. In this case, the individual central waveguide structures are not connected to each other. They can be spaced apart in the z-direction by a gap or directly connected to each other, which increases the compactness of the arrangement.In further embodiments, the first quantum gate of the series is provided on the input side and the last quantum gate of the series is provided on the output side with connections for optical fibers on the first waveguide structure and on the second waveguide structure. The object underlying the invention is also achieved by an optical quantum computer comprising at least one previously described optical quantum gate and / or at least one previously described series of previously described optical quantum gates. Such a quantum computer has a high level of robustness and efficiency due to the design of the quantum gates as non-adiabatic holonomies. Finally, the object underlying the invention is also achieved by using at least one previously described optical quantum gate and / or at least one previously described series of previously described optical quantum gates in an optical quantum computer.Further features of the invention will become apparent from the description of embodiments according to the invention together with the claims and the attached drawings. Embodiments according to the invention can fulfill individual features or a combination of several features. Within the scope of the invention, features marked with "in particular" or "preferably" are to be understood as optional features. The invention is described below, without limiting the general inventive concept, using exemplary embodiments with reference to the drawings, wherein express reference is made to the drawings for all details of the invention not explained in more detail in the text. They show: Fig. 1 an experimental setup with an optical quantum gate, Fig. 2 a)-d) various embodiments of quantum gates according to the invention, Fig.3 a), b) experimental results on the interference contrast for a quantum gate according to the invention and a fiber-integrated beam splitter, Fig. 4 a), b) rows of quantum gates in a first embodiment, Fig. 5 a), b) the rows of quantum gates shown in Fig. 4 a) and b) in a second embodiment, Fig. 6 a) - d) further rows of quantum gates in a third embodiment, Fig. 7 a) - c) further rows of quantum gates in a modification of the third embodiment and Fig. 8 a) - c) different configurations of the waveguides in a cross-sectional plane transverse to the z-direction. In the drawings, identical or similar elements and / or parts are provided with the same reference numerals, so that a repeated introduction is omitted in each case. Fig. 1 shows an optical experimental setup for investigating optical quantum gates according to the present disclosure.On an optical table 10, a coherent light beam from a continuous-wave laser 12 is focused into a bismuth borate crystal (BiB3O6) 14, where two indistinguishable photons are generated by parametric fluorescence (SPDC). The herald photon is guided via a delay fiber 16 to one of several detectors, which may be embodied as avalanche photodiodes 26, while the signal photon is coupled via a standard fiber arrangement with an optical fiber 18 into a photonic chip 20, into which a waveguide arrangement comprising a first waveguide structure, a second waveguide structure, and a central third waveguide structure is embedded, according to the present disclosure. In all subsequent examples, the three waveguide structures are embodied as individual waveguides.After the signal photon has passed through the waveguide structure in the photonic chip 20, it is collected by an optical fiber array whose optical fibers 21 are each optically coupled to the first waveguide, the second waveguide, and, in this case, the central third waveguide, and then detected by further avalanche photodiodes 26. These detectors are connected to a correlation card (not shown) that performs a coincidence measurement between the signal and herald photons. It is sufficient to read the outputs of the two outer waveguides of the waveguide arrangement in the photonic chip 20. In fact, it is also not possible to read the third central waveguide if it is not continuous.In the experiments shown in the following figures, the waveguides in each photonic chip 20 were introduced into a Corning 7980 fused silica substrate using focused femtosecond laser pulses, with an accuracy of approximately 225 nm. This is not the only material suitable as a substrate, and there are other techniques suitable for producing the structures. Due to the significantly higher refractive index, significant reduction in size can be achieved, for example, by using silicon as a substrate. Corresponding techniques for producing photonic waveguides in silicon substrates are known as silicon photonics and can achieve accuracies in the range of single nanometers.For the experiments and results shown below, a continuous wave laser 12 was used, which generated laser light with a wavelength of 407 nm, whereby the photons had a wavelength of 814 nm after SPDC. At the input and output of the photonic chip 20, the distance between the waveguides was increased to 82 μm to allow the connection of optical fibers with a certain diameter. The increase was carried out to minimize losses due to bending in a cosine form. For the used combination of wavelength, substrate and incorporated waveguide structure, a distance-dependent coupling strength of ^ ≈ 1.3 ^^ was determined using a tight-binding coupling scan. ^^ exp(−140 ^^ ^^ × ∆) (Fig. 2) or of ^ ≈ 2.3 ^^ ^^ exp(−160 ^^ ^^× ∆) (Figs. 4 to 7), where ∆ is the distance between the fibers in mm. The differences are explained by fluctuations in the laser system and minimally different properties of different glass samples. In all of the following examples, the three waveguides are in a planar configuration, which is particularly favorable both for the separation of the left and right waveguides and for manufacturing purposes. The following Figures 2 and 4 to 7 show various waveguide shapes that implement different optical quantum gates. The waveguide shapes each define the enveloping function Ω, which is used in the cyclicity condition (formula (3)) for non-adiabatic holonomies. All shapes shown fulfill the cyclicity condition and represent non-adiabatic holonomies.Figures 2 a) to d) show several examples of individual non-adiabatic holonomic quantum gates, each with a first waveguide 22.1 (top, "R"), a second waveguide 22.2 (bottom, "L"), and a central third waveguide 22.3 (middle, "C"). This nomenclature also applies to Figures 2 b) to 2 d) and 4 to 7 and is not repeated there for the sake of clarity. The horizontal axis corresponds to the z-direction of propagation in millimeters, while the vertical axis represents the separation between the three waveguides, also in millimeters. For simplicity, this transverse direction can also be called the x-direction or y-direction, depending on the convention. The first two examples (Figs. 2 a) and 2 b)) represent a so-called negative Pauli X-gate ^. ^ ^A Pauli-X gate has the property that a photon entering the left waveguide exits the right waveguide, while a photon entering the right waveguide exits the left waveguide. In matrix notation, a (negative) Pauli-X gate, based on the formula (4) shown above with ^ = ^ and ^ = 0 the form The two embodiments differ in that in Fig. 2 a) both outer waveguides follow a cosine shape, while in Fig. 2 b) the total 70 mm long cosine shape is interrupted by a straight, 29 mm long segment. Fig. 2 c) shows another standard gate, namely a Pauli-Z gate. ^ ^ . In the corresponding matrix notation with ^ = 0 and ^ = 0, this has the form In the Pauli-Z gate, the photons remain in the respective waveguide into which they initially entered. For this purpose, the right waveguide is designed with a sequence of half-cosine waveforms (35 mm long), a linear section (29 mm), and the complementary half-cosine waveform (35 mm long), while the left waveguide is at a constant distance from the central waveguide. Fig. 2d shows a third standard gate, a so-called ^ ^^ (negative) Hadamard gate ^ ^ (^ = ^ and ^ = 0), whose matrix notation The actual gate begins and ends at the horizontal bars shown in Fig. 2 d). This quantum gate features a 10 mm long connecting section at the input and output, where the waveguides are spaced apart in a fan-out pattern to create the required distance for connecting optical fibers. In these widely spaced regions, photons effectively do not transfer from one waveguide to the other. The quantum gate itself comprises a sequence of a 15 mm long half cosine shape, a 49 mm long straight section, and another 15 mm long half cosine shape in the left and right waveguides. The following experimental results were achieved with the four optical quantum gates of Figure 2: Quantum Transfer Matrix Reproduction Figure Gate Theoretical Experimental Fidelity 2a) Pauli-X ^0 1^ ^0.008 0.992^ 1 0 0.99 0.010 ^ ^= 99.1% 2b) Pauli-X ^0 1^ ^0.046 0.954 1 0 0.951 0.049 ^ ^ ^ = 95.3% 2c) Pauli-Z ^1 0^ ^0.998 0.002^ 0 1 0.004 0.996 ^ ^ = 99.7% 2d) Hadamard ^0.5 0.5^ ^0.577 0.423^ ^ ^ = 99.2% 0.5 0.5 0.403 0.597 Table 1 In the matrices, the values ​​are transition probabilities p kj from state j to state k. The values ​​on the diagonal of the matrix thus describe the probability that a photon remains in the left or right waveguide, while the two values ​​above and below the diagonal describe the probability that a photon switches from the left to the right waveguide and vice versa. The average fidelity ^ ^ (mean fidelity) in the last column of Table 1 and the following tables describes the agreement between theoretical and experimentally determined transition probability and is defined as It can be seen that high fidelity can be achieved with the cosine shape from Fig. 2a), as well as with the shape with a straight section. For the Pauli-Z gate and the Hadamard gate, high fidelity is also achieved with straight sections. The transition matrix of the ideal Hadamard gate is the same as that of a conventional beam splitter. For the example of the Hadamard gate from Fig. 2d), an experimental result for the interference contrast in an interference measurement in a Hong-Ou-Mandel configuration, in which the Hadamard gate was used as a beam splitter, is shown in Fig. 3a) for further verification. In Hong-Ou-Mandel interference, two indistinguishable photons from two different directions are passed through a beam splitter. The two photons can each be either transmitted or reflected, i.e., they can change waveguides or not.Since each of the two photons is either transmitted or reflected independently of the other, there are four possible final states, of which the final states in which both photons are measured in the left or right waveguide are directly distinguishable. Two further final states, namely those in which either both photons are transmitted or reflected, i.e., one photon is detected in each of the two output channels, are indistinguishable due to the indistinguishability of the photons. Their contributions cancel each other out due to their different signs, so this case does not occur with perfect quantum mechanical interference.This result is shown in the measurement plots of Figure 3, where the number of coincident photon pairs measured in the left and right waveguides ("coincidence counts") is shown as a function of the temporal shift ("translation") between the arrival times of the two photons. The experimental data points with their error bars were fitted with a Gaussian function. Fig. 3 a) shows that when using the Hadamard gate from Fig. 2 d), the coincidence is indeed strongly suppressed at a temporal shift of 0 ps. This results in a measured interference contrast of approximately 95.0%. The test experiments show that the realized structure is indeed a Hadamard gate, not only in terms of the probabilities, but also in terms of the relative phase. For comparison, Fig. 3b) shows the case in which a regular fiber-integrated beam splitter was used.Due to its design, the number of observed coincidences in the observed time window is significantly higher. In this case, the interference contrast is 98.7%. Fig. 4 shows, as a proof of principle, sequences or rows of three optical quantum gates each based on non-adiabatic holonomies in two different embodiments. Fig. 4 a) shows the sequence Hadamard – Pauli-X – Hadamard, and Fig. 4 b) the sequence Hadamard – Hadamard – Pauli-X. The optical quantum gates each have the basic structure of cosine shapes that are divided in half by straight sections (so-called cos-straight-cos shape), whereby the straight sections, as in Fig. 2, have the shortest distance to the central third waveguide in the course of the two outer waveguides.In addition, to match the outer waveguides from one gate to the next, an interstitial cosine-shaped connecting section was added not only at the front and end facets, but also between any two different gates, to ensure a smooth transition and accessibility of all three waveguides for measurements. No further connecting section was inserted between the Hadamard gates in the configuration shown in Fig. 4 b). The two gate rows achieved the following results: Quantum Transfer Matrix Reproduction Figure Gate Theoretical Experimental Fidelity Hadamard – 4a) Pauli-X – ^1 0^ ^0.990 0.010^ ^. ^ = 99.2% 0 1 0.007 0.993 Hadamard Hadamard – 4b) Hadamard – ^0 1^ ^0.088 0.912^ 1 0 0.917 0.083 ^ ^ = 91.5% Pauli-X Table 2 The high average fidelity of ^ ^= 99.2% for the first gate sequence shows that the additional coupling between the waveguides in the fan-shaped sections is minimal. Since these two gate sequences effectively replace the commutator ^^ ^^ realize, they elegantly demonstrate the non-Abelian nature of holonomic gates. So far, all gates and gate sequences have been designed so that the central waveguide is both straight and continuous. This allows the detection of any photons in the central waveguide. If the cyclicity condition of Eq. (3) is met, the probability of detecting a photon in the central waveguide is theoretically zero. Consequently, the probability that photons are detected in the central waveguide is a measure of how well the cyclicity condition is met. In all previous examples, this probability was below 10%. Due to the non-adiabatic nature of the presented gates, it is not absolutely necessary that the central waveguide is continuous across all gates in a sequence. In another example, shown in Fig.In the design approach shown in Fig. 5 a) and b), the same two gate sequences were implemented, with each individual gate again consisting of a cos-even-cos shape, but the central waveguide being discontinuous. This eliminates the need for interstitial connecting sections. Furthermore, a 1 mm gap in the propagation direction was inserted between two consecutive central waveguides to remove any photons remaining there due to possible malfunctions of the previous gate. With the exception of the transition between two identical gates, such as the two Hadamard gates in Fig. 5 b), where the central waveguides are already aligned due to the similarity of the gates, the gaps ensure that a photon probability remaining in the central waveguide at the end of a gate is not injected into the central waveguide of the next gate.This separation suppresses cumulative errors and increases the overall performance of the gate sequence. In this design, it is not necessary to measure the photon probability of the central waveguide, since it only reflects the deviation from the cyclicity condition of the last gate. Therefore, it can be omitted from the connecting sections. The two gate series shown in Fig. 5 achieved the following results: Quantum Transfer Matrix Reproduction Figure Gate Theoretical Experimental Fidelity Hadamard – 5a) Pauli-X – ^1 0^ ^0.976 0.024^ 0 1 0.031 0.946 ^. ^ = 97.2% Hadamard Hadamard – 5b) Hadamard – ^0 1^ ^0.063 0.994 1 0 0.971 0.029 ^ ^ ^= 98.3% Pauli-X Table 3 The final embodiments in Figures 6 and 7 present a radically simplified design approach for the realization of non-adiabatic holonomic gates and gate sequences. In this design approach, the gates are realized exclusively with straight waveguides. In the previous embodiments in Figures 1, 2, 4, and 5, the length of each gate and each section of the gate were chosen a priori, while all transverse distances were used as free parameters to satisfy the underlying equations (2) and (3). In this new approach, the left and right waveguides are completely straight throughout the entire gate sequence, and their spacing is chosen a priori. The position of the central waveguide is chosen such that equation (2) is satisfied, i.e., the position of the central waveguide determines the gate to be realized.The length of each gate is used as a free parameter to satisfy the cyclicity condition of Equation (3). As shown in Fig. 6, the influence of the coupling between adjacent central waveguides in the Hadamard-Pauli-X-Hadamard gate sequence was first investigated by reducing the distance between adjacent gates. The distances are 5 mm in Fig. 6 a), 3 mm in Fig. 6 b), 1 mm in Fig. 6 c), and 0 mm in Fig. 6 d; thus, there are no gaps between two consecutive gates in Fig. 6 d). The uniform distance between the outer waveguides in the gates was 45 μm. The results of the gate sequences realized in Fig.6, Hadamard – Pauli-X – Hadamard, are listed in the following table with the corresponding gaps: Fig. Quantum Gap Transfer Matrix Replay Gate Width Theoretical Experimental Supervised Hadamard 6a) – Pauli-X – 5 mm ^1 0^ ^0.985 0.016^ ^. ^= 98.7% Hadamard 0 1 0.009 0.991 Hadamard 6b) – Pauli-X – 3 mm ^ 1 0^ ^0.986 0.014^ ^ ^ = 98.5% Hadamard 0 1 0.015 0.985 Hadamard 6c) – Pauli-X – 1 mm ^1 0^ ^0.985 0.015^ ^ Hadamard 0 1 ^ = 98.7% 0.012 0.988 Hadamard 6d) – Pauli-X – 0 mm ^ 1 0^ ^0.967 0.033^ 0 1 0.029 ^ ^= 96.9% Hadamard 0.971 Table 4 In general, the presence of a gap between consecutive gates increases the average fidelity of a gate sequence (cf. difference in average fidelity in Table 4 between the first three rows and the last row). A larger spacing does not, however, reduce the average fidelity, since the additional coupling between the left and right waveguides is minimal. But even in the complete absence of a gap, very high fidelity values ​​can be achieved (Fig. 6 d)), which shows that the probability of photons being in the central waveguide is small for each individual gate. Fig. 7 shows several examples of rows with a larger number of 7 to 8 gates, where the outer waveguides are each spaced apart by approximately 35 μm. Figure 7 a) shows a row of 7 gates with alternating Hadamard and Pauli-X gates. Fig.Figure 7 b) shows a series of 8 Pauli-X gates in a row, and Figure 7 c) a sequence of arbitrarily and randomly selected gates, where θ has the following values: 1.701, 1.1557, 1.618, 1.0584, 0.74656, 0.9272, 0.8653. The measurements with these gate sequences yielded the following results: Fig. Quantum transfer matrix re-gate theoretical experimentally supervised 7 gates 7a) (3.5x Hadamard – ^0 1^ ^0.174 0.826^ ^ 1 0 0.807 0.193. ^ = 81.7% Pauli-X 7b) 8x Pauli-X ^1 0^ ^0.987 0.013^ 0 1 0.013 0.987 ^ ^ = 98.7% 7 gates with 7c) arbitrary ^0.047 0.953^ ^0.043 0.957 0.95 ^ ^ ^ = 99,8%chem θ 3 0.047 0.922 0.078 Table 5 It can be seen that the simple design approach for non-adiabatic holonomic quantum gates shown in Fig. 7 is suitable for realizing arbitrary gate sequences with high fidelity. The number of gates in a sequence is only limited by the length of the silica glass sample, since small distances between the waveguides can lead to detuning due to pre-existing strain fields and disrupt the tight-binding approximation. The detuning could be compensated by deliberately introducing detuning between neighboring waveguides in the laser writing process. In the following, the example of a Hadamard gate will be used to illustrate how the equations (1), (2), (3) and (5) introduced above can be used to construct a corresponding optical quantum gate.For example, a non-adiabatic holonomic quantum gate can consist of three straight, parallel waveguides. To create a (negative) Hadamard gate, for example, wird ^ = 3^ ^ 4 and ^ = 0. From equation (1) follows ^ ^ 1 ^^^^^^^^ = −^1 1 ^, √ 2 1 −1 which corresponds (except for the minus sign) to the definition of the Hadamard gate. From equations (2) and (5) we then have and where Δ ^ (^) is the distance between the left and the central waveguide and Δ ^ (^) is the distance between the right and central waveguides. The parameters ^ and ^ are the coupling parameters known from calibration measurements. Since all three waveguides are supposed to be straight and parallel, Δ ^ and Δ ^ constant and no longer dependent on ^. As a result, the coupling functions ^ are also simplified. ^ and ^ ^to constants, as is the enveloping function Ω. From these two equations for Ω follows for the distances between the waveguides: l n^tan^ ^ ^ ^^ ln^tan^ 3^ ^ ^^ where ln is the natural logarithm. One of the distances Δ ^ and Δ ^ can therefore be freely selected. In the embodiments shown in Fig.6, for example, the total distance between the left and right waveguides was set to Δ ^ + Δ ^ = 45 µm. To generate a non-adiabatic holonomy that implements the Hadamard gate described above, the cyclicity condition of equation (3) must be satisfied. Since the enveloping function Ω is constant in this example, the cyclicity condition simplifies to: This directly determines the total length ^ of the gate. The embodiments shown in Figure 6 are based on calibration measurements that determine the parameters ^ and ^ as ^ ≈ 2.3 ^^ ^^ and ^ ≈ 160 ^^ ^^ . This results, for example: Δ ^ ≈ 25.2 μm, Δ ^ ≈ 19.8 μm and ^ ≈ 30.9 mm. A non-adiabatic holonomic Hadamard gate can also be implemented with other geometries. For example, the left and right waveguides can be cosine-shaped, while the central waveguide is straight. In this case, the distance between the left and central waveguides can be generally described as Δ ^ (^) = ^ cos(^ ^ − ^) + ^, where A, B, C and D are real parameters yet to be determined. For example, one could assume that the cosine over the distance Z runs through a full period of 2^, so that ^ = 2^^ ^, where the total length Z of the gate can be freely chosen and, for example, can be based on the length of the photonic chip used. Additionally, it could be required that the distance between the left and central waveguide is maximal exactly at the front and end facets of the photonic chip, which leads to ^ = 0. Similar to before, an expression for the enveloping function is first formulated: To determine the remaining free parameters ^ and ^, one could apply a further geometric constraint. This could be that the distance between the left and central waveguide at the beginning and end of the path takes on a certain value to enable a seamless transition to optical fibers, for example, Δ ^ ( 0 )= 82 µm. This additional geometric condition is combined with the cyclicity condition to form a system of equations with the two unknowns ^ and ^: Δ ^ ( ^ ^ ) = ^ + ^ = 82 µm. This system of equations can be solved numerically, for example. From the now known distance between the left and central waveguide Δ ^ ( ^ ) In the following step, the distance between the right and central waveguide Δ ^ ( ^ ) According to equation (2), the couplings are 0.41 ^ ^ (^) and equation (5) allows an expression for the distances: Fig. 8 a) to c) show three different configurations of waveguides 22.1, 22.2 and 22.3 in a photonic chip 20 or a substrate or medium, which can also be air. Fig. 8 a) shows the planar arrangement in which all three waveguides lie in one plane, which is illustrated by the imaginary connecting line 24. In Fig. 8 b), the three waveguides 22.1, 22.2 and 22.3 are arranged in a triangular configuration in cross-section, with the imaginary connecting lines 24.1 and 24.2, which run from the first waveguide 22.1 and from the second waveguide 22.2 to the central third waveguide 22.3, forming an angle there. This means that the distance between the two outer waveguides 22.1 and 22.2 is greater than that to the central third waveguide 22.3, so that direct coupling between the two outer waveguides 22.1, 22.2 is suppressed. In Fig.Fig. 8 c) again shows a triangular configuration which, in contrast to Fig. 8 b), has a coupling-suppressing structure 23 in the line of sight (not shown) between the two outer waveguides 22.1, 22.2, which suppresses coupling between these two waveguides 22.1, 22.2. This structure can, for example, be a defect structure in the substrate or an inclusion of an opaque material. The lines of sight or imaginary connecting lines of the two outer waveguides 22.1, 22.2 to the central third waveguide 22.3 are not blocked, so that the coupling in this combination occurs according to the design of the respective gate. In summary, the disclosed technical teaching presents a robust and versatile platform for the realization of error-resistant photonic quantum gates.The realization of each individual gate as a non-adiabatic holonomy increases its performance and reduces errors due to manufacturing defects. Furthermore, this platform is very versatile and adaptable, as the gates can be realized with a variety of different waveguide geometries. The design concept shown in Figures 6 and 7 is particularly promising for industrial implementation, as it consists only of straight waveguides in a two-dimensional geometry, fully exploits the advantages of non-adiabaticity, and reduces the design effort by separating the gate definition (Eq. (2)) from the cyclicity condition (Eq. (3)). All mentioned features, including those that can be taken from the drawings alone as well as individual features that are disclosed in combination with other features, are considered essential to the invention, alone and in combination.Embodiments according to the invention can be fulfilled by individual features or a combination of several features within the scope of the patent claims.

[0002] List of reference symbols 10 Optical table 12 CW laser 14 BiB3O6 crystal 16 Delay fiber for Herold photons 18 Optical fiber 20 Photonic chip 21l Optical fibers 22.1 First optical waveguide 22.2 Second optical waveguide 22.3 Central third optical waveguide 23 Coupling-suppressing structure 24 Connecting line 24.1, 24.2 Connecting line 26 Avalanche photodiodes

Claims

Optical quantum gate, series of optical quantum gates, optical quantum computer and use 1. An optical quantum gate based on a non-adiabatic and non-Abelian holonomy for photons of a predetermined wavelength and polarization, comprising a medium with a predetermined refractive index and an arrangement having a first waveguide structure (22.1), a second waveguide structure (22.2), and a central third waveguide structure (22.3), wherein the central third waveguide structure (22.3) is arranged between the first waveguide structure (22.1) and the second waveguide structure (22.2), which each run between a respective input and a respective output, wherein the first waveguide structure (22.1) and the second waveguide structure (22.2) are spaced from one another such that transition probabilities for a photon of the predetermined wavelength directly from the first waveguide structure (22.1) to the second waveguide structure (22.2) and vice versa are negligible, wherein the courses of the first waveguide structure (22.1) on the one hand and the second waveguide structure (22.2) on the other hand with respect to the central third. Waveguide structure (22.3) are each selected so that for photons of the predetermined wavelength and polarization, the conditions of a non-adiabatic and non-Abelian holonomy ^ ^ are satisfied, from which transition probabilities from the first waveguide structure (22.1) to the second waveguide structure (22.2) and vice versa result.

2. Optical quantum gate according to claim 1, wherein the non-adiabatic holonomy ^ ^ the shape where θ and ^ are constant, real factors, with a coupling function ^ ^ (^) between the first waveguide structure (22.1) and the central third waveguide structure (22.3) and a coupling function ^ ^(^) between the second waveguide structure (22.2) and the central third waveguide structure (22.3) as and ^ ^ (^) = Ω(^) cos^^^ 2 ^, where Ω(^) is an enveloping function and the coupling functions ^ ^ (^), ^ ^ (^) depend on the refractive index of the medium, the respective distance of the first or second waveguide structure (22.1, 22.2) from the central third waveguide structure (22.3) at the location z within the distance Z of the course of the waveguide structures (22.1, 22.2, 22.3) as well as the predetermined wavelength and polarization, wherein additionally a cyclicity condition is fulfilled.

3. Optical quantum gate according to claim 1 or 2, wherein the first, the second and the third waveguide structure (22.1, 22.2, 22.3) extend in a common main direction z over a distance Z between the respective input and the respective output, wherein in particular a length of the distance Z as well as the distances and distance profiles of the first and second waveguide structures (22.1, 22.2) with respect to the central third waveguide structure (22.3) are each selected such that for photons of the predetermined wavelength and polarization, the conditions of non-adiabatic and non-Abelian holonomy ^ ^are met.

4. Optical quantum gate according to one of claims 1 to 3, wherein the first waveguide structure (22.1), the second waveguide structure (22.2), and the central third waveguide structure (22.3) are each designed either as a single waveguide or as a group of two or more evanescently coupled waveguides.

5. Optical quantum gate according to one of claims 1 to 4, characterized in that the waveguides of the waveguide structures (22.1, 22.2, 22.3) are designed as optical fibers or as waveguides embedded in a medium whose predetermined refractive index is smaller than a refractive index in the center of the waveguides.

6. Optical quantum gate according to one of claims 1 to 3, wherein the waveguide structures (22.1, 22.2, 22.3) are arranged in a cross-sectional plane perpendicular to the z-direction in a planar or triangular configuration with the central third waveguide structure (22.3) between the first waveguide structure (22.1) and the second waveguide structure (22.2).

7. Optical quantum gate according to claim 6, wherein in the triangular configuration of the waveguide structures (22.1, 22.2, 22.3), connecting lines (24, 24.1, 24.2) from the first waveguide structure (22.1) to the central waveguide structure (22.3) on the one hand and from the second waveguide structure (22.2) to the central waveguide structure (22.3) on the other hand form an angle in the cross-sectional plane perpendicular to the z-direction, which deviates from 180°, wherein the first waveguide structure (22.1) and the second waveguide structure (22.2) are spaced further apart from one another than from the central third waveguide structure (22.3) and / or ^ a light-absorbing or light-reflecting structure (23) is arranged in a direct connecting line (24, 24.1, 24.2) between the first waveguide structure (22.1) and the second waveguide structure (22.2), which suppresses coupling between the first waveguide structure (22.1) and the second waveguide structure (22.2).

8. Optical quantum gate according to one of claims 1 to 7, wherein at least one of the first waveguide structure (22.1), the second waveguide structure (22.2), and the central third waveguide structure (22.3) has a rectilinear course parallel to the z-direction.

9. Optical quantum gate according to one of claims 1 to 8, wherein at least one of the first waveguide structure (22.1), the second waveguide structure (22.2) and the central third waveguide structure (22.3) has a curved course, in particular a course of a cosine curve, which approximates it at least in sections to an immediately adjacent waveguide structure (22.1, 22.2; 22.2, 22.3).

10. The optical quantum gate according to claim 3, wherein, at the input and / or output of the path Z, in the further course in the negative or positive z-direction, respectively, the first waveguide structure (22.1) and the second waveguide structure (22.2) are each spaced apart from one another in a curved shape, in particular a cosine shape, at a distance that allows connection of optical fibers (18, 21) to the first waveguide structure (22.1) and / or the second waveguide structure (22.2). 11.Optical quantum gate according to one of claims 1 to 10, wherein the central third waveguide structure (22.3) has a connection for an optical fiber (18, 21) at its output, and in particular also at its input.

12. A series of optical quantum gates in which a plurality of optical quantum gates are coupled in series, wherein for each pair of successive optical quantum gates, a first waveguide structure (22.1) of the leading optical quantum gate of the pair is connected to a first waveguide structure (22.1) of the subsequent optical quantum gate of the pair, and a second waveguide structure (22.2) of the leading optical quantum gate of the pair is connected to a second waveguide structure (22.2) of the subsequent optical. Quantum gate of the pair, wherein at least one of the optical quantum gates in series is formed according to one of claims 1 to 11 and has a non-adiabatic holonomy 13. A series of optical quantum gates according to claim 12, wherein the central third waveguide structures (22.3) of the individual quantum gates in the series are connected to one another or at least partially unconnected to one another.

14. A series of optical quantum gates according to one of claims 10 to 12, wherein the first quantum gate in the series is provided on the input side and the last quantum gate in the series is provided on the output side with connections for optical fibers (18, 21) to the first waveguide structure (22.1) and to the second waveguide structure (22.2).

15. An optical quantum computer comprising at least one optical quantum gate according to one of claims 1 to 11 and / or at least one row of optical quantum gates according to one of claims 12 to 14.

16. Use of at least one optical quantum gate according to one of claims 1 to 11 and / or at least one row of optical quantum gates according to one of claims 12 to 14 in an optical quantum computer.

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