Method for determining at least one total charge transfer of a charge stability diagram, computer program and computer-readable medium

Abstract

The present invention relates to a method for determining at least one total charge transition (3) of a charge stability diagram (1) of a quantum dot system comprising at least two quantum dots, wherein S1) to which at least one total charge transfer (3) parameters are provided and / or determined, S2) based on the parameters at least one geometric representation (11) for the at least one total charge transition (3) is created, wherein the at least one geometric representation (11) comprises two linear sections (11a, 11c, 11e, 11g) and a curve (11b, 11d, 11f) connecting the two linear sections and optionally one or more further curves (11b, 11d, 11f) and one or more further linear sections (11a, 11c, 11e, 11g), and S3) preferably uses at least one constructed geometric representation (11) to assess the state of the quantum dot system.

Description

[0001] The invention relates to a method for determining at least one total charge transition of a charge stability diagram of a quantum dot system comprising at least two quantum dots. Furthermore, the invention relates to a computer program and a computer-readable medium.

[0002] Charge stability diagrams are used in particular to analyze the charge states of quantum dots based on structures contained in the diagrams. Charge stability diagrams are usually recorded using charge measurement methods and as a function of two voltages. For the metrological recording of a charge stability diagram for a quantum dot system, in particular a double quantum dot, another quantum dot can be used, for example, which is capacitively coupled to the quantum dot system or the double quantum dot. This is often also referred to as a sensing dot. The conductance through the sensing dot reacts to the change in the charge distribution in the quantum dot system, in particular the double quantum dot, resulting in structures in the form of edges in the charge stability diagrams.Charge stability diagrams required for calibrating the total charge are often measured as a function of two gate voltages. These are preferably the so-called plunge gates, which control the electrochemical potential of the quantum dots. So-called barrier gates can optionally be placed before, between, and after the plunger gates. These can serve to shield the quantum dots from the environment and from each other. They can, in effect, prevent electrons from entering or exiting the quantum dots.

[0003] It is known to use partial structures of charge stability diagrams to investigate various problems.

[0004] Among other things, the typical pattern of so-called triple dots is exploited, as described in the articles "Computer-automated tuning of semiconductor double quantum dots into the single-electron regime" by TA Baart et al., Applied Physics Letters 21, 108 (2016), DOI: 10.1063 / 1.4952624 and "Computer-automated tuning procedures for semiconductor quantum dot arrays" by AR Mills et al., Applied Physics Letters 115, 11:113501 (2019), DOI: 10.1063 / 1.5121444 and "Automated tuning of inter-dot tunnel coupling in double quantum dots" by CJ van Diepen et al., Applied Physics Letters 113, 033101 (2018), DOI: 10.1063 / 1.5031034 and "Tuning Methods for Semiconductor Spin Qubits" by T. Botzem et al., Phys. Rev. Applied 10, 054026 (2018), DOI: 10.1103 / PhysRevApplied.10.054026.

[0005] It is also known to exploit the parallelism of the line-shaped charge transitions, as can be seen from the articles "Automatic virtual voltage extraction of a 2x2 array of quantum dots with machine learning" by GA Oakes et al., arXiv:2012.0368 (2020), DOI: 10.48550 / arXiv.2012.03685 and the aforementioned article "Computer-automated tuning procedures for semiconductor quantum dot arrays" and "Image Processing Algorithms for Tuning Quantum Devices and Nitrogen-Vacancy Imaging" by CP Monical et al. (2020), DOI: 10.2172 / 1662017 and "Algorithm for automated tuning of a quantum dot into the single-electron regime" by M. Lapointe-Major et al., Phys. Rev. B 102, 085301 (2020), DOI: 10.1103 / PhysRevB.102.085301.

[0006] Simulating charge stability diagrams is also known. Two essentially known methods are used for this.

[0007] On the one hand, there is the so-called capacitive model, a model of classical physics that neglects any quantum effects. This is discussed, for example, in the article "Electron transport through double quantum dots" by WG van der Weil et al., Rev. Mod. Phys. 75, 1 (2002), DOI: 10.1103 / RevModPhys.75.1.

[0008] Another approach is the so-called Hubbard model, which incorporates quantum effects such as tunnel coupling between quantum dots. This can be found, for example, in the articles "Generic Hubbard model description of semiconductor quantum-dot spin qubits" by S. Yang et al., Phys. Rev. B 83, 161301(R) (2011), DOI: 10.1103 / PhysRevB.83.161301 and "Hubbard model description of silicon spin qubits: Charge stability diagram and tunnel coupling in Si double quantum dots" by S. das Sarma et al., Phys. Rev. B 83, 235314 (2011), DOI: 10.1103 / PhysRevB.83.235314 and "Quantum theory of the charge-stability diagram of semiconductor double-quantum-dot systems" by X. Wang et al., Phys. Rev. B 84, 115301 (2011), DOI: 10.1103 / PhysRevB.84.115301.

[0009] A disadvantage of the known approaches for simulating charge stability diagrams is that the number of required parameters grows quadratically with the isotropic expansion of the simulation space. Furthermore, the known methods do not account for deformation due to quantum effects. They have limitations regarding the representation of significant variations.

[0010] For example, when using the capacitive model, all geometric quantities remain constant throughout the charge stability diagram, contrary to reality. Furthermore, physical effects (e.g., tunnel coupling), which significantly influence the structure(s) of the diagrams, are not taken into account. The Hubbard model does consider some of these effects. However, the maximum simulation space is generally significantly limited, in particular to a maximum of two electrons per quantum dot. The complexity of a complete physical model is very high, so calculating a single diagram requires enormous computational effort.

[0011] It is therefore an object of the present invention to provide a way of describing structures in charge stability diagrams in a simple, flexible and at the same time precise manner, in particular in order to simulate charge stability diagrams based on such a description.

[0012] This object is achieved according to the invention by a method for determining at least one total charge transition of a charge stability diagram of a quantum dot system comprising at least two quantum dots, in which S1) parameters belonging to at least one total charge transfer are provided and / or determined, S2) at least one geometric representation for the at least one total charge transition is created on the basis of the parameters, wherein the at least one geometric representation comprises two linear sections and a curve connecting the two linear sections and optionally one or more further curves and one or more further linear sections, and S3) preferably the at least one created geometric representation is used to assess the state of the quantum dot system.

[0013] In other words, the invention proposes a combination of linear sections and one, preferably several curves, in other words curved sections or curve components, for the coherent representation of total charge transitions. The geometric representation(s) with a combination of linear sections and connecting curve(s) enables a simple, flexible, and at the same time precise description of those structures, in particular edges, in charge stability diagrams that are formed therein by total charge transitions. The curve(s) of the respective geometric representation can describe the deformation due to the tunnel coupling, in particular of two quantum dots, preferably a double quantum dot, with just a few parameters. The linear sections can be defined, for example, by just two points or by their lengths and gradients and can thus be described.

[0014] The at least one geometric representation created according to the invention for determining the at least one total charge transfer is a low-complexity representation of the respective total charge transfer and, in particular, makes it possible to obtain sufficiently realistic simulations of charge stability diagrams within a reasonable computing time. The computing time can, in particular, be shorter than the computing time of physical models, which typically require solving a comparatively complex system of equations.

[0015] In the representation proposed according to the invention via the geometric representation(s) with linear sections and curve(s), the number of parameters increases only in the main direction of the total charge increase and thus linearly.

[0016] The respective geometric representation, or in the case of multiple representations, represents a particularly coherent, continuous description, representation, or reconstruction of a total charge transition. Each linear segment is connected to one or two curves, and each curve is connected to one or two linear segments. Linear segments and curve(s) alternate in the geometric representation of a total charge transition.

[0017] The respective geometric representation is, in particular, a description or representation of the or—in the case of several—the respective total charge transfer and can, for example, be represented or given mathematically and / or pictorially. It can also be said that it represents a reconstruction of the respective total charge transfer or serves to reconstruct it. The method can thus also be referred to as a method for reconstruction. A pictorial representation can, for example, be in the form of a continuous line, in particular, with straight and one or more curved sections that alternate.

[0018] The respective geometric representation may comprise one or more mathematical formulas or functions, or may be given by one or more mathematical formulas or functions. The formula / function is / are then created based on the parameters provided and / or determined in step S1, in particular derived from them. Alternatively or additionally, at least one formula or function may also be or have been obtained by a fitting method.

[0019] The at least one geometric representation may alternatively or additionally also comprise a plurality of points or the coordinates of a plurality of points that lie on the linear sections and the one or more curves or that form them. It may be in the form of a table or a sub-region of a table with coordinates of a plurality of points or may comprise one such table. A plurality of points or associated coordinates may have been obtained by solving at least one mathematical formula or function for a given parameter range or given parameter ranges, in particular for two voltages.

[0020] Software or a software module can be used to create, in particular calculate, the at least one geometric representation based on provided parameters that can be transferred to the software or software module, for example, via a suitable interface. Such software or such a software module can be part of simulation software for obtaining simulated charge stability diagrams. If parameters are (also) determined in step S1, the software or software module can also be designed to determine parameters.

[0021] To obtain a geometric representation with at least two linear segments and at least one curve connecting two linear segments, the gradients and lengths of the two linear segments, as well as two anchor points of the connecting curve, can be used. The two anchor points are then conveniently the endpoints of the curve, where the curved line ends and the respective linear segment transitions. A further, third anchor point for the curve, conveniently located between the two (end) anchor points, can then also be calculated, for example, based on the two (end) anchor points and the gradients of two linear segments.

[0022] It is useful to assume that the slope of the curve at the two (end) anchor points corresponds to the slope of the adjacent linear segment. Then the geometric representation is continuous and differentiable.

[0023] The quantum dot system is, in particular, one with at least one double quantum dot. In other words, the quantum dots or at least two quantum dots of the quantum dot system preferably form a double quantum dot. This refers to two adjacent quantum dots that can interact with each other due to their spatial proximity. It is preferred that they jointly serve as a computing unit for subsequent quantum algorithms.

[0024] A total charge transition represents a boundary in the voltage space that separates two total charge states of the system. Thus, the total charge of a quantum dot system, especially a double quantum dot system, is exactly one more electron on one side of a total charge transition than on the other.

[0025] One embodiment of the method according to the invention is characterized in that in step S2 at least one geometric representation with two or more curves is created, wherein all curves have the same shape and extent. It can be provided that a curve or its course is determined or created and then it is assumed that all curves have this course. In other words, the periodicity can be used. This advantageous simplification makes it possible, in particular, for the complexity of the representation to be or become independent of the size of the overall charge transfer. It should be noted that the curves with the same course will generally not all have the same alignment or orientation in the (voltage) space.

[0026] According to a particularly preferred embodiment, at least one geometric representation with two or more curves is created in step S2, wherein at least one curve is obtained, in particular calculated, by a particularly affine transformation of at least one other curve. Further preferably, the particular affine transformation includes or is provided by a translation and in particular a rotation by 180° of the other curve. The particular affine transformation is or was expediently carried out while maintaining continuity.

[0027] In other words, starting from a (first) curve, the or each further curve is obtained by rotating it by 180° and shifting it (in stress space), preferably while maintaining continuity.

[0028] The total number of curves of a charge transition from n to n+1 electrons is 2*(n+1)-1. Furthermore, the linear section following the ends of a first and last curve of a total charge transition is not limited and represents the so-called single quantum dot region.

[0029] A further embodiment is characterized in that, in step S2, at least one geometric representation with two or more curves is created, and for the two curves, or for each two curves connected by a linear segment, they are rotated by 180° relative to each other. In other words, each curve is followed by a curve that is rotated by 180° relative to it, which particularly well accounts for the basic structure of total charge transitions with multiple curves.

[0030] In the event that at least one geometric representation is created with three or more linear sections, it is expedient that the gradient of one linear section corresponds to the gradient of the next but one linear section.

[0031] The parameters required for creating the geometric representation can be provided and / or determined in step S1. They can either all be provided or all be determined. It is also possible for some of the parameters to be provided in step S1 and another part to be determined. In other words, the method according to the invention can utilize already known parameters and / or the method can include the determination of parameters.

[0032] It is possible to determine the parameters required to create the geometric representation in different ways.

[0033] For example, it may be provided that at least some of the parameters, preferably all of the parameters, are determined in step S1 using a measured charge stability diagram of the quantum dot system. In other words, measurement data can be used to determine the parameters.

[0034] Furthermore, it can be provided that several points are identified in the measured charge stability diagram that lie at least approximately on a total charge transition contained therein, and that the several points and / or variables derived from the several points are used as parameters for step S2. The coordinates of the identified points are expediently determined and used.

[0035] Total charge transitions are usually recognizable in measured charge stability diagrams as more or less pronounced gradients, one could also say "edges," with a characteristic progression of alternating straight and curved sections. Several consecutive total charge transitions usually form a "honeycomb" structure or pattern in the charge stability diagram. For example, a user can visually identify a gradient or "edge" that constitutes a measured total charge transition in a measured charge stability diagram and select suitable points on it, in particular reading the coordinates of the points. Alternatively or additionally, it is possible to use software-based line or edge detection tools for point identification and, in particular, coordinate determination.

[0036] The plurality of points, or at least one of the plurality of points, can be used directly as a parameter for step S2, for example. Alternatively or additionally, it is possible to derive quantities from the plurality of points or a portion of these, which are then used as parameters for step S2. In other words, the points themselves can form or represent the parameters and / or the points can be used to determine suitable parameters from them. Purely as an example, the coordinates of the points are determined and, if they lie on a linear section, the gradient and / or length of the linear section is determined and used as a parameter for step S2.

[0037] The identification of the multiple points can also be achieved in various ways. On the one hand, it is possible for a user to identify and, in particular, mark the points in the measured charge stability diagram. Alternatively or additionally, the points can be identified using at least one suitable image analysis software tool, preferably in the form of an edge and / or line detector.

[0038] Preferably, eight or nine or ten or eleven parameters are provided and / or determined in step S1.

[0039] For example, the gradients and lengths of two linear sections and the coordinates of two anchor points of a curve connecting them can be provided and / or determined as parameters. This would be an example of eight parameters (two gradients and two lengths of the linear sections, as well as two X-coordinates and two Y-coordinates of the anchor points). The anchor points are expediently endpoints of the curve. In the event that points are identified in a measured charge stability diagram (with or without software support), a first and a last point of a curved section of the edge representing the overall charge transfer, particularly due to a deformation caused by tunnel coupling, are then expediently identified as the first and second anchor points.In other words, the two points at which the curvature ends and the subsequent linear section begins are chosen as anchor points.

[0040] Alternatively or in addition to accessing measured charge stability diagrams for parameter determination, it is also possible to derive them from the physical system. Accordingly, a further embodiment of the method according to the invention is characterized in that the parameters are determined in step S1 by deriving them from properties of the quantum dot system. The derivation of the parameters can be based on a model that requires or includes essential physical quantities, e.g., all applied voltages and the strength of the influence (the coupling) of the gates on the potentials of the quantum dots and barriers, although this is to be understood purely as an example.

[0041] A further embodiment is characterized in that, in step S2, at least one geometric representation is created that extends over a stress space that differs from the stress space spanned by the measured charge stability diagram. Then, preferably, the stress space over which the at least one geometric representation extends is larger in at least one dimension than the stress space spanned by the measured charge stability diagram. In other words, it is also possible, for example, to make predictions for neighboring areas by extrapolation based on the geometric representation(s) from comparatively small measurement areas.

[0042] It can also be provided that the creation of the at least one geometric representation in step S2 includes fitting at least one fitting function to at least some of the parameters provided or determined in step S1 or variables derived therefrom. A fitting can be performed, for example, at a plurality of points that were previously identified, in particular, in a measured load stability diagram.

[0043] A further particularly preferred embodiment of the method according to the invention is characterized in that the or the respective curve of the at least one geometric representation is given or represented by a Bézier curve, preferably by a Bézier curve of the 2nd degree.

[0044] Bézier curves can be defined by comparatively few parameters, which in turn makes it possible, within the scope of the present invention, to describe a deformation caused by the tunnel coupling of two quantum dots with particularly few parameters. A Bézier curve, especially of the second order, can be described, for example, by two (end) anchor points and a third, middle anchor point located between these two.

[0045] For the respective Bézier curve(s), a third, in particular middle anchor point of the respective Bézier curve(s) can be determined, preferably calculated, from the two associated (end) anchor points and the gradients of two linear sections connected by the respective Bézier curve(s).

[0046] It has proven particularly suitable if, in step S1, the gradients and lengths of two linear sections and the coordinates of two anchor points of a curve connecting the two linear sections and, in particular, if two or more additional curves are present, an identifier for the total number of curves are determined and / or provided as parameters. This is particularly true if a measured load stability diagram is used for parameter determination, from which these variables can then be taken or derived.

[0047] An identifier for the total number of curves can, in particular, represent a ninth parameter, which can be determined or provided in addition to the eight parameters mentioned above (two gradients, two lengths, two X-coordinates, and two Y-coordinates). The identifier is, in particular, an integer that specifies how many curved sections, i.e., curves, the total charge transfer comprises. It is sometimes not always possible to determine exactly how many curves a total charge transfer, in particular of a measured stability diagram, has. A total number can therefore also be assumed or estimated. In other words, the identifier for the total number of curves can also be an identifier for an assumed or estimated total number of curves of a particularly measured total charge transfer.

[0048] Another particularly preferred embodiment is characterized in that the voltage space spanned by the measured charge stability diagram is affinely transformed such that the gradients of the linear sections are in a rational number space, in particular, are non-zero or infinite. In other words, such a transformation allows the gradient to always be specified as a rational number. This facilitates the handling of the representation. For example, the linear representation Y(x) = m*x + n can be used to equate the two linear components and calculate a third, middle anchor point.

[0049] In a preferred development, a computer simulation is performed to obtain a simulated charge stability diagram for the quantum dot system, and the at least one geometric representation created in step S2 is used within the framework of the computer simulation. In other words, the creation of at least one geometric representation of an overall charge transfer provided according to the invention can form a basic building block for simulation software for obtaining simulated charge stability diagrams. Such simulations are an important building block for the development and fair comparison of algorithms for automating qubit calibration, and the present invention can also make a contribution here.

[0050] The stress space spanned by the simulated charge stability diagram can differ from the stress space spanned by the measured charge stability diagram. Preferably, the stress space spanned by the simulated charge stability diagram is larger in at least one dimension than the stress space spanned by the measured charge stability diagram. In other words, even in the case of simulations, for example, it is possible to make predictions for neighboring areas from comparatively small measurement ranges by extrapolation.

[0051] The optional step S3 concerns the assessment of the state of the quantum dot system, for which the at least one geometric representation created according to the invention can be used. The method according to the invention can therefore also be used to assess the state of the quantum dot system, which has proven to be a particularly suitable application.

[0052] Various physical quantities and / or system properties can be derived from the geometric representation of the total charge transfer or the geometric representations of the total charge transitions.

[0053] For example, the sum of the electrons present in the quantum dot system can be determined on the basis of the at least one geometric representation, in particular the sum of the electrons present in a double quantum dot comprised by the quantum dot system or forming the quantum dot system.

[0054] Alternatively or additionally, it is possible to determine the charge transition between two quantum dots (interdot transition) of the quantum dot system, in particular between two quantum dots of a double quantum dot, based on the at least one geometric representation. This is preferably done from opposite third anchor points of two curves of the geometric representations of two adjacent total charge transitions. These can, in particular, form so-called triple points.

[0055] Furthermore, alternatively or additionally, an approximate strength of the tunnel coupling between two, in particular adjacent, quantum dots of the quantum dot system can be determined based on the at least one geometric representation. This is preferably determined from the width of at least one curve of the at least one geometric representation.

[0056] It is also possible that approximated capacitive couplings are determined on the basis of the at least one geometric representation, in particular from the gradients of linear sections of the at least one geometric representation.

[0057] As noted, it is in principle possible to create a geometric representation for only one total charge transition. However, it will generally be particularly expedient to do this for multiple total charge transitions of a charge stability diagram. Accordingly, a further embodiment is characterized in that, in step S1, parameters associated with two or more total charge transitions of a charge stability diagram are provided and / or determined, and in step S2, based on the parameters, a geometric representation is created for each total charge transition, which comprises two linear sections and a curve connecting the two linear sections and optionally one or more further curves and one or more further linear sections, and in step S3, the multiple geometric representations are preferably used to assess the state of the quantum dot system.

[0058] The present invention can also be used for system calibration. A further advantageous embodiment is characterized in that the at least one geometric representation created in step S2 is used for a particularly automatic calibration of the quantum dot system. The calibration preferably includes converting the quantum dot system into a desired charge configuration. This is achieved, in particular, using the total charge and the charge transitions between two quantum dots.

[0059] It is also possible for the at least one geometric representation created in step S2 to be used for or taken into account in the operation of a quantum dot system, in particular a quantum dot system comprising at least two quantum dots, preferably at least one double quantum dot. In other words, the method according to the invention can also be used for the operation of a quantum dot system. The at least one geometric representation created in step S2 can be used in particular for the control of a quantum dot system, in particular a quantum dot system comprising at least two quantum dots, preferably at least one double quantum dot.

[0060] As noted, in an advantageous further development, the method according to the invention can be implemented with software support. In particular, at least one piece of software can be used to carry out the steps of the method according to the invention.

[0061] A further subject matter of the invention is accordingly a computer program comprising instructions which, when the program is executed by a computer, cause the computer to carry out the steps of the method according to the invention.

[0062] The software, in other words the computer program, may comprise a library for calculating curves, preferably Bezier curves.

[0063] The invention also relates to a computer program product comprising instructions which, when the program is executed by a computer, cause the computer to carry out the steps of the method according to the invention.

[0064] Further features and advantages of the present invention will become apparent from the following description of embodiments of the invention with reference to the accompanying drawings. Fig. 1 a measured charge stability diagram of a quantum dot system, Fig. 2 the charge stability diagram Fig. 1, in which several points lying on a total charge transition are identified, Fig. 3 a geometric representation of the total charge transfer from Fig. 2, on which the points were identified, Fig. 4 another geometric representation for a total charge transfer shown in an affinely transformed voltage space to ensure that the gradients of the linear sections are in the rational number space, Fig. 5 the geometric representation from Fig. 4 shown in the original stress space, and Fig. 6 the geometric representation of the total charge transfer from Fig. 5 together with the geometric representations of three other total charge transitions from the same charge stability diagram.

[0065] Identical or corresponding elements and components are provided with the same reference numerals in the figures.

[0066] The Fig. Figure 1 shows a measured charge stability diagram 1 of a double quantum dot (not shown). To measure the charge stability diagram 1, another quantum dot, referred to as a sensor quantum dot, was used. This quantum dot is capacitively coupled to the double quantum dot. The conductance through the sensor dot reacts to the change in the charge distribution in the double quantum dot, resulting in structures in the form of edges in the charge stability diagram 1.

[0067] In charge stability diagram 1, the measured sensor response is plotted in grayscale against the two voltages V1 and V2. In this case, it was measured using RF reflectometry, but this should be understood as an example. The voltages V1 and V2 are the voltages that are or were applied to the gates of the double quantum dot, known as plunger gates. These gates are used to control the potential, and thus the number of electrons, of the respective quantum dot. Since a plunger gate influences not only the associated quantum dot but also the surrounding quantum dots, it is advantageous to adjust the two quantum dots forming a double quantum dot together. A charge stability diagram as a function of the two plunger gate voltages thus offers the possibility of calibrating both quantum dots independently of each other, in particular of determining the appropriate plunger gate voltages.

[0068] The measured charge stability diagram 1 shows several total charge transitions, of which Fig. Four of these can be seen in Figure 1. These are designated by the reference numbers 2, 3, 4, and 5. The total charge transitions 2-5 are clearly defined by linear structures or are recognizable as edges. Each total charge transition 2-5 comprises alternating straight sections, in other words, linear sections, with a largely fixed length and gradient, which are connected to one another by curved sections, in other words, curves. The curved sections of the total charge transitions 2-5 are due in particular to the deformation caused by the tunnel coupling of the quantum dots.

[0069] For example, in the total charge transfer 3 one finds - in Fig. 2, viewed from left to right, shows a linear section 3a, followed by a curve 3b, followed by a linear section 3c, another curve 3d, and another linear section 3e. The total charge transfer 4 shows a linear section 4a, a curve 4b, a linear section 4c, another curve 4d, and a third linear section 4e.

[0070] It should be emphasized that the indices ae were chosen only as examples in light of the given voltage space. For example, the linear section 4a does not necessarily have to be the first linear section of the total charge transition 4; rather, further sections (with arbitrary indices) can follow to the left of it. It should also be noted that stability diagram 1 can, in principle, contain further total charge transitions that lie outside the voltage space spanned by V1 and V2, and thus outside the image space.

[0071] The embodiment of the method according to the invention described below enables, among other things, a simple, flexible and at the same time precise description of the total charge transitions forming structures in charge stability diagrams.

[0072] In a first step S1, parameters belonging to at least one of the total charge transitions 2-5 are provided and / or determined, and in a step S2, based on parameters provided and / or determined in step S1, at least one geometric representation 11 (cf. Fig. 3) for the at least one total charge transition 3 in the measured charge stability diagram 1.

[0073] In the embodiment described here, in step S1, several points in the charge stability diagram recorded by measurement technology, i.e. containing measured values, are determined from Fig. 1. It should be noted that, alternatively or in addition to providing and / or determining parameters using measurement data, it is also possible in principle to derive parameters for one or more total charge transitions from properties of the quantum dot system.

[0074] In the following, the identification of several points is described as an example, which are located on the total charge transfer marked with the reference number 3 of the stability diagram 1 from the Fig. 1 and Fig. 2. It is understood that, in a completely analogous manner, several points can be identified on one or more of the other total charge transitions 2, 4 and 5.

[0075] In this case, five points 6-10 are identified on the total charge transfer 3, which is also to be understood as an example. The points 6-10 are in the Fig. 2, which again shows the charge stability diagram from Fig. 1 but with the five points 6-10 marked therein.

[0076] The following applies to points 6-10. Points 6 and 7 are the two endpoints of the Fig. 2 of the leftmost curved section 3b of the total charge transfer 3. At these points, where the curved course ends, the two adjacent linear sections 3a and 3c adjoin section 3b. Points 6 and 7 are therefore also end points of the adjacent linear sections 3a, 3c. Point 8 is the other end point of the linear section 3c, which adjoins the curved section 3b to the right and lies at the end of the linear section 3c facing away from section 3b. Points 9 and 10 are the two end points of the next - or the next but one, starting from the first linear section 3a - linear section 3e. Between the two linear sections 3c and 3e lies another curved section 3d. The two points 8 and 9 are the two end points of the curved section 3d. At point 10 begins the next curved section of the total charge transfer 3, which is in the Fig. 2, however, is no longer fully recognizable. The gradient of a linear section corresponds to the gradient of the next but one linear section, or a corresponding assumption is made for the creation of at least one geometric representation 11. In the case of the total charge transfer 3, this applies, for example, to the linear sections 3a and 3e. The same can apply or be assumed for the lengths.

[0077] The identification of points 6-10 can be carried out, for example, by a user who observes the charge stability diagram 1 and identifies the points 6-10, which are located at the ends of the linear or curved sections of the total charge transfer 3, and draws them by hand if necessary.

[0078] From the vectors spanned between the pair of points 7 and 8, on the one hand, and between the pair of points 9 and 10, on the other hand, the length and gradient of the linear section 3c, and on the other hand, the length and gradient of the linear section 3e—and thus 3a—can be determined. In other words, these quantities can be derived, in particular calculated, from points 7-10 or their coordinates on the X and Y axes using the voltages V1 and V2.

[0079] It should be noted that the lowest total charge transition (between 0 and 1 electrons in the system) of a charge stability diagram 1 has only one curved section and two linear sections, with the curved section being open towards the origin, i.e. like sections 3b and 4d in Fig. 2. For total charge transitions with more than one curved section, it also applies that they begin and end with a curved section of this orientation.

[0080] Points 6 and 7 or points 8 and 9 are considered or used as anchor points for the reconstruction of a curve.

[0081] In addition to the identified points and the parameters derived from them, another parameter is provided or determined, namely the total number of curves of the total charge transition 3. This total number cannot always be determined exactly, since further curved sections of the total charge transition 3 outside the image space - in the example described here of Fig. 2 - could be present. In this case, three curved sections of the total charge transfer 3 can be seen, so this number is assumed to be the total number here.

[0082] In step S2, based on parameters that were provided and / or determined in step S1, the at least one geometric representation 11 (cf. Fig. 3) for the at least one total charge transition 3 in the measured charge stability diagram 1. In this case, at least one continuous, in other words connected, geometric representation is created, which comprises two linear sections 11a, 11c and a curve 11b connecting the two linear sections 11a, 11c, as well as optionally one or more further curves 11d, 11f and one or more further linear sections 11e, 11g (cf. also Fig. 4). Each curve 11b, 11d, 11f is connected to two linear segments 11a, 11c, 11e, 11g. Linear segments 11a, 11c, 11e, 11g and curve(s) 11b, 11d, 11f alternate in the respective geometric representation 11.

[0083] In the embodiment described here, the curves 11b, 11d, 11f of the geometric representation 11 are or will be represented specifically by Bézier curves.

[0084] One can also say that the geometric representation 11 is a reconstruction of the total charge transfer 3 from the measured stability diagram 1 or for it or is understood as such.

[0085] The creation of a geometric representation 11 is again shown here using the example of the total charge transfer 3 from Fig. 1 and Fig. 2, whereby it is clear that for one or more further total charge transitions 2, 4, 5 from the Fig. 1 and Fig. 2 or other charge stability diagrams 1 can be used in a completely analogous manner.

[0086] Of the parameters described above, the gradients and lengths of the two linear sections 3c and 3e, the X and Y coordinates of points 6 and 7 (or point 8 and point 9), and the total number of three curves are used to create the geometric representation 11. In addition to the two (end) anchor points 8 and 9, a third, middle anchor point is calculated from the two anchor points 6 and 7 (or 8 and 9) and the gradients of the two adjacent linear sections 3c and 3a (or 3e, whose gradient matches that of 3a).

[0087] It should be noted that for the Fig. 1 and Fig. In the example of a measured charge stability diagram 1 shown in Figure 2, the gradients of all linear sections of all total charge transitions 2-5 are in a rational number space. In particular, they are neither zero (none of the linear sections are parallel to the X-axis) nor infinite (none of the linear sections are parallel to the Y-axis). However, this does not always have to be the case. Therefore, it can be provided that the voltage space V, which is spanned by V1 and V2 (cf. Fig. 1 and Fig. 2), is first affinely transformed so that the gradients are in a rational number space. It is possible that such a transformation is always carried out as a precaution. Fig. Figure 4 shows an example of a geometric representation 11 for a total charge transition in such a transformed voltage space V', which is spanned by voltages V1' and V2'. This is for points 12-17, which were identified in another measured total charge transition from another measured charge stability diagram (not shown). Fig. 5 shows the geometric representation of Fig. 4 in the original stress space V corresponding to the transformed stress space V'.

[0088] The middle anchor points 18, 19, 20 are also only in Fig. 4, there for all three Bézier curves 11b, 11d, 11f of the geometric representation 11 of a total charge transfer shown here. As can be seen, the third, middle anchor points 18, 19, 20 are each located at the intersection point of Fig. 4 imaginary extensions of the two adjacent linear sections 11a, 11c, 11e, 11g shown with dashed lines beyond the two respective anchor points 12, 13, 14, 15, 16, 17.

[0089] It should be noted that, although in Fig. 4 a total of six (end) anchor points 12, 13, 14, 15, 16, 17 and three middle anchor points 18, 19, 20 for three Bézier curves 11b, 11d, 11f are shown, it is in principle sufficient to determine only a third, middle anchor point for only one Bézier curve 11b, 11d, 11f, in Fig. 4, for example, only for the Bézier curve 11b.

[0090] For the geometric representation 11 from Fig. 3, for example, a central anchor point can only be calculated for points 6 and 7 (or 8 and 9) and thus the Bézier curve 11b (or 11d). Each additional Bézier curve, e.g., curve 11d (or starting from 11d, curve 11b) from Fig. 3, is then determined, in particular calculated, by an affine transformation while maintaining continuity. The affine transformation to obtain one or more additional curves 11d (or 11b) comprises a combination of rotation by 180° and translation, or even just one translation.

[0091] By simplifying the process by obtaining one or more additional curves through affine transformation, the complexity of the geometric representation, in other words the representation, is independent of the size of the total charge transition 3. The total number of Bézier curves of the charge transition from n to n+1 electrons is 2*(n+1)-1. The linear section following the ends of the first and last Bézier curve of a total charge transition 2-5 is not limited and represents, in particular, the so-called single quantum dot region. This requires no further parameters.

[0092] How to Fig. 4, two curves of a geometric representation 11 of a total charge transfer, which are adjacent to each other and connected by a linear section, are rotated by 180° relative to each other and translated, i.e. shifted, relative to each other in the voltage space. Fig. 4, this applies to curves 11b and 11d as well as to the two curves 11d and 11f. For any two curves with another curve between them, it is true that they are not rotated relative to each other, but are only shifted relative to each other. Fig. 4, this is clearly true for curves 11b and 11f. Accordingly, 11d can be obtained from 11b by rotation by 180° and translation, and 11f from 11b by simple translation.

[0093] In the figures, the geometric representation 11 or the plurality of geometric representations 11 are each depicted pictorially by continuous lines with alternating straight sections 11a, 11c, 11e, 11g and curves 11b, 11d, 11f. It should be noted that such a pictorial representation is not absolutely necessary. Rather, one or more geometric representations 11 can also be depicted mathematically. Thus, the respective geometric representation can comprise one or more mathematical formulas or functions or can be given by one or more mathematical formulas or functions. In particular, it can be given by a function defined in sections or can comprise such a function. The formula / function is or are then created on the basis of the parameters provided and / or determined in step S1, in particular derived from these.Function parameters are preferably calculated from these.

[0094] Alternatively or additionally, at least one formula or function can also be or have been obtained using a fitting method. Purely as an example, any points located on, for example, the total charge transition 3 are selected and a sectionally defined function is fitted to these points. For example, the function parameters are iteratively adjusted and an error measure for the distance between the points is evaluated. The goal would then be, in particular, to minimize the errors and gradually approach the actual course of the total charge transition 3.

[0095] For example, if it is not possible to determine the beginning and end of a curved section, e.g. sections 3b and 3d in Fig. 2, to precisely identify the points, a rough estimate can also be chosen. Based on this, function parameters can be optimized by fitting to other identified points.

[0096] It is also conceivable to determine a first approximation for function parameters of a mathematical function, particularly one defined in sections, from identified points and then to improve this in a fitting procedure.

[0097] Structures 2-5 of the charge stability diagram 1 can be fitted robustly and without prior knowledge, especially about the physical system. This is particularly due to the flexibility of the geometric representation(s) 11, which is characterized by the ability to represent both very small and very large regions.

[0098] Within the scope of the present invention, in particular any or any sections of a total charge transfer can be fitted, regardless of which part of the total charge transfer is contained or recognizable in a measured charge stability diagram.

[0099] The at least one geometric representation 11 can alternatively or additionally also comprise a plurality of points or the coordinates of a plurality of points that lie on the linear sections and the one or more curves or form them. It can be in the form of a table or a sub-area of ​​a table with coordinates of a plurality of points or can comprise one such table. A plurality of points or associated coordinates can have been obtained by solving at least one mathematical formula or function for a given parameter range or given parameter ranges, in particular for two voltages.

[0100] The geometric representation 11 is in Fig. 3 for the same voltage range as the measured charge stability diagram 1 according to the Fig. 1 and Fig. 2. This is to be understood purely as an example and it is of course also possible that at least one geometric representation is created for a stress space which differs from the stress space of a measured charge stability diagram 1 from which parameters were determined in step S1, which in particular is larger than the stress space of the measured stability diagram 1. This is even particularly advantageous since in this case a prediction for neighboring regions can be made from small measurement ranges by extrapolating the representation(s).

[0101] As a rule, it will be useful to create geometric representations 11 for several total charge transitions 2-5 of a charge stability diagram 1. This is exemplified in Fig. 6 shown. The Fig. 6 also contains the total charge of the individual regions bounded by the geometric representations 11 of the total charge transitions, expressed in number of electrons. The abbreviations "0 E, 1 E, 2 E, 3 E, 4 E" stand for a charge of 0 electrons, one electron, two electrons, three electrons, and four electrons. Fig. 6 you can also see the typical “honeycomb” structure, which can also be seen in the Fig. 1 and Fig. 2 sees.

[0102] Optionally, in a step S3, the at least one created geometric representation can be used to assess the state of the quantum dot system.

[0103] In principle, various physical quantities and system properties can be derived from the geometric representation(s) 11. One example is the sum of the electrons in a double quantum dot (system). Also, for example, from opposite central Bezié anchors 18, 19, 20 of neighboring geometric representations 11 (cf. in particular Fig. 6) the charge transfer between the two quantum dots (interdot transition), in particular of a double quantum dot, can be derived. The approximate strength of the tunnel coupling between the two quantum dots, in particular of a double quantum dot, can be derived from the width of at least one Bézier curve 11b, 11d, 11f of at least one geometric representation. Approximate capacitive couplings can be derived from the gradients of linear sections 11a, 11c, 11e.

[0104] Beyond the characterization of qubit samples, the derived quantities also offer applications in the automated calibration of quantum dot systems. For example, using the total charge and the charge transitions between the two quantum dots, the quantum dot system can be converted into a desired charge configuration.

[0105] It is also possible to perform a computer simulation to obtain a simulated charge stability diagram for the quantum dot system, and to use the at least one geometric representation created in step S2 within the framework of the computer simulation. One or more geometric representations 11 for one or more total charge transitions 2-5 can, for example, be used to calculate the charge states of a double quantum dot. The voltage space can thus first be divided into fixed total charges in the system. Subsequently, the transition of electrons from one of the quantum dots to the respective other quantum dot can be added. This can be calculated from the opposite curves 11b, 11d, 11f of two geometric representations 11 of total charge transitions 2-5.When the boundary drawn between these two curves 11b, 11d, and 11f is crossed, an electron is shifted from one quantum dot to the other. The charge states thus generated can serve as the ground truth for the simulation. Based on this, a sensor response (of a sensor point) can be calculated, which corresponds to the measured value or measurement. List of reference symbols 1 Charge stability diagram 2 Total charge transfer 3 Total charge transfer 3a linear section 3b curve 3c linear section 3D curve 3e linear section 4 Total charge transfer 4a linear section 4b curve 4c linear section 4d curve 4th linear section 5 Total charge transfer 5a linear section 6-10 point in the load stability diagram 11 geometric representation 11a linear section 11b Curve 11c linear section 11d curve 11e linear section 11f curve 11g linear section 12-20 anchor point QUOTES CONTAINED IN THE DESCRIPTION

[0000] This list of documents submitted by the applicant was generated automatically and is included solely for the convenience of the reader. This list is not part of the German patent or utility model application. The DPMA assumes no liability for any errors or omissions. Cited non-patent literature

[0000] Computer-automated tuning of semiconductor double quantum dots into the single-electron regime” by TA Baart et al., Applied Physics Letters 21, 108 (2016), DOI: 10.1063 / 1.4952624

[0004] Computer-automated tuning procedures for semiconductor quantum dot arrays" von A. R. Mills et al., Applied Physics Letters 115, 11:113501 (2019), DOI:10.1063 / 1.5121444

[0004] „Automated tuning of inter-dot tunnel coupling in double quantum dots" von C.J. van Diepen et al., Applied Physics Letters 113, 033101 (2018), DOI: 10.1063 / 1.5031034

[0004] Tuning Methods for Semiconductor Spin Qubits" von T. Botzem et al., Phys. Rev. Applied 10, 054026 (2018), DOI: 10.1103 / PhysRevApplied.10.054026

[0004] Automatic virtual voltage extraction of a 2x2 array of quantum dots with machine learning" von G. A. Oakes et al., arXiv:2012.0368 (2020), DOI: 10.48550 / arXiv.2012.03685

[0005] Image Processing Algorithms for Tuning Quantum Devices and Nitrogen-Vacancy Imaging" von C. P. Monical et al. (2020), DOI: 10.2172 / 1662017

[0005] Algorithm for automated tuning of a quantum dot into the single-electron regime" von M. Lapointe-Major et al., Phys. Rev. B 102, 085301 (2020), DOI: 10.1103 / PhysRevB.102.085301

[0005] Electron transport through double quantum dots" von W. G. van der Weil et al., Rev. Mod. Phys. 75, 1 (2002), DOI: 10.1103 / RevModPhys.75.1

[0007] Generic Hubbard model description of semiconductor quantum-dot spin qubits" von S. Yang et al., Phys. Rev. B 83, 161301(R) (2011), DOI: 10.1103 / PhysRevB.83.161301

[0008] Hubbard model description of silicon spin qubits: Charge stability diagram and tunnel coupling in Si double quantum dots" von S. das Sarma et al., Phys. Rev. B 83, 235314 (2011), DOI: 10.1103 / PhysRevB.83.235314

[0008] Quantum theory of the charge-stability diagram of semiconductor double-quantum-dot systems" von X. Wang et al., Phys. Rev. B 84, 115301 (2011), DOI: 10.1103 / PhysRevB.84.115301

[0008]

Claims

[1] Method for determining at least one total charge transition (3) of a charge stability diagram (1) of a quantum dot system comprising at least two quantum dots, in which S1) parameters belonging to at least one total charge transfer (3) are provided and / or determined, S2) at least one geometric representation (11) for the at least one total charge transfer (3) is created on the basis of the parameters, wherein the at least one geometric representation (11) comprises two linear sections (11a, 11c, 11e, 11g) and a curve (11b, 11d, 11f) connecting the two linear sections and optionally one or more further curves (11b, 11d, 11f) and one or more further linear sections (11a, 11c, 11e, 11g), and S3) preferably the at least one created geometric representation (11) is used to assess the state of the quantum dot system. [2] Method according to claim 1,characterized by , in step S2 at least one geometric representation (11) with two or more curves (11b, 11d, 11f) is created, wherein all curves (11b, 11d, 11f) have the same shape and extent. [3] Method according to claim 1 or 2, characterized by that in step S2 at least one geometric representation (11) with two or more curves (11b, 11d, 11f) is created, wherein at least one curve (11b, 11d, 11f) is obtained, in particular calculated, by a particularly affine transformation of at least one other curve, preferably, wherein the particularly affine transformation comprises or is given by a translation and preferably rotation by 180° of the other curve (11b, 11d, 11f) and / or the particularly affine transformation is or was carried out while maintaining the continuity. [4] Method according to one of the preceding claims, characterized bythat in step S2 at least one geometric representation (11) with two or more curves (11b, 11d, 11f) is created and for the two or two curves (11b, 11d, 11f) which are connected to one another via a linear section (11a, 11c, 11e, 11g), it applies that they are rotated by 180° relative to one another. [5] Method according to one of the preceding claims, characterized by that at least some of the parameters are determined in step S1 by reference to a measured charge stability diagram (1) of the quantum dot system. [6] Method according to claim 5, characterized bythat in the measured charge stability diagram (1) a plurality of points (6-10) are identified which lie at least approximately on a total charge transition (3) contained therein, and the plurality of points (6-10) and / or variables derived from the plurality of points (6-10) are used as parameters for step S2, in particular, wherein the plurality of points (6-10) are identified by a user and / or by means of at least one suitable image analysis software tool, preferably in the form of an edge and / or line detector. [7] Method according to claim 5 or 6, characterized bythat in step S2 at least one geometric representation (11) is created which extends over a voltage space which differs from the voltage space spanned by the measured charge stability diagram (1), preferably, wherein the voltage space over which the at least one geometric representation (11) extends is larger in at least one dimension than the voltage space spanned by the measured charge stability diagram (1). [8] Method according to one of the preceding claims, characterized by that the parameters are determined in step S1 by deriving them from properties of the quantum dot system. [9] Method according to one of the preceding claims, characterized by that in step S2 the at least one geometric representation (11) is created in the form of a mathematical and / or a pictorial representation. [10] Method according to one of the preceding claims, characterized by that the creation of the at least one geometric representation (11) in step S2 comprises that at least one fit function is fitted to at least some of the parameters provided and / or determined in step S1 and / or to variables derived therefrom. [11] Method according to one of the preceding claims, characterized by that the respective curve (11b, 11d, 11f) of the at least one geometric representation (11) is given by a Bézier curve, preferably by a Bézier curve of the 2nd degree. [12] Method according to one of the preceding claims, characterized bythat in step S1 the gradients and lengths of two linear sections (11a, 11c, 11e, 11g) and the coordinates of two anchor points (6, 7) of a curve (11b, 11d, 11f) connecting the two linear sections and, if one or more further curves (11b, 11d, 11f) are present, an identifier for the total number of curves (11b, 11d, 11f) are determined and / or provided as parameters. [13] Method according to one of claims 5 to 7 and according to claim 11, characterized by that the voltage space spanned by the measured charge stability diagram (1) is affinely transformed such that the gradients of the linear sections (11a, 11c, 11e, 11g) are in a rational number space, in particular are not equal to zero or infinite. [14] Method according to claim 11 and claim 12 or 13, characterized bythat for the respective Bézier curve (11b, 11d, 11f) a third anchor point (18, 19, 20) is determined, preferably calculated, from the two associated anchor points (6, 7) and the gradients of two linear sections (11a, 11c, 11e, 11g) connected in particular by the respective Bézier curve (11b, 11d, 11f). [15] Method according to one of the preceding claims, characterized by that a computer simulation is carried out to obtain a simulated charge stability diagram for the quantum dot system, and that the at least one geometric representation (11) created in step S2 is used in the context of the computer simulation. [16] Method according to one of claims 5 to 7 and according to claim 15, characterized bythat the voltage space spanned by the simulated charge stability diagram differs from the voltage space spanned by the measured charge stability diagram (1), wherein the voltage space spanned by the simulated charge stability diagram is larger in at least one dimension than the voltage space spanned by the measured charge stability diagram (1). [17] Method according to one of the preceding claims, characterized bythat the at least one determined geometric representation (11) is used in step S3 to assess the state of the quantum dot system, wherein the assessment comprises that the sum of the electrons present in the quantum dot system is determined on the basis of the at least one geometric representation (11), and / or that the charge transfer between two quantum dots of the quantum dot system is determined on the basis of the at least one geometric representation (11), and / or that an approximated strength of the tunnel coupling between two, in particular, adjacent, quantum dots of the quantum dot system is determined on the basis of the at least one geometric representation (11), in particular from the width of at least one curve (11b, 11d, 11f) of the at least one geometric representation (11), and / or that approximated capacitive couplings are determined on the basis of the at least one geometric representation (11),in particular from the gradients of linear sections (11a, 11c, 11e, 11g) of the at least one geometric representation (11). [18] Method according to one of the preceding claims, characterized by that in step S1 parameters belonging to two or more total charge transitions (2-5) of a charge stability diagram (1) are provided and / or determined and in step S2, on the basis of the parameters for each total charge transition (2-5), a geometric representation (11) is created which comprises two linear sections (11a, 11c, 11e, 11g) and a curve (11b, 11d, 11f) connecting the two linear sections (11a, 11c, 11e, 11g) and optionally one or more further curves (11b, 11d, 11f) and one or more further linear sections (11a, 11c, 11e, 11g), and in step S3, preferably the plurality of geometric representations (11) are used to assess the state of the quantum dot system. [19] Method according to one of the preceding claims, characterized by that the at least one geometric representation (11) created in step S2 is used for a particularly automatic calibration of the quantum dot system, wherein the calibration comprises converting the quantum dot system into a desired charge configuration. [20] A computer program comprising instructions which, when executed by a computer, cause the computer to carry out the steps of the method according to any one of claims 1 to 19. [21] A computer program product comprising instructions which, when executed by a computer, cause the computer to carry out the steps of the method according to any one of claims 1 to 19.

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