Method and apparatus for deep learning-based reconstruction of internal and external parameters of a Hamiltonian system

Abstract

A method for determining the internal and external parameters of a Hamiltonian system (112, 120) using a neural network, wherein training the neural network comprises: generating (110) a set of N internal and external parameters; computing (114) the M observables using the Hamiltonian H and the generated set of parameters; training (116) the neural network to obtain a mapping from the observables to the corresponding parameters; and repeating (118) the preceding steps until sufficient accuracy is achieved. Also provided is a computer-readable storage medium and a device for determining parameters of a Hamiltonian system with Hamiltonian H (112; 120) using a neural network (124).

Description

The present invention relates to the reconstruction of the parameters of a physical system from measured data. More particularly, the present invention relates to the reconstruction of internal and external parameters of the hamiltonian of a system based on measured transition frequencies using a neural network.Hamiltonian systems are systems whose dynamics are determined by the Hamiltonian equation or with hamiltonian function H for a classic system or for a quantum system by using hamiltonian (hamiltonian) H. Examples of Hamiltonian systems are quantum spin systems such as defects in diamonds, for example nitrogen void center (NV) based defects, qubits, coupled spin systems (ising models), and other coupled and / or isolated systems. These quantum systems are typically described by a set of equations representing a hamiltonian in which some parameters are unknown. In the case of external excitation, e.g. illumination, magnetic field, etc., such a system exhibits specific measurable transition energies (frequencies or energy levels). For example, in the case of NV-based systems, the external parameters are the three directions of the magnetic field [Bx, By, Bz] (3 parameters) and the internal parameters are the voltages on each of the NV axes and in each of the directions (4 axes, 3 directions of the respective voltage, 12 parameters), and the transition energies are measured as spectra of the optically acquired magnetic resonance (ODMR).The general form of hamiltonian is commonly known and the transition energies for each set of internal and external parameters are easily calculated by substituting the same parameters into the hamiltonian.For practical purposes, however, the inverse problem is of interest: starting from given transition frequencies, it is necessary to reconstruct the internal parameters and / or the external stimuli of a quantum system in order to be able to use this system as a sensor. In the case of an NV-based magnetometer, in which the external magnetic field and the voltages in the diamond define the transition frequencies, it is thus necessary to determine the magnetic field and voltage parameters from measured (e.g. by ODMR) transition frequencies. In contrast to the "forward" problem of calculating the transition frequencies, the solution to this inverse problem is typically very computationally expensive, often requires approximations and simplifications of the system, and is not analytically obtainable in most cases. Therefore, a general method for quickly solving inverse problems in Hamiltonian systems and quantum systems is needed for reconstruction of internal parameters and external parameters / stimuli.Prior ArtUS 2019 / 024757 A1 discloses a method for identifying molecular parameters including input of observed transition frequencies, generation of transition frequency sets and spectral parameter sets, training one or more artificial neural networks (ANN) by analysis of the transition frequency sets and the spectral parameter sets, analysis of the observed transition frequencies using the one or more trained artificial neural networks to predict estimated spectral parameters, and identification of molecular parameters by analysis of the estimated spectral parameters. A molecular parameter identification system may include a rotation spectrometer, a user interface, and a spectral analysis application that retrieve observed transition frequencies, identify a Hamiltonian type by a neural network that analyzes the observed transition frequencies, select a second trained artificial neural network based on the identified Hamiltonian type, analyze observed transition frequencies using the second artificial neural network to identify estimated spectral parameters, and identify molecular parameters. A method for identifying molecular parameters based on the analysis of observed transition frequencies with a trained neural network is disclosed. The method may include receiving observed transition frequencies. The method may further include generating transition frequency sets and / or spectral parameter sets. These sets may consist of "imaginary" values which, according to the laws and / or theorem of physics and chemistry, are within a valid range, but do not correspond to molecules whose existence is experimentally known. Since they contain valid physical attributes, they do not describe reality, but are useful for training an ANN to recognize rules that are theoretically possible, and thus predict one based on the other in the presence of new information. The method may include structuring and / or training the ANN. The method may further include analyzing the observed transition frequencies using the trained ANN. The trained ANN may be provided with the observed transition frequencies as input and may predict a set of estimated spectral parameters based on the previous training with the imaginary values. The method may comprise identifying one or more molecules based on the analysis of the output of the ANN including the estimated spectral parameters.CN 115327451 A discloses a deep learning-based NV color center magnetometer magnetic field calculation method and an application thereof. The method comprises the steps of: 1, acquiring and preprocessing a dataset; 2, constructing a neural network for magnetic field solution, the neural network comprising a feature extraction network, a magnetic field calculation network and an error adjustment network; and 3, training the neural network to obtain an optimal combined network, training an error adjustment network and outputting predicted values of three magnetic induction intensity components under a Cartesian coordinate system. Specifically, there is disclosed: step 1.1: applying an external magnetic field to an NV color center magnetometer and substituting three components of the magnetic induction strength of the obtained external magnetic field in a Cartesian coordinate system into the hamiltonian to solve, resulting in eight resonance frequencies forming a first data set D1={d1,k|k=1,2,---m} and serving as training data; wherein d1, k represents the kth data set in the first data set D1, and d1, k includes eight resonance frequencies F1, K={f1, k,i|i=1,2,---8} and standard values of magnetic induction intensity components in a Cartesian coordinate system {B1,k,j|j=x,y,z}; f1, k, i represents the i-th resonant frequency value of the k-th data d1, k in the first data set D1, B1, k, j represents the standard values of the j-th magnetic induction intensity components of the k-th data set d1, k in the first data set D1, m represents the total number of groups of the first data set D1; step 1. 2: Generating a second data set - - n} as test data using the method of step 1. wherein d2, g represents the g-th data of the second data set D2, and d2, g contains eight resonant frequencies F2, k'={f2, g, i|i=1,2,---n}8} and the standard values of the magnetic induction intensity components in a Cartesian coordinate system. Step 2: Construction of a neural network for magnetic field solution, comprising: a feature extraction network, a magnetic field calculation network and an error matching network; Step 3: Training of the neural network.Disclosure of the InventionThe aim of the invention is to provide a reliable and fast solution to the inverse problem for Hamiltonian systems in which the general form of Hamiltonian is known and the transition frequencies depend on internal parameters and external parameters / stimuli. The object achieved by the invention is the need for an accurate, fast, non-computationally intensive and efficient reconstruction of the parameters of the Hamiltonian system from measured energy levels. The methods in the current state of the art are based on simplifications and approximations of the hamiltonian to obtain a system of equations that connect the parameters of the hamiltonian to the transition frequencies. This system of equations is then solved by means of further linearizations and / or the least squares method and / or other known approximations. These methods can result in inaccuracies in the reconstructed parameters. In the case that the Hamiltonian system is used as a sensor, these approximations can lead to losses in absolute accuracy and / or sensitivity.In the specific case of a nitrogen void-based (NV-based) defect in a diamond for use as a magnetometer, the "forward problem" is as follows:• The hamiltonian has 15 parameters consisting of the 3-dimensional magnetic field (3 parameters) and the three-dimensional voltages along each of the crystal axes of the diamond (4 axes, 12 parameters).• The temperature may be assumed to be 16th external or internal parameters.• The hamiltonian allows a fast and uncomplicated calculation of 24 different transition frequencies.In contrast to the forward problem, the inverse problem is as follows:• Up to 24 transition frequencies are measured, e.g. by optically acquired magnetic resonance (ODMR)• 15 (16) Hamiltonian parameters must be reconstructed• Examination phase: The parameters found, used in the hamiltonian, must give exactly the same 24 transition frequencies.While the forward problem involves only applying a formula, the inverse problem requires at least solving a system of 9x9 nonlinear equations. This system is more commonly simplified as described in Balasubramanian et al., "Nanoscale imaging magnetometry with diamond spins under ambient conditions", Nature 445, 648-651 (2008) and Weggler et al., "Determination of the Three-Dimensional Magnetic Field Vector Orientation with Nitrogen Vacany Centers in Diamond", Nano Lett. 20, 2980-2985 (2020) for reconstructing only the external parameters (magnetic field), assumptions being made about the internal parameters. These assumptions and simplifications lead to inaccuracies which are evident in the checking phase (if implemented) of the solution method for the inverse problem.The present invention aims to solve the described problems in the prior art by using artificial intelligence (AI) and deep learning (DL), in particular by training a deep neural network (DNN) to solve the inverse problem for reconstructing the parameters of a Hamiltonian system from the transition frequencies.This object is achieved by the subject matter of the independent claims.One aspect of the invention describes a method for determining the parameters of a Hamiltonian system, the method comprising: providing a Hamiltonian H to the system, wherein the Hamiltonian H N has internal and external parameters; providing M observables for each of a set of known Hamiltonian systems, wherein each Hamiltonian system consists of the set of known Hamiltonian systems from the Hamiltonian H together with a set of values of the N parameters; training a neural network with the set of known Hamiltonian systems and their associated observables, and using the trained neural network to determine the N internal and external parameters of an unknown Hamiltonian from observables measured m≤M, wherein training a neural network with the set of known Hamiltonian systems and their associated observables includes: generating a set of N external and internal parameters; calculating the M observables using the Hamiltonian H and the set of N external and internal parameters; training the neural network to generate a mapping from the observables to the associated parameters; Use of the trained neural network to reconstruct the N internal and external parameters from m≤M observables; calculate M observables using the hamiltonian H and the reconstructed set of parameters; compare the M calculated observables with the m≤M observables used to reconstruct the parameters; and repeat the preceding steps until sufficient accuracy is achieved.A second aspect of the invention describes a computer readable storage medium including information which, when executed by one or more processors, performs the method of the first aspect.A third embodiment of the invention describes an apparatus for determining the parameters of a Hamiltonian system with Hamiltonian H, the apparatus comprising a neural network which is configured to map measured observables to parameters of the Hamiltonian system, wherein the neural network has been trained by: generating a set of N internal and external parameters; calculating M observables with the aid of the Hamiltonian H and the set of N parameters; training the neural network to generate a mapping from the calculated observables to the corresponding parameters; using the trained neural network to reconstruct the N internal and external parameters from m≤M observables; calculating M observables using the hamiltonian H and the reconstructed set of parameters; comparing the M calculated observables with the m≤M observables used to reconstruct the parameters; and repeating the preceding steps until sufficient accuracy is achieved.The training data of the neural network, preferably a deep neural network, DNN, can thus be generated in any quantity if necessary by using a set of internal parameters and external parameters / stimuli in the hamiltonian H with known general form and calculating the transition frequencies resulting therefrom. Similarly, the test and validation data sets may be calculated as needed, thereby ensuring that there is no information leak between training and verification data. In contrast to a method that trains DNNs with measured data, the size of the training dataset is potentially unlimited for Hamiltonian (quantum) systems. The present approach avoids the typical problems of deep learning based neural networks, lack of training data, overfitting, data leaks, and other problems of a limited dataset. Further, the structure and size of the DNN and the size of the data set may be selected according to the present inverse problem.Once this DNN is trained, it allows a fast and low computational solution to the inverse problem by forward propagation of the DNN.The present approach obviates the need for approximations, linearizations, or simplifications of the hamiltonian in solving the inverse problem and thus results in a more precise reconstruction of the parameters. Furthermore, the use of deep learning allows a more flexible and adaptable approach for the reconstruction of the parameters, since the model can be refined at any time with newly generated data, if necessary.In this respect, the central advantage of the present invention is the possibility of solving the complete inverse problem of a Hamiltonian system and reconstructing internal parameters and external parameters / stimuli from transition frequencies in an accurate and efficient manner by training a neural network with a dataset whose size can be increased until the desired accuracy is achieved. Another core advantage is that after training is completed, the computational effort for finding a solution to the inverse problem is relatively low and only involves the forward propagation (matrix calculation) of the DNN. Furthermore, the training of the DNN does not necessarily have to be implemented on the actual device (for example a quantiser sensor), but can be swapped out to an external computer or a data centre. Training need only be performed once and can be used for all devices that operate within the parameter range used for training.Preferably, the Hamiltonian H describes a magnetometer and / or the measured observables are transition frequencies measured with the aid of optically detected magnetic resonance, ODMR.Preferably, Hamiltonian H describes a nitrogen void (NV)-based defect in a diamond.Preferably, the external parameters describe a magnetic field and the internal parameters describe an internal stress in the diamond. More preferably, the parameters describe a three-dimensional magnetic field having 3 components and the three-dimensional stress along each of the crystal axes of the diamond having 12 parameters. Most preferably, the parameters also describe a temperature.Preferably, the neural network is a deep neural network (DNN).Preferably, the calculation of the M observables with the aid of the hamiltonian H and the generated set of parameters includes at least one of the following possibilities: calculating the natural frequencies of the hamiltonian H; using data from classical simulations of the system or a similar system or its components; using data from digital reconstructions of the system or similar systems such as digital twins, full models, empirical models, time range and frequency range models, surrogate models, quantum dynamic simulations, quantum computer simulations, Monte Carlo simulations, stochastic simulations, deterministic simulations or other physical, chemical and mathematical simulations.The connection of using deep learning and transition frequencies to reconstruct the parameters of the hamiltonian, including the internal parameters and external parameters / stimuli, is particularly advantageous in the case of NV-based magnetometers, as this combination allows the input parameters, including crystal voltage parameters, to be reconstructed from the ODMR spectra using the DNN.Summary of the Figures: FIG. 1 ashows a flow diagram of the training phase of a neural network according to an embodiment of the invention. FIG. 1 b shows a flow diagram of the use of a trained neural network to reconstruct the parameters of a Hamiltonian system, according to an embodiment of the invention. Figure 1c shows an example of the reconstruction of the parameters of a Hamilton's system according to an embodiment of the invention.As shown in Figure 1, the invention uses neural networks, particularly deep neural networks, DNNs, to reconstruct parameters of Hamiltonian systems. The basic structure of the invention is a neural network trained with a dataset of internal and external parameters of the hamiltonian and the respective observables, in particular transition frequencies. This data set can be created by substituting the parameters into the hamiltonian as needed.Figure 1a shows the training of the neural network. In a first step 110, a random set of internal (e.g., voltages) and external (e.g., magnetic fields) parameters of a known hamiltonian 122 is generated. After inserting the same parameters into the hamiltonian 122, observables of the Hamiltonian system, e.g. transition frequencies 114, are calculated on the basis of the hamiltonian 122 and the parameters. The same observables 114 are then used in a further step 116 to train a neural network and the process repeats 118 until a sufficient accuracy is achieved.The trained model is then used to predict the parameters of the model based on the input observables, e.g., transition frequencies. Training may take place on an external computer or cluster or in a data center. The reconstruction of the parameters based on the input observables may be done externally or in an embedded system (on the software and hardware containing the Hamilton's system) to provide a sensor or characterize the system itself.In order to implement the invention, in addition to or instead of a DNN, other possible network structures such as a convolutional neural network, CNN or a recurrent neural network, RNN can also be used or the DNN can be embodied as one of these network structures. The specific architecture here depends on the characteristics and complexity of the data set used for training and the physical Hamiltonian system studied.FIGS. 1 band 1 c show the reconstruction of the internal parameters with the aid of a neural network trained with the method shown in FIG. 1 a. A physical Hamiltonian system 120, i.e., a system, e.g., magnetometer 130, which may be modeled with a Hamiltonian of known form, is used to measure observables, e.g., transition frequencies 122, particularly those measured by ODMR 132. The measured values 122, 132 are then fed into the neural network 124, 134 trained with the method explained above to obtain a real-time output 126 of the internal and external parameters of the associated hamiltonian, for example in the case of an NV-based magnetometer 130, the 3D components of the magnetic field and the components of the voltage along each of the 4 crystal axes 136.If necessary, this modeling can be checked with the aid of the hamiltonian 112 by inserting the output 126, corresponding to the internal and external parameters of the hamiltonian system 112, into the hamiltonian system 112 according to the method shown in FIG. 1 ato calculate observables 114, in particular transition frequencies, which can then be compared with the measured transition frequencies 122 in order to check the accuracy of the reconstruction.In the case of an NV-based magnetometer (see FIG. 1 c ), the method according to the invention involves generating a training dataset by generating random parameter sets (voltages, temperature, magnetic field vector) and inserting them into the known hamiltonian to calculate transition frequencies or ODMR spectra. A portion or the whole of the data set may also consist of physically measured and / or simulated values. This data set is then used to train a deep neural network, DNN, to predict the ground truth of the inverse problem, i.e., the input parameters. After the neural network has been trained, it can be used to predict the parameters from newly measured transition frequencies or ODMR spectra in real time and with low computational complexity by forward propagation of the neural network. An unlimited amount of training data can be generated from the hamiltonian, which simplifies the training of the neural network.With respect to the interaction of the deep learning algorithm, the hamiltonian describing the system and the actual elements of the physical system, the present model is based on: a) the generation of a high quality and large data set of transition frequencies and corresponding parameters of the hamiltonian, b) a full hamiltonian describing the physical system being examined sufficiently accurately, and c) the quality of the measurement data used in the forward propagation of the neural network for the reconstruction of the parameters.Improving each of these requirements (a, b, c) for the precision, resolution and size of the data set results in a more accurate and precise real-time reconstruction of the parameters of the hamiltonian.A possible alternative to the described method is the use of traditional machine learning methods such as random forest or support vector machine (SVM). However, these traditional methods are often not as efficient as deep neural networks when dealing with the handling of large and complex data sets, and they are often unable to find complex patterns and relationships in the data as needed to predict the internal and external parameters of Hamiltonian systems.Another alternative is to use hybrid methods that combine traditional machine learning methods with deep learning, which can potentially improve the accuracy of the prediction. Further, it is possible to use various types of deep learning models such as autoencoders or generative adversarial networks, GANs instead of the DNN, CNNs, or RNNs to generate the deep learning model.Possible alternatives for directly computing the observables from the hamiltonian to generate the training dataset include one or more of the following options:• Use data from classical simulations of the system or similar systems or their components.• Use of data from quantum simulations of the system or similar systems or their components.• Use of data from digital reconstructions of the system or similar systems or their components such as digital twins, full models, empirical models, time domain and frequency domain models, surrogate models, quantum dynamic simulations, quantum computer simulations, Monte Carlo simulations, stochastic simulations, deterministic simulations or other physical, chemical and mathematical simulations.• Use of measured experimental data of the system or similar systems or their components.References included in the specificationThis list of documents cited by the applicant has been produced in an automated manner and is only included for the better information of the reader. The list is not part of the German patent application or utility model application. The DPMA does not take any adhesion for any faults or omissions.Patent Literature citedUS 2019 / 024757 A1

[0005] CN 115327451 A

[0006] Cited Non-Patent LiteratureBalasubramian et al., "Nanoscale imaging magnetometry with diamond spins under ambient conditions", Nature 445, 648-651 (2008) and Weggler et al., "Determination of the Three-Dimensional Magnetic Field Vector Orientation with Nitrogen Vacany Centers in Diamond", Nano Lett. 20, 2980-2985 (2020

[0010]

Claims

A method for determining the parameters of a Hamiltonian system (112,120), the method comprising: providing a Hamiltonian H to the system, wherein the Hamiltonian H includes N internal and external parameters; providing M observables for each of a set of known Hamiltonian systems, wherein each of the known Hamiltonian systems consists of the Hamiltonian H along with a set of the N parameters; training (116) a neural network with the set of known Hamiltonian systems and associated observables; and utilizing the trained neural network (124, 136) to reconstruct the N internal and external parameters (126, 136) of a physical system from observables measured by m≤M, wherein training a neural network with the set of known Hamiltonian systems and associated observables includes: generating (110) a set of N internal and external parameters; calculating (114) the M observables using the hamiltonian H and the generated set of parameters; training (116) the neural network to achieve a mapping from the observables to the associated parameters; using the trained neural network to reconstruct the N internal and external parameters from m≤M observables; Calculating M observables using the hamiltonian H and the reconstructed set of parameters; comparing the M calculated observables with the m≤M observables used to reconstruct the parameters; and repeating (118) the preceding steps until sufficient accuracy is achieved.The method of claim 1, wherein the hamiltonian H describes a magnetometer (130), and / or wherein the measured observable transition frequencies are spectra (132) detected by optically detected magnetic resonance.The method according to claim 1 or 2, wherein the hamiltonian H describes a nitrogen void-based defect in a diamond, wherein preferably the external parameters describe an external magnetic field and the internal parameters describe internal voltages of the diamond, and particularly preferably the parameters describe a three-dimensional magnetic field with three parameters and the three-dimensional voltage along each of the crystal axes of the diamond with four parameters and preferably a temperature.The method of any preceding claim, wherein the neural network (124, 134) is a deep neural network.The method of any preceding claim, wherein calculating the M observables (114) using the hamiltonian H and the generated set of parameters includes at least one of the steps of: calculating the natural frequencies of the hamiltonian H; using data from classical simulations of the system or similar systems or components thereof; using data from quantum simulations of the system or similar systems or components thereof; or using data from digital reconstructions of the system or similar systems or components thereof such as digital twins, full models, empirical models, time domain and frequency domain models, surrogate models, quantum dynamic simulations, quantum computer simulations, or other physical, chemical and mathematical models.A computer readable storage medium containing information which, when executed by one or more processors, performs the method of any one of claims 1 to 5.An apparatus for determining parameters of a Hamiltonian system (112; 120) having Hamiltonian H, the apparatus comprising: a trained neural network (124; 134) configured to predict internal and external parameters of Hamiltonian H from measured observables by forward propagation, wherein the neural network (124; 134) has been trained by: generating (110) a set of N internal and external parameters; calculating (114) the M observables using Hamiltonian H and the generated set of parameters; training (116) the neural network to achieve a mapping from the observables to the associated parameters; Use of the trained neural network to reconstruct the N internal and external parameters from m≤M observables; calculate M observables using the hamiltonian H and the reconstructed set of parameters; compare the M calculated observables with the m≤M observables used to reconstruct the parameters; and repeat (118) the preceding steps until sufficient accuracy is achieved.The apparatus of claim 7, wherein the hamiltonian describes a magnetometer (130), and / or wherein the measured observable transition frequencies are spectra (132) detected by optically detected magnetic resonance.The device according to claim 7 or 8, wherein the hamiltonian H describes a nitrogen void-based defect in a diamond, and / or wherein the parameters describe a three-dimensional magnetic field with three parameters and the three-dimensional stress along each of the crystal axes of the diamond with four parameters and preferably a temperature, and / or wherein the neural network is a deep neural network.The apparatus of any of claims 7 to 9, further comprising: means for obtaining transition frequencies measured as measured observables from a physical system; and means for outputting the predicted internal and external parameters, wherein the Hamiltonian H describes the physical system, wherein preferably the apparatus comprises the physical system configured as a nitrogen void-based magnetometer (130).

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