Determining exchange-correlation functionals for molecular systems using a quantum computer

Abstract

A method of determining an exchange-correlation functional for computing electronic properties of a molecular system is described A hybrid data processor comprising a classical computer and a quantum computer is used to determine the exchange functions. The method may include receiving or determining reference data comprising synthetically computed and / or real-world measured reference electron-orbital features, preferably reference electron densities and associated reference ground state energies, for a plurality of grid points, the reference electron-orbital features being associated with different molecular reference systems; determining a first electron density of a molecular system for which reference electron-orbital features are available; computing a second electron density for each of the grid points based on: the first electron density, an exchange correlction energy computed by a variational quantum model using the first electron density as input; and, a density functional algorithm, preferably a self-consistent density functional algorithm; and, if the difference between the second electron density and the first electron density is smaller than a predetermined delta value, then updating variational parameters of the variational quantum model based on a training loss, wherein a loss function is used to compute the training loss based on the second electron density and a reference electron density respectively.

Description

[0001]Determining exchange-correlation functionals for molecular systems using a quantum computer Technical field The disclosure relates to exchange-correlation functionals for molecularsystems, and in particular, though not exclusively, to methods and systems fordetermining exchange-correlation functionals for molecular systems using a quantumcomputer and methods and systems for determining electronic properties of a molecularsystem using such exchange-correlation functionals, and a computer program product forexecuting such methods. Background Density Functional Theory (DFT) has emerged as a major direction of research in the modern quantum chemistry, enabling the efficient prediction of the elec- tronic structure and properties of molecules and materials. It has become the dominant method for solving the Schrodinger equation in physics-based atomistic simulations. DFT’s effectiveness largely stems from the ability to express the ground-state density of interacting electrons via a system of non-interacting electrons, the so-called Kohn-Shamsystem. However, DFT can yield exact ground state energy and electron density only ifthe exact exchange-correlation (XC) functional is provided. Despite the widespreadapplicability of the technique, the development of accurate and versatile DFT XC functionals remains a challenge. Consequently, DFT faces problems such as underestimation of reaction barriers, bandgaps, and poor handling of degenerate states potentially due to delocalization and static correlation errors. The universal functional capable of solving the aforementioned problems and capturing the diverse electronic properties of molecules, solids and surfaces remains unknown. In its pursuit, hundreds of approximate XC functionals have been devised. Many of these XC functionals are tailored for certain types of chemical systems and properties thereof, and often require prior knowledge of the targeted data to ensure the success of the method. An additional challenge when constructing such functionals lies in achieving a good balance between accuracy and computational efficiency. Anumber of recent studies propose the use of Machine Learning (ML)techniques as a solution for the efficient design of XC functionals that can be used for generic DFT calculations. ML algorithms, ranging from deep neural networks to sophisticated regression models, have demonstrated their potential to unlock hiddenpatterns in vast datasets of chemical systems, allowing for the estimation of a universalXC functional. This combination of quantum chemistry and artificial intelligence heralds a promising era for XC functional design with many demonstrations for different systems indicating the increase of the reach of the DFT methods. The majority of these ML-based techniques leverage differentiable programming as a widely used paradigm in Deep Learning (DL) that involves optimizing parameters using gradient-based methods. For example, WO2022 / 148847 and an associated article by Kirkpatrick et al, Pushing the frontiers of density functionals by solving the fractional electron problem,science 374 (6573) 2021, descibe a scheme to compute exchange-correlation energies ofatomic systems using neural networks. Similarly, Li et al., describe in their aticle Kohn-Sham equations as regularizer: building prior knowledge into machine-learned physics (Phys. Rev. Lett.126, 036401 (2021)), a scheme for computing exchange-correlationsenergies using neural networkds. Training neural XC functionals based a DFT procedureshow improved generalization capabilities in 1D compared to the previous approachhowever to date useful general XC functionals based on deep neural networks have notbeen realized yet. Hence, from the above, it follows that there is a need in the art for improvedmethods and systems for determining exchange-correlation functionals for computingelectronic properties of molecular systems.Summary As will be appreciated by one skilled in the art, aspects of the present invention may be embodied as a system, method or computer program product. Accordingly, aspects of the present invention may take the form of an entirely hardware embodiment, an entirely software embodiment (including firmware, resident software, micro-code, etc.) or an embodiment combining software and hardware aspects that may all generally be referred to herein as a "circuit," "module" or "system." Functions described in this disclosure may be implemented as an algorithm executed by a microprocessor of a computer. Furthermore, aspects of the present invention may take the form of a computer program product embodied in one or more computer readable medium(s) having computer readable program code embodied, e.g., stored, thereon. Any combination of one or more computer readable medium(s) may be utilized. The computer readable medium may be a computer readable signal medium or a computer readable storage medium. A computer readable storage medium may be, for example, but not limited to, an electronic, magnetic, optical, electromagnetic, infrared, or semiconductor system, apparatus, or device, or any suitable combination of the foregoing. More specific examples (a non-exhaustive list) of the computer readable storage medium would include the following: an electrical connection having one or more wires, a portable computer diskette, a hard disk, a random access memory (RAM), a read-only memory (ROM), an erasable programmable read-only memory (EPROM or Flash memory), an optical fiber, a portable compact disc read-only memory (CD-ROM), an optical storage device, a magnetic storage device, or any suitable combination of the foregoing. In the context of this document, a computer readable storage medium may be any tangible medium that can contain, or store a program for use by or in connection with an instruction execution system, apparatus, or device. A computer readable signal medium may include a propagated data signal with computer readable program code embodied therein, for example, in baseband or as part of a carrier wave. Such a propagated signal may take any of a variety of forms, including, but not limited to, electro-magnetic, optical, or any suitable combination thereof. A computer readable signal medium may be any computer readable medium that is not a computer readable storage medium and that can communicate, propagate, or transport a program for use by or in connection with an instruction execution system, apparatus, or device. Program code embodied on a computer readable medium may be transmitted using any appropriate medium, including but not limited to wireless, wireline, optical fiber, cable, RF, etc., or any suitable combination of the foregoing. Computer program code for carrying out operations for aspects of the present invention may be written in any combination of one or more programming languages, including an object- oriented programming language such as Java(TM), Smalltalk, C++ or the like and conventional procedural programming languages, such as the "C" programming language or similar programming languages. The program code may execute entirely on the user's computer, partly on the user's computer, as a stand-alone software package, partly on the user's computer and partly on a remote computer, or entirely on the remote computer or server. In the latter scenario, the remote computer may be connected to the user's computer through any type of network, including a local area network (LAN) or a wide area network (WAN), or the connection may be made to an external computer (for example, through the Internet using an Internet Service Provider). Aspects of the present invention are described below with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the invention. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions may be provided to a processor, in particular a microprocessor or central processing unit (CPU), of a general purpose computer, special purpose computer, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer, other programmable data processing apparatus, or other devices create means for implementing the functions / acts specified in the flowchart and / or block diagram block or blocks. These computer program instructions may also be stored in a computer readable medium that can direct a computer, other programmable data processing apparatus, or other devices to function in a particular manner, such that the instructions stored in the computer readable medium produce an article of manufacture including instructions which implement the function / act specified in the flowchart and / or block diagram block or blocks. The computer program instructions may also be loaded onto a computer, other programmable data processing apparatus, or other devices to cause a series of operational steps to be performed on the computer, other programmable apparatus or other devices to produce a computer implemented process such that the instructions which execute on the computer or other programmable apparatus provide processes for implementing the functions / acts specified in the flowchart and / or block diagram block or blocks. Additionally, the Instructions may be executed by any type of processors, including but not limited to one or more digital signal processors (DSPs), general purpose microprocessors, application specific integrated circuits (ASICs), field programmable logicarrays (FP- GAs), or other equivalent integrated or discrete logic circuitry.The flowchart and block diagrams in the figures illustrate the architecture, functionality, and operation of possible implementations of systems, methods and computer program products according to various embodiments of the present invention. In this regard, each block in the flowchart or block diagrams may represent a module, segment, or portion of code, which comprises one or more executable instructions for implementing the specified logical function(s). It should also be noted that, in some alternative implementations, the functions noted in the blocks may occur out of the order noted in the figures. For example, two blocks shown in succession may, in fact, be executed substantially concurrently, or the blocks may sometimes be executed in the reverse order, depending upon the functionality involved. It will also be noted that each block of the block diagrams and / or flowchart illustrations, and combinations of blocks in the block diagrams and / or flowchart illustrations, can be implemented by special purpose hardware-based systems that perform the specified functions or acts, or combinations of special purpose hardware and computer instructions. In an embodiment, the embodiments in this disclosure relate to a method ofdetermining an exchange-correlation functional for computing electronic properties of a molecular system using a hybrid data processor comprising a classical computer and a quantum computer, wherein the method may comprise: receiving or determiningreference data comprising synthetically computed and / or real-world measured referenceelectron-orbital features, preferably reference electron densities and associated referenceground state energies, for a plurality of grid points, the reference electron-orbital featuresbeing associated with different molecular reference systems; determining a first electrondensity of a molecular system for which reference electron-orbital features are available;computing a second electron density for each of the grid points based on: the first electrondensity, an exchange correlction energy computed by a variational quantum model usingthe first electron density as input; and, a density functional algorithm, preferably a self-consistent density functional algorithm; and, if the difference between the second electrondensity and the first electron density is smaller than a predetermined delta value, thenupdating variational parameters of the variational quantum model based on a training loss, wherein a loss function is used to compute the training loss based on the secondelectron density and a reference electron density respectively.Hence, the method involves generating a molecular dataset, selecting a quantum model, and optimizing it using specific loss functions. The dataset may beconstructed using accurate chemical calculations or experimental results that include theenergy and electron density for different molecular systems. A quantum neural network (QNN) comprising a feature embedding block, a parametrized ansatz block, and a measurement block for determing an expectation value of a parametrized observable may then be used. The electron density is loaded into the quantum circuit via the feature map, and a parametrized ansatz may produce a feature vector used to compute the expectation value of an observable to represent the XC energy at a grid point. The DQC-based XCfunctional may be embedded in a differentiable density functional algorithym, e.g. adifferentiable KS-DFT program (e.g., PySCFAD) to estimate the corresponding energy and density. To train the QNN, a cost function may be introduced. In an embodiment, the cost function may be based on the Mean Squared Error (MSE) between the CCSD(T) energies / density and the DQC-based ones. Gradient-based optimization methods, such as L-BFGS-B or Adam, may be used to vary the parameters and minimize the cost function, however the embodiments in the disclosure are not limited to gradient-based optimizers and may also be applied with any optimizer. The optimized parameters are used to define the quantum-based XC- functional, which can be used for arbitrary molecular structures for enabling more accurate quantum chemistry simulations. The embodiments provide improved XC functionals compared to traditional XC functional design and aims to create reliablequantum-based XC functionals. The quantum-based XC functionals may be used insolving a density functional theory problem in molecular chemistry.The embodiments use the synergy between QML and DFT for designing XC functionals for computing electronic energies of atomic systems in chemistry. Techniques arising from the QML domain are used to process empirical data from DFT (or general quantum chemistry) calculations within the supervised ML approach to create robust and versatile XC functionals that can be applied in a wider range of problems in the quantum chemistry domain. In an embodiment, the variational quantum model may be represented by aquantum circuit defining gate operations, the quantum circuit defining a quantum neuralnetwork comprising one or more feature maps and one or more parameterized variational circuits. In an embodiment, the variational quantum model may furhter comprises one or more classical neural networks configured, wherein the updating of the variational parameters includes updating the weights of the classical neural network. In an embodiment, computing the exchange correlation energy may include: using the one or more feature maps to encode the first electron density at at least part of the grid points in the Hilbert space; executing the quantum circuit on quantum elements of the quantum computer; and, measuring the state of the quantum elements to obtain an exchange correlation energy. In an embodiment, executing the quantum circuit may include translatingquantum operations of the quantum circuit into control signals; controlling the quantumelements of the quantum computer based on the control signals.In an embodiment, executing the quantum circuit may include: determiningclassical measurement data by measuring the state of the quantum elements of the quantum computer; and, determining the exchange-correlation energy based on the classical measurement data. In an embodiment, computing predicted electron-orbital features may include: determining an exchange-correlation potential based on the derivative of the predicted exchange correlation energy with respect to the electron density; and, determining Coulomb and Hartee potentials using the first electron density; computing the second electron-orbital features based on the self-consistent density functional algorithm using the Coulomb and Hartee potentials and the exchange-correlation potential as input. In an embodiment, the one or more feature maps may be trainable feature maps, the trainable feature maps including one or more unitary operators defining a time evolution over a Hamiltonian, preferably a generator Hamiltonian, applied to the quantum elements of the quantum register, the Hamiltonian evolution being parameterized by variational parameters. Preferably, updating the variational parameters may include updating the variational parameters of the trainable feature maps. In an embodiment, the quantum circuit may comprise digital quantum gateoperations, preferably digital single quantum gate operations, and one or more analog quantum gate operations configured to entangle different quantum elements of thequantum computer and to evolve a Hamiltonian associated with quantum elements intime; In an embodiment, the method may comprise: if the difference between the second electron-orbital features and the first electron-orbital features is larger than the predetermined delta, computing third electron-orbital features for each of the grid pointsbased on: the second electron-orbital features, a predicted exchange correlction energycomputed by the quantum model using the second electron-orbital features as input; and, the density functional algorithm. In an embodiment, the method may comprise: computing a ground state energy associated with the second electron density; and, further computing the loss value based on the ground state energy and a reference ground state energy associated with the reference electron density. In an embodiment, at least part of the variational parameters may beadjusted using a classical optimizer, preferably a gradient-based optimizer. In an embodiment, the one or more electron densities may be encoded based rotation operations and a non-linear encoding function, preferably the non-linear encoding function including a Chebyshev polynomial. In another embodiment, the data may be encoded into amplitudes ofcomputational basis states of the quantum computer.In a further embodiment, the quantum model for XC functional may berepresented as a projection: ^^^(^) = ^^(^) = ^^ |^^^ where the state |^^ represents theinput data (e.g., electron density) that is encoded in the Hilbert space of the quantumcomuter using a Chebyshev or Fourier encoding, and wherein the state |^^^ = ^(^)|0^represents the quantum state after application of a parametrized ansatz ^(^) on anarbitrary initial state |0^. In an embodiment, the variational quantum model may comprise one or more differentiable quantum circuits (DQCs), the one or more differentiable quantum circuits including one or more derivative features maps for encoding the one or more derivatives of the exchange-correlation functional. In an embodiment, the quantum computer may be implemented based onat least one of: a neutral atom quantum device, an ion-based quantum device, an opticaland photonic quantum device, a superconducting quantum device, a silicon based quantum device, a diamond N-V centre quantum device and / or a gaussian boson sampling device. In a further aspect, the embodiments may relate to a hybrid data processor for determining an exchange-correlation functional for computing electronic properties of a molecular system, the hybrid data processing comprising a classical computer and a quantum computer, the hybrid data processing being configured to: receiving or determining reference data comprising synthetically computed and / or real-world measured reference electron-orbital features, preferably reference electron densities and associated reference ground state energies, for a plurality of grid points, the reference electron-orbital features being associated with different molecular reference systems; determining a first electron density of a molecular system for which reference electron-orbital features are available; computing a second electron density for each of the gridpoints based on: the first electron density, an exchange correlction energy computed by avariational quantum model using the first electron density as input; and, a densityfunctional algorithm, preferably a self-consistent density functional algorithm; and,if the difference between the second electron density and the first electron density issmaller than a predetermined delta value, then updating variational parameters of the variational quantum model based on a training loss, wherein a loss function is used tocompute the training loss based on the second electron density and a reference electrondensity respectively. The embodiments may also relate to a computer program or suite of computer programs comprising at least one software code portion or a computer program product storing at least one software code portion, the software code portion, when run on a hybrid data processing system comprising a classical computer and a quantum computer comprising a hardware quantum computer, being configured for executing the method steps according any of the above-described embodiments. Brief description of the drawings Fig. 1 depicts a hybrid computer processor comprising a classicalcomputer and a quantum computer;Fig. 2A and 2B illustrates different quantum models for implementingquantum XC functionals; Fig. 3 illustrates illustrates classical MPL-based neural networkarchitectures for implementing classical XC functionals; FIG.4A and 4B depicts a process for computing electronic properties of amoleculair system using a hybrid computing system according to an embodiment; Fig. 5 depicts a flow chart for determining XC functionals for moleculairsystems according to an embodiment;Fig. 6 depicts the process of computing an electron density based on aself-consistent density functional algorithm according to an embodiment; Fig. 7A and 7B depict a scheme for executing differentialbe quantumcircuits (DQCs) using a quantum computer; Fig. 8A–8C depict quantum circuits which may be used in theembodiments in this disclosure; Fig. 9A illustrates depict another scheme for executing differentialbequantum circuits (DQCs) using a quantum computer; Fig. 10 depicts experimental regarding computation of electronic states of amolecular system using an density functional algorithm based on classical XC functions; Fig.11A-11C depict experimental regarding computation of electronicstates of a molecular system using an density functional algorithm based on quantum-enhanced XC functions; Fig. 12 schematically depicts a hardware-level schematic illustrating theapplication of logical operations to qubits using a quantum circuit; Fig. 13 schematically depict hardware-level schematics illustrating theapplication of logical operations to qubits using a quantum circuit; Fig. 14 schematically depicts a quantum circuit based on a digital-analogimplementation; Fig. 15 schematically depicts a quantum circuit based on a further digital-analog implementation; Fig. 16 schematically depicts a quantum circuit based on another digital-analog implementation; Fig. 17 schematically depicts a quantum circuit based on yet anotherdigital-analog implementation; Fig. 18 schematically depicts a quantum circuit based on a further digital-analog implementation. Description of the embodimentsThe embodiments in this disclosure generally relate to improved schemesfor efficient training of quantum models as XC functionals for computing electronicproperties of molecular systems. Classical neural network schemes for determining XC functions that are used in density functional algorithms for computing electronic propertisfor molecular system so far have not provided good results. The embodiments in thisdisclosure use quantum models to determine exchange-correlation functionals that are used in density functional schemes for quantum chemistry. The embodients furtherdescribe methods and systems for determining exchange-correlation functions formolecular systems using a quantum computer and methods and systems for determining electronic properties of a molecular system using quantum models representing exchange-correlation functions. The quantum models described with reference to the embodiments in thisapplication may be executed by a hybrid computer system comprising a quantumcomputer and a classical computer. Fig.1 depicts an example of such hybrid computersystem comprising a classical computer 102 comprising one or more classical processorsand a quantum computer system 104 wherein the classical computer may be configuredto train a quantum model that is executed on the quantum computer as a functional that is used in a so-called density theory algorithms which are used to compute electron-orbital features, such as an electron density and a ground state energy of a molecular system.As shown in the figure, the quantum computer system may comprise one or morequantum registers 106, e.g. a gate-based qudit quantum register. Such quantum registermay also be referred to as a quantum processor or a quantum processing unit (QPU). The quantum register may be implemented based on one of the well-known quantum hardware platforms including but not limited to superconducting circutis, quantum dots, neutral atoms, ions, optical qubits or NV centers. The system may further include a memory storage for storing arepresentation of a quantum circuit 114. A quantum circuit may represent a set ofinstructions to be executed on the quantum computer. The instructions are transformedinto control signals for executing gate operations or unitaries on the qubits of the quantumcomputer, wherein the control signals are sent by a classical computer to the quantumcomputer. In practice, the execution of a quantum circuit may involve a sequence ofoperations executed on the quantum registers by a controller system 108 comprising inputoutput (I / O) devices which form an interface between the quantum register and theclassical computer. For example, the controller system may include an optical and / orelectromagnetic pulse generating system for generating pulses, e.g. optical, voltageand / or microwave pulses, representing gate operations to be applied to the qudits of thequantum register in accordance with the quantum circuit.The controller system may include readout circuitry for readout of thequdits. At least a part such readout circuitry may be located or integrated with the chip that includes the qudits. The system may further comprise a (purely classical information)input 110 an (purely classical information) output 112. The input and output may be part ofan interface, e.g. an interface such as a user interface or an interface to another system.Input data may include training data associated with molecular systems. The trainingdataset may comprise reference data, in particular reference electron-orbital features such as reference electronic densities and associated energies, e.g. ground stateenergies, of atomic and / or molecular systems. The reference data may include syntheticdata reference data which may be computed using accurate ab initio methods such as full configuations interation (FCI) calculations. Further, the reference data may include experimental data, regarding electron-orbital features such as electron densities and ground state energies for the different atomic and / or molecular systems. Other information may include information about the quantum model, including parameters and / or conditions that are needed for training process. The input data may be used by the system to classically calculate values,e.g. parameter settings, which may be used to initialize a quantum circuit that isimplemented on the quantum processor. Similarly, output data may include loss function values, sampling results, expectation values, optimization convergence results, optimizedquantum circuit parameters and hyperparameters, and other classical data. The systemmay include a training module 116 that is configured to train a quantum model based ongate operations of a parameterized quantum circuit 114 which may be stored in a memoryof the system. In some embodiment, the trainable quantum model may represent aquantum neural network (QNN). The system may also include one or more classicalneural networks which may be used in combination with the QNN to form a hybridquantum neural network. Information about the quantum circuit and training data may beprovided via input 110 to the classical computer.The embodiments in this disclosure describe density functional schemes,such as the Kohn-Sham Density Functional Theory (KS-DFT) scheme, that use quantummodels that are trained as XC functionals. More generally, the embodiments in thisdisclosure describe quantum machine learning (QML) schemes for training XC functionalsand ways of using (embedding) such neural XC functionals in a density functionalscheme, e.g. the KS-DFT scheme, for computing electronic properties of molecularsystems. Hence, the system in Fig.1 may include a density functional algorithum120 that uses a trained quantum model or a trained hybrid quantum model to computeelectronic properties such as electron-orbital features of a new molecular system.Electron-orbital features such as the electronic ground state of an atomic ormolecular system may be determined based on self-consistent resolution of the KSequations, The Hamiltonian operator ^^ may be defined as ^^[^](^)^^(^) = −∇^ / 2 + ^^^⌈^⌉(^)wherein ^(^), is the electron density which may be define as the sum of probabilitydensities over all ^^^^ occupied electron-orbitals, ^(^) = |^,. Here, ^^mayrepresent an electron-orbital, in particular a one-electron KS orbital, with correspondingenergy ^^. The KS potential ^^^ may be defined based on the external ionic Coulombpotentials ^(^), the Hartree potential ^^[^](^) and the XC potential:^^^[^](^)^^^[^](^) = ^(^) + ^^[^](^) + ^^^[^](^). The XC potential may be computedas the functional derivative of the XC energy, ^^^, with respect to the electron density, i.e.^^^[^](^) = ^^^^[^] / ^^(^), wherein the XC energy ^^^ is defined asand wherein ^^^[^](^) is the XC energy per electron. Hence, in this scheme a XC functionmay be used to compute energies associated with electron-orbitals wherein the integralover all energies may defined the XC energy. The total energy of the atomic or molecularsystem may then be obtained as the sum, of the kinetic energy ^^[^], the externalpotential energy ^[^], the Hartree energy ^^[^] and the XC energy ^^^[^]: ^[^] =^^[^] + ^[^] + ^^[^] + ^^^[^].Given a finite basis set , a linear combination of ^^ basis functions,^^(^) = ξ^(^), may be optimized to produce suitable electron-orbitals ^. Thisinvolves the solution of Roothan’s equation defined aswhere ^ = (^^, ... , ^^^)^ are the electron-orbital coefficients, ∗^ = ∫ d^^^(^)ξ^(^) is anoverlap matrix and ℱ[n] is the Fock matrix as a functional of the electron density profile ^,wherein elements of the Fock matrix may be defined as ℱ^^ = ∫ d^^^(^)^^[^]ξ∗^(^). Eq. (3) may then be solved, e.g., numerically, by an eigen decomposition ofthe Fock matrix to obtain the orbital coefficients {^^} that correspond to an electron densityprofile ^. Hence, given a basis of electron-orbitals, a corresponding electron density maybe determined based on a self-consistent iterative calculation. Self-consistency may beachieved when the electron density no longer changes within a specified tolerance for anumber ^^^ of KS iterations (solutions of Eq.(3)). The energy ^ associated with thecomputed electron density can then be compared to validated data, e.g. experimentaland / or computed reference data. Here, the quality of the computed electron density and energy may dependent on the choice of the functional used to represent E^^[n]. If theexact XC functional was known, the Hohenberg-Kohn theorem guarantees that the self-consistent solution yields the exact ground-state energy and density. In an embodiment, a quantum model, e.g. a trainable quantum neuralnetwork QNN, may be trained according to Eq. (2), so that it represents a XC functional.The quantum model may represent a differentiable function f^: (n, r) ^ ^^^(^) withtrainable parameters ^. Hence, in this embodiment, the trained quantum model ^^(^, ^)representing an XC functional may be used to determine an energy for a given machine learned XC energy. Fig. 2A illustrates a quantum model representing a XC functional as acombination of differentiable functions. As shown in the figure a quantum modelrepresenting an XC functional may be implemented as one or more QNNs and – in someembodiments – one or more classical NNs to form a composition of differentiable functions202, which can be implemented based on a quantum neural network (QNN) 204. The QQNmay be represented as a quantum circuit including “layers” of gate operations including singlegate operations 206 associated with one or more feature maps which may be used forencoding input data in the Hilbert space of the quantum computer. The encoding of the inputdata may be realized via the the gate parameters of the gate operations of the feature map.The quantum circuit may further include one or more variational quantum circuits 208 forenabling the qubits to interact with each other. The XC energy may be expressed in general as ^[^] ≈ ^ ^^^ [^] =∫ ^^^ ^^(^, ^)^(^), where ^^^ [^] approximates the exact correlation energy E^^[n]. A so-called integration grid may be used to compute the integral of equation (2) . Theintegration grid may be a finite numerical integration grid of ^^^^^ grid points. The gridpoints may be grid points of any grid scheme suitable for three-dimensional numericalintegration. In some embodiments, the plurality of grid points may include grid points on areal-space quadrature grid. Embodiments of specific grid schemes include the Treutler, Mura-Knowles, Delley, and Gauss-Chebyshev schemes. The grid points may be associated with a coordinate system, e.g. aCartesian coordinate system or the like. In an embodiment, a grid generation algorithmbased on the atom-centered grids (ACGs) approach may be used. This algorithm isconfigured to generate a set of radial and angular grids for each atom in a molecule. In anembodiment, a radial grid may be designed to be denser near the nucleus and sparser inthe outer regions, while the angular grids may be designed to be denser near the polesand sparser near the equator. In an embodiment, the XC energy may be computed as an integral over aneural functional ^^ at each of the grid points: Here, ^^ defined grid weights which may be computed for each grid type andmolecular system. The article by Dsgupta et al, Standard grids for high-precisionintegration of modern density functionals: SG-2 and SG-3, Journal of ComputationalChemistry 38(12), 2017 describes gridpoints that may be used for the embodiments in thisclosure. Here, denote Cartesian coordinates associated with the grid points sothat ^(^^) defines an electron density at a grid point coordinate ^^. The computation of thecorrelation exchange energy according to equation (4) may be referred to as local energyembedding since it involves computation of a local XC energy density ^^^(^) at each gridpoint. In another embodiment, the total XC energy may be determined directlyusing a neural XC functional ^^[n], bypassing the integration as explained above withrefeence to equation (4), as where during training the neural XC functional ^^[n] learns implicitly the relation betweenthe XC energy and the grid points during training through input density values provided to the input of the neural XC functional. Hence, in this embodiment, a neural network is trained to receive ^^^^^input density value and to compute the XC energy.This approach may be referred to as the global energy embedding since the modelrequires the whole density profile to infer the total XC energy. The local and the globalenergy embedding (represented by Eqs. (4) and (5)), provide distinct ways of using atrained quantum model as an XC functional in a density functional scheme for computingelectronic properties of a molecular system. Despite avoiding grid integration in the globalenergy embedding, both approaches (local and global) require the computation of thelocal XC potential ^^^[^](^) at every grid point because the local XC potential is used toconstruct the Fock operator as defined by Eq.3, which shows that the local XC potentialis in the Hamiltonian and Fock operator is the integral of that Hamiltonian with the atomic orbitals.The quantum model representing an XC functional may be implemented asone or more QNNs and – in some embodiments – one or more classical NNs to form acomposition of differentiable functions, comprising trainable parameters ^ = the total number of layers of themodel. When combining a QNN with one or more classical NNs the model may be referred toas a hybrid quantum model. As described above, either trained quantum models for localembodding or quantum models for global energy embeddings may be used in a densityfuctional scheme, such as KS-DFT, for computing electronic properties of a molecular system. Fig. 2B illustrates different quantum neural network (QNN) architectures forimplementing quantum XC functionals. The quantum models may be characterized byvarious degrees of locality that represent the number of electron density points providedat input of the quantum model. The architectures include a local QNN 210 wherein dataare embedded via the gate parameters of the feature map into the QNN, a global QiQNN212 wherein data are embedded in the QNN using an amplitude encoding scheme and ahybrid quantum model 214, in which a QNN is combined with a classical NN, e.g. an MPL-based NN, to form a QNN / MPL model. QNN / MPL models may be combined to forma network of QNN / MLP models.The local QNN 210 as depicted in Fig. 2B may be used to representfuctionals where the quantum model only has access to a single localdensity information value at a grid point. To compute the total XC energy, the XCfunctional may be applied in succession to each point of the ^^^^^ points. Such functionmay be embedded in the density functional scheme using Eq. (4). In another embodiment,a ^-local quantum model may be used, which may be represented by the functions where the quantum model has access to only ^ neighbouring density points (kernel size)at a time. Hence, to compute the XC energy, the quantum model is applied in successionon, e.g., ⌈^^^^^ / ^⌉ points. Similarly to the one-local quantum models this quantum modelmay be used in the DFT scheme based on Eq. (4).In further embodiment, global models 212 and 214 may be used, which areconfigured to receive the whole electron density, i.e. all electron density values at all gridpoints, so that the quantum model can be trained to represent the whole integrated XC energy.A global quantum model may be used, which can be embedded in in the DFT schemeusing Eq. (5).In other embodiment, the global quantum model may (also) be embeddedin the DFT scheme using Eq. (4) as wherein the local XC energy depends on the whole electron density.Known classical models for computing exchange-correlation energies ofatomic systems, as e.g. described in WO2022 / 148847, may be based on a MultilayerPerceptron (MLP) model, wherein the layers are given by ^ ^^^ (^, ^) = ^(^^ + ^) includingactivation function ^ and trainable matrix ^ and vector ^. Given enough layers, a MLPmodel would be theoretically capable of representing any function according to theuniversal approximation theorem. However, practical use of an MLP model is hindered bythe large number of parameters and the lack of inductive biases. This leads to overfittingon the training data and poor generalization to data outside the training set. MLPs areused in known schemes for computing exchange-correlation energies of atomic systemsusing classical neural networks. In this disclosure, such classical neural networks areused for comparative studies involving quantum models and hybrid quantum models asdescribed with reference to the embodiments in this disclosure. Classical MLP models that are used for these comparative studies aredepicted in Fig.3. As shown in the figure, depending on their implementation the modelsmay have different number of inputs and ^^ layers consisting of ^^ neurons, wherein theLocal MLP (LMLP) model may be defined as using Eq. (7), where the model function has access only to the local density informationand is applied on ^^^^^ points. Fig.3 shows an example of a local MPL model 302 (L=1).This function may be embedded in DFT via Eq. (4). In a similar way, ^-local MLP, wherenot a single density value but a collection of ^ neighbouring densities {^^}^are taken asinput, may be defined. MPL model 304 of Fig. 3 shows an global MLP (GMLP) neuralnetwork which may be defined as a functional of the whole density,using Eq. (8). Ths model may be embedded in the DFT theory using Eq. (5). More elaborated classical NNs may include the Global Kohn-ShamRegularizer (GKSR) model as described in the article by Li et al, Kohn-Sham equations asregularizer: building prior knowledge into machine-learned physics (Phys. Rev. Lett. 126,036401 (2021). The GKSR model is currently one of the best models for computing theXC energy. The GKSR model combines local and global convolutional layers, andincorporates inductive biases such as a self-interaction gate and a negativity transform, addressing common issues like self-interaction errors and imposing the negativitycondition on the XC energy. In an implementation, the GKSR model may be representedby the function: using Eq. (9). Despite representing a global functional, it is embedded in DFT via Eq. (4)unlike GMLP as defined above. This allows comparision of the results of embedding theseXC models in DFT using both Eq.(4) (local / global) and Eq.(5) (global). As already described above, useful general XC functionals based on trained classical neural network have not been obtained yet. In other words, useful general XC functions for computingelectronic properties of molecular systems cannot be achieved based on current classicalneural network models.The embodiments in this application aim to address this problem by using aquantum model, such as a quantum neural network (QNN) model, to learn XC functionals,wherein a quantum model may be represented as a quantum circuit (which may includefurther (sub) blocks of quantum circuits) comprising qubit operations which need to beexecuted on qubits or qudits of a quantum coputer. As will be described below in moredetail, different quantum models may be used for implementing the embodimentsdescribed in this application this application. Hence, in an embodiment, XC functionals may be learned by optimizing aquantum model, e.g. training a quantum neural network (QNN), comprising blocks ofquantum circuits. These blocks may include a feature map block ^^ for embeddeding real-world data (i.e. electron-orbital features such as electron densities) into the Hilbert spaceof the quantum register, a parameterized quantum circuit block (sometimes referred to asan ansatz ^^) which may be configured to act on the embedded data using single andmultiple qubit operations. Further, a measurement block may be used to determine theexpectation value of an observable representing an XC energy. Typically, the QNN maybe implemented on a ^^-qubit quantum register that is initialized at some initial state |0^.For example, in an embodiment, the zero computational basis state for the qubits may beused as the initial state.A feature map may be used to embed the data (e.g., electron-orbitalfeatures such as the electron density ^) in the Hilbert space, ^^(^)|0^ in a differentiablemanner so that the XC potential ^^^ may be computed. As will be described below inmore detail, different feature map architectures may be used. In an embodiment, the feature map may be used to embed one or more discretized electron-orbital features,such as one or more discretized electron density valuesin the QNN as gateparameters of the feature map. In an embodiment, the feature map may be a trainablefeature map. The parameterized block ^^(θ^) having trainable parameters ^^ may beused to act on the embedded data with local qubit operations and entangling gates. Manydifferent feature maps and / or variational quantum circuit blocks blocks may be used totrain a quantum model to efficiently represent an XC functional. Embodiments of such feature maps and variational DQC blocs are disclosed in this application. In an embodiment, the output of the model may be extracted based onprojective measurements (typically referred to as “shots”) of an cost operator. In an embodiment, a cost function may be based on Pauli operators. For example, the costfunction may be expressed as ^^ = θ^^ ^^^ , where ^^ are fixed or trainable parametersassociated to Pauli operators ^^ ∈ ^^, ^^, ^^, ^^^^^^ . In an embodiment, the cost operatormay be configured as the total magnetization ^^ = , which was shown to avoid cost- function-induced barren plateaus. In this embodiment, a complete representation of the quantum model maybe given by the following expression is the general representation of the modelwith ^^ ansatz layers and ^^ total repetition of ansatz and feature maps, which mayimplement the data re-uploading strategy. Repetitions of layers may be used to augmentthe expressivity of the model in order to provide a good approximate of a general XCfunction. In an embodiment, the so-called Hardware-Efficient Ansatz (HEA) knownfrom variational quantum eigensolver (VQE) algorithms that are used in quantumchemistry applications may be used as an ansatz ^^ layer. The structure of the HEA maybe used to represent a general quantum model. The HEA may comprise one or more^blocks of single-qubit rotations (i.e., ^^(^) = ^^ ^^^^). The single-qubit rotations may beplaced on qubits of a quantum register with distinct angles ^^, which enable the representation of most general rotation on a Bloch sphere. The HEAmay further comprise one or more entanglement blocks. The entanblement blocks mayinclude an arbitrary pattern of (parametrized) multi-qubit gates that introduce interactionsbetween qubits. For example, in an embodiment, an entanglement block include aplurality of CNOT operations wherein ^ represents an ordered set of pairs of qubit indices. In an embodiment, a CNOToperation on qubit pair ^, ^ in equation C2 may be defined as ^^^^(^, ^) = In an embodiment, an “alternate ladder” set of qubit pairs may be used. Such alternateladder may be defined as: ^^^ = ^^ ∪ ^^ with ^^ = ((0,1), (2,3), ... , (^^ − 1, ^^)) and ^^ =((1, 2), (3, 4), ... , (^^ − 2, ^^ − 1)).In an embodiment, a parametrized entanglement layer may be definedbased on a sequence of parameterized ZZ gates. For example, a parameterizedentanglement layer may be defined = ^^^^(^, ^), where ^^^,^^represents the parametrized ZZ gatebetween qubits. This allows the model to cancel interaction ^^^,^ ^^^ = 0^ between qubits ifnecessary. Afeature map may be designed to provide expressiveness to a trainedquantum model representing an XC functional. In an embodiment, electron-orbitalfeatures such as electron energy densities may be encoded in the parameters (e.g.angles) of a quantum circuit, typically a feature map. In another embodiment, the data may be encoded into amplitudes of computational basis states . A general angleembedding unitary may be represented by a set of feature maps that may be expressedas with generators ^^ and a data transformation function ^: ℝ → ℝ . In an embodiment, atleast part of the generators may be defined as multi-qubit operators. Non-limitingexamples of feature maps that may be used to embed data in quantum circuits includeproduct feature maps, Chebyshev feature maps and amplitude data encoding schemes.In an embodiment, a 1-local product feature map may be used. The fullylocal version (1-local) of the product feature map may be configured to embed the datavalue ^ in the rotation angles of ^^ gates In another embodiment, a 1-local Chebyshev tower feature map may beused, which utilizes an efficient polynomial basis and which is defined byIn yet a further embodiment, a plurality of data points (input data) may beencoded by a feature map. For example, in an embodiment, an ^-local version of aproduct feature ^^,^^ may be realized by embedding different data points into single-qubitgates angles associated with available qubits. Here, the parameter ^ represents thenumber of data points that can be encoded in the circuit simultaneously. Since the featuremaps presented in this disclosure can embed at most 1 ≤ ^ ≤ ^^ data points, there are^^^^^ / ^ different embeddings for grid points to perform (model executions). Theseemboddigns can be executed in parallel since they are independent.In a further embodiment, amplitude encoding may be used for mapping thedensity vector ^ to a quantum state as |^^ = ^(^^^^^)^,^(^)|0^. Afinal quantum state |^^ may be prepared by repetitive applications of oneor more feature map blocks and one or more variational circuit blocks as described above.The final state may be represented by the following expression: This final state may subsequently be used in Eq. (13) to define a functional ^^(^) =^^(^, ^)|^^|^(^, ^)^.As described above, the XC energy may be computed based on integrationof a local functional (eqation (4)) or based on a global functional (equation (5)).In an embodiment, a quantum model implemented based on a (fully) localQNN (LQNN) f^^^^^ , may be trained to take a single density point and providean expectation value of a local XC energy density ^^^^,^at its output: as defined through Eq. (7). This quantum neural functional may be used in the DFTscheme based on Eq. (4). Local QNNs may include product feature maps and / orChebyshev feature maps as e.g. described with reference to Eq. C5 and C6 (referred to as local PrQNN and local ChQNN) respectively. An ansatz may be used that has astructure as defined by Eq. (C3) above.In a further embodiment, the quantum model may be implemented basedon a ^-local QNN, f ^^^^^ : {n^}^ ^ ^ ^^^,^ = ^0|^^(^, θ)C^^(^, θ)|0^ wherein a collection of ^neighbouring densities are embedded as input data in gate operation parameters of afeature map.In another embodiment, the quantum model may be configured as a globalQNN. Typically a global model (e.g., ^ = ^^^^^) may provide the best model forapproximating exact XC functionals. However this would require a qubit register with ^^^^^number of qubits, which – for a large number of gridpoints – could be a problem forcurrent quantum registers. To broach this problem, in an embodiment, a hybrid neuralnetwork including at least a classical neural network and a quantum neural network maybe used. Such hybrid neural network may combine advantages of classical NNs andquantum QNNs. Notably, the number of elements ^^^^^ in the electron density ^ scaleslinearly with the number of atoms. Even for small di-atomic molecules, there are hundreds of grid points for the smallest grid density available with the best integration grids. Thismay pose an problem when using feature maps based on angle encoding since in thatcase, the number of qubits ^^ required to embed the entire density ^ equals the size ofthe grid ^^^^^. This poses severe requirements for implementing a global QNN on a near-term quantum computer. Therefore, in an embodiment, a Global QNN (GQNN) may bedefined that includes a classical neural network and quantum neural network: where f^ and f^ represent the classical and quantum networks with associated trainingparameters θ^ and θ^, respectively. An example of such hybrid quantum model is shownin Fig.2B. In principle, any differentiable classical neural network may be used to mapoutputs of the quantum neural network f^ to an XC energy. Here, the quantum neuralnetwork may comprise ^-local QNNs defined as including a feature map block, e.g. a product feature map, and an parametized circuitblock, e.g. an alternate ladder ansatz. The input may be partitioned in ^^ = [^^^^^⁄ ^ ]batches of input electron densities and each batch of input electron densities may beembedded in the feature map of the QNN ^^^ resulting in ^^ outputs which may be usedas input of the classical neural network ^^ which outputs the exchange energy:In an embodiment, the classical neural network may be implemented using a standardMLP with ^^input neurons and one output neuron (potentially with additional MLPnetwork layers). In general, any classical neural network model architecture may be usedto represent the classical neural network. XC functionals may also be designed based on simulated quantum modelsas quantum-inspired (Qi) solution. The Global QiQNN, which incorporates density throughamplitude embedding (following Eq. (C4)) and which may utilize a standard hardwareefficient VQE ansatz as described above. The Global QiQNN may be represented by thefunctionwherein an amplitude embedding feature map is used. A global quantum-inspiredconvolutional NN (GQiCNN) architecture may be used that implements the composition of quantum layers with angle embedding as follows where the product feature map (Eq. (C5)) is throughout all the layers. Similarly to the hybrid architecture (GQNN), the quantum layers may comprise QNNs, forexample, ^-local QNNs, as defined in Eq. (16). Further, a product feature map andalternate ladder ansatz may be used to define a QNN. The input may be partitioned in^^ = [^^^^^⁄ ^ ] batches and then ^^^ is applied to each batch. This may be repeated ^^times until the output is a single scalar, which represents ^^^. If not (i.e., odd number ofinputs), the output of quantum layers may be combined with an MLP as in Eq. (17) tooutput ^^^. Hereunder, embodiments will be discussed for efficiently training QNN-based XC functions for differentiable chemistry frameworks as for example described withreference to Fig. 4-6 below.To evaluate the capabilities of quantum computers for machine learning XCfunctionals, different differentiable frameworks may be used. Li et al proposed in theirarticle "Kohn-Sham equations as regularizer: Building prior knowledge into machine-learned physics." Physical review letters 126.3 (2021): 036401, a method to facilitate thegeneralization of neural XC functionals wherein the KS iterative procedure is used as aregularizer. In this method, the energy loss may be computed iteratively at every KSiteration and electron density loss may be computed at the last KS iteration only. At eachKS iteration, the loss function is differentiated with respect to the trainable parameters ofthe QNN. Repeating this process for ^ KS iterations (solutions of Eq. (1)) enables fast,i.e. in fewer KS iterations, convergence of the quantum model to a reference, whileregularizing its parameters. To implement this method, a fully differentiable chemistry code in 1D inJAX was developed with the aim to demonstrate the regularization capacity of the framework. Since it reduces the number of dimensions (to 1D) and hence simplifies theproblem, this is an suitable setup for examining QNN-based models as XC functionals.However, realistic chemistry simulations require a full 3D treatment of molecular systems.To that end, the well-known chemistry framework, PySCF may be used. PySCF has andifferentiable version, referred to as PySCF with Automatic Differentiation (PySCFAD).PySCFAD, which is described in the article by Zhang et al, J. Chem. Phys.157, 204801(2022), is based on JAX but offers the capability for the accurate treatment of chemical systems in 3D, making it suitable for a wide range of chemical simulations. In an embodiment, to facilitate the training of neural XC functionals withinKS-DFT in 3D, the spin components of the electron density may be constrained to beequal, i.e., ^↑(^) = ^↓(^) and ^(^) = ^↑(^) + ^↓(^), thus defining the so-called spin-restricted case of KS-DFT (RKS-DFT). In a further embodiment, the scheme may beextended to the unrestricted case (UKS-DFT), allows the treatment of open-shell systems,which means that one has to transver from a conventional electron density ^(^) to adescription that is based on an electron density that has two spin components(^↑(^), ^↓(^)) with corresponding potentials ^^^,^[^↑, ^↓](^) = ^^^^[^] / ^^^(^), where spinstates are denoted by ^ = {↑, ↓}. This modification doubles the number of inputs for theneural XC model. Although the examples in this disclosure are based on the RKS-DFTapproach, the embodiments are not limited thereto and may also include schemes basedother density functional theory algorithms such as UKS-DFT, RKS-DFT or furtherimprovements thereof. The exact density of an interacting molecular system, which may becomputed exactly using a Full Configuration Interaction (FCI) algorithm, allows forfractional occupation numbers, which is preferably for accurate computation of the groundstate and accurate training of the XC model. Therefor, in an embodiment, the algorithm ofChai, J. Chem. Phys.136, 154104 (2012) for fractional occupation may be used to enablefractional occupation numbers. This algorithms provides a computational efficient solutionof introducing fractional occupation numbers. Other approaches however may also beemployed with an increased computational cost. The algorithm for fractional occupation as developed by Chai et al. is basedon the premisse that the distribution of occupation numbers is independent of the specificmany-body system and uses a universal form described by the Fermi-Dirac distribution. Inthat case, the electron density can be expressed a sum of the electron-orbitals ^^(^):^(^) = ∑^^^^ ^^|^^(^)|^where the occupation number ^^is the Fermi-Dirac function ^^={1 + ^^^[[(^ − ^) ^^^ / ^]} Here, the occupation number ^^ obeys the following twoconditions, ^^ = ^ and 0 ≤ ^^ ≤ 1, where ^^ is the orbital energy of the ithorbital ^^(^)and ^ is the chemical potential chosen to maintain a constant number of electrons ^.Hence, taken into account the fractional occupation of the electron-orbitals, theexpression for the total energy may then be given byWith temperature (^) dependent XC energy given by ^^ ^^ [^] = ^^^[^] + ^^[^], where^^[^] represents the difference between the non-interacting kinetic free energy at zerotemperature and that at temperature ^. In that case, the entropy contribution may be given by − Given a suitable electronictemperature ^, the chemical potential ^ is set by ∑^ ^^^^^ {1 + ^^^[[(^^ − ^) / ^]} = ^ toconserve the number of particles. Hence, in an embodiment, in order to take account offractional occupation numbers, only the entropy contribution needs to be added to anexisting DFT implementation so that ^ ^ ^^ [^] can be learned by the QNN. This approach isparticular advantageous as it does not impact the overall computational scaling. Theimproved density representability requires the inclusion of a single free parameter, thetemperature ^ as described above.FIG.4A and 4B depict a process for computing electronic properties of amolecular system using a hybrid computing system according to an embodiment.As shown in Fig.4A, the process may include providing and / or computinga dataset 402 for training a quantum neural network (QNN). The training dataset maycomprise reference data, in particular reference electron-orbital features such as reference electronic densities ^^^^and associated ground state energies of atomic and / or molecular systems. The reference data include synthetic data reference data which maybe computed using accurate ab initio methods such as full configuations interation (FCI)calculations. Alternatively and / or in addition, the reference data may include experimental data, regarding electron-orbital features such as electron densities and ground state energies for the different atomic and / or molecular systems. These data may be obtained by techniques that can be used to probe electronic properties of atomic and molecular systems. For example information about the relative position of atoms in molecular systems may be obtained using X-ray crystallography and information about electron densities may be obtained by (but are not limited to) techniques that atomic force microscopy (AFM), scanning tunnelling microscopy (STM) or transmission electron microscopy (TEM) and spectroscopy techniques such as angle-resolved photoelectron spectroscopy. Based on these data, information about the quantum mechanical wave function of an atomic or molecular system can be obtained. Then, training process 404 may be executed, wherein a differentiablequantum neural network (QNN) 406 is trained based on the reference data. In particular,the training process may be used to train the trainable parameters ^ of the QNN based onthe reference data of the different atomic and / or molecular systems.The training scheme may include selecting an initial (first) electron density^ of a atomic or molecular system for which reference data are available. Then, a new(second) electron density and associated ground state energy may be computed basedon the initial (first) electron density ^ and based on a differentiable density functionalalgorithm 408 and an XC energy computed by a trainable QNN associated with firsttrainable parameters using the first electron density as input data. In an embodiment,the differentiable density functional algorithm may be a Density Functional Theory (DFT)algorithm such as the KS-DFT algorithm.The computation of the new (second) electron density and the associatedground state energy may be computed using an self-consistent density functionalalgorithm, such as the KS-DFT algorithm. This computation may include determining anestimate of an XC energy which is generated by the differentiable QNN using the initial(first) electron density as input data. A corresponding XC potential ^^^ may be computedby differentiation of the computed XC energy with respect to the electron density: ^^^ =^^^^ / ^^(^). The XC potential may then be used by the differentiable density functionalalgorithm to compute new electron-orbitals that can be used to determine the new(second) electron density. For example, Fock equations of the KS-DFT algorithm may beused to compute new orbitals from which the new (second) density may be determined. Ifthe difference between the newly computed electron and the earlier (initial) electrondensity is lager than a certain delta, then the new (second) electron density for computinga new electron density and associated ground state energy. If the difference between the newly computed electron and the earlier (initial) electron density is smaller than a certain delta, the computed electron density and associated ground state energy may be used for computing a loss. In particular, a loss function ℒ(^) 410 may be used to evaluate thecomputed density and the associated ground state energy using the reference data, forexample a reference electron density and a reference ground state energy of the selectedatomic or molecular system. Based on the computed loss, it may be determined to updatethe trainable parameters of the QNN using a classical optimizer 412 wherein the updatingof the parameters may include determining a gradient of the loss with respect to thetrainable parameters ∇^ℒ. Hence, after the update of the parameters, the QNN isassociated with second trainable parameters ^^. Subsequently, the self-consistently computed electron density and theassociated ground state energy may be used as an input to the self-consistent densityfunctional algorithm for computing a further energy density and associated groundstate.The training process 404 may be repeated for the different atomic and / or molecularsystems for which reference data 402 are available.Fig. 4B further illustrates the use of the trained QNN as an XC functional. Ifconvergence of the trainable parameters have been reached , the converged parameters^∗ may define the trained functional ^^∗ 418. i.e. the trained QNN representing the XCfunctional. This trained QNN representing a quantum-enhanced XC functional, may thenbe used to evaluate electronic properties, such as the ground state energy or the electrondensity of a new atomic or molecular system (step 420). These electronic properties maybe computed based on a density functional computation cycle as described with reference to Fig.4A, using the trained QNN presenting the functional ^^∗to compute the XC energy. The thus obtained function provides improved performance for computing or predicting electronic properties of atomic or molecular systems, which is particular relevant in quantum chemistry in which the computed or predicted properties asre used in determining a decision about a chemistry process in the real-word, e.g. fabricating or synthesying a particular new molecular system based. For example, the process may be used to compute the ground state energyfor a new molecular system or for an intermediate molecular systems that are part of achemical synthesis scheme. The ground state energy can be used to determine if a(intermediate) molecular system is stable or if atoms in a new molecular system can bind.In an embodimenmt, a supervised ML approach may be used to train aQNN to represent an XC functional. A training dataset ^ = ^^ ^^^^ , ^^^^^^^^^^^^^ (also referredto as a refenrece dataset) may be generated using an accurate ab initio method (forexample FCI) for ^^^^ different moleculair systems (which may also include differentgeometries of same molecules). Alternatively and / or in addition the training data set mayinclude experimentally determined data. In an embodimemt, in case of a 1Dimplementation, the training data may comprise training data as determined by Li et al."Kohn-Sham equations as regularizer: Building prior knowledge into machine-learnedphysics." Physical review letters 126.3 (2021): 036401. This dataset may includereference data for hydrogen-based molecules (H , H ^^ ^ and H^). For a 3D implementation,computational schemes such as Python-based Simulations of Chemistry Framework(PySCF) and / or Coupled Cluster Singles Doubles (CCSD) may be used to generate datafor hydrogen type molecular system.Fig. 5 depicts a flow chart for training a quantum model as an exchange-correlation functional according to an embodiment. In particular, the figure depicts amethod of training a quantum model, e.g. a QNN, as an XC functional as e.g. described with reference to Fig.4, wherein, in a first step 502, the method may include receiving or determining a training dataset comprising synthetically generated and / or real-worldmeasured reference electron densities and reference energies for a plurality of grid points,the reference electron densities and reference energies being associated with differentmolecular reference systems. In a further step 504, the method may include receiving or determining information regarding a quantum circuit defining gate operations of a parameterizedquantum model, wherein the quantum circuit may comprise one or more feature maps forembedding an electron density into the Hilbert space of the quantum computer. Thequantum circuit may further comprise one or more parameterized variational quantumcircuits having trainable parameters may be used to act on the embedded data using localqubit operations and entangling gates. Then, the parameterized quantum model may be trained as an exchange-correlation functional (step 506), wherein the training may include embedding for at leastpart of the grid points first electron density in the one or more feature maps. The gateoperations of the quantum circuit may be executed using quantum elements of thequantum computer in order to compute an exchange-correlation energy. Further, asecond electron density may be computed based on the exchange correlation energy anda self-consistent density functional algorithm, such as a DFT algorithm. Then, theparameters of the variational quantum circuit may be updated based on a training loss,wherein the training loss may be computed based on the second electron density and areference electron density. Hence, the method provides an efficient way of training a QNN as a XCfunctional that is used in a density functional theory scheme for computing electronicproperties of a molecular system. Fig. 6 depicts the process of computing an electron density based on aself-consistent density functional algorithm according to an embodiment. In particular, thefigure depicts a process of computing an electron density based on a self-consistentdensity functional algorithm as described in Fig. 4A in greater detail.The process may start with providing or determining training data of aplurality of different moleculair systems (step 602), wherein the training data may includereference electronic-orbital features, such as reference energy and reference electron density of these different molecutlar systems. The training data may further include atomiccoordinates forming a discrete grid of ^^^^^ points in space to which electron-orbitalfeatures may be assigned. In a next step 604, an initial electron density may be estimated(e.g. using a classical computation and / or using experimental data) which can be used asinput to a density functional algorithm for computing an estimate of the energy density of amolecular system and the associated ground state energy. In particular, the initial electrondensity ^ estimate may be used to compute the Coulomb and Hartree potentials ^,^^(step 512). Further, the electron density ^ estimate may be provided to the input of aQNN, including one or more feature maps and one or more variational circuits, which istrained as a XC correlation function ^^ (step 606) to compute an XC energy. Then, theelectron density may be embedded in the QNN (step 608), different ways of embedding the electron density values in the QNN. In an embodiment, the QNN may be configured toreceive ^ density values as inputs wherein 1 ≤ ^ ≤ ^^^^^ . In case of ^ = 1 (i.e. one pointof the integration grid) the electron density is fully locally embedded in the gateparameters of the feature map. In case of ^ = ^^^^^ (i.e. the full integration grid) theembedding of the input data is globally embedded in the gate parameters of the featuremap. Other values of ^ may be used for the embedding.After the embedding of the input data, the gate operations of the quantumcircuit defining the QNN may be executed on the quantum computer to compute the XCenergy (step 610). In case of an QNN with local embedding, the output of the QNN will bean energy value ^^ for a particular electron density value on a point of the integrationgrid. Hence, in that case, the QNN needs to be executed multiple times so that the sum ofall computed values at all grid points of the integration grid will result in the XC energy. AnQNN with global embedding will output an energy value representing the total XC energyfor the whole integration grid. Thereafter, the XC potential ^^^ may be computed as thederivative of the XC energy with respect to the energy density (step 611). Based on thecomputed Coulomb and Hartree potentials ^,^^, the computed XC potential ^^^ and thecomputed energies ^^ at the grid points, a Fock operator may be constructed and the KSequations may be solved to obtain electron-orbital coefficients ^^(step 614). Theseelectron-orbital coefficients may be used to compute a new density value ^^^^ (step 616),which may be compared with the earlier electron value (step 618).If the difference between the old and new density value is larger than acertain delta value ∆, the newly computed electron density value may be used as the newinput value (step 613) to start a next computation cycle (i.e. steps 606-618) until thedifference between subsequently computed electron densities is smaller than the deltavalue. If that condition is met, the self-consistency is reached and the density functionalgorithm may be stop. The computed estimated energy and electron density associatedwith a molecular system (step 620) may then be used in a training process as explainedwith reference to Fig.4A, wherein the variational parameters of the QNN may be updated based on a loss function, the computed electron density and associate ground state energy and the associated reference data from the training data. Fig. 7A and 7B depict a scheme for executing a QML model that may beused for solving a density functional theory problem in molecular chemistry as describedwith reference to the embodiments in this disclosure. In an embodiment, a QML model may be on differentiable quantum circuits(DQC) may be used. DQC schemes are described in WO2022 / 101483 with title Solving aset of (non)linear differential equaitons using a hybrid data processing system comprisinga classical computer and a quantum computer system, which is hereby incorporated byreference into this application. In the DQC scheme, the QML may include a quantumcircuits for modelling a surrogate solution ^(^) e.g. an XC functional that is configuredto receive input data ^, e.g. electron-orbital features such as electron densities asdescribed with reference to the embodiments in this application. The quantum circuit may include one or more feature maps and one or more variational quantum circuits. In an embodiment, only the variational quantum circuit may be parameterized for training. In another embodiment, both the one or more features maps and the one or more variational quantum circuits may be parameterized for training. In particular, in an embodiment, the quantum circuit representing the QMLmodel may include one or more variational feature maps ^^^,⃗ ^^⃗ ^^ 705 with aparameterized generator Hamiltonian ^^^(^)⃗, ^,⃗ ^^⃗ ^^ and one or more parameterizedansatze 708 for computing function values ^(^^) for different input values of ^^,e.g. electron densities. Hence, in this case the feature map may also include variationalparameters which may be updated when training the QNN. In a further embodiment,QML models based comprising one or more trainable feature maps are described in EPapplication no. 23186558.5 with title Solving computational problems using trainablefeature maps which is hereby incorporated by reference into this application.In some embodiments, the quantum circuits may also include derivativequantum circuits 707 which may be determined using auto-differentation and theparameter-shift rule. The derivative quantum circuits 707 and associated variationalquantum circuits 709 may be used to determine a value of a function derivativefor different values of ^^. As shown in Fig.7B, the quantum circuits may be executed on aquantum computer comprising a quantum register 718 comprising quantum elements,such as qubits or qudits. The quantum circuits including one or more variationalfeatures maps 720 and variational ansatze 723 may comprise a sequence of singleand / or multiple operations 721,722,725 that are applied to the quantum elements,followed by a read-out operation 724 to read-out the state of the qubit register. Whenexecuting the quantum circuits, the operations are translated into a sequences of pulses tho control the quantum elements and readout the quantum elements. The quantum circuits may be executed based on an optimization schedule, wherein the execution may include execution of the quantum circuits andmeasuring the state of the quantum regiter 704. A set of points {^} (a regular or arandomly-drawn grid) may be specified for each equation variable ^ 706. Variationalparameters ^^⃗ , ^^⃗ may be set to initial values (e.g. as random angles or apredetermined set of intial anges) and the operations may be executed by the quantumregister as shown in Fig. 7B and the state of the quantum register may be measured.Using a cost function ^^, expectation values with respect to the variational quantum state be determined (steps 710-711), for the chosen point ^^to represent the surrogate solution at that point ^(^^). In another embodiment, using a cost quantum wavefunction ^^^^^^^ theoverlaps can be determined (steps 710-711), for the chosen point ^^to represent the surrogate solution. The cost wave function can be parameterized by the encoding of a second point ^^and additional action of othertrainable parameters, or a fixed reference state (e.g., |^(^^)^ = |000 … ^, thewavefunction where all qubits are in the computational state |0^). In both embodiments,a potential solution at this point may be constructed taking into account the boundary conditions. Further, the expectation values ^^^(^, ^^⃗ , ^^⃗)^ / ^^ associated with thederivative quantum circuits may be estimated 711 for a predetermined cost function atpoint ^^. Repeating the procedure 706 for all points ^^ in {^}, function values andderivative values may be computed. Further a loss function and other quantitiesrequired by the optimizer 712 may be determined (forming required polynomials andcross-terms by classical post-processing). In some embodiments, regularization pointsmay be added, forcing the solution to take specific values at these points. The lossfunction may be used to determine a loss value, which may be used to determine ascore ^ (a metric) which is indicative to how well the QNN (parametrized by variationalangles ^^⃗ , ^^⃗) is capable of computing the XC energy, matching derivative terms andthe function polynomial to minimize the loss. With the aim to increase the score (and decrease the loss function), thegradient of the loss function 712 with respect to variational parameters ^^⃗ , ^^⃗ may becomputed. Using classical optimization procedure 714, such as the gradient descentprocedure, the variational parameters may be updated from iteration ^^ = 1 into thenext one ^^ + 1 in step 716: ^^^^^ ← ^^^^^^^ − ^∇^ℒ (with ^ being here a `learning' rate).The above-described steps may be repeated until one or more exit conditions arereached. Suitable exit condition may include: 1) the maximal number of iterations ^iterreached; 2) loss function value is smaller than pre-specified value; and 3) loss gradient is smaller than a certain value. Once the classical loop is exited, the solution may be chosen as a circuitwith angles that minimize the loss. Finally, the full solution may be extractedby sampling the cost function for optimal angles Notably, this can be done for any point ^, as DQC constructs the solution valid also beyond (and between) the points at which loss is evaluated originally. In a further embodiment, XC functionals may also be evaluated byevaluating quantum kernels and derivatives thereof. A QML model based on differentialquantum kernels may be used. Such differential quantum kernel schemes are describedin WO2023170003 with title “Quantum kernel based regression”, which is herebyincorporated by reference into this application. In that case, kernel values and akernel derivative values ∇^^ ^^^^^ may be computed for different input values ^^. Thesequantum feature maps, which may be referred to quantum kernel feature maps, cancomprise unitary operations that are similar to unitary operations of the trainable featuresmap described with reference to the embodiments in this application. Application ofoperations on the quantum register will results in a variational quantum state Subsequently, in an embodiment, kernel values may be determined as a wavefunctionoverlap of the a variational quantum state and another reference state Note that in other parts of this document and figures, the reference state |^^^ is alsoreferred to as a cost state|^^, which may arbitrarily depend on another input point orparameterization such that it is written as ^^^^^^^. A kernel value ^ may be measuredusing a well-known Hademard test configured to measure the real and imagingary part ofthe overlap: Similarly, derivatives of the kernel function may be be computed on thequantum computer using similar operation sequences resulting in a measurement of akernel derivative value. The measurement of a kernel value typically involves repeated measurements on the same circuit, the quantity of interest being an average of the repeated measurements. Thus, from the above, it follows that a broad class of feature maps can bedescribed as the tensor product of an arbitrary number of sub-feature-maps, each represented by the time evolution of a generator Hamiltonian applied on an arbitrary sub- dimension of the qubits where for the sub-feature-map ^, ^^^(^)⃗ is the generator Hamiltonian that depends onnon-trainable parameters ^⃗ and is an encoding function that depends on the inputfeatures ^⃗. Note that ^^^(^⃗) can be applied several times across the quantum circuit,interleaved with layers, for example in data re-uploading or serial feature maps. Indata re-uploading, ^^^(^⃗) is applied more than once in non-parallel blocks separated byother quantum operations which could be so that the data is “re-uploaded” into adifferent qubit basis. This is understood to increase the number of unique frequencies ofthe model. In serial feature maps, multi-dimensional data may be encoded by non-parallel blocks that are separated by other quantum operations which each encode one dimension Introducing trainable parameters in the generator Hamiltonian ofthe quantum model, produces a measured output which can be expressed as a Fourierseries, where the frequencies of each mode depend explicitly on the parameterization ofthe FM. For example, for a QNN with a cost ^^, the output of the model can be written as: Where ^ contains the spectral gaps of all generators and ^^ are the correspondingcoefficients. Thus, introducing trainable parameters into the generator, which allows theeigenvalue spectrum to change over the course of training, leads to a quantum model withtrainable eigenfrequencies ^^(^^⃗). This - in turn - allows the quantum model to “discover”a choice of spectral modes not known a priori that may minimize the loss function more efficiently than non-trainable FMs. Since the number of frequencies accessible to the model|Ω|scales with the Hilbert space dimension, parameterization of each frequency individually would require exponentially many parameters. The effectiveness of a simpleparameterization that grows only linearly with the number of qubits is described withreference the example below. The advantage of trainable feature maps (TFMs) stemsfrom two important technical effects. Firstly, TFMs are able to be trained such that their frequency spectra contain non-uniform gaps, producing non-orthogonal basis functions. Fig. 8A–8C depict quantum circuits which may be used in theembodiments in this disclosure. Fig. 8A shows a basic form of a quantum feature map,which is here illustrated as an example of a ‘product’ type feature map. In this example, single qubit rotations 804 (here chosen as ^^(^(^))) act on each qubit individually andare parametrized by a function ^ of variable ^. The function ^ is preferably a non-linearfunction. For a non-linear feature map encoding, the non-linear function ^(^) may beused as an angle of rotation. The application 802 of one rotation operation to each qubit ina register may be referred to as a single layer of rotation operations. A product feature map can be further generalized to several product layers, and different functions {^}; for example, several feature maps may be concatenated to represent a multivariable function. For a quantum kernel, a single layer (as in Fig.8A) or a double layer (as in Fig. 8B) is generally sufficient. Fig. 8B illustrates an example of a derivative quantum circuit for theproduct feature map of Fig.7A. Differentiation over variable ^ follows the chain rule ofdifferentiation, including qubit operations 812 with shifted phases ^^(^(^) ± ^ / 2). Here,the expectation value of the derivative is written as a sum of separate expectations with shifted phases, repeated for each ^-dependent rotation 8101–4. Fig. 8C depicts an example of a generalized product feature map, wherethe layer of rotations is followed by the unitary evolution generated by Hamiltonian 820.For complicated multiqubit Hamiltonians, the encoded state may comprise exponentiallymany unique ^-dependent amplitudes. The time interval ^ can be set variationally, orannealed from zero to a finite value during the optimization procedure. Preferably, the product feature map has a non-linear dependence on the encoded variable ^. In the simplest case, this may correspond to a single layer of rotations as shown in Fig.8A. Such product feature map may be described by the following expression: where ^^ ≤ ^ is a number of qubits that is used by the quantum computer for theencoding and the symbol denotes the tensor product. Further, ^^,^(^) = is a Pauli rotation operator for Pauli matrices ^^^,^ = ^^^, ^^^ ^^ ^^^, (^ =^, ^, ^, respectively) acting at qubit ^ for phase ^. This type of feature map circuit mayalso be used in Quantum Circuit Learning (QML).The next step is to assign a non-linear function for rotation. In anembodiment, the non-linear function may be selected as ^(^) = arcsin(^) and ^ = ^,such that only real amplitudes are generated. The unitary operator may then be rewrittenas (for ^^ = ^): ^ ^^(^) = ^ exp ^−iarcsin(^) 2^^^^ ,^^^ leading to amplitudes that depend on the encoded variables assin(arcsin(^) / 2) and cos(arcsin(^) / 2). Acting on the initial state |0 ^, this feature mapmay encode the variable as an ^-th degree polynomial formed by {1, ^, √1 − ^^} andproducts thereof. The product feature map can be generalized to several layers of rotationsℓ = 1, 2, … , ^, various non-linear functions ^ℓ and specific subsets of qubits ^ writtenas: In a further embodiment, a quantum model based on a parallel DQCscheme may be used. Methods and systems based on the parallel DQC scheme aredescribed in EP application no.23186602.1 with title Quantum circuits, which is herebyincorporated by reference into this application. Fig. 9 depicts the use of another scheme for executing an QML model thatmay be used for solving a density functional theory problem in molecular chemistry asdescribed with reference to the embodiments in this disclosure. In this DQC scheme aChebyshev or Fourier feature map may be used.In particular, in this embodiment, the quantum model for XC functional maybe represented as a projection: ^^^(^) = ^^(^) = ^^ |^^^ where the |^^ represents theinput data (e.g., electron density) that is mapped via Chebyshev or Fourier embedding inthe Hilbert space, and |^^^ = ^(^)|0^ the quantum state after application of aparametrized ansatz ^(^) on an arbitrary initial state |0^. The gradients with respect to theinput data ^ then may be computed via the projection ^′^(^) = ^ ^ |^|^^^ where ^represents a non-unitary differentiation operator. This model may be referred to as theparallel DQC model. To train the model, a mean squared error loss may be used withrespect to the reference data ^^ (e.g., CCSD(T) electron densities) as In a first step 902, information regarding a density function theory problemin molecular chemistry may be provided. As shown in the figure this step may includeproviding molecular data, e.g. reference electron densities and / or energies associatedwith different molecular systems. Further, (step 904) a parametric quantum model basedon the parallel DQC model may be selected wherein each term of a XC functional may bewritten as one or more terms in a latent space. This may comprise a step 920, comprisingrepresenting the unknown XC functional as a parametrized wavefunction, a step 922comprising determining one or more (first-order and / or higher-order) derivatives of theunknown XC functional with respect to the independent variable as modified parametrizedwavefunctions, and – optionally - a step 924 comprising determining one or more knownfunctions as wavefunctions. A subsequent step (not shown) may comprise receiving ordetermining one or more quantum circuits defining operations of the parameterizedquantum model, wherein the operations effectuate the encoding determined in theprevious step. Different quantum circuits defining operations of the parameterizedquantum model may be found in EP application no.23186602.1. Then, the XC functional may be determined by optimizing the model in thelatent space by running a DFT scheme, wherein DFT scheme uses the model during KSiterations and computation of loss gradients. The optimization of the model may include avariational solver 916 and a step of measuring overlaps on a quantum processor 918.After optimization of the model, the DFT scheme (DFT solver) and theoptimized quantum model may be used to evaluate electronic densities and energies of anew (unknown) molecular system that differs from the reference molecular system thatwere used during optimization (training) of the quantum model. Fig. 10 depicts experimental data regarding computation of electronicstates of a molecular system using a density functional algorithm based on a classical and quantum-enhanced XC functionals. The experimental data are computed based on a classical MPL-type neural network (as described with reference to equations (10)-(12), which may be used as a reference for comparing the experimental data associated with the quantum models, Fig. 11A-11C depict experimental data regarding computation of electronicstates of a molecular system using a density functional algorithm based on quantum-enhanced XC functionals. In particular, Fig. 11A-C depict experimental data computedbased on a fully quantum model, a hybrid quantum model or a quantum-inspired model asdescribed with referencs to embodiments in this application. The experimental data in thefigures illustrate the of training neural XC functionals on different geometries of hydrogenmolecule in 1D KS-DFT framework of Li et al.. Local models improve but qualitative follow the LDA shape. Best results are obtained for global models when the whole electrondensity is considered as model input. The reference data (ref) depicted in the figures aregenerated using FCI. LDA results are shown for comparison with local quantum (hybrid) models. All neural functionals are trained on eight training data points (circles) and thefinal model parameters are selected based on best loss on the validation results. Themodel architectures that were use to compute these data are described with reference toequations (6)-(19) above.When more than two electrons are present, the CCSD with approximateTriples (CCSD[T]) method is used, which is the “golden standard” in computationalchemistry. This method is described in detail in the article by R. J. Bartlett and M. Musial,Reviews of Modern Physics79, 291 (2007). Gaussian-type orbital basis set is used allsystems, in particular 6-31G. As integration grid, SG-3 with Becke scheme may be usedto combine multi-atomic grids, such that for a given molecule at all geometries the samenumber of grid points are obtained. The grid density may be set to minimum to reduce thenumber of neural XC functional inputs, thereby reducing the complexity of the problem.Given that models are trained that allow a fixed number of inputs (^^^^^), grids have to beadapted (e.g., pruning) to have the same number of grid points for all molecules in the dataset. However, this is only a limitation imposed by simple prototypical (quantum) models that are investigated in this work. To train the models, a loss function may be used given by where the expectation is taken on ^^^^molecules of the training set and ^^weights thecontribution of every energy at each KS iteration. Hence, generally, in an embodiment, theloss function may include a density loss ℒ^computed based on the predicted electrondensities (associated with parameter ^) and the reference densities at each point of thegrid and an energy loss ℒ^computed based on the predicted energies (associated with parameter ^) and the reference energies at each point of the grid. For the energy loss a weighting scheme as described by Li et al. may beused ^^ = 0.9^^^^^^(^ − 10), where ^ is the Heaviside function to tune the contributionof every loss term. Derivatives of the loss may be computed via JAX automaticdifferentiation. On quantum hardware, exact differentiation schemes such as parameter-shift rule or its generalized version (as described with reference to Fig.18) can be used.Gradient evaluation is needed for training NNs in general. Since QNNs canbe thought of as a function ^^, approaches for differentiation may be used. In classicalML, automatic differentiation (AD) is used, which that exploits the chain rule and storesthe intermediate values of sub functions (the layers) of the ML model to evaluate thegradients of ^^. This approach has an advantageous scaling that enables training very large classical models and, hence, is always used for the differentiation of classical layersin this work. In case of QNNs, an efficient approach has been developed, beyond finitedifferences, for gradient evaluation. Due to the trigonometric relations within the unitaries^associated with single-qubit rotations, ^^^^ = ^^ ^^^^(that include single-qubit Pauli generators ^^), it is possible to derive a simple and exact derivative formula which issimilar to finite differences, referred to as the Parameter-Shift Rule (PSR): The Generalized PSR (GPSR) rule, which is needed for general multi-qubit generators, isdescribed hereunder with reference to Fig.18. The loss function may be minimized to obtain the optimal modelparameters ^^^^ = ^^^^^^^ℒ(^) that define the optimal XC correlation functional. Aclassical optimizer such as Adam of L-BFGS-B may be used to find the optimzal parameters. It is noted however that the embodiments in this application may also be performed based on gradient free optimization methods. At each converged SCF calculation, the gradient of the loss ^ may becomputed based on automatic differentiation or the parameter-shift rule when using aquantum computer. The gradient is then provided to the classical optimizer (Adam or L- BFGS-B). The learning rate for the Adam optimizer can be decreased from 10−2to 10−4as the training progresses to fine tune the results. Otherwise, L-BFGS-B and Adam are used with the standard hyperparameters. To converge the training, it is possible to use a combination of Adam at first to ensure the escape of local minima due to the stochastic direction of the gradient in Adam and then switch to L-BFGS-B to follow the actual gradient, in order to achieve accurate results. The experimental data illustrating the performance of the different modelsare illustrated in Fig. 10 and 11. All models have been evaluated using a 1D framework.The training and test dataset from DMRG simulations of Li et al. have been used, where^^^^^ = 513 fixed for all the systems. For all the results, a density error in terms of themean squared error (MSE) has been defined.For the MLP neural networks that were used for the experimental datashown in Fig. 10, an expressive model that has been used to evaluate the approaches oflocal and global embeddings. To that end, in all MLP models, 2 hidden layers with 513neurons have been used due to the number of grid points to achieve enough expressivity.The Local MLP model has a single input and output neuron to infer the local XC energy based on corresponding density. In Fig.10, the local MLP improves on the LDA DFT results in terms of overall energy error, however is not sufficient to improve on the density. The restriction local information is limiting the capacity of the model to improve. Hence,the best trade-off of energy-density error that this model is capable of is observed.The experimental results of the quantum models are shown in Fig.11A-11C. In order to evaluate if local quantum models can improve on the local MLP of Fig.10, the quantum models (local QNNS) employ one or more Chebyshev feature maps(ChQNN) or one or mor product feature maps (PrQNN), respectively. Here ^^ = 4 qubits,^^ = 1 (no data reuploading is used) and ^^ = 8 layers for the variational ansatz. Asshown in Fig.11A, the results in terms of non-parallelity error (NPE) of the energy is similar to the local MLP model. The energy and density profile of local quantum and classical models are qualitatively similar, as expected due to locality of both models. It is therefore necessary to verify if including more information in the quantum models bringsbenefits in terms of lower energy error |∆^| and MSE of the density profile compared toXC functionals that are determined based on classical MLP type neural networks.As shown in Fig. 10 to that end, a global version of the MLP neural networkwith ^^^^^ = 513 input neurons has been evaluated. This allows the model to consider thewhole density to infer the XC energy via the global embedding. Hence, the globa MPLneural network has one output neuron. This results in ∼ 25 · 104 parameters. As shown inFig.10, the energy profile of the global MPL neural network is significantly improved inthat it closely follows the reference (lower error with respect to the reference) incomparison to all other models. However, the KSR model may be tailored to any locality.Similarly to MLP, the global version of KSR model (see equation (12) for global KSRmodel) performs best and generalizes in a single molecule even based pm only 2 datapoints (in this example 8 samples were used). Using the same density initialization as forglobal MLP, global KSR model struggled to generalize correctly, leading to MSE on theelectron density comparable to the local density approximation (LDA) models, despiterendering chemically accurate energies. The global KSR model is however significantlymore parameter-efficient, having only ∼ 103 parameters versus ∼ 25 · 104 compared tothe global MLP models, which are generally known to overfit the training data. In addition,given a good initialization, the global KSR model is capable to learn the dissociationbased on only 2 data points. Generally, one may conluded based on the experimentaldata that models with global embedding, the results are similar or better compared to localLDA-like functionals, at least for the total energies. The models provide smooth energy profiles due to regularization capabilities of the used differentiable density functional algorithm. It is clear that 8 data points, sampling low to strongly correlated regimes of the hydrogen molecule, are sufficient to generalize correctly within a single molecule provided the model has enough expressivity. In general, all the models show characteristic dips in energy errors at the training data. Since local QNNs are not capable of correctly fitting the training data,architectures may be used that allow to embed ^ data points and hence approach non-local functionals. To that end, a global QNN model is used, which is adapted to embed ^data points in a single QNN with ^^ = 6 qubits, ^^ = 1 (no data reuploading) and ^^ =8 variational layers. This model is then applied onto ^^^^^ / ^ different batches. The resultsare combined via a single-layer MLP with 171 input neurons and 1 output for the globalembedding. The experimental data are shown in Fig. 11B. This architecture allows toembed chunks of data into a quantum model with limited number of qubits. By combiningsuch outputs, a fully quantum-enhaned global functional may be obtained. The model hasonly ∼ 1000 trainable parameters, which is orders of magnitude smaller than the globalMLP for comparison since both are composed from most general quantum / classicallayers. Note that an accurate objective comparision of the hybrid quantum model with theKSR model is not possible as the latter possesses numerous inductive biases such as theself-interaction gate and the softly imposed negativity of XC energy. Neverthless, theglobal QNN of Fig. 11B show that good results are obtained compared to the othermodels.. Fig. 11C depicts a quantum model using amplitude embedding as a featuremap. Here a DQC is used with ^^ = 9 qubits, ^^ = 1 and ^^ = 10. This model may bereferred to as the global QiNN model since differentiating amplitude embedding exactly isnot feasible and in general hard to embed the classical data (density) exactly. The numberof qubits was chosen to fit all the elements of the electron density ^ exactly as a state-vector. The data in Fig.11C show that the results for the global QiNN model are in goodagreements with the reference both in chemical accuracy and overall density error whichshows that this model provides a significant improvement over the LDA models. Notably,the outcome is very similar to Global QNN, which is simple to implement on near-term devices. This model architecture, despite producing satisfactory results, does not producea significant advantage over the hybrid QNN of Fig.11B. It can however serve fortheoretical studies since only quantum layers are employed, dissociating it completely from classical models. Fig. 12 is a hardware-level schematic of the actions effectuating the logicaloperations as defined in a quantum circuit, e.g. DQC quantum circuit, as described with reference to the embodiments of this applications. Unitary operators, e.g. those to encode the feature map and derivatives thereof, can be decomposed into a sequence of logical gate operations. These logical gate operations are transformations in a quantum Hilbertspace over the qubits that encode an input parameter. In order to transform the internalstates of these qubits, a classical control stack may be used to send pulse information toa pulse controller that affects one or more qubits. The controller may send a sequence of such pulses in time and for each qubit independently. An initialization pulse is used toinitialize the qubits into the |0^ state 1202. Then, for example a series of single-qubitpulses may be sent to the qubit array in 1204, which may represent a single-layer featuremap. Additionally, two-qubit or multi-qubit pulse sequences can be used to effectivelyentangle multiple qubits of the feature map. The duration, type, strength and shape ofthese pulses determine the effectuated quantum logical operations.1208 indicates a‘break’ in the depicted timeline, which means the sequence of gates may be repeated in a similar fashion in the direction of the time axis 1212. At the end of the pulse sequences, one or more of the qubits are measured 1210. Fig. 13A is a hardware-level schematic of a photonic / optical quantumprocessor which can be used for implementing quantum models comprising trainablefeature maps as described with reference to the embodiments in this application. As known from Gan et al, Fock state-enhanced expressivity of quantum machine learning models, EPJ Quantum Technology 2022, 022-00135-0 feature maps for encoding classical features or data may also be implementedbased on photonic circuitry. Unitary operators, e.g. those used to encode the quantum feature map and derivatives thereof, can be decomposed into a sequence of optical gate operations. These optical gate operations are transformations in the quantum Hilbert space over the optical modes. In order to transform the internal states of these modes, a classical control stack is used to send pulse information to a pulse controller that affects one or more modes. The controller may formulate the programmable unitary transformations in a parametrised way. Initially the modes 1314 may all be in the vacuum state 1316, which arethen squeezed to produce single-mode squeezed vacuum states 1318. The duration, type, strength and shape of controlled-optical gate transformations determine theeffectuated quantum logical operations 1320 which may define one or more feature mapsassociated with a first generator Hamiltonian ^^(^(^)⃗) and, optionally, one or morevariational quantum circuits (parameterized ansatze) associated with a parameterizedHamiltonian ^^^^^⃗ ^^. At the end of the optical paths, one or more modes may be measuredusing a known optical measurement technique 1322, including but not limited to photon-number resolving, a Fock basis measurement, tomography or threshold detectors.Fig. 13B is a hardware-level schematic of the actions effectuating thelogical operations shown in circuit diagrams specified for a Gaussian boson sampling device. Unitary operators, e.g. those used to encode the quantum feature map and derivatives thereof, can be decomposed into a sequence of optical gate operations. These optical gate operations are transformations in the quantum Hilbert space over the optical modes. In order to transform the internal states of these modes, a classical control stackmay be used to send information to optical switches and delay lines. The controller mayformulate the programmable unitary transformations in a parametrised way. Initially the modes 1326 may all be in a weak coherent state, which ismostly a vacuum state with a chance of one or two photons and negligibly so for highercounts. Subsequently, the photons travel through optical waveguides 1328 through delaylines 1330 and two-mode couplers 1332 which can be tuned with a classical control stack,and which determines the effectuated quantum logical operations, which may which may define one or more feature maps associated with a first parameterized generator Hamiltonian and, optionally, one or more variational quantum circuits (parameterized ansatze) associated with a second parameterized Hamiltoning. At the end of the optical paths, one or more modes are measured with photon-number resolving 1334, or threshold detectors. Fig. 13C is a hardware-level schematic of the actions effectuating thelogical operations specified for a photonic / optical quantum processor. The quantum model can be decomposed into a sequence of optical gate operations. These optical gate operations are transformations in the quantum Hilbert space of the photons. In order to transform the internal states of these photons, a classical control stack is used to send information to a universal multiport interferometer. The controller may formulate theprogrammable unitary transformations in a parameterized way. Initially the photons 1355are in Fock states, weak coherent states or coherent states. The duration, type, strength and shape of controlled-optical gate transformations determine the effectuated quantumlogical operations 1356 , which may which may define one or more feature mapsassociated with a first parameterized generator Hamiltonian and, optionally, one or more variational quantum circuits (parameterized ansatze) associated with a secondparameterized Hamiltoning. At the end of the optical paths, the modes are measured withphoton-number resolving, Fock basis measurement 1357, tomography or threshold detectors. Fig.14 is a hardware-level schematic of actions effectuating the logicaloperations shown in a quantum circuit which may be executed on a so-called neutral-atom-based quantum computer. On this type of hardware, unitary operators, e.g. those used to encode the quantum feature map and derivatives thereof, can be decomposed in two different kinds of operations: digital or analog. Both of these operations are transformations in the quantum Hilbert space over atomic states. Schematic (a) of Fig.14 depicts a digital quantum circuit 1438, whereinlocal laser pulses may be used to individually address neutral atoms to effectuate transitions between atomic states which effectively implement sets of standardized or‘digital’ rotations on computational states in accordance with the quantum circuit. Digitalgates may include any single-qubit rotations, and a controlled-pauli-Z operation with arbitrary number of control qubits. Additionally, such digital gate operations may also include 2-qubit or even multi-qubit operations. Schematic (b) of Fig.14 depicts an analog mode 1446 of operation of aneutral-atom-based quantum computer, wherein a global laser light pulse may be applied to groups of, or all, atoms at the same time, with certain properties like detuning, Rabi frequencies and Rydberg interactions to cause multi-qubit entanglement therebyeffectively driving the evolution of a Hamiltonian 1444 of the atomic array in an analogway. The combined quantum wavefunction evolves according to Schrödinger’s equation,and particular, unitary operators ^^ = ^, where ℋ^ denotes the Hamiltonian and ^ thetime, can be designed by pulse-shaping the parametrised coefficients of the Hamiltonian in time. This way, a parametric analog unitary block can be applied, which entangles the atoms and can act as a variational ansatz, or a feature map, or other entanglement operation. The digital and analog modes shown in this figure may be combined oralternated, to yield a combination of the effects of each as shown in schematic (c) of Fig.14. This figure depicts an example of such digital-analog quantum circuit, including digitalblocks 12461-3 of digital qubit operations (single or multi-qubit) and analog blocks 14481-3,wherein the operations of the digital and / or analog blocks may be used to implement oneor more trainable feature maps as described with reference to the embodimebts in thisapplication. It can been proven that any computation can be decomposed into a finite set of digital gates, including always at least one multi-qubit digital gate (universality of digital gate sets). This includes being able to simulate general analog Hamiltonian evolutions, by using Trotterization or other simulation methods. However, the cost of Trotterization is expensive, and decomposing multi-qubit Hamiltonian evolution into digital gates is costly in terms of number of operations needed. Digital-analog circuits define circuits which are decomposed into both explicitly-digital and explicitly-analog operations. While under the hood, both are implemented as evolutions over controlled system Hamiltonians, the digital ones form a small set of pre-compiled operations, typically but not exclusively on single-qubits, while analog ones are used to evolve the system over its natural Hamiltonian, for example in order to achieve complex entangling dynamics. It can be shown that complex multi-qubit analog operations can be reproduced / simulated only with a relatively large number of digital gates, thus posing an advantage for devices that achieve good control of both digital and analog operations, such as neutral atom quantum computer. Entanglement can spread more quickly in terms of wall-clock runtime of a single analog block compared to a sequence of digital gates,especially when considering also the finite connectivity of purely digital devices. Hence, inan embodiment, the quantum models described in this disclosure for representing XCfunctionals may include digital-analog quantum circuits.Further, digital-analog quantum circuits for a neutral quantum processor that are based on Rydberg type of Hamiltonians can be differentiated analytically so that they can be used in variational and / or quantum machine learning schemes, including the differential quantum circuit (DQC) schemes and the kernel methods as described in thisapplication. In order to transform the internal states of these modes, a classical controlstack is used to send information to optical components and lasers. The controller may formulate the programmable unitary transformations in a parametrised way. At the end of the unitary transformations, the states of one or more atoms may be read out by applying measurement laser pulses, and then observing the brightness using a camera to spot which atomic qubit is turned ‘on’ or ‘off’, 1 or 0. This bit information across the array is then processed further according to the embodiments. Fig.15A depicts a system describing a digital-analog implementation of a quantum circuit as described with reference to the embodiments in this application. Inparticular, Fig. 15A represents a digital-analog implementation where the qubitinteractions can be switched on and off and there exists single-qubit addressability for the application of single-qubit gates. In order to transform the internal states of these qubits, a classical control stack may be used to send information to a controller that affects one or more qubits. The controller may send multiple information carriers (e.g., pulses) in timeand for each qubit independently. An initialization protocol may be used to initialize thequbits, in this example into the |0^ state 1502. The current example shows only fourqubits, but the same principles may be extended in a straightforward manner to systems with more (or less) than four qubits. Then, a quantum feature map may be applied 1504 to encode quantuminformation, e.g., an input variable, into a Hilbert space associated with the quantumprocessor. Following application of the feature map, a variational Ansatz may be applied,implemented as a variational quantum circuit 1506. In the current example, the variationalAnsatz comprises three single-qubit gates ^^ – ^^ – ^^ with different rotational angles ^^. These single-qubit gates are typically sets of standardized or ‘digital’ rotations on computational states applied to different qubits with different rotation angles / parameters. These digital gates include any single-qubit rotations according to the ^^argument of the rotation. A single rotation / single-qubit gate is not sufficient to perform arbitrary rotation regardless of the parameter / angle since it can only rotate over a single axis. In order to accomplish a general rotation block, three gates in series with different rotation angles / parameters can be used, in this example represented by a block of single-qubit gates 1506. Then, entanglement in this digital-analog approach may be generated by awavefunction evolution 1508, described by the block e^^ ℋ ^. During this evolution, the qubits are interacting amongst themselves, letting the system evolve for a specified amount of time. This process produces the necessary entanglement in the system. The combined quantum wavefunction evolves according to Schrödinger’s equation, andparticular unitary operators ^^ = e^^ ℋ ^ in which is the Hamiltonian of the system (forexample, for neutral atoms the Hamiltonian that governs the system is = ^ being the state occupancy of the Rydberg atoms), and ^ isthe evolution time. In this way, a parametric analog unitary block can be applied, which entangles the atoms and can act as a variational Ansatz. After the evolution of the wavefunction 1508, another set of single-qubitgates may be applied, similar to the process described in the block of gates ^^ – ^^ – ^^1506. Then, the wavefunction may be evolved once more, and finally a measurement inthe computational basis occurs as described in 1510. Optionally, additional steps may be included, represented by the ellipses, e.g., before, after, or in between the blocks shown.For example, a different initial state than |0^ may be prepared prior to application of thefeature map. Furthermore, the blocks can be executed in a different order, for instance, insome embodiments, block 1508 might precede block 1506. One or more of the blocks1504–1508, or variations thereof, may be repeated one or more times prior to themeasurement 1510. Fig. 15B represents a trainable quantum feature map 1512 which appliessingle-qubit rotations to qubits 1502 with (typically) different rotation angles 1504. In thisexample, single qubit rotations 1504 (here chosen as ^^(^^ (^))) act on each qubitindividually and are parametrized by a variable ^ which may be encoded according to oneor more enconding functions ^^(^). For a non-linear feature map encoding, the variable ^ may be used as an angle of rotation. The application of one rotation operation to each qubit in a register may be referred to as a single layer of rotation operations. This type ofencoding of classical information into quantum states wherein a data vector isrepresented by the angles of a quantum state. This type of encoding can be found inmany quantum machine learning (QML) algorithms, with the main advantage being that itonly requires ^ = log ^^ qubits to encode a dataset of ^ inputs with ^ features each.Thus, considering an algorithm that is polynomial in ^, it has a logarithmic runtime dependency on the data size. Fig.15C illustrates different pulses (potentially differing in pulse duration and / or amplitude / height) that can be sent to the quantum circuit (different qubits) to encode the classical information. The angle of the rotation that is described by the^^(^) dictates the duration and geometry of the pulse.Fig. 16A presents a quantum circuit based on a digital-analogimplementation 1600 that has no single-qubit addressability and uses a wavefunctionevolution that can be turned on and off for the entanglement generation. Fig.16A differsfrom Fig.15A in that it shows a technique known as data re-uploading that alters thetrainable quantum feature maps 1604, 1610. Loading classical data into a quantumsystem is a non-trivial task, and uploading large amounts of data makes it an even more critical issue. However, when the case of big-data is not considered, techniques like data re-uploading can be utilized. As the name suggests, this entails uploading the classicaldata multiple times, and encode them via rotational angle / parameter encoding into single-qubit rotations throughout the quantum circuit. Such a data re-uploading technique may start by setting the quantum register to an initial state, e.g., the all-zero state 1602, followed by angle encoding that is based on a feature map 1604, in this case a tower feature map. Following that, as already explained above with reference to 1606, three single-qubit gates in series are used to perform a general rotation block that acts as theAnsatz 1606. In this example the encoding is done through the ^^ variable of the ^^ – ^^– ^^ rotations; however, any type of combined rotations of the type (^^, ^^, ^^) could beapplied here. Since there is no single-qubit addressability in this example, each ^^and ^^gate is applied at all qubits simultaneously with the same rotation angle ^^. Subsequently, the wavefunction is allowed to evolve for a specified amount of time 1608. Then the classical data are re-uploaded through angle encoding into single- qubit rotations 1610. However, each time the (same) data are uploaded, the information isencoded using different angles than in the previous data uploading steps. For example,the amount of rotation may be doubled in each data (re-)uploading step 1604,1610 (i.e., the rotational angle of the might be increasing as 1–2–4, etc.). Once again, the rotations ^^ – ^^ – ^^ are applied to all qubitssimultaneously with the same rotation angle for each single qubit gate, but different than 1606. The classical information can be repeatedly encoded into the quantum feature map through the data re-uploading technique, while changing the angle of the single qubit rotational angle (1604, 1610, etc.) every time after the wavefunction evolution. The resulting quantum feature map can become more expressive as a tower feature map by doing this process of serial data re-uploading with different levels of rotational angles every time the re-uploading is done. Fig. 16B illustrates a different (and at current technology levels morerealistic) form of a digital-analog implementation which may be used in the embodimentsdescribed in this application. In this implementation, the system has single-qubit addressability, but the wavefunction evolution cannot be turned off, thus staying always on. This system requires less control over the qubit interactions (since they are always on)and only requires single-qubit addressability. The single-qubit gates are applied while thewavefunction evolution (qubit interaction) is also happening. In the shown example, thequantum circuit 1618 starts by encoding information in the |0^ quantum state 1602,followed by a trainable quantum feature map 1605. Then, a variational Ansatz is applied, comprising both wavefunctionevolution e^^ ℋ ^ 1609 and simultaneous application of a block of single-qubit gates 1616.After multiple repetitions of block 1616, each time with different rotational angles at thesingle-qubit gates, the circuit ends with a measurement in the computational basis 1620.Fig.17A illustrates a quantum circuit 1700 that implements a digital-analog implementation with single-qubit addressability, wherein the wavefunction evolution (e.g., atom interactions) can be turned on and off. This quantum circuit is equivalent to thequantum circuit in Fig. 16A, with the time of the wavefunction evolution 1608 being ^ = ^(in dimensionless coordinates), while all other operations are the same as the ones described in 1500. The same technique of data re-uploading with a tower feature map1704, 1510 and the same Ansatz comprising single-qubit gates 1706,1712 followed bywavefunction evolution 1708 are used.The reason that such a specific time period for the wavefunction evolution1708 is selected, is due to the fact that such evolution results in the application ofControlled-Z (CZ) gates between pairs of qubits. The benefit of describing the wavefunction evolution as a set of CZ gates lies in the fact that this Digital-Analog implementation of a quantum circuit resembles the structure of the equivalent Digitalquantum circuit, which instead of the operation 1708 performs a set of CNOT gatesbetween pairs of qubits included in the quantum circuit. This allows more straightforward comparisons to be drawn between the Digital and Analog implementation of such feature maps, and to evaluate their relative performance. Fig.17B illustrates a quantum circuit that implements a digital-analog implementation with single-qubit addressability 1718, similar to 1700, but with the entanglement generated via CZ gates between the qubits, resembling the atom interaction / entanglement generation. Fig. 18 presents a workflow for generalized circuit differentiation. InQuantum Machine Learning (QML) algorithms, various quantum feature maps and Parametric Quantum Circuits (PQCs) are typically utilised to express the circuit required to solve a particular problem. These quantum circuits contain tuneable parameters that use a classical optimizer to find the optimal values of the parameters that exist for that formulation. The differentiation required for most parameter optimization methods can be achieved in various ways, with the most common one being the Parameter Shift Rule (PSR). An example of such kind of differentiation for quantum feature maps is presented in Fig.18. One way to calculate analytic derivatives of quantum circuits is by measuring overlaps between quantum states. Analytic derivatives can be used fordifferentiating unitaries like the one presented in 1804, e.g., ^^ = e^^ ^ ^^(^,^) / ^ generated byarbitrary Hermitian generator ^^. However, in order to have a less resource-intensive method for differentiation, the parameter shift rule (PSR) was proposed. The PSR algorithm can provide analytic gradient estimations through measurement of the expectation values, with gate parameters being shifted to different values. The PSRalgorithm is much used to perform differentiation for QML algorithms; however, it is onlyvalid for a specific type of generators, viz., generators that are involutory or idempotent. This is because the full analytic derivative only requires 2 measurements of expectation values (2 unique eigenvalues). Although this simplifies the differentiation protocol, it also restricts it to only work for a certain type of generators. Therefore, as an improvements and generalizations of the PSR algorithm,in 1800 a scheme for a Generalized Parametric Shift Rule (GPSR) algorithm is shown.Such approach allows for differentiating generic quantum circuits with unitaries generated by operators with a rich spectrum (i.e., not limited to idempotent generators). The GPSR algorithm is based on spectral decomposition, and showcases the role of the eigenvalue differences (spectral gaps) during differentiation for the generator spectrum. Such approach works for multiple non-degenerate gaps, in contrast to the PSR that is only valid for involutory and idempotent operators with a single unique spectral gap. A workflow for generalized circuit differentiation can be described asfollows: create a quantum circuit that encodes the function ^ similar to 1800 includingsingle- and multi-qubit gates 1802. Pick a unitary operator such as ^^(^) = exp^−i ^ ^^ / 2^1804 that is parametrized by some tuneable parameter ^, and study the spectrum of itsgenerator ^. Then, using unique and positive spectral gaps, a system of equations that includes the spectral information can be created, alongside the calculated parameter shifts, and the measured function expectation values as shifted parameters. The solution of such system can provide the analytical derivative for any general generator G and thus perform GPSR. GPSR is described in more detail in Kyriienko et al., ‘Generalizedquantum circuit differentiation rules’, Phys. Rev. A 104 (052417), which is herebyincorporated by reference. The techniques of this disclosure may be implemented in a wide variety of devices or apparatuses, including a wireless handset, an integrated circuit (IC) or a set of ICs (e.g., a chip set). Various components, modules, or units are described in this disclosure to emphasize functional aspects of devices configured to perform the disclosed techniques, but do not necessarily require realization by different hardware units. Rather, as described above, various units may be combined in a hardware unit or provided by a collection of interoperative hardware units, including one or more processors as described above, in conjunction with suitable software and / or firmware. The terminology used herein is for the purpose of describing particular embodiments only and is not intended to be limiting of the invention. As used herein, the singular forms “a,” “an,” and “the” are intended to include the plural forms as well, unless the context clearly indicates otherwise. It will be further understood that the terms“comprises” and / or “comprising,” when used in this specification, specify the presence ofstated features, integers, steps, operations, elements, and / or components, but do not preclude the presence or addition of one or more other features, integers, steps, operations, elements, components, and / or groups thereof. The corresponding structures, materials, acts, and equivalents of all means or step plus function elements in the claims below are intended to include any structure, material, or act for performing the function in combination with other claimed elements as specifically claimed. The description of the present invention has been presented for purposes of illustration and description, but is not intended to be exhaustive or limited to the invention in the form disclosed. Many modifications and variations will be apparent to those of ordinary skill in the art without departing from the scope and spirit of the invention. The embodiment was chosen and described in order to best explain the principles of the invention and the practical application, and to enable others of ordinary skill in the art to understand the invention for various embodiments with various modifications as are suited to the particular use contemplated.

Claims

CLAIMS 1. Method of determining an exchange-correlation functional for computingelectronic properties of a molecular system using a hybrid data processor comprising aclassical computer and a quantum computer, the method comprising:receiving or determining reference data comprising synthetically computedand / or real-world measured reference electron-orbital features, preferably referenceelectron densities and associated reference ground state energies, for a plurality of gridpoints, the reference electron-orbital features being associated with different molecularreference systems; determining a first electron density of a molecular system for whichreference electron-orbital features are available;computing a second electron density for each of the grid points based on:the first electron density, an exchange correlction energy computed by a variationalquantum model using the first electron density as input; and, a density functionalalgorithm, preferably a self-consistent density functional algorithm; and,if the difference between the second electron density and the first electrondensity is smaller than a predetermined delta value, then updating variational parametersof the variational quantum model based on a training loss, wherein a loss function is usedto compute the training loss based on the second electron density and a referenceelectron density respectively.

2. Method according to claim 1 wherein the variational quantum model isrepresented by a quantum circuit defining gate operations, the quantum circuit defining aquantum neural network comprising one or more feature maps and one or more parameterized variational circuits.

3. Method according to claim 2 wherein the variational quantum modelfurther comprises one or more classical neural networks configured, wherein the updatingof the variational parameters includes updating the weights of the classical neural network.

4. Method according to claims 2 or 3 wherein computing the exchangecorrelation energy includes: using the one or more feature maps to encode the first electron density atat least part of the grid points in the Hilbert space; and,executing the quantum circuit on quantum elements of the quantum computer; measuring the state of the quantum elements to obtain an exchangecorrelation energy.

5. Method according to claim 4 wherein executing the quantum circuitincludes: translating quantum operations of the quantum circuit into control signals;controlling the quantum elements of the quantum computer based on thecontrol signals.

6. Method according to claims 4 or 5 wherein executing the quantum circuitincludes: determining classical measurement data by measuring the state of the quantum elements of the quantum computer; determining the exchange-correlation energy based on the classical measurement data.

7. Method according to any of claims 1-6 wherein computing predictedelectron-orbital features includes: determining an exchange-correlation potential based on the derivative of the predicted exchange correlation energy with respect to the electron density; and, determining Coulomb and Hartee potentials using the first electron density; computing the second electron-orbital features based on the self-consistentdensity functional algorithm using the Coulomb and Hartee potentials and the exchange-correlation potential as input.

8. Method according to any of claims 1-7 wherein the one or more featuremaps are trainable feature maps, the trainable feature maps including one or more unitary operators defining a time evolution over a Hamiltonian, preferably a generator Hamiltonian, applied to the quantum elements of the quantum register, the Hamiltonian evolution being parameterized by variational parameters.

9. Method according to any of claims 1-8 wherein quantum circuitcomprises digital quantum gate operations, preferably digital single quantum gateoperations, and one or more analog quantum gate operations configured to entangledifferent quantum elements of the quantum computer and to evolve a Hamiltonianassociated with quantum elements in time;10. Method according to any of claims 1-9 further comprising:if the difference between the second electron-orbital features and the firstelectron-orbital features is larger than the predetermined delta, computing third electron-orbital features for each of the grid points based on: the second electron-orbital features,a predicted exchange correlction energy computed by the quantum model using thesecond electron-orbital features as input; and, the density functional algorithm.

11. Method according to any of claims 1-10 wherein further comprises:computing a ground state energy associated with the second electron density; and, further computing the loss value based on the ground state energy and areference ground state energy associated with the reference electron density.

12. Method according to any of claims 1-11 wherein at least part of thevariational parameters are adjusted using a classical optimizer, preferably a gradient- based optimizer.

13. Method according to any fo claims 1-12 wherein the one or moreelectron densities are encoded based rotation operations and a non-linear encodingfunction, preferably the non-linear encoding function including a Chebyshev polynomial.

14. Method according to any of claims 1-13 wherein the variationalquantum model comprises one or more differentiable quantum circuits (DQCs), the one or more differentiable quantum circuits including one or more derivative features maps forencoding the one or more derivatives of the exchange-correlation functional.

15. Method according to any of claims 1-14 wherein the quantum computeris implemented based on at least one of: a neutral atom quantum device, an ion-basedquantum device, an optical and photonic quantum device, a superconducting quantum device, a silicon based quantum device, a diamond N-V centre quantum device and / or a gaussian boson sampling device.

16. A hybrid data processor for determining an exchange-correlationfunctional for computing electronic properties of a molecular system, the hybrid dataprocessing comprising a classical computer and a quantum computer, the hybrid dataprocessing being configured to: receiving or determining reference data comprising synthetically computedand / or real-world measured reference electron-orbital features, preferably referenceelectron densities and associated reference ground state energies, for a plurality of gridpoints, the reference electron-orbital features being associated with different molecularreference systems; determining a first electron density of a molecular system for whichreference electron-orbital features are available;computing a second electron density for each of the grid points based on:the first electron density, an exchange correlction energy computed by a variationalquantum model using the first electron density as input; and, a density functionalalgorithm, preferably a self-consistent density functional algorithm; and,if the difference between the second electron density and the first electrondensity is smaller than a predetermined delta value, then updating variational parametersof the variational quantum model based on a training loss, wherein a loss function is usedto compute the training loss based on the second electron density and a referenceelectron density respectively.

17. A computer program or suite of computer programs comprising at least one software code portion or a computer program product storing at least one software code portion, the software code portion, when run on a hybrid data processing system comprising a classical computer and a quantum computer comprising a hardware quantum register, being configured for executing the method steps according any of claims 1–15.

Images