Apparatus for providing control signals for controlling a quantum computing system

Abstract

The invention refers to an apparatus (710) for providing control signals for controlling a quantum computing system. The quantum computing system comprises at least two quantum computers with different fidelities. A problem providing unit (711) provides a problem description indicative of a problem. A sub-problem deriving unit (712) derives sub-problems from the provided problem description based on the fidelities of the at least two quantum computers. A control signal generation unit (713) generates control signals for controlling the quantum computing system to cause a quantum computation of the derived sub-problems on the at least two quantum computers, wherein the sub-problems are distributed to the at least two quantum computers based on the fidelity of the respective at least two quantum computers.

Description

[0001] Apparatus for providing control signals for controlling a quantum computing system

[0002] FIELD OF THE INVENTION

[0003] The invention refers to an apparatus, a method and a computer program product for providing control signals for controlling a quantum computing system. Further, the invention refers to a quantum computing system that can be controlled by the control signals, a problem solving apparatus and method for solving a problem utilizing a quantum computer and a system for calculating a solution of a problem.

[0004] BACKROUND OF THE INVENTION

[0005] Quantum computing is an emerging technology that exploits quantum mechanical phenomena to perform quantum computational tasks. Quantum computers are expected to solve certain computational problems substantially faster than classical computers. However, applying quantum computers to real world problems is still a challenging task.

[0006] SUMMARY OF THE INVENTION

[0007] It is an object of the present invention to provide apparatuses, methods, systems and computer program products that allow to increase the efficiency and effectivity, i.e. allow to decrease the computational time and required computational resources with the same resulting quality, for calculations of problems related to chemical products, for instance, electronic structure problems, utilizing a quantum computer. Moreover, the chemical industry sells chemical products with specific technical application properties. The core of any chemical product is the chemical molecule or the combination of chemical molecules, e.g. the chemical product includes one or more chemical molecules, which determines the properties of the chemical product. Properties of chemical monomers or oligomers may be determined to determine the properties of chemical polymers e.g. through extrapolation.

[0008] Chemical products with one or more chemical molecules of one or more types are highly complex real-world systems containing multiple interacting electrons that make up the properties of the chemical molecules and thus the properties of the chemical products. Hence, it is utterly important to capture the complexity of chemical molecules in calculations by reflecting the atom types, the chemical bonds between the atoms of the same and / or different types, the three-dimensional structure of the atom arrangement, the interactions between multiple atoms in that three-dimensional arrangement and in particular the subatom structure made up by electrons and the electron density. Such highly complex molecular structures are commonly described by way of an electronic structure problem, such as by utilizing a Schrodinger equation utilizing a Hamiltonian operator. To allow for a solution of such complex problems, like such complex electronic structure problems, approximations can be used. However, through approximation the properties stemming from complex correlations, for instance, electron correlations, may be inadequately reflected in most systems. Hence there is a need to include as much of the complexity, for example, of the electron correlations, of the real-world system as possible to arrive at results that are as close to the real world technical application properties of the chemical molecule as possible. This is particularly important for the chemical industry, which produces chemical products based on the generation of chemical molecules, their properties and the production recipes leading to a respective real-world chemical product with respective properties. However, also in other industries and applications solving complex problems with many variables is a challenging task, for instance, in cryptography, logistics, etc.

[0009] Calculations of problems associated with a chemical product, for example, electronic structure calculations, are essential in a process for designing an experimental setup, wherein often the number of experiments that lead to the real-world chemical molecule with desired properties depends on the accuracy of the result of the calculated electronic structure problem and the accuracy of the properties determined by the result. Obtaining a highly accurate solution can be a NP hard problem - meaning more than multiple ten years for a classical computer to solve. Hence, it is impossible in many real-world scenarios of developing practically relevant chemical molecules with enhanced or new properties to generate such accurate solutions. This leads to more intense laboratory work, which is inefficient and increases the time for developing new chemical molecules. With the environmental impacts of chemical developments and the speed at which chemical industries have to adapt their chemistry backbones and chemical products, it is advantageous to improve the development of new or enhanced chemical products specifically via more accurate solutions to the electronic structure problems.

[0010] Embedding quantum computing into such development cycles may provide for a solution to such complex problems. However, the embedding of quantum computing is a challenge on its own. As a result, the apparatuses, systems, methods and computer program products disclosed herein allow for more efficient and effective use of quantum computing to enhance the development of new or enhanced chemical products.

[0011] By deriving sub-problems from a provided problem description based on the fidelities of to be utilized quantum computers and to cause a quantum computation of the derived subproblems, wherein the fidelity determines on which of the quantum computers the respective sub-problem computation is performed, quantum computing can be advantageously embedded in development of new or enhanced chemical products in a more efficient and effective way. In particular, taking the fidelity into account for distributing the sub-problems on different quantum computers, the different computational requirements of the sub-problems can be taken into account and respective parts of the problem, for instance, referring to a molecular substructure of a chemical product, can be solved with a higher efficiency leading to the higher overall efficiency in solving the respective problem.

[0012] In a first aspect an apparatus is presented for providing control signals for controlling a quantum computing system for performing a quantum computation of a problem, in particular, associated with a chemical product, wherein the quantum computing system comprises at least two quantum computers with different fidelities, wherein the apparatus comprises a) a problem providing unit for providing a problem description indicative of the problem, b) a sub-problem deriving unit for deriving sub-problems from the provided problem description based on the fidelities of the at least two quantum computers, and c) a control signal generation unit for generating control signals for controlling the quantum computing system to cause a quantum computation of the derived sub-problems on the at least two quantum computers, wherein the sub-problems are distributed to the at least two quantum computers based on the fidelity of the respective at least two quantum computers.

[0013] Generally, the apparatus can be realized in form of software or hardware or a combination thereof, wherein the hardware can refer to any known dedicated or general classical computer hardware. For example, the apparatus can be realized as any known computational device, like a PC. However, the apparatus can also be realized as a cloud environment, computational network, etc., such that at least parts of the apparatus can also be realized as a network solution and thus can be spread over a plurality of computational devices.

[0014] Generally, a quantum computer is a computer that exploits quantum mechanical phenomena for performing calculations. For example, a quantum computer can utilise during a quantum computation effects like superposition and entanglement to perform the quantum computation. Quantum computers are based on quantum elements that adhere to the physics of quantum mechanics, for instance, superconductors, ions, atoms, quantum dots, photons, particle spins, etc. To perform a quantum computation quantum elements adhering to the physics of quantum mechanics can be manipulated in a controlled manner according to a predetermined algorithm.

[0015] The fidelity of a quantum computer is a measure defining the quality with which the quantum computer performs a respective quantum computation, for instance, the operations, e.g. quantum gates, of an algorithm performed during a quantum computation. The fidelity of a quantum computer can be based on quantum gate fidelities and / or decoherence times. A quantum gate is a basic quantum operation usually generated through one or more control pulses applied to a respective small number of qubits, wherein a plurality of quantum gates build a quantum circuit and thus can be regarded as an analogue to classical logical gates that build conventional digital circuits. Any multi-qubit operation can be generated from a sequence of one- and two-qubit operations. In this context a fidelity measures the quality with which a quantum gate realised on the quantum computer hardware resembles the theoretical quantum gate it is supposed to implement in the quantum computer Thus, the quantum gate fidelity measures the difference between the real operation performed by the real hardware and the perfect theoretical operation. A decoherence time describes how long quantum information can be storedon the quantum computer before it is lost to the environment. Thus, the decoherence times also have an influence on the quantum gate fidelities that are determined, for instance, by all small imperfections in the build, calibration and control of the quantum computer and also based on the specific decoherence time of the quantum computer. Generally, the fidelity of a quantum computer can be determined utilizing known methods and measurements, for example, cross-entropy benchmarking, randomized benchmarking or full state tomography. More details on possible fidelity determination methods can be found, for instance, in “Quantum supremacy using a programmable superconducting processor.”, Arute, F., Arya, K., Babbush, R. et al., Nature 574, 505- 510 (2019). In most cases the fidelity will be provided as general information by a provider of the quantum computer. The problem can generally refer to any problem that is translatable into a problem description that can be computed on a quantum computer. In a preferred embodiment the problem refers to an electronic structure problem. In an example, the problem is related to a chemical product. For instance, a solution of the problem can be utilized for determining a technical application property of a chemical product or the solution of the problem can be utilized for optimizing a chemical reaction for producing a chemical product. In an example, the problem is an electronic structure problem. Generally, electronic structure problems refer to problems that are associated with determining an electronic structure of an atom, molecule, crystal, or amorphous solid, etc. For example, electronic structure problems can be utilized for determining energetic ground states of atoms and / or molecules, etc. Based on solutions of the electronic structure problem, further properties of the atom, molecule, crystal, or amorphous solid can be derived. Generally, the electronic structure problem can be mathematically represented utilizing different classes of underlying electronic structure Hamiltonians. For example, Born-Oppenheimer Hamiltonians, non-Born-Oppenheimer Hamiltonians, Hamiltonians with additional one-electron potentials representing, e.g., electrostatic fields, two-component Hamiltonians including spin-orbit coupling, fully relativistic four-component Dirac-Hamiltonians, etc.

[0016] The problem providing unit is adapted to provide a problem description indicative of the problem. The problem providing unit can refer to a storage unit on which the problem description is already stored. However, the problem providing unit can also comprise an input unit with which, for instance, a user can indicate a problem description to the problem providing unit. Generally, the problem description can referto any form of description of the problem that allows to determine a quantum mechanical form of the problem and the quantities defining the problem, wherein the quantum mechanical form indicates the interaction between these quantities. Preferably, the problem description refers to a mathematical description of the problem, for example, to a mathematical description of an electronic structure problem utilizing an electronic structure Hamiltonian, as described above. However, the problem description can also refer to any other unambiguous form of notation of the problem. In an example, the problem description is related to a chemical product. The chemical product to which the problem description is related to can define an electronic structure problem and the quantities of the electronic structure problem. For example, if the chemical product refers to a specific molecule for which a technical application property should be determined, the respective molecule and its structure define the quantities and interactions of the quantum mechanical electronic structure problem indicated by the problem description. The problem description can comprise or can be indicative of atomic positions, a basis set, charge, and spin multiplicity depending on the specific chemical product and / or quantum mechanical electronic structure problem. In an embodiment the problem is an electronic structure problem associated with a molecular structure of a chemical product, wherein the solution of the electronic structure representation is indicative of a property associated with this chemical product. The electronic structure problem can be represented by an electronic structure representation, wherein the electronic structure representation represents the problem such that it can be processed by the respective apparatus, for example, represents the electronic structure in a mathematical formulation. The chemical product can be any chemical product that includes, for example, consists of, one or more molecular structures. Molecular structures are structures that form at least a part of a molecule. For example, in a polymer consisting of a plurality of monomers, the molecular structure can be one of these monomers, but can also refer to a plurality of these monomers or even the complete polymer. Moreover, the molecular structure can also be an atom in a molecule forming the chemical product. Generally, the molecular structure comprises an electronic structure that can be described by an electronic structure representation.

[0017] Further, the apparatus comprises a sub-problem deriving unit configured for deriving subproblems from the provided problem description based on the fidelities of the quantum computers. At least some of the sub-problems can be dependent sub-problems. Sub-problems are considered as being dependent if at least one of the sub-problems depends on at least one other sub-problem. For example, at least two sub-problems can be coupled such that the solution of one sub-problem depends on the solution of another sub-problem. However, the sub-problems can also be interacting such that both sub-problems each depend on the solution of the other sub-problem. Generally, sub-problems can be defined as problems whose solution contributes to the solution of the problem to be solved. In most cases, dependent on the problem, a plurality of different sub-problems or combination of subproblems can be formulated that allows to solve the overall problem. Sub-problems are problems that are less complex than the original problem and thus easier to solve. For example, the overall problem can be split in smaller problems, e.g. sub-problems, with less unknown quantities than the overall problem and thus into problems that are easier to calculate.

[0018] The sub-problems are derived based on the fidelities of the quantum computers utilised for calculating the problem. Since the fidelity determines the quality and accuracy of a quantum computation performed on the respective quantum computer, the sub-problems can be determined based on this quality criterion. For example, if a quantum computer with a higher fidelity and a quantum computer with a lower fidelity are part of the quantum computing system, sub-problems can be determined for which it is sensible to be solved on a quantum computer with a higher fidelity, for instance, a fidelity above a predetermined threshold, and sub-problems can be derived that can adequately be solved on quantum computers with lower fidelity, for instance, a fidelity below a predetermined threshold. In an embodiment, sub-problems can be derived based on the fidelities and predetermined rules. For example, the rules can determine the size of the derived sub-problems and the number of sub-problems based on the available quantum computers and the known fidelities of these quantum computers. In an embodiment candidate sub-problems can be derived, for example, based on the determined rules, and a fidelity threshold can be determined for each sub-problem, wherein the fidelity threshold indicates the lowest fidelity with which a respective sub-problem can be calculated. The fidelity threshold can be determined based on expert knowledge, known relations and / or general rules. For example, the size of the sub-problem, the type of sub-problem, the target accuracy, the intended quantum algorithms, etc. can be taken into account. Generally a rough estimation of the fidelity threshold is already suitable. The fidelities of the available quantum computers can then be matched with the fidelity thresholds. If all sub-problems can be matched with a respective quantum computerthe candidate sub-problems are utilized as sub-problems. If the one or more subproblems cannot be matched to a quantum computer, the candidate sub-problems can be amended. For example, a size or number of the candidate sub-problems can be changed and again a match determined for the new candidate sub-problems. Moreover, also other criteria can be taken into account when deriving the sub-problems. In an example, the problem is represented by an electronic structure representation and the sub-problems are sub-problems derived from the electronic structure representation. In a preferred example, the problem is represented by an electronic structure representation of a chemical molecule and the sub-problems refer to an active space problem and an inactive space problem, wherein the active space problem is determined to be calculated on a higher fidelity quantum computer and the inactive space problem is determined to be calculated on a lower fidelity quantum computer, wherein the active space problem represents a part of the electronic structure of the molecule that comprises quantities that are correlated above a predetermined correlation threshold and the inactive space problem represents the remaining electronic structure of the molecule. In another preferred example, the problem is an optimization problem comprising a plurality of degrees of freedom that are to be optimized and the sub-problems refer to a correlation problem representing degrees of freedom that are correlated above a predetermined threshold and a remaining problem representing the remaining degrees of freedom, wherein the correlation problem is determined to be calculated by a higher fidelity quantum computer and the remaining problem is determined to be calculated by a lower fidelity quantum computer.

[0019] The control signal generation unit is adapted for generating control signals for controlling quantum computing systems to cause a quantum computation of the sub-problems on the quantum computers of the quantum computing system. The control signals are generated such that the fidelity determines on which of the at least two quantum computers the quantum computation of a sub-problem is performed. Thus, the control signals are generated based on the fidelities of the respective at least two quantum computers. For example, a sub-problem derived to be calculated on a quantum computer of the quantum computing system with a high fidelity, for instance, a fidelity above a predetermined threshold, can lead to control signals that are generated such that the respective sub-problem is calculated on a quantum computer with a high fidelity, for instance, a fidelity above the predetermined threshold. Thus, the control signals are generated such that a sub-problem is computed on the quantum computer with a respective fidelity for which the sub-problem has been derived.

[0020] Generally, in case of control signals referring to the controlling of a quantum computing system, the control signal generation unit can be configured to generate control signals for controlling the quantum computer, for example, by controlling a manipulation part of a quantum computer configured to manipulate the states of the quantum elements, in accordance with a corresponding sequence of manipulations. However, if the quantum computer itself already provides a controlling unit that is adapted to control the manipulation part such that the states of the quantum elements are manipulated, the control signal generation unit of the apparatus can be adapted to provide the control signals to the controlling unit of the quantum computer. In this case, for instance, the control signals can simply refer to a representation of a sequence of manipulations that can be interpreted by the controlling unit of the quantum computer to provide the respective control signals for controlling the parts of the quantum computer accordingly. However, the control signals can, in this case, also refer to generally known and interpretable control signals that are translated by the controlling unit of the quantum computer to respective dedicated control signals for controlling the specific hardware of the quantum computer. Thus, the control of the control signal generation unit of the apparatus can be directly or indirectly, depending on the respective realization of the quantum computing system. The control signal generation unit of the apparatus alone or together with an optional controlling unit of the one or more quantum computers can hence be regarded as referring to an interface between a quantum computing system, in particular, the hardware of the quantum computing system, and the software for solving a respective problem running on a generally known classical computer.

[0021] Further, the control signal generation unit can also be adapted for controlling a readout of the quantum computing system to measure, after the application of a sequence of predetermined quantum manipulations comprising quantum gates and operations, at least one observable of the quantum mechanical state of the prepared representation of a solution of one or more sub-problems. In particular, the control signals can be adapted to control a readout part of the quantum computer such that the one or more observables are measured, i.e. read out, after the preparation of the respective quantum state solving a respective sub-problem has been completed. In particular, the control signal generation unit can be adapted to receive the readout of the readout part and provide the readout, for instance, to another apparatus for further processing the readout, for instance, in a classical computational environment. However, as described above, also here the control signal generation unit can interact optionally with a controlling unit of a quantum computer to act as an interface between a classical computer environment and the quantum computer.

[0022] The quantum computation of the derived sub-problems performed by the quantum computers can be performed parallel, sequential or iterative. For example, the control signals can be generated such that they control the quantum computing system to first cause a quantum computation of a first derived sub-problem on a first quantum computer and then secondly cause a quantum computation of a second derived sub-problem on the same or another quantum computer based on the results of the computation of the first sub-problem. Thus, for example, if more sub-problems have been derived than quantum computers available, different sub-problems can also be solved sequentially on the same quantum computer. However, sub-problems can also be calculated in parallel on the quantum computers, sequentially on different quantum computers or iteratively. For example, even dependent sub-problems can be computed in parallel during an iteration. In this case for starting the computation of the sub-problems in parallel instead of using the solution of a sub-problem as input for the computation of the dependent sub-problem a respective starting condition can be used and in the next iteration the respective solution of the previous calculation can then be utilized. Preferably, the sub-problems are derived such that each subproblem corresponds exclusively to one available quantum computer, i.e. no two sub-problems are computed on the same quantum computer. Thus, the number of derived subproblems is preferably equal or less than the number of available quantum computers. This allows to most effectively utilize the available quantum computer hardware.

[0023] In an embodiment at least two of the quantum computers are configured to be entangled and the control signals are generated for further controlling the entanglement between the at least two quantum computers of the quantum computers of the quantum computing system during a parallel quantum computation of the sub-problems to distribute and share information between the at least two quantum computers during the quantum computation, wherein the entanglement of two quantum computers is defined by at least one quantum element of each of the quantum computers being entangled. An entanglement between two quantum computers can be realised, for instance, by configuring the quantum computers such that at least one quantum element of one quantum computer can interact with at least one quantum element of another quantum computer. For achieving this a photonic link or respective entanglement bus or another qubit can be utilized. The details of the realisation strongly depend on the nature of the respective quantum computers, for instance, superconducting quantum computers, trapped-ion quantum computers, photonic quantum computers, etc. The quantum states of the entangled quantum elements depend on each other such that the quantum elements can only be described together as if they were one object. This state dependency of the entangled quantum elements leads to the distribution and sharing of information between the at least two quantum computers during the quantum computation during which the states of the quantum elements utilised in the quantum computation are changed, for instance, due to respective quantum operations performed during the quantum computation. Thus, an operation performed on one of the entangled quantum elements also has an effect on the state of the other of the entangled quantum elements influencing the calculation of the other quantum computer. In this way basis operations, intermediate results, and final results of a quantum computation performed on the first quantum computer can directly, i.e. without being read out and processed by a classical computer and then prepared on the other quantum computer, influence the quantum computation on the other quantum computer. This allows for a true parallel computation of two sub-problems even if the sub-problems depend on each other. For realizing the entanglement different hardware solutions can be utilized depending on the respective type of quantum computer. For example, for superconducting based quantum computers waveguides and transmission lines can be utilized, as described, for example, in the article “Quantum computer with superconducting circuits in the ultrastrong coupling regime”, Stassi, R., Cirio, M. & Nori, F. Scalable, npj Quantum Inf 6, 67 (2020) incorporated herein by reference. Further, photonic links as described in the article “Modular entanglement of atomic qubits using photons and phonons.”, Hucul, D., Inlek, I., Vittorini, G. et al., Nature Phys 11 , 37-42 (2015) incorporated herein by reference can be used. For ion trap based quantum computing architectures the ion can be transported between respective quantum computers as disclosed, for instance, in the article “A high-fidelity quantum matter-link between ion-trap microchip modules.”, Akhtar, M., Bonus, F., Lebrun-Gallagher, F.R. et al., Nat Commun 14, 531 (2023), or electron shuffling can be utilized as described in the article “Conveyor-mode single-electron shuttling in Si / SiGe for a scalable quantum computing architecture.”, Seidler, I., Struck, T., Xue, R. etal., npj Quantum Inf 8, 100 (2022) both incorporated herein by reference.

[0024] In an embodiment the deriving of the sub-problems from the problem description based on the fidelities of the at least two quantum computers comprises identifying a sub-problem that requires a higher solution accuracy than another sub-problem, wherein the control signals are generated such that the sub-problem requiring a higher solution accuracy is computed on a quantum computer with a higher fidelity and wherein the other sub-problem is computed on a quantum computer with a lower fidelity. The higher fidelity can be regarded as being higher with respect to a) the fidelity of at least one other quantum computer of the at least two quantum computers, b) the fidelity of all other quantum computers of the at least two quantum computers and / or c) a predetermined fidelity threshold. The lower fidelity can be regarded as being lower with respect to a) the fidelity of at least one other quantum computer of the at least two quantum computers, b) the fidelity of all other quantum computers of the at least two quantum computers and / or c) a predetermined fidelity threshold. For example, a sub-problem requiring higher solution accuracy can be a sub-problem that is more crucial to the solution of the problem. For example, when the problem is an electronic structure problem the sub-problem can refer to an active space describing electrons and electron orbitals actively participating in a binding or chemical reaction of the respective atom or molecule. In contrast thereto a sub-problem describing an inactive space of an atom or molecule mostly refers to electrons and electron orbitals that remain fully occupied with electrons or completely unoccupied during a chemical reaction and thus only have a weak influence on the chemical reaction of the molecule or atom. Such sub-problems that need a higher solution accuracy can be identified automatically, for instance, based on a predefined set of rules defining in accordance with a to be utilised solution algorithm subproblems that have a higher influence on the solution of the problem. However, such subproblems can also be identified by a user. For instance, derived sub-problems can be proposed to a user and a user can then select the sub-problems that are to be computed with higher accuracy.

[0025] In an embodiment the sub-problems are further derived from the problem description based on a predetermined quantum algorithm class that is defined to be utilised for solving the problem. For example, a quantum algorithm class can be predefined by a user, for instance, in a selection step selecting the respective quantum algorithm class for a respective input problem or can be predefined for all problems. However, a quantum algorithm class can also be selected automatically for respective problems based on a predetermined rule set indicating which problem class is to be solved with which quantum algorithm class. The quantum algorithm classes can comprise at least one of a variational algorithm, shallow algorithms, annealing algorithms, non-variational algorithms, deep algorithms.

[0026] In an embodiment, the control signal generation unit is adapted for generating control signals further based on a computation algorithm to be utilized for solving a respective subproblem, wherein the computation algorithm to be utilized determines a threshold for the fidelity of the quantum computer on which the computation algorithm can be performed, wherein the control signal generation unit is configured to take the respective threshold into account when generating the control signals. For example, computation algorithms to be performed generally on higher fidelity quantum computers, since they are associated with a higher fidelity threshold, can refer to quantum Fourier transform, quantum phase estimation, Grover-type search algorithms, Shor-type factorization algorithms and Harrow-Hassi- dim-Lloyd-type algorithms. Examples for relevant computation algorithms for lower fidelity quantum computers, since they are associated with a lower fidelity threshold compared with the above algorithms, are variational algorithms, such as the variational quantum ei- gensolver and the quantum approximate optimization algorithm. Based on a derived subproblem the user can be presented with a set of possible algorithms that can be utilized for solving the sub-problem. The user can then determine the respective algorithm that should be utilized by a respective selection and the control signal generation unit then utilizes this selection as described above. However, also pre-set algorithms can be utilized for respective derived sub-problems.

[0027] In an embodiment the deriving of sub-problems is further based on the number of logical quantum elements provided by the respective at least two quantum computers. Logical quantum elements are quantum elements that can be operated during a quantum computation. For example, depending on the quantum algorithm utilised some quantum elements are utilised as ancillary quantum elements and thus cannot be utilised for representing quantities and aspects of a respective sub-problem. Additionally taking the number of logical quantum elements on a respective quantum computer into account when deriving the sub-problems allows to specifically adapt the sub-problems to the respective quantum computers on which they are to be calculated allowing to utilise more efficient and effective quantum algorithms for the computation of the sub-problems on the respective quantum computers.

[0028] In an embodiment, the problem description is an electronic structure representation associated with the molecular structure of the chemical product and the derived sub-problems refer to i) a first part of the electronic structure representation indicative of an active space including a part of the electronic structure associated with a molecular structure and ii) a second part of the electronic structure representation indicative of an inactive space including another part of the electronic structure associated with a molecular structure, iii) an interaction representation associated with a combination of the solution of the first part and the solution of the second part. Preferably, the interaction representation is generated based on a reference state associated with a superposition of electron configurations generated based on the solution of the first part and the solution for the second part. Generally, the electronic structure representation can refer to problems that are associated with determining an electronic structure of an atom, molecule, crystal, or amorphous solid, etc. For example, electronic structure representations can be utilized for determining energetic ground states of atoms and / or molecules, etc. Based on solutions of the electronic structure representation, further properties of the atom, molecule, crystal, or amorphous solid can be derived. Generally, the electronic structure representation can be mathematically represented utilizing different classes of underlying electronic structure Hamiltonians. For example, Born-Oppenheimer Hamiltonians, non-Born-Oppenheimer Hamiltonians, Hamiltonians with additional one-electron potentials representing, e.g., electrostatic fields, two-component Hamiltonians including spin-orbit coupling, fully relativistic four-component Dirac-Hamiltonians, etc.

[0029] From the electronic structure representation a first part of the problem and a second part can be derived. The first part of the electronic structure representation is indicative of an active space including a part of the electronic structure associated with the molecular structure. The second part of the electronic structure representation is indicative of an inactive space including another part of the electronic structure representation associated with the molecular structure. Preferably, the part of the electronic structure representation associated with the molecular structure included in the second part is the remaining part of the electronic structure representation without the first part. Preferably, the active space and the inactive space of the electronic structure representation refer to subspaces of the Hilbert space of the electronic structure representation. In particular, the basis of the Hilbert space of the electronic structure representation can be formed by electron orbitals and mathematically represented by antisymmetrized tensor products of the electron orbitals and the inactive space and active space are defined by the electron orbitals to which they refer. However, the active and the inactive space can also be defined in other mathematical representations, for instance, if the electronic structure representation is formulated in another space. The first part and the second part are preferably defined based on the respective active or inactive space, i.e. are defined in the respective basis ofthe respective space, but can also comprise terms that are related to the orbitals representing the respective other space, for example, terms that are related to electrons or electron orbitals that form the basis of the respective other space. The first part and the second part of the electronic structure representation can be determined based on the chemical product and the molecular structure to be described by the electronic structure representation in addition to the fidelities. In particular, since the molecular structure of the chemical product defines the quantities and interrelations between the quantities of the electronic structure representation, it also defines how these quantities and interrelations are described with respect to the first part and the second part of the electronic structure representation. In particular, since the chemical product defines the electron orbitals in the electronic structure representation, and in particular, which orbitals have the strongest correlations such that for an accurate solution of the electronic structure representation of this molecular structure of the chemical product these correlations should be accounted for more accurately, the chemical product determines how the first part and the second part are defined based on the available fidelities of the quantum computers. For example, the first part of the electronic structure representation is provided such that it allows to calculate parts of the electronic structure of the chemical product that are most relevant for a specific application with high accuracy using a higher fidelity quantum computer. In such cases, the first part is provided such that it is based on electron orbitals that are most relevant for a specific application, since these electron orbitals exhibit the strongest correlations in the molecular structure of the chemical product during a chemical reaction of the chemical product and / or change during the specific application, e.g. are involved in bond-breaking and / or -forming processes during a chemical reaction of the molecular structure of the chemical product.

[0030] The control signal generation unit can be adapted to generating control signals for causing a quantum computation of the first part based on the respective fidelities of the quantum computers. For example, the first part can be derived such that the control signal generation unit controls a higher fidelity quantum computerto perform the computation of the first part. Alternatively, first part can be derived such that the control signal generation unit controls a lower fidelity quantum computer or even a classical computer to perform the computation of the first part. Depending on the respective electronic structure representation, utilizing a lower fidelity quantum computer or a classical computer for solving the first part of the electronic structure representation can be more effective and efficient. For example, for electronic structure representations with first parts that only represent a small part of the electronic structure, for instance, only below a predetermined number of orbitals and electrons, it can be more resource efficient to utilize a lower fidelity or classical computer. In such a case methods like full configuration interaction, density matrix renormalization group theory, M0ller-Plesset perturbation theory or coupled cluster theory can be utilized.

[0031] The control signal generation unit can be adapted to generating control signals for causing a quantum computation or classical computation of the second part based on the respective fidelities of the quantum computers. For example, the second part can be derived such that the control signal generation unit controls a lower fidelity quantum computer or a classical computer to perform the computation of the second part. For example, the solution of the second part can be a representation of an electron configuration of the inactive space of the respective molecule. The solution of the second part can be any kind of contribution of the second part to the solution of the electronic structure problem. For example, the solution can refer to any contribution of the inactive space that is taken into account for the interaction representation. For example, the solution of the second part can be determined based on respective known algorithms like Hartree-Fock or density functional theory (DFT). In a preferred embodiment, the solution of the second part is determined based on the solution of the first part. Thus, already in this step an influence of the first part on the second part, i.e. of the active space on the inactive space, can be taken into account. In a preferred embodiment a SCF solution is determined for the second part. The solution of the second part hence is a better approximation to the overall solution and the interaction representation thus is based on an already quite accurate solution. In particular, it is preferred that the electronic structure representation is already provided such that the solution of the second part can be calculated on a classical computing device utilizing respective known calculation methods. Alternatively, the second part can be derived such that the control signal generation unit controls a higher fidelity quantum computer to perform the computation of the second part.

[0032] The computationally combining includes generating an interaction representation associated with the combination of active and inactive space. The interaction representation is based on a reference state associated with a superposition of electron configurations and is generated based on the solution of the first part and the solution of the second part. The reference state refers to a representation of the superposition of electron configurations that are utilized as basis for the determination of correlations between the active space and the electronic structure of the inactive space and optionally also within one or both of these spaces. Thus, the reference state based on the solution of the first part and the solution of the second part represents the electron configurations in such a way that it can form a basis forthe further calculations of the solution of the interaction representation. For example, for generating the reference state, solutions of the first part and solutions of the second part that are generated on a quantum computer refer to direct measurements of the electronic configuration states that are then directly constructed on the respective classical and / or quantum computer that is utilized for solving the interaction representation. Thus, a solution for a first part and / or a second part prepared on a quantum computer is not measured by measuring, for instance, characterizing quantities of the respective state, for instance, an energy, a reduced density matrix or any other observable, but refers directly to the states measured for the quantum elements that represent the respective states of the electrons in the electronic structure. The reference state therefore can be regarded as a mathematical representation of the electron configuration of the molecular structure of the chemical product after solving the first and the second part of the electronic structure representation. The reference state can be generated as a superposition of the electron configurations being part of the solution of the first and the second part, wherein each electron configuration is associated with a weight for weighting the contribution of the electron configuration to the overall solution of the electronic structure problem. Determining the weights refers to determining the overall solution. For example, if the reference state is represented utilizing configuration interaction basis states the weights refer to configuration interaction coefficients associated with respective configuration interaction basis states. The utilized electron configurations can comprise a part that represents the active space and a part that represents the inactive space. For example, an electron configuration can comprise electron orbitals that are part of the first part, i.e. active space, and electron orbitals that are part of the second part, i.e. inactive space. Depending on the respective problem, the respective parts of the electron configurations utilized with the superposition in the reference state can be treated differently. For example, the parts of the electron configurations representing the active space can be more numerous than the parts of the electron configurations representing the inactive space. In particular, the parts of the electron configurations representing the inactive space can be identical, whereas parts of the electron configurations representing the active space can be different. Thus, the reference state can refer to a superposition of electron configurations wherein the different active parts of the electron configurations are combined, for instance linear combined, and the inactive parts are the same. Based on this reference state the interaction representation can then, during the respective computation, also allow for the generation of different inactive parts resulting from the correlation between the active parts and the inactive part. The interaction representation generated based on the reference state then generally represents the correlation between the parts of the electronic structure representation represented by the first part and represented by the second part and further represents correlations within one or both parts, in particular, within the second part. The interaction representation can include correlations between electron orbitals among which at least one electron orbital is a member of the inactive space.

[0033] Preferably, the interaction representation includes correlations a) within the inactive space, and / or b) between the active space and the inactive space. Correlations occurring within the active space are determined when determining the solution of the first part. Generally, correlations between spaces are defined as correlations between quantities defined in these spaces. Thus, the quantities refer to the quantities as defined in the first part of the electronic structure representation and the second part of the electronic structure representation. The above preferred embodiment can therefore also be formulated as the inter- action representation including correlations a) within the second part of the electronic structure representation, and / or b) between the first part and the second part of electronic structure representation.

[0034] The control signal generation unit can be adapted to generating control signals for causing a quantum computation of the interaction representation. For example, the control signals can control a higher fidelity quantum computer to compute the solution of the interaction representation. However, also lower fidelities quantum computers can be utilized. In particular, a fidelity threshold can be determined for the interaction representation and the control signals can be generated to control a quantum computerto perform the computation that comprises a fidelity above the determined threshold.

[0035] In an embodiment the quantum computing system comprises at least one fault-tolerant quantum computer and one noisy intermediate-scale quantum computer, wherein the fault- tolerant quantum computer has a higher fidelity than the noisy intermediate-scale quantum computer. Noisy intermediate-scale quantum computers are generally sensitive to the environment and prone to quantum decoherence such that they can only apply a finite number of gates or operations before too many errors have accumulated such that too much information is lost due to the noise and the computation becomes irrelevant. This limits the number of operations and thus also the size and complexity of sub-problems that can be computed on such a quantum computer. Fault-tolerant quantum computers perform a quantum computation with a physical error rate below a predetermined threshold defined by the quantum threshold theorem such that through the application of quantum error correction the logical error rate can be suppressed to arbitrary low levels. This allows to perform arbitrarily long quantum computations, i.e. to apply an arbitrary number of gates and operations, leading to much higher precision and accuracy of calculation and allowing to calculate more complex sub-problems with more variables. Fault-tolerant quantum computers can be realized by applying quantum error correction codes utilizing at least three physical quantum elements to represent one logical qubit. An example is a surface code described in the article “Google Quantum Al. Suppressing quantum errors by scaling a surface code logical qubit.”, Nature 614, 676-681 (2023). However, also other codes exist that exploit this principle. Many of these error correction codes are only applicable to quantum computers with a respective high number of qubits. However, also codes which utilize only three physical quantum elements for representing one logical qubit exist, like the Steane code. These might not lead to fully fault-tolerant quantum computers for all cases but strongly increase the fidelity, and can thus lead to a higher fidelity quantum computer. In a further aspect a quantum computer system is presented comprising a) a controlling unit for receiving control signals generated by an apparatus as described above, and b) at least two quantum computers with different fidelities, wherein a quantum computation of a sub-problem, in particular, associated with a chemical product, is performed on the at least two quantum computers based on the received control signals based on the different fidelities of the at least two quantum computers.

[0036] In an embodiment, at least two quantum computers of the quantum computers are configured for an entangled quantum computation, wherein the entanglement between the at least two quantum computers is controlled based on the control signals, and wherein the entanglement of the two quantum computers is defined by at least one quantum element of each of the quantum computers being entangled.

[0037] In an embodiment, at least one of the at least two quantum computers is a fault-tolerant quantum computer and at least one of the at least two quantum computers is a noisy inter- mediate-scale quantum computer.

[0038] In a further aspect of the invention a problem solving apparatus is presented for solving a problem, in particular, associated with a chemical product, wherein the apparatus comprises a) a receiving unit for receiving quantum computation results of sub-problems performed by at least two quantum computers of a quantum computing system in accordance with control signals generated by an apparatus as described above, and b) a problem solving unit configured for determining a solution of the problem based on the results of the quantum computations of the sub-problems.

[0039] In a further aspect of the invention a system is presented for generating a solution of a problem, in particular, associated with a chemical product, wherein the system comprises a) an apparatus as described above, b) a quantum computer system as described above, and c) a problem solving apparatus as described above.

[0040] In a further aspect, a computer implemented method is presented for providing control signals for controlling a quantum computing system for performing a quantum computation of a problem, in particular, associated with a chemical product, wherein the quantum computing system comprises at least two quantum computers with different fidelities, wherein the method comprises a) providing a problem description indicative of the problem, b) deriving sub-problems from the provided problem description based on the fidelities of the at least two quantum computers, and c) generating control signals for controlling the quantum computing system to cause a quantum computation of the derived sub-problems on the at least two quantum computers, wherein the sub-problems are distributed to the at least two quantum computers based on the fidelity of the respective at least two quantum computers. The controlling of the quantum computing system can comprise also causing the providing of the results of the quantum computation such that a respective solution can be determined. The providing can refer to providing the results to an apparatus configured for determining the respective solution as described above. The apparatus can be the same or a different apparatus than the apparatus utilized for controlling the quantum computing system.

[0041] In a further aspect a problem solving method is presented for solving a problem, in particular, associated with a chemical product, wherein the method comprises a) receiving quantum computation results of sub-problems performed by at least two quantum computers of a quantum computing system in accordance with control signals generated by a method as described above, and b) determining a solution of the problem based on the results of the quantum computations of the sub-problems.

[0042] In a further aspect of the invention, a use of an apparatus as described above for solving problems referring to at least one of a chemical reactivity, spectra and spectroscopic properties, and molecular properties derivable from electronic structure problem calculations of a chemical product is presented.

[0043] In a further aspect of the invention, a use of an apparatus as described above is presented, for solving problems referring to at least one of metalorganic compounds containing transition metals including lanthanides and actinides, chelating agents interacting with metals, catalysts, biomolecules with active centers, macromolecular systems and transition metal compounds in solution or embedded in an environment.

[0044] In a further aspect of the invention, a use of an apparatus as described above is presented for determining activation and / or reaction energies for a predetermined chemical reaction. An apparatus as described above can be used for determining activation and / or reaction energies for a predetermined chemical reaction. Generally, an activation energy refers to an energy difference between a transition state and reactants. A reaction energy refers to an energy difference between products and reactants. The chemical reaction can be part of a complex reactive network, e.g. of a catalytic cycle. Utilizing an apparatus as described above is in particular advantageous in cases where at least one chemical species of this reactive system a) contains one or more transition metal atoms, lanthanide atoms and / or actinide atoms with unpaired electrons or b) exhibits an electronic structure with a small energetic gap between occupied and unoccupied electron orbitals, i.e., an energetic gap that is equal or smaller than the gap of at least one of the molecules ozone, pentacene or para-quinodimethane calculated with the same electronic structure approach, i.e. same basis set, same self-consistent field (SCF) method, e.g. Hartree-Fock, etc. or c) exhibits a multi-reference diagnostic that exceeds a predetermined limit, e.g. ^(CCSD) greater than 0.02 and / or D^CCSD) greater than 0.05 and / or £>1(MP2) greater than 0.04 and / or £>2(MP2 / CCSD) greater than 0.18 and / or Zs(1)greater than 0.1 and / or %TAE greater than 10, wherein the T diagnostic is determined based on the Frobenius norm of the amplitudes for single excitations of a CCSD wave function based on a Hartree-Fock reference state scaled by the square root of the number of correlated electrons in the CCSD calculation, and wherein the D, diagnostic is determined based on the matrix 2-norm of the amplitudes for single excitations of a CCSD or MP2 wave function based on a Hartree-Fock reference state, and wherein the D2diagnostic is determined analogously to the D, diagnostic but refers to double excitations, and wherein the Zs(1)diagnostic is determined based on orbital-entanglement information obtained from an approximate correlated wave function such as a partially converged but qualitatively correct density matrix renormalization group (DMRG) wave function, and wherein the %TAE diagnostic is determined based on the difference of total atomization energies obtained with CCSD(T) and CCSD relative to the total atomization energy obtained with CCSD(T).

[0045] In an aspect of the invention, a use of an apparatus as described above is presented for determining activation energies for a predetermined catalytic cycle and / or for determining reaction energies for a predetermined chelating agent.

[0046] In an aspect of the invention, an apparatus for determining a technical application property of a chemical product is presented, wherein the apparatus comprises a) a providing unit for providing a quantum mechanical electronic structure problem description related to the chemical product, wherein a solution of the quantum mechanical electronic structure problem is indicative of the technical application property, b) a solution unit for using an apparatus as described above for determining a solution of the quantum mechanical electronic structure problem, and c) a property determination unit for determining based on the solution the technical application property of the chemical product.

[0047] In a further aspect of the invention, a computer implemented method for determining a technical application property of a chemical product is presented, wherein the method comprises a) providing a quantum mechanical electronic structure problem description related to the chemical product, wherein a solution of the quantum mechanical electronic structure problem is indicative of the technical application property, b) using an apparatus as described above for determining a solution of the quantum mechanical electronic structure problem, and c) determining based on the solution the technical application property of the chemical product.

[0048] In a further aspect a computer program product is presented for determining a technical application property of a chemical product, wherein the computer program product comprises program code means for causing the apparatus as described above to execute the method as described above.

[0049] In a further aspect of the invention, an apparatus for determining a target chemical product comprising a target technical application property is presented, wherein the apparatus comprises a) a target property providing unit configured to provide a target technical application property and a potential chemical product, b) a property determination unit utilizing an apparatus as described above and / or a method as described above for determining a technical application property of the potential chemical product, c) an iteration unit configured to compare the determined technical application property of the potential chemical product with the target technical application property and, based on the comparison, either i) determining the potential chemical product as the target chemical product, or ii) providing a new potential chemical product and repeating the determination of the technical application property utilizing the new potential chemical product, and d) a control data generation unit configured to generate control data for producing the determined target chemical product.

[0050] In a further aspect of the invention, a computer implemented method for determining a target chemical product comprising a target technical application property is presented, wherein the method comprises a) providing a target technical application property and a potential chemical product, b) utilizing an apparatus as described above and / or a method as described above for determining a technical application property of the potential chemical product, c) comparing the determined technical application property of the potential chemical product with the target technical application property and, based on the comparison, either i) determining the potential chemical product as the target chemical product, or ii) providing a new potential chemical product and repeating the determination of the technical application property utilizing the new potential chemical product, and d) generating control data for producing the determined target chemical product. In a further aspect a computer program product is presented for determining a target chemical product comprising a target technical application property, wherein the computer program product comprises program code means for causing the apparatus as described above to execute the method as described above.

[0051] A technical application property can generally refer to any property of a chemical product that allows to assess a technical applicability of the respective chemical product as provided after its production. The generating or determining of the property associated with the chemical product refers to determining or deriving and providing the respective property. The property can be a property of a chemical product that is related to a solution of a problem being an electronic structure problem provided as electronic structure representation, for instance, refers to or can be derived from a solution of the electronic structure representation, of a molecular structure forming the chemical product. The property can be an electronic property of the electronic structure representation. The property can refer to a property of a chemical product that allows to assess a technical applicability of the respective chemical product as provided after its production. Preferably, the technical application property comprises any of a chemical reactivity, spectra and spectroscopic property, molecular property, activation and reaction energies for a predetermined chemical reaction. Preferably, the technical application property comprises at least one of mechanical properties, spectroscopic properties, physicochemical properties, chemical properties and biological properties. Generally, mechanical properties can refer to any of adhesion, tensile strength, stiffness, hardness, shrinkage, elongation, split tear, tear-strength, rebound, compressibility, abrasion, spillage, morphology, haptic properties, stress at break, elongation at break, granulometry and a degree of filling. A spectroscopic property can generally comprise any of coloration, turbidity, opaqueness, lucidity, reflection, appearance, absorption, scattering, color strength, colour hue, colour saturation, colour intensity, cloud point, matting degree, optical density, spectra, refractive index, IR spectra, Raman spectra, NMR spectra, ESR spectra, and UVA / is spectra. Moreover, a physicochemical property can refer to any of density, viscosity, K-value, molar weight, dispersity, molar mass distribution, particle size distribution, solubility, partition coefficients, interfacial properties, surface tension, dispersibility, storage stability, odor, segregation, coagulation, electric conductivity, electric capacity, surface area, flow time, vapor pressure, VOC, solid content, hygroscopicity, magnetism, miscibility, thixotropy, phase transition properties, glass transition temperature, corrosion inhibition, solvent separation, aggregation, self-heating ability, impact sensitivity, loss on drying, angle of response, electrostatic charge, minimum film-forming temperature, charge density, electrostatic multipole moments, and thermal conductivity. The chemical property can comprise any of reaction thermodynamics, reaction kinetics, chemical resistance, reaction timing, demolding time, growing, hard / soft segment content, crystallinity, reaction temperature, reaction pressure, decomposition, thermal decomposition, photodegradation, acidity, pKa, pH, moisture / water content, flammability, burning rate, self-ignition, flash point, formation of flammable gases, reaction to fire, deflagration rate, residual monomer count, side product formation, degree of polymerization, salt content, temperature tolerance, oxidizing properties, reduction properties, reactivity, ash content, nonvolatile matter content, stability, chelating ability, calorific value, saponification value. Further, the biological property can comprise any of biodegradability, biological resistance, in particular, resistance against a pathogenic virus, bacterium, fungus, plant or animal or developmental stage of said pathogen, tolerance against environmental parameters, e.g. drought tolerance, resistance against enzymatic degradation, e.g. protease resistance, lipase resistance, amylase resistance, hydrolase resistance, pesticide resistance, toxicity, biotransformation, ecotoxicology, sensitization, in particular, allergenicity, bacterial count, enzyme activity, substrate specificity, cofactor dependence, product specificity, substrate and / or product inhibition, dissociation constant, Michaelis-Menten-kinetics values, activity / stability at or in different: pH, temperature, pressure, organic solvent concentration, carrier formulations, encapsulation formulations; distribution in environment, compartimentalisation, bioaccumulation, biological exposure LD50, mutagenicity.

[0052] The problem providing unit is in the case of determining a property of a chemical product adapted to provide an electronic structure representation associated with at least one of the one or more molecular structures of the chemical product. In particular, the problem providing unit can refer to a storage unit on which the electronic structure representation is already stored. However, the problem providing unit can also comprise an input unit with which, for instance, a user can indicate an electronic structure representation as problem to the problem providing unit. Generally, the electronic structure representation can be provided in any form that allows to determine the form of the electronic structure problem to be solved and the quantities defining the electronic structure problem and the form of interaction between these quantities. Preferably, the electronic structure representation refers to a mathematical description of an electronic structure problem to be solved, for example, to a mathematical description of the electronic structure problem utilizing an electronic structure Hamiltonian. However, the electronic structure representation can also refer to any other unambiguous form of notation of a respective electronic structure problem. Generally, the electronic structure representation is related to a molecular structure of the chemical product. In particular, the molecular structure of the chemical product to which the electronic structure representation is related to can define the electronic structure problem and the quantities of the electronic structure problem. For example, if the chemical product refers to a specific molecule for which a technical application property should be determined, the respective molecule and its structure define the quantities and interactions of the quantum mechanical electronic structure problem indicated by the electronic structure representation. Preferably, the electronic structure representation comprises or is indicative of atomic positions, a basis set, charge, and spin multiplicity depending on the specific chemical product and / or quantum mechanical electronic structure problem.

[0053] Generally, an electronic structure representation can refer to problems that are associated with determining an electronic structure of an atom, molecule, crystal, or amorphous solid, etc. For example, electronic structure representations can be utilized for determining energetic ground states of atoms and / or molecules, etc. Based on solutions of the electronic structure representation, further properties of the atom, molecule, crystal, or amorphous solid can be derived. Generally, the electronic structure representation can be mathematically represented utilizing different classes of underlying electronic structure Hamiltonians. For example, Born-Oppenheimer Hamiltonians, non-Born-Oppenheimer Hamiltonians, Hamiltonians with additional one-electron potentials representing, e.g., electrostatic fields, two-component Hamiltonians including spin-orbit coupling, fully relativistic four-component Dirac-Hamiltonians, etc. From the solution of the electronic structure representation a plurality of properties of a respective chemical product can be derived.

[0054] In an embodiment, the problem for which the apparatus as described above generates control signals for controlling a quantum computer can refer to any problem for which a solution can lead to deriving the respective technical application property of a chemical product, for example, one of the technical application properties mentioned above. In particular, the problem can refer to an electronic structure problem, wherein based on the solution of the electronic structure problem a plurality of further characteristics of a respective chemical product can be derived. Examples for such problems that can advantageously be solved in this invention can refer to calculating a ground state energy of a molecule or general electronic system. This allows, in particular, in the context of determining the ground state energy of all molecular species occurring in a chemical reaction to predict the reaction end products, the thermodynamic properties of the reaction, and the kinetic properties of the reaction. This understanding and the properties of the reaction can then again be utilized to optimize chemical production processes, for prediction of a microstructure of polymers, for optimizing material properties, etc. Further, the problem can also refer to calculating multipole moments of a chemical product. Such a calculation can be relevant for determining properties of the chemical product related to electrical and other technical application properties, e.g. the dielectric behavior of the chemical product, but also properties resulting from intermolecular interactions which are strongly dependent on the polar nature of the chemical product - such technical application properties can range from solubility and compatibility with certain media to complexation behavior of ions or impacts on spectroscopic properties, e.g. color. Accordingly, the determining of the value of the technical application property based on the result of the quantum mechanical calculation is based on the respective technical application property that should be determined and further based on the information that is provided by the solution of the problem indicated by the results of the quantum mechanical calculation.

[0055] The providing of the target application property can refer to receiving the target application property from an input of a user applying, for instance, a respective input unit. Moreover, the providing can also refer to accessing a storage unit on which a target application property is already stored and providing the same. Further, the providing can also comprise receiving a target application property, for instance, via a network connection from other sources and providing the received target application property. Generally, the target application property can referto one target value, for instance, a specific hardness of a chemical product, or can refer to a value range that should be met by the chemical product. Moreover, the target application property can also referto any kind of target function, for instance, a timely sequence of a property under a changing environmental condition, like a hardness under changing temperature conditions. Such more complex target application properties can be advantageous in cases in which the application of the chemical product comprises different environmental conditions, for example, different temperatures. The target chemical product then refers to a chemical product that provides the respective target technical application property, i.e. meets the target technical application property within predetermined limits, when provided in a respective form, for example, as pure substance, or mixture. In particular, the target chemical product provides the respective target technical application property when produced according to a corresponding recipe.

[0056] A potential chemical product can be provided in any digital representable format such that the potential chemical product and / or characteristics of the potential chemical product can be processed by the apparatus. Moreover, the providing of the potential chemical product can also comprise providing a respective problem for determining the technical application property of the potential chemical product. However, the respective problem can also be selected automatically, for example, by the apparatus based on the potential chemical product and the provided target technical application property. However, also a user can, for example, select a respective problem preferably based on a selection of a plurality of possible problems presented to the user based on the potential chemical product and / or the target technical application property. The comparison of the determined technical application property with the target technical application property allows to determine whetherthe determined technical application property fulfils a predetermined criterion, for example, that the determined technical application property meets the target technical application property within predetermined limits. If such a criterion is fulfilled, the potential target chemical product is determined as the target chemical product and the method proceeds to the next step. However, if the comparison indicates that the determined technical application property does not meet the target technical application property within the predetermined limits, a next iteration step utilizing a new potential chemical product can be performed. In particular, for each iteration step of the iteration a new potential chemical product is determined, preferably based on the previous potential chemical product, for instance, by amending one or more features of the previous chemical product, e.g. one or more constituents or other characteristics. However, a new potential chemical product can also be generated by arbitrarily choosing a new potential chemical product, for example, from a large amount of previously generated potential chemical products. Moreover, also more sophisticated methods can be utilized for selecting a new potential chemical product from a plurality of previously already generated potential chemical products. Based on the new potential chemical product, in each iteration step again the quantum computer is utilized for determining the technical application property and the such determined technical application property is again compared with the target technical application property such that the comparison can again lead to a further iteration step or if the respective criterion is fulfilled the respective new potential chemical product can be selected as the target chemical product. Moreover, also an additional abortion criterion for the iteration can be selected, for instance, a number of iteration steps can be determined before the iteration is aborted with a notification to a user that no target chemical product could be found for the respective target technical application property. However, alternatively, after a predetermined amount of iteration steps the method can further comprise amending the target technical application property, for instance, by increasing the predetermined limits around the target technical application property and to repeat the iteration while utilizing the increased limits during the comparison. This can allow to find a target chemical product that meets the target technical application property as much as possible, even if a meeting of the original goal might not be possible. After the target chemical product has been determined as described above, the target chemical product can be provided, for instance, to a user via an output unit. Preferably, a recipe of the target chemical product is utilized to generate control data that can be utilized for controlling a production system for producing the target chemical product.

[0057] It shall be understood that the apparatuses as described above, the methods as described above, the systems as described above and the computer program products as described above have similar and / or identical preferred embodiments, in particular, as defined in the dependent claims.

[0058] It shall be understood that a preferred embodiment of the present invention can also be any combination of the dependent claims or above embodiments with the respective independent claim.

[0059] These and other aspects of the invention will be apparent from and elucidated with reference to the embodiments described hereafter.

[0060] BRIEF DESCRIPTION OF THE DRAWINGS

[0061] In the following drawings:

[0062] Fig. 1 illustrates a state representation of a qubit as used in a quantum computing device,

[0063] Fig. 2 illustrates a schematic example of a quantum computing device with qubits as calculation unit,

[0064] Fig. 3 illustrates a schematic example method for generating a control signal to perform operations on a quantum computing device and for processing measurement signals from the quantum computing device,

[0065] Fig. 4 illustrates a schematic example of a hybrid system including a classical and a quantum computing device,

[0066] Fig. 5 illustrates a schematic example of a quantum computing device based on superconductors,

[0067] Fig. 6 illustrates a schematic example of a quantum computing device based on trapped ions,

[0068] Fig. 7 shows schematically and exemplarily an embodiment of a system for determining a solution of a problem related to a chemical product,

[0069] Fig. 8 shows schematically and exemplarily a flow chart of a method for determining a solution of a problem related to a chemical product, Fig. 9 shows schematically and exemplarily examples of quantum computing systems with quantum computers comprising different fidelities,

[0070] Figs. 10, 11 show schematically and exemplarily flowcharts of exemplary implementations of computational methods on the quantum computer system,

[0071] Fig. 12 shows schematically and exemplarily an illustration of an electron orbital energy diagram,

[0072] Fig. 13 shows schematically and exemplary a quantum circuit diagram of a Hadamard overlap measurement,

[0073] Fig. 14 shows schematically and exemplary an overview over different embodiments of methods using the quantum computer system, and

[0074] Fig. 15 - 17 show schematically and exemplarily further applications of the method for determining a solution.

[0075] DETAILD DESCRIPTION OF DRAWINGS

[0076] In the following first a short introduction into the general basic principles of quantum computers and the performance of calculations of quantum computers will be provided. Further, general principles can also be found in “Quantum Computation and Quantum Information: 10thAnniversary Edition”, M. A. Nielsen and I. L. Chuang (2010).

[0077] Classical computing devices use processors which are based on transistors. The state of each transistor has two controllable states 1 or 0 representing a digital binary or a bit. To perform operations on a classical computing device a human readable program code is translated via a compiler into machine-readable instructions. Machine-readable instructions are control signals, e.g. voltage settings, for each transistor. Representations of the machine-readable instructions may include binary or hexadecimal representations. Based on such machine-readable instructions, the operations are performed on the processor of a classical computing device.

[0078] Quantum computation is a relatively new computation method that uses quantum effects, such as superposition and entanglement, to perform certain computations more efficiently than classical digital computers. In contrast to digital computers, which represent information in the form of bits (e.g., “1 ” or“0”), as described above, quantum computing devices, i.e. quantum computers, use qubits, i.e. quantum bits, to represent information. Quantum computing devices are based on quantum elements adhering to the physics of quantum mechanics, such as superconductors, ions, atoms, quantum dots, photons, particle spins, bosons or the like. These quantum elements may be manipulated in a controlled manner to perform operations.

[0079] Although qubits and their manipulation may be described in terms of their mathematical properties, each such qubit may be implemented in a physical quantum element in any of a variety of different ways. Examples of such quantum elements include superconducting materials, trapped ions, photons, optical cavities, individual electrons trapped within quantum dots, point defects in solids (e.g., phosphorus donors in silicon or nitrogen-vacancy centers in diamond), molecules (e.g., alanine, vanadium complexes), or any medium that exhibits qubit behavior comprising quantum states and transitions therebetween that can be controllably induced or detected.

[0080] Generally, for any given physical quantum element that implements a qubit, any of a variety of properties of that physical unit may be chosen to implement the qubit. For example, if electrons are chosen to implement qubits, then the x, y or z component of an electron spin degree of freedom can be chosen as the property of such electrons to represent the states of such qubits. For any particular degree of freedom, the physical quantum elements can be controllably put in a state of superposition or entanglement and measurements can then be taken in the chosen degree of freedom to obtain readouts of qubit values.

[0081] In contrast to transistors of classical computing devices each quantum element of quantum computing devices can not only take the basis states |1) or |0) but also any superposition of such basis states, such as state |X). The state of each quantum element is represented by a state of a quantum bit, i.e. qubit, as illustrated in the two-dimensional simplification of Fig. 1. To represent such states Dirac notation is commonly used in quantum mechanics. In Dirac notation a state in a n dimensional, complex vector space, such as a Hilbert space, is represented in braket notation, for example |X). According to conventional terminology, the superposition of “0” and “1 ” states in a quantum computing device can be represented as <z|0) + ?|l) . The states “0” and “1 ” or bits of the classical computing device are similar to the basis states |0) and |1) or quantum bits of the quantum computing device, respectively. The value |<x|2represents the probability that the qubit will be measured in the |0) state, while the value | / ?|2represents the probability that the qubit will be measured in the |1) state. If more than one qubit is present, two or more qubits may be entangled. Entanglement means that the state of one qubit is dependent on the state of at least one other qubit and vice versa, wherein further in the entangled state the respective qubits cannot be regarded as individual qubits anymore. Generally, a register of / V qubits in a quantum computer can be put into a superposition of basis states at once whereas a register of N classical bits can only be in a single basis state at once. Thus, in contrast to classical computing devices on a quantum computing device 2Nbasis states can be manipulated and processed simultaneously allowing for exponential intrinsic parallelism.

[0082] To perform operations on the quantum computing device the computational method to solve a given problem may be translated into qubit manipulations, which may be translated into control signals for manipulating qubits. Representations of the machine-readable instructions may include common quantum mechanical representations of operations in the Hilbert space. Depending on a specific realization of the quantum computer different representations of the qubit states may be chosen. Any state preparation on the quantum computing device may be represented by a manipulation acting on the qubit states. A manipulation may be translated into control signals to control a respective part of the quantum computer, which depend on the type of quantum computing device used. This way based on the manipulation acting on the qubit states, operations may be performed on the quantum equivalent of a classical processor as part of the quantum computing device.

[0083] In gate-based quantum computer systems the manipulations acting on the qubit states may generally be one- or multi-qubit operations. A one-qubit operation may change the state of one qubit e.g., into a specific superposition which corresponds to a rotation of the vector |X) as illustrated in Fig 1. For example, in a superconducting quantum computer this can be accomplished by microwave pulses or in a trapped-ion quantum computer by irradiation of the ion with a laser beam. A multi-qubit operation may create entanglement between two or more qubits. For example, in a superconducting quantum computerthis may be achieved by connecting qubits via an intermediate electrical coupling circuit or in a trapped-ion quantum computer via controlling the collective vibrations of the trapped ions.

[0084] Generally, to prepare manipulations for solving a given problem a respective quantum mechanical representation of the problem may be translated into qubit manipulations, which are carried out to prepare a solution of the given problem. After the preparation of the predetermined solution, i.e. afterthe application of the operations to the qubits of the quantum computer, a projective measurement of all individual qubits is carried out returning either 0 or 1 for each qubit. This projection usually happens in the ozeigenbasis of the qubits which is also used to define the computational basis stats “0” and “1 ” of the qubit. This means that only operators that are products of ozPauli operators or can be directly transformed into such operators can be measured concurrently. On the quantum computing device this measurement is achieved by applying a hardware-specific readout protocol of a series of readout manipulations including control pulses and monitoring the response to control pulses. For example, a superconducting qubit may be coupled to a hardware resonator. The measured shift of the resonator frequency allows to determine the state of the qubit as this shift depends on the state of the coupled qubit. In case of trapped ions, for example, an optical readout may be used, e.g. the state of the qubit is 1 if the ion emits light or 0 if the ion does not emit light or vice versa. This way qubits may be used to implement logical circuits or gates as in classical computing devices.

[0085] In Fig. 2 a schematic example of a quantum computer is illustrated. The quantum computing device 100 shown in Fig. 2 includes a quantum register 104 configured to perform the quantum computation, a manipulation part 106 configured to manipulate the quantum register, in particular, quantum elements forming the qubits, and a readout part 108 configured to collect measurement signals from the quantum register 104 for reading out the qubits after a quantum mechanical calculation. The manipulation part 106, in particular, provides manipulation signals for manipulating the quantum register, wherein the manipulation signals are generated based on received control signals that are determined based on the respective operations that should be performed on the qubits. In some embodiment a feedback loop between the manipulation part 106 and measurement part 108 can be provided. In contrast to classical computing, where one measurement cycle provides the state of a transistor, quantum computing includes performing multiple measurement cycles to provide a probability density or a probability for the qubit states in case of gate-based quantum computers.

[0086] The quantum register 104 can be based on different quantum elements representing the qubits. In some embodiments of gate-based quantum computers the qubits may be implemented by photons as quantum elements. Such optical quantum computing devices may include lasers that generate photons that are provided to a waveguide. A beam splitter can be provided for manipulating the photon states based on manipulation signals such as a mechanical rotation applied to a mirror. The measurement part 108 can in such an embodiment be a photon detector, and the measurement signals can be photons.

[0087] In other embodiments of gate-based quantum computers the qubits can be implemented by electronic states of ions trapped in a magnetic field. The manipulation part 106 can in such a case utilize a laser, and the manipulation signals can cause the providing of control laser pulses. Moreover, in this case, the readout part 108 can be a photon detector combined with read-out laser pulses, and the measurement signals 102 may be photons. Other qubit implementations may be based on superconductors as quantum elements, semiconducting material with anyons as quantum elements, or the like. Fig. 3 illustrates a schematic exemplary method for generating a control signal to perform manipulations on the quantum computing device and for processing measurement signals from the quantum computing device. In most embodiments of quantum computing devices known to date, the control signals for the quantum computing device are prepared on a classical computing device and the measurement signals provided by the quantum computing device are further processed on the classical computing device. Other embodiments are, however, conceivable as quantum computing devices mature. In the following example, the quantum computer refers to a gate-based quantum computer for which the manipulations refer to operations on the quantum elements of the quantum computer.

[0088] For generating the control signal to perform operations on the quantum computing device, the problem to be solved with the aid of the quantum computing device is provided in step S10, preferably, in a mathematical description. Such problem may for instance include determining a material property based on the mathematical description of the material’s electronic structure. Other problems may include optimization problems and associated objective functions. Based on the problem to be solved, an operation description of the problem or a sub-problem may be generated in step S12, wherein the operation description comprises the operations to be applied to the qubits of the quantum computer to solve the problem in the quantum mechanical calculation. Further, the operation description can include a reference state that allows to generate a representation of an initial qubit state on the quantum computer on which the further operations are then applied by manipulating the qubit states. Based on the operation description control signals can then be generated in step S14 to control the quantum computer, for instance, by providing the control signals to the manipulation unit that can then manipulate the qubit states based on the control signals. In step S16 the manipulation unit then applies the manipulation operations to individual or multiple qubits of the quantum computer, wherein based on the manipulation operations the qubits perform the quantum mechanical calculation. After the manipulation, measurement signals can be generated to determine the result of the quantum mechanical calculation in step S18. This step can include a read-out, i.e. measurement, of the qubit states after applying the manipulation operations to the initial qubit states. The measurement signals can in step S20 then be translated into a measured quantity on the classical computer and in case of a sub-problem fed back into the problem to be solved. Finally, the result of the problem calculation including the quantum mechanical calculation can be provided on the classical computing device in step S22.

[0089] Fig. 4 illustrates a schematic example of a hybrid system including a classical and a quantum computing device. As described with respect to the method illustrated in Fig. 3, quantum computing devices are often used in connection with classical computing devices. As shown in Fig. 4 a problem preparation system, e.g. control signal generation apparatus, can be realized as a classical computing device 110 performing, for instance, steps S10, S12, S20, S22 of the method illustrated in Fig. 3. A controlling unit can then be provided as interface between the classical computing device 110 and the quantum computer 100, wherein the controlling unit can also be a classical computing device, for instance, performing step S14. The control unit can then be communicatively coupled with the manipulation part 106 that can control the manipulators of the quantum computing device. Also, the manipulation part 106 can be realized as a classical computing device, for instance, a classical controlling hardware for the control of specific hardware components of the quantum computer that perform the manipulation of the qubits. However, the manipulation part 106 is generally regarded as part of the quantum computer, since it directly influences the quantum register. The quantum computing device 100 is adapted to perform the quantum operation S16, in particular, by the manipulation of the qubits of the quantum register. The measurement part 108 that is also generally regarded as part of the quantum computing device can then perform the step S18 by utilizing classical hardware. The measurement part 108 can then be communicatively coupled to the preparation system 110 for further processing of the measurement signals.

[0090] Fig. 5 illustrates a schematic example of a quantum computing device based on superconductors. Superconducting quantum computing devices are one of the solid-state quantum computing technologies. Here the quantum register 104 can include superconducting circuits 520, 522, 524 based on Josephson junctions. The qubits can then, for instance, refer to charge, flux, transmon, or phase qubits depending on the quantity of the superconducting circuits that are chosen to represent the qubits. Fig. 5 refers to a simplified illustration of a superconducting quantum computer utilizing charge qubits. For charge qubits the different states of the qubit are represented by an integer number of Cooper pairs on a superconducting island. In case of gate-based quantum computing quantum manipulations can then be performed by manipulating the qubits through microwave pulses. Resonators 512, 514, 516 can be utilized to manipulate the state of the qubits by applying the microwaves or for reading out the state of the qubits by measuring respective microwaves, wherein generally different resonators are used for the manipulation of the state of the qubits and the readout of the qubits. Moreover, resonator 518 can be utilized for applying microwaves that entangle the qubits. However, instead of resonator 518 the entanglement can also be achieved by an inductive or capacitive coupling of the superconducting circuits or even by providing another qubit, here a superconducting circuit, between the to be entangled qubits. On an operational level such systems are maintained at extremely low temperatures, e.g., in the tens of mK. The extreme cooling of the systems keeps superconducting materials below their critical temperature and helps to avoid unwanted state transitions. To maintain such low temperatures, the quantum information processing systems may be operated within a cryostat, such as a dilution refrigerator. In some implementations, control signals are generated in higher-temperature environments, and are transmitted to the quantum computer using shielded impedance-controlled GHz capable transmission lines, such as coaxial cables. In some implementations, the state measurement of superconducting qubits is achieved using a dispersive detection scheme. In order to read out or detect the state of any qubit, a probing signal, e.g., a travelling microwave, may be excited along a readout transmission line coupled to the qubit via a respective readout resonator. The frequency of the probing signal can be in the vicinity of the resonance frequency of the readout resonator. Depending on the internal quantum mechanical state of the qubit, the intensity or phase of the probing signal transmitted along the readout transmission line may be altered because the reflectivity of the readout resonator coupled to the qubit changes depending on the state of the qubit. This allows for the state detection of the qubits, wherein during the readout of a qubit state the state of the qubit collapses, i.e. is projected with the respective probability onto one of the basis states. By performing the quantum mechanical calculation and the readout a plurality of times the respective probabilities can be determined. Further details for superconducting quantum devices are described e.g. in documents EP 3830867 A1 , EP 3449427 A1 , US 2020272925 A1 , CN 212061223 U and US 2019019099 A1.

[0091] Fig. 6 illustrates a schematic example of a quantum computing device based on ions in an ion trap. Similar to neutral atom traps ion traps with, e.g. positively charged Calcium ions, can be used to implement the quantum computing device. Here ions 626 are trapped in an oscillating electromagnetic field 624 inside a high or ultra-high vacuum. The ions 626 are laser cooled and held in the oscillating electrical field 624. For qubit manipulation such as superposition or entanglement laser light 628 at different frequencies may be used.

[0092] Generally, based on the above described quantum computer realizations gate-based type calculations can be performed on a quantum computer hardware architecture. The gatebased type calculation is based on quantum gates. In contrast to classical gates, there is an infinite number of possible single-qubit quantum gates that can change the state vector of a qubit. Changing the state of a qubit state vector typically is referred to as a single qubit rotation, and may also be referred to herein as a state change or a single-qubit quantum gate operation. A rotation, state change, or single-qubit quantum gate operation can be represented mathematically by a unitary 2 x 2 matrix with complex elements. A rotation corresponds to a rotation of a qubit state within its Hilbert space, which can be conceptualized as a rotation of a vector on the Bloch sphere, wherein the Bloch sphere is generally known as a geometrical representation of the space of the pure states of a qubit. Multiqubit gates alter the quantum state of a set of qubits. For example, two-qubit gates rotate the state of two qubits as a rotation in the four-dimensional Hilbert space of the two qubits, wherein, as generally known, the Hilbert space is an abstract vector space possessing the structure of an inner product that allows length and angle to be measured. Furthermore, Hilbert spaces are complete, i.e. there are enough limits in the space to allow the techniques of calculus to be used.

[0093] In the following the term operation description refers to a representation of a problem that comprises a sequence of quantum operations that should be applied during a quantum mechanical calculation of the problem. The term “quantum operation” can include in the context of this invention all types of quantum gates as described above and more generally all manipulations known for quantum elements on any quantum computer hardware. Moreover, the term can also include operations performed on components of the quantum computer representing a coupling between the quantum elements forming the qubits and bosonic fields and, optionally, components representing the bosonic fields themselves. These operations then refer to any kind of change of the state of the coupling or bosonic field representing components, for example, a turning of a coupling on and off, or the change of a field frequency, etc. Further, in some applications the quantum operations can also include measurement operations. This allows to implement algorithms using a measurement feedback. For example, in such an algorithm a quantum computer can execute the quantum gates defined by the sequence of quantum operations and then measure only a subset, i.e., fewer than all, of the qubits or other calculation elements, like the bosonic field states, in the quantum computer, and then decide which further quantum manipulations to execute next based on the outcome of the one or more measurements. In particular, measurement feedback can be useful for performing quantum error correction, but is not limited to use in performing quantum error correction.

[0094] Not all quantum computers are gate-based quantum computers. Embodiments of the present invention are not limited to utilizing gate-based quantum computers. As an alternative example, embodiments of the present invention can also utilize, in whole or in part, a quantum computer that is implemented using a quantum annealing paradigm which is an alternative to the gate-based quantum computing paradigm. More specifically, quantum annealing is a metaheuristic for finding the global minimum of a given objective function over a given set of candidate solutions (candidate states), by a process using quantum fluctuations. In particular, quantum annealing is closely related to adiabatic quantum computing. Generally, also quantum annealing procedures start with utilizing a classical computer providing or generating an initial Hamiltonian and a final Hamiltonian based on a computational problem to be solved, and providing the initial Hamiltonian, the final Hamiltonian and an annealing schedule as input to a quantum computer. In case of an annealing procedure in which an optimization problem is solved, preferably, the final Hamiltonian refers to an Ising Hamiltonian representing the optimization problem, orto a good approximation of the ground state of the Ising Hamiltonian representing the optimization problem. The quantum computer is then adapted, for instance, by utilizing a respective control unit controlling a manipulation part of the quantum computer, to prepare a relatively easy to prepare initial state, such as a quantum-mechanical superposition of all possible states, e.g. candidate states, with equal weights, based on the initial Hamiltonian. After the preparation of the initial state on the quantum computer, the initial state is then evolved according to the annealing schedule following a time-dependent Schroedinger equation referring to a natural quantum-mechanical evolution of the physical system of the quantum computer. More specifically, the state of the quantum computer undergoes time evolution under a time-dependent Hamiltonian, which starts from the initial Hamiltonian and terminates at the final Hamiltonian. If the evolution rate is slow enough, the system stays close to the ground state of the instantaneous Hamiltonian. At the end of the time evolution, the set of qubits, i.e. quantum elements, on the quantum annealer is in a final state, which is expected to be close to the ground state of the Ising Hamiltonian that corresponds to a solution to the original problem, referring, for instance, to an optimization problem. The final state of the quantum computer can then be measured, thereby producing results that can be utilized for solving the original problem. The measurement operation can be performed, for example, in any of the ways described already above. A classical computer can then perform postprocessing on the measurement results to produce an output representing a solution to the original computational problem. A quantum annealer as described above can, for instance, be realized on a superconducting quantum computer hardware.

[0095] Moreover, embodiments of the present invention can also utilize, in whole or in part, quantum computers that are implemented using a one-way quantum computing architecture, also referred to as a measurement-based quantum computing architecture. More specifically, the one-way or measurement-based quantum computer refers to a method of quantum computing that first prepares an entangled resource state, usually a cluster state or graph state, then performs single qubit measurements on it. It is "one-way" because the resource state is destroyed by the measurements. In such an architecture, the outcome of each individual measurement is random, but they are related in such a way that the computation always succeeds. In general, the choices of basis for later measurements need to depend on the results of earlier measurements, and hence the measurements cannot all be performed at the same time.

[0096] Fig. 7 shows schematically and exemplarily a system for generating a solution of a problem associated with a chemical product. For example, the system 700 can be configured for determining a property of a chemical product, wherein the property can then be utilised for monitoring and / or controlling a production of a chemical product or performing of an experiment comprising the chemical product. However, in other examples the problem that is solved by the system 700 can also refer to other applications, for instance, applications in logistics, scheduling, cryptography, engineering, etc.

[0097] The system 700 comprises a control signal generation apparatus 710, a quantum computer system 720 and a problem solving apparatus 730 that will be described in detail in the following. The control signal generation apparatus 710 as well as the problem solving apparatus 730 can be realised in form of any software and / or hardware of a classical computing system. In particular, the functions performed by the units of the respective apparatuses can be performed by one or more computing systems, for instance, in distributed computing like cloud computing. The control signal generation apparatus 710 is configured for providing control signals for controlling a quantum computer system 720. The quantum computer system 720 comprises at least two, in this example three quantum computers 721 , 722, 723. Further, the quantum computing system 720 can comprise the hardware and / or software utilised for controlling, managing and monitoring the respective quantum computer. The quantum computers 721 , 722, 723 of the quantum computing system 720 can be independent of each other, for instance, can be located at different locations and can be controlled and managed by different providers. However, at least some of the quantum computers of the quantum computer system 720 can also allow a direct interaction between the quantum computers. Such a direct interaction can refer to exchanging information from quantum calculations of the quantum computers in different ways. In an embodiment such information can be exchanged in a classical way, for instance, using a communicative coupling between the classical computers controlling, monitoring and / or managing the quantum computers. In this case the results of quantum computations can be exchanged on the classical level of a quantum computer, i.e. utilizing classical computer communication of the classical computer parts associated with the quantum computers, and then prepared or integrated into the quantum computation of another quantum computer. However, in an embodiment at least two of the quantum computers can also be directly connected on the quantum level, i.e. without having to measure, process and derive information from one quantum computer to send the information to the other quantum com- puter where it has to be processed by, implemented into and prepared on the other quantum computer. Such a direct exchange on the quantum level can be achieved, for instance, by allowing for an entanglement between at least two of the quantum computers, wherein an entanglement of the quantum computers is defined at least one quantum element of each quantum computer being entangled with each other. Since an entanglement couples the states of the respective quantum elements, manipulations performed on one quantum element and one quantum computer can be directly related to the quantum element of the other quantum computer leading to a direct exchange of information on the quantum level. Further details to such an embodiment will be described with respect to Figs. 9 to 14.

[0098] The quantum computers of the quantum computing system 720 are configured such that at least two of the quantum computers, for instance, quantum computer 721 and quantum computer 722 comprise different fidelities. For example, the quantum computer 721 can comprise a higher fidelity than the quantum computer 722. The fidelity of a quantum computer is generally dependent on the respective construction of the quantum computer, for instance, on the hardware and / or controlling system of the quantum computer. Thus, the fidelity is determined individually for each quantum computer.

[0099] The apparatus 710 comprises a problem providing unit 71 1 , a sub-problem deriving unit 712 and a control signal generation unit 713. The problem providing unit 711 can be realized, for instance, as a user interface where a user can input a respective problem. However, the problem providing unit 711 can also be realized as an interface to a storage on which a respective problem is already stored. The problem can be any problem that can be translated for a quantum computation. In an embodiment the problem is an electronic structure problem that allows to derive a property of a chemical product based on a solution of the electronic structure problem. The problem description provided by the problem providing unit 711 can thus referto any form of digital representation of the problem, for instance, to a mathematical representation, an identification number that allows to retrieve the problem from a storage, a digital code that encodes the respective problem, etc.

[0100] The sub-problem deriving unit 712 is configured for deriving sub-problems from the provided problem description based on the fidelities of the at least two quantum computers. The solution of the sub-problems then allows to calculate a solution of the provided problem. Moreover, sub-problems are problems that only solve parts of the overall problem, for instance, that comprise less quantities or a less complexity in other ways, such that in most cases sub-problems can be easier solved than the overall problem. At least some of the derived sub-problems are in most cases dependent problems, wherein the solution of one sub-problem depends on the solution of another sub-problem. However, in some cases at least one of the sub-problems can be independent in such a way that the sub-problems can be solved independently of each other. The sub-problems are derived based on the fidelities of the quantum computers of the quantum computing system. For example, for respective problem classes rules can be implemented, that can be utilised by the sub-problem deriving unit 712 to identify and derive the sub-problems of a problem taking the fidelities of the quantum computers into account. For example, the fidelities of the quantum computers can determine a maximum size of a respective sub-problem, for instance, the number of maximum variables of a sub-problem. Also the complexity of a sub-problem can be influenced by the available fidelities of the quantum computers, for instance, the maximal number of interacting variables can be determined based on the fidelities. However, the sub-problem driving unit 712 can also comprise an interface to a user, wherein the subproblem deriving unit 712 in this case can provide suggestions for respective sub-problems to a user and the user can then confirm or amend the respective sub-problems suggested by the sub-problem deriving unit 712. Moreover, in addition to the fidelities of the quantum computers available in the quantum computing system 720 also other criteria can be taken into account for deriving respective sub-problems. For example, a solution algorithm that should be utilised for solving the problem can be predetermined or selected by a user and the sub-problem deriving unit 712 can then be configured for deriving the sub-problems further based on the predetermined solution algorithm. Moreover, also the number of logical quantum elements provided by the quantum computers of the quantum computer system 720 can be taken into account when deriving the respective sub-problems. Further details and embodiments with respect to example for deriving of the sub-problems are provided with respect to Figs. 9 to 14.

[0101] The control signal generation unit 713 is configured for generating control signals for controlling the quantum computing system, in particular, the quantum computers of the quantum computing system. The control signals cause one or more of the quantum computers of the quantum computing system 720 to perform a quantum computation of at least one of the derived sub-problems. The control signals can refer to any signal that cause one or more of the quantum computers to perform a quantum computation on one or more of the sub-problems. For example, the control signals can be configured to directly control the quantum computer, i.e. the operations performed on the quantum computer. However, the control signals can also include information on the sub-problem and which quantum computer should compute the sub-problem, wherein further adaptations to the quantum computer hardware, for instance, the determination of a specific operation algorithm for performing the quantum computation of the sub-problem, can be performed by the classical computer acting as a control unit of the quantum computer. Thus, the generating of the control signals can comprise translating a respective sub-problem into a respective operation algorithm comprising one or more operations for preparing and manipulating quantum states of the quantum elements on a quantum computer for performing the quantum computation. However, if the control of a respective quantum computer is configured to perform such a translation of a provided problem such a step can also be omitted.

[0102] Moreover, the control signals cause a distribution of the sub-problems to the respective quantum computers of the quantum computing system 720 based on the fidelity of the respective quantum computers. In particular, since the fidelity has been taken into account when deriving the sub-problems, the sub-problems can be assigned to the respective quantum computers with the appropriate fidelity for the respective derived sub-problem. For example, if a sub-problem has been derived for being computed on a quantum computer of the quantum computing system with a high fidelity the sub-problem will be assigned by the control signal to the respective quantum computer of the quantum computing system. Subproblems derived for lower fidelities can either be computed by respective quantum computers with a respectively lower fidelity but can also be computed on quantum computers with a higher fidelity, if necessary. Generally, the higher fidelity can be regarded as being higher with respect to a) the fidelity of at least one other quantum computer of the at least two quantum computers, b) the fidelity of all other quantum computers of the at least two quantum computers and / or c) a predetermined fidelity threshold. The lower fidelity can be regarded as being lower with respect to a) the fidelity of at least one other quantum computer of the at least two quantum computers, b) the fidelity of all other quantum computers of the at least two quantum computers and / or c) a predetermined fidelity threshold.

[0103] The control signals can cause the quantum computation of the sub-problems in a sequential, parallel or iterative manner or a combination thereof. The result of the quantum computation, for instance, measurements of the quantum elements on the quantum computer or a solution of the sub-problem determined based on the measurements of the quantum elements after the quantum computation are then provided by the quantum computers of the quantum computing system 720 to the problem solving apparatus 730.

[0104] The problem solving apparatus 730 comprises a receiving unit 731 and problem solving unit 732. The receiving unit can be realized as an interface with a quantum computing system 720 for receiving the result of the quantum computation. The received results of the quantum computation of the sub-problems are then provided to the problem solving unit 732 that is configured for determining a solution of the problem based on the results of the quantum computation of the sub-problems. The determination of the solution of the prob- lem based on the results of the sub-problems generally refers to a combination of the results of the sub-problems. The combination is determined by the kind of sub-problems and how the sub-problems have been derived from the problem. Thus, for different problems the determination of the solution of the problem based on the sub-problem results is different. Respective rules for respective sub-problems and problems can be implemented and utilised respectively. The solution can then be utilized to control, for instance, a production of a chemical product in a chemical plant 740 or a series of experiments in a laboratory 750. For example, the solution can refer to a property of a chemical product that can be utilized for controlling a chemical reaction in the chemical plant 740 or planning a respective experiment in the laboratory 750.

[0105] In the example shown in Fig. 7 the problem is an electronic structure problem of a chemical product, for instance, of a molecule, crystal or atom and a solution of a problem is a solution of the respective electronic structure problem. From a solution of the electronic structure problem properties of the respective chemical product can be derived based on respective known calculations. Such properties can then be utilised, for instance, for controlling a production of the chemical product or controlling laboratory equipment to perform respective experiments including the chemical product.

[0106] Fig. 8 shows schematically and exemplarily a flowchart of a method for generating a solution of a problem related to a chemical product, wherein the method can be performed, for instance, by the system described with respect to Fig. 7. The method comprises providing a problem description indicative of a problem and deriving sub-problems from the provided problem description based on the fidelities of quantum computers of a quantum computing system. Further, the method comprises generating control signals for controlling the quantum computing system to cause a quantum computation of the derived sub-problems on the quantum computers, wherein the sub-problems are distributed to the quantum computers based on the fidelity of the respective quantum computers. Further, the method comprises performing the quantum computations of the respective sub-problems. In a next step the results of the quantum computations are received. Optionally, based on the received results an iteration can be performed wherein based on the results again control signals are generated for again computing the sub-problems based on the results of the previous quantum computation of the sub-problems. Moreover, also the sub-problems can also be computed sequentially, wherein the control signals are generated based on the solution of one sub-problem to control the computation of the next sub-problem based on the solution. For example, the previous results of the quantum computations can now be utilised as starting points or as input for other sub-problems. Based on the result of the quantum computations, optionally after one or more further iterations, a solution of the problem can be determined based on the results of the sub-problems. This solution of the problem is generally applicable to control and / or monitor a respective application of the problem. For example, in case of a problem being an electronic structure problem the solution of the electronic structure problem can be utilised to determine a property of a chemical product and then to control and / or monitor a production of the chemical product based on the determined property.

[0107] Further details and embodiments are described in the following. Quantum computing is an emerging technology that exploits quantum-mechanical phenomena to perform computational tasks. Quantum computers are expected to solve certain computational problems substantially faster than classical computers. They can be used for example to simulate quantum mechanical problems such as electronic structure problems, including but not limited to molecules, crystals, and amorphous solids. In an industrial environment such simulations of electronic structure problems are important for the discovery of new materials and chemicals, the improvement of chemical processes, the tailoring of molecules, solids and materials to desired properties and, generally, for making research activities more efficient by reducing the number of required laboratory and production trials that are often resource intensive. Additionally, optimization problems, machine learning and artificial intelligence are further exemplary application fields of quantum computing. Quantum computing is expected to significantly surpass classical computing with regard to treatable problem size, required computation time, in terms of energy costs, and / or achievable accuracy in the above mentioned fields.

[0108] The fundamental processing unit of a quantum computer is a quantum mechanical bit (qubit) represented physically by one or more quantum elements of the quantum computer. By executing a suitable quantum circuit via control pulses that act on the qubits a solution to one of the abovementioned problems or a specific sub-problem can be prepared on the qubit register and finally measured within a hardware-specific readout protocol. Both the number of qubits, e.g. number of logical quantum elements, and their quality, e.g. error rates or fidelities, mainly determine the maximum size, e.g. dimension, and maximum complexity of the problem that can be solved by a respective quantum computer.

[0109] Currently both the number and the fidelities of the qubits are rather restricted resulting in so-called noisy intermediate-scale quantum (NISQ) computers. Typically, those NISQ computers are combined with classical computers, like a classical high-performance computer (HPC). Within this so-called hybrid quantum-classical framework components of algorithms are run on the computing architecture which can best resolve the specific components of the problem. In this context typically variational approaches are used, wherein the quantum computer is controlled by a classical computer. For example, after preprocessing and preparation of the problem on a classical computer, in a second step a parameterized quantum state is prepared on the NISQ computer, in a third step a target function is measured on the NISQ computer and in a fourth step the parameters are optimized on the classical computer to minimize or maximize the target function. This procedure is repeated with the updated parameters until the value of the target function converges and thus becomes minimal or maximal, respectively. In the last step post-processing can be carried out on the classical computer.

[0110] Both the number and fidelities of the qubits are expected to continuously increase. As soon as a certain threshold is reached, several hundred or thousands, the exact number depends on the specific error-correction code, with two examples being the “surface code” and the “color code” for error correction, of physical qubits can be entangled to form a single error-free logical qubit. A quantum device comprised of several of those logical qubits is termed a fault-tolerant (FT) quantum computer. It is noted, that the FT quantum computers mentioned in the following embodiments are not restrict to fully FT quantum computers, but also include such FT quantum computers that still exhibit sufficiently small errors. The error rate of such a FT quantum computer depends on the error-correction code and the number of physical qubits used. As a rule of thumb, the more physical qubits of a certain quality are used the more errors are suppressed.

[0111] Such a FT quantum computer is able to accommodate deep quantum circuits that allow for highly accurate or exact solutions of many highly relevant computational problems which cannot be obtained on classical computers in reasonable time or with reasonable energy cost. However, due to the massive overhead in physical qubits required to build one logical qubit, some FT quantum computers remain rather restricted with regard to the number of logical qubits and thus the size, e.g. dimension, of the computational problem that can be solved. As a consequence, for many practically relevant computational problems the problem dimension will exceed the dimension of the Hilbert space that is spanned by the quantum computer. In such cases, the computational problem cannot be solved on a FT quantum computer in a straightforward manner. Further, more specific examples of problems from the area of electronic-structure problems that can be solved utilizing the invention as described herein are described below.

[0112] Although fault-tolerant quantum computers, e.g. fault-tolerant quantum processing units (FT-QPUs), outperform noisy intermediate-scale quantum computers, e.g. noisy intermediate-scale quantum processing units (NISQ-QPUs), with respect to accuracy as they are able to accommodate deeper quantum circuits, i.e. more gate operations, NISQ-QPUs in many cases are able to accommodate wider quantum circuits, i.e. utilizing more qubits but fewer gate operations than in case of FT-QPUs, to provide approximate but still accurate solutions for many problems. Moreover, even without FT-QPUs NISQ-QPUs with a wide range of fidelities exist that allow for calculations with different accuracies or for quantum circuits with different depths.

[0113] The invention refers to a quantum computing system with a plurality of quantum computers with different fidelities. For example, the quantum computing system can be a triple hybrid quantum computing system that combines FT-QPUs, NISQ-QPUs and classical central processing units (CPUs) or other conventional computational devices wherein computational operations are carried out wherever most advantageous for solving the problem. However, the quantum computing system can also combine different NISQ-QPUs with different fidelities and CPUs or other conventional hardware without using FT-QPUs.

[0114] In Fig. 9 two examples of respective quantum computing systems are shown. The examples are shown with respect to quantum computing systems comprising FT-QPUs and NISQ-QPUs. However, additionally or alternatively to FT-QPUs also NISQ-QPUs with higher fidelities can be used with the same exemplary principles described below. In the following the term “higher fidelity NISQ-QPUs” is utilized with the meaning that a higher fidelity NISQ-QPU can have a higher fidelity compared to a) all other NISQ-QPUs of the quantum computing system, b) higher fidelity than at least on other NISQ-QPU of the quantum computing system and / or c) higher fidelity than a predetermined threshold. In particular, any combination of these three criteria can be defined a NISQ-QPU that can be utilized for computing a sub-problem as described in the following. The threshold can be based for instance, on previous consideration, experience with a respective problem type, an algorithm to be utilized, a specific problem type, a to be achieved accuracy of a solution, etc. However, in most cases the higher fidelity NISQ-QPU simply refers to the NISQ-QPU with the highest fidelity of the NISQ-QPUs of the quantum computing system, wherein in this case the respective sub-problem is derived such that it can be computed by the higher fidelity NISQ-QPU.

[0115] In the following examples, the FT-QPU(s) referred to are optional and can also be replaced by higher-fidelity NISQ-QPU(s) or can be used together with higher-fidelity NISQ-QPU(s). One example comprises controlling the FT-QPU(s), and / or higher-fidelity NISQ-QPU(s), and the NISQ-QPU(s) and exchange data via the CPU(s). Another example comprises a quantum computing system where FT-CPU(s), and / or higher-fidelity NISQ-QPU(s), and NISQ-QPUs are additionally entangled in a way that quantum information is distributed and shared among the entangled QPU(s) and can also be directly transferred between the respective QPU(s). In this case it is preferred, that the FT-CPU(s), or higher-fidelity NISQ- QPU(s), and NISQ-QPU(s) are on the same chip to facilitate entanglement. However, by using a quantum bus such as optical fibers and photons, spatially separated QPUs can also be entangled. Both setups depicted in Fig. 9 in principle allow to solve practically relevant computational problems with a problem size, e.g. dimension, that exceeds the dimension of the Hilbert space of a FT-QPU or higher-fidelity NISQ-QPU by optimally distributing computational tasks.

[0116] An exemplary detailed method for solving a problem according to the invention is shown in Fig. 10 for an unentangled quantum computing system and in Fig. 11 for an entangled quantum computing system. In a first step the problem can be defined on a classical computing system, for instance, by a user using a user interface. Typical problems are for example electronic structure problems, such as for molecules, solids and materials, or optimization problems. In a second step the problem can be divided on the classical computing system into sub-problems taking into account the fidelities of the FT-QPU(s), and / or NISQ- QPU(s) of the quantum computing system on which the sub-problems are to be solved. FT- QPUs or higher-fidelity NISQ-QPU(s) are best used to solve smaller computational subproblems for which exact or highly accurate solutions are desired that can be efficiently obtained neither on lower-fidelity NISQ-QPUs nor on CPUs. Also sub-problems for which the solution requires deeper quantum algorithms exceeding the capabilities of lower-fidelity NISQ-QPU(s) and CPUs can be derived to be solved on the higher-fidelity NISQ-QPU(s) or FT-QPU(s). The dimension of the sub-problem can accordingly be chosen to be less or equal than the dimension of the Hilbert space spanned by the qubits of the FT-QPU(s) or higher-fidelity NISQ-QPU(s). The lower-fidelity NISQ-QPU(s) are best used to solve larger computational sub-problems where accurate solutions are desired that cannot be efficiently obtained on classical computers, e.g. CPU(s), or where the solution requires wide rather than deep quantum algorithms utilizing a larger number of qubits. CPUs are used for all remaining tasks and controlling of the entangled or unentangled quantum computers. In a third step control signals are provided that cause to prepare and run respective quantum circuits on the quantum computing system for calculating and solving the respective subproblems. Optionally, the quantum circuits can contain parameters, which are typically used in the context of variational procedures. In this case an initial set of parameters is provided in the first iteration step. In the next step, the sub-problem is encoded into qubits, the quantum circuit is compiled, and control signals are provided to either the FT-QPU(s), higher- fidelity NISQ-QPU(s) or the NISQ-QPU(s), all simultaneously or to the entangled FT- QPU(s), higher-fidelity NISQ-QPU(s), and NISQ-QPU(s) systems. This is also done in all subsequent iteration steps with an appropriately updated set of parameters. In a fourth step the quantum circuit of the current iteration step is executed on eitherthe FT-QPU(s), higher- fidelity NISQ-QPU(s) or the NISQ-QPU(s), all simultaneously or on the entangled FT- QPU(s), higher-fidelity NISQ-QPU(s), and NISQ-QPU(s) to prepare the respective subproblem solutions. Examples for relevant algorithms for FT-QPU(s) or higher-fidelity NISQ- QPU(s) are quantum Fourier transform, quantum phase estimation (QPE), Grover-type search algorithms, Shor-type factorization algorithms and Harrow-Hassidim-Lloyd-type algorithms. Examples for relevant algorithms that can also be used on higher-fidelity NISQ- QPU(s) or NISQ-QPU(s) are the variational quantum eigensolver (VQE) and the quantum approximate optimization algorithm (QAOA). Respective quantum gate operations are carried out on the qubit registers. In case of an entangled FT- or higher-fidelity NISQ- and NISQ-QPU system the solutions of sub-problems can be directly transferred between the two types of QPUs. In a fifth step relevant observables of the sub-problems can be measured on either the FT-QPU(s), the higher-fidelity NISQ-QPU(s), or the NISQ-QPU(s), all simultaneously orthe entangled FT-, or higher-fidelity NISQ-, and NISQ-QPU systems and provided to the CPU(s). Examples for observables that can be measured are qubit occupation numbers, reduced density matrices and energies. In a sixth step the target function and / or quantity and if needed its derivatives or other dependent quantities can be evaluated on the CPU(s) based on the measured observables. Examples for target functions and / or quantities are the total energy of the electronic structure problem and classical loss functions. Then, feedback calculations connecting iteration steps of the algorithm like self-consistency conditions can be invoked, or, in case of variational procedures, optimization approaches such as quasi-Newton methods can be invoked to obtain a new set of parameters to minimize or maximize the target function and / or quantity. Finally, the iteration can return to the second orthird step or if one or more stop criteria are fulfilled, e.g. the target function and / or quantity has converged, the iteration can terminate the feedback loop and proceed with the next step. In the next step after convergence has been achieved, optional postprocessing steps, e.g. to enhance the overall accuracy or further process the result, e.g., error mitigation or error correction, may be carried out on the CPU(s). If necessary for this, further observables can be measured on the QPU(s). The results of the sub-problems are further combined either in this post-processing or during the iteration, depending on the interactions between the sub-problem results. Lastly, the final result is provided to the user as solution of the processed problem.

[0117] In the following it is described in detail how the methods and systems described above can be used for solving electronic-structure problems, e.g. the simulation of quantum systems such as molecules. A basis for the following embodiments is a method that partitions an electronic structure problem, such as a problem describing a molecule or a solid, into a so- - M - called “active space” and an “inactive space” based on its electronic states, e.g., electronic orbitals, as shown in Fig. 12.

[0118] In the following an example of a method is described utilizing a hybrid quantum-classical approach with respect to the above described quantum computer system and the method for deriving and distributing sub-problems. The following exemplary method refers to an “active space” that is solved on a quantum computing system and wherein result is then used to adjust electronic orbitals, preferably, both in an “inactive” and the “active space”, on a classical computer. However, also other distributions are possible depending on the problem and the determined “active space” and “inactive space” as sub-problems. For example, also an “inactive space” orthe influence of the “active space” on the “inactive space” can be solved on a quantum computer system. The procedure is iteratively repeated until convergence of the electronic orbitals and total energy can be achieved. This method can be referred to as “complete active space self-consistent field” (CASSCF) method. A related method is “complete active space configuration interaction” (CASCI), which can be viewed as a special case of CASSCF that neglects the adjustment of the electronic orbitals to the “active space” solution. In the following both methods are referred to as “CAS methods”.

[0119] By means of the described partitioning into an “active space” and an “inactive space”, the method allows solving electronic-structure problems with a size, e.g. dimension, that would otherwise exceed the dimension of a quantum computer, e.g., in terms of available qubits onto which the electronic orbitals are mapped and / or in terms of gate operations that are needed to prepare the solution, as only the electronic orbitals comprising the “active space” are explicitly treated on the quantum computer. The method is particularly advantageous for the simulation of so-called statically correlated molecular systems, such as transition metal compounds that are relevant for improving and designing, e.g., new catalysts and chelating agents, which is typically very challenging for traditional methods utilizing CPUs both with respect to computational effort and accuracy.

[0120] Generally, the procedure for solving the “active space” and the “inactive space” parts, i.e. the first and the second part, respectively, of the problem is compliant with the general procedure depicted in Fig. 10 above wherein the type of quantum computer utilized can be determined based on the fidelity of the respective available quantum computers before the quantum computation and can remaining fixed throughout the whole iterative procedure. In summary, for the specific case of the hybrid quantum-classical CAS method the steps in Fig. 10 are detailed in the following. In the first step the specific electronic-structure problem to be solved is provided as a molecular system specified by atomic positions, basis sets, electric charge and spin on a CPU. This is in analogy to the first step in the general process in Fig. 10. Further, an initial set of electronic orbitals is specified, e.g. from a Hartree-Fock calculation on a CPU. Along with these electronic orbitals, a fixed “active space” is defined for example by electronic orbitals and electrons that constitute the “active space”, wherein the active space defines a first sub-problem. These steps are performed on a CPU, in analogy to the second step in the general process in Fig. 10. In a next step, the matrix elements of the effective Hamiltonian of the “active space” in the electronic orbital basis are set up on a CPU defining the first sub-problem further. This is in analogy to the third step in the general process in Fig. 10. Moreover, the effective “active space” Hamiltonian operator, which is fermionic / electronic in nature, is encoded, i.e. translated, in qubit operators, e.g., via the Jordan-Wigner or Bravyi-Kitaev transformation. The quantum circuit to solve the “active space” sub-problem is then compiled on a CPU such that it can be performed on a quantum computer. The “active space” sub-problem is then prepared and solved on a quantum computer, for example, a higher-fidelity NISQ-QPU, or FT-QPU, for example, by a so-called quantum phase estimation algorithm (QPE). This way the exact or a nearexact solution, for example, an exact or near-exact eigenfunction, of the “active space” subproblem, for example, the “active space” Hamiltonian, is prepared on the quantum computer. This is in analogy to fourth step in the general process in Fig. 10. The one- and two- particle reduced density matrices can then be measured on the quantum computer as solution of the sub-problem. This is in analogy to the fifth step in the general process in Fig. 10. Alternatively and also according to step four in Fig. 10, the “active space” sub-problem can be solved on a NISQ-QPU, e.g. by a so-called variational quantum eigensolver (VQE), in the step above. Depending on the chosen ansatz, typically approximate but accurate solutions, for instance as eigenfunctions of the “active space” sub-problem can be prepared on the NISQ-QPU. According to step five the one- and two-particle reduced density matrices would then be measured on the NISQ-QPU in the step above. The energy can then be calculated from the one- and two-particle reduced density matrices on a CPU. Furthermore, the gradient with respect to variation of the electronic orbitals can be calculated on a CPU, wherein this step is only performed in case of CASSCF but not for CASCI. This is in analogy to step six in the general process in Fig. 10. In analogy to step seven in Fig. 10, if energy and gradient are converged, it can be proceeded with the final step. If not converged yet, it can be proceeded with updating the electronic orbitals on the CPU as in step eight in Fig. 10, such that the energy is minimized until convergence of the iterative procedure is obtained. In case of CASCI step seven can be omitted and there is no self-consistency loop, and thus it can be proceeded directly to the next step in that case. Further steps performed to determine a solution of the problem are described in the following.

[0121] Although these CAS-like approaches are well suited to capture static correlation in an “active space”, for real-world applications the accuracy of the results is often not sufficient, even if the “active space” is solved exactly on a FT-QPU, because the “inactive space” has been treated only at a very basic level often comparable to the Hartree-Fock level that completely neglects correlations of the electronic orbitals in the “inactive space” and between electronic orbitals in the “active space” and “inactive space”. In the following, this problem is solved by utilizing multireference dynamic correlation methods which account for dynamic correlation of the “inactive space” and among “active space” and “inactive space” both more systematically than e.g. multiconfiguration pair-density functional theory (MC-PDFT) and more efficiently than e.g. second-order n-electron valence state perturbation theory (NEVPT2). Moreover, these methods are in particular suited to be utilized with the method and quantum computing system described above taking the fidelities of the quantum computers into account when solving sub-problems.

[0122] In all approaches discussed below, the first step is described above. The result then is a CAS wavefunction, |<I’CAS). In the following it is described how to proceed from this CAS wavefunction utilizing the principles of the invention. First it is described how the CAS wavefunction can be read out in order to be reconstructed on a different hardware device, for example, in cases in which the quantum computing system does not provide the possibility to entangle different quantum computers. Then it is explained how to add dynamic correlation on top of the CAS wavefunction utilizing the quantum computing system comprising quantum computers with different fidelities and respective derived sub-problems. For an overview see also Fig. 14.

[0123] For all multireference dynamic correlation methods that will be discussed below that leverage unentangled quantum computing systems, relevant specifics of the CAS wavefunction determined as described above can be read out from the QPU(s) in order to be reconstructed or approximated on a different hardware device, for instance, CPU(s) or a different QPU(s), e.g. with a different fidelity, that is most suited for the multireference dynamic correlation treatment. In case of entangled quantum computing systems a reconstruction of the CAS wavefunction on a different hardware device is not required and thus the methods for the readout of the CAS wavefunction can be omitted.

[0124] The CAS wavefunction obtained as described above can be expanded as s states, e.g. Slater determinants, adhering to the “active space” definition. More precisely, those basis states can be so-called configuration interaction (Cl) states given by a fixed distribution of the electrons among the electronic orbitals in the “active space”. The index / runs over all possible electron distributions among the electronic orbitals in the “active space” sub-problem. The expansion coefficients are then the so-called Cl coefficients, which are utilized to reconstruct the CAS wavefunction on a different QPU or CPU that is most suited for the next computational step, for instance, a multireference dynamic correlation treatment.

[0125] The following method can be utilized to determine the Cl coefficients of a CAS wavefunction that has been prepared on a quantum computer, for instance, as described above. In a first step an absolute value of Cl coefficients can be measured on the respective QPU. The absolute value of the Cl coefficients can be measured by determination of an overlap of the CAS wavefunction and the defined basis states, for instance according to: ized. The CAS wavefunction, |<I’CAS),canbe prepared on the QPU as the result of the preceding CAS calculation. Thus, in a previous step, the CAS wavefunction can be prepared on the QPU and a projective measurement on all qubits can be performed. Since no basis rotations of the qubit states are required, all qubits can be measured simultaneously. The measurement result (either 1 or 0 for each qubit) directly corresponds to the distribution of the electrons among the electronic orbitals in the “active space”, wherein the measurement is 0 if an electronic orbital is unoccupied and 1 if an electronic orbital is occupied in case the Jordan-Wigner transformation is used. From this the corresponding basis state, e.g. Slater determinant, can be easily deduced. The count of the respectively measured basis state, e.g. Slater determinant, can be increased by one and the process is repeated until a histogram of how often which basis state, e.g. Slater determinant, has been measured can be recorded. According to Eq. (2) the probability to measure a basis state, e.g. Slater determinant, is given by and the histogram translates directly to these absolute values. A significant advantage of running the CAS method and measuring the result using a quantum computer is that it directly provides the most important basis states, e.g. Slater determinants, to span a relevant portion of the Hilbert space. This information is used in the following steps below. On a CPU, the CAS method would need to search for the relevant basis states, e.g. Slater determinants, which are not known a priori.

[0126] Optionally, if the absolute values of selected, important Cl coefficients are required to be determined with higher precision than achievable by the step above alone, the absolute value can be additionally directly measured in a second step where the CAS wavefunction is projected onto the basis states for instance, via the projection operator: 3) being the unitary operator that prepares the basis state from the vacuum state |vac), i.e. the state where each qubit is in state 0, according to = X |vac> by exciting some qubits to state 1 such that the distribution of the electrons among the electronic orbitals in the “active space” is reflected in the qubit register, where a qubit in state 0 represents an empty electronic orbital in the “active space” and a qubit in state 1 represents an electronic orbital occupied by one electron in the “active space” in case of the Jordan-Wigner transformation. Furthermore, in Eq. (3),vac= n7| (l7+ Zj)- The unitary elementary operators Xtand Zj directly translate into unitary elementary operations acting on the qubits.

[0127] In a further step for reconstructing the CAS wavefunction the phase of the Cl coefficients can be determined in a post-processing step on a CPU. The above described step only determined the absolute value of the complex-valued Cl coefficients but not the phase, which, can be crucial to reconstruct the CAS wavefunction according to Eq. (1). To determine the phase of the Cl coefficients the following steps can be carried out. Since the absolute value |c2can be measured on the QPU(s) it can be determined which basis states have the largest Cl coefficient and are therefore most important for the reconstruction of the CAS wavefunction. Those basis states can then be selected. The number of determined Cl coefficients can be predetermined, for example, such that the measurements lead to a manageable number of basis states and also to a sufficient representation of a sufficiently large portion of the CAS wavefunction. Truncating the summation in Eq. (1) to a predetermined number of basis states leads to a more efficient calculation of the problem. In a next step a representation of a Hamiltonian H, for example, which defines the molecule that is simulated, is reconstructed in a small subspace spanned by the most important basis states selected in the previous step. The Hamiltonian in this sufficiently small subspace has a reduced size compared to the full Hamiltonian and can therefore be efficiently diagonalized, e.g. through power iterations or by the Davidson method. As a result, the Cl coefficients with their absolute value and phase within that subspace can be obtained. Optionally, to increase the accuracy, if needed, further Cl coefficients that have not been included in the previous step can be obtained self-consistently using the results of the previous step as input via:

[0128] Alternatively, as the absolute values |c2are already known from the preceding CAS calculation on the QPU(s) and the measurement of the Cl coefficients described above, in case of real Cl coefficients Cg, a variant of Eq. (4) could comprise fixing the absolute values | | and only modifying the signs, e.g. in the framework of a Monte-Carlo-like procedure that flips the signs randomly: define C as a vector of the values on the left-hand side of Eq. (4), which are obtained by evaluating the right-hand side of Eq. (4) with a given set of Cl coefficients C. One possible target of the sign-finding procedure is to minimize the deviation between the vectors C and C, for example by minimizing |c - c| with respect to the signs,

[0129] = ±|c . Another possible target of the sign-finding procedure is to minimize the energy expectation value of the Hamiltonian with respect to the signs.

[0130] Instead of the process described above a direct measurement of the Cl coefficients with absolute value and phase is possible using, e.g., a Hadamard overlap measurement as shown in Fig. 13. The Hadamard test requires controlled application of the entire quantum state preparation on the QPU. This algorithm is thus not feasible on NISQ-QPU devices but can be executed on a FT-QPU if part of the quantum computer system. This expensive measurement can be combined with predetermination of important determinants as described above. Alternatively, there are other elaborate measurement schemes like shadow tomography that can be used to measure the absolute value and phase of the Cl coefficients.

[0131] After a sufficiently large number of Cl coefficients has been determined to adequately represent the CAS wavefunction according to Eq. (1), the CAS wavefunction can be reconstructed on a different computing device, i.e. a different QPU or a CPU, that is most suited for the subsequent multireference dynamic correlation treatment. To determine how many of the most important Cl coefficients are sufficient to adequately represent the CAS wavefunction the following rules can be implemented. For example, it can be monitored how the energy, i.e. the expectation value of the Hamiltonian operator, converges with the number of included Cl coefficients and thus basis states. For example, if an additional Cl coefficient leads to a change in the energy below a predetermined threshold, then it can be determined that enough Cl coefficients have been included. Additionally and optionally, the stability of the probability distribution and thus the |c2can be also be obtained as described in the following. During the measurement of the Cl coefficients it can be determined if the Cl coefficients change when the number of measurements is increased and / or if the ratios of important Cl coefficients change when the number of measurements is increased. Depending on the result it can be determined which Cl coefficients to include. Additionally or alternatively, it can be determined how much the absolute values of the Cl coefficients determined after the reconstruction of the Hamiltonian in a reduced subspace as described above differ from the absolute values of the Cl coefficients obtained during the original measurement on the QPU(s). Additionally or alternatively, it can be determined if the absolute values are kept fixed in Eq. (4), how much the values resulting from evaluating the right-hand side of Eq. (4) differ from the values C . Based on the respective results it can be determined if the relevant Cl coefficients are included in the reconstructed CAS wavefunction.

[0132] Alternatively, to the method described above utilizing a quantum computer the CAS wavefunction of a problem can also be determined by a traditional CAS, RAS (“restricted active space”, such as RASSCF or RASCI) or GAS (“generalized active space”, such as GASSCF or GASCI) calculation running entirely on CPU(s) without any quantum computing part. In this case the QPU(s) can be utilized for the dynamic correlation treatment of the “inactive space” in a further computational step. However, using traditional CAS, RAS or GAS methods utilizing CPU(s) can be computationally more expensive than using the method described above utilizing a quantum computer in combination with the procedure describe above to determine the Cl coefficients.

[0133] In the following the multireference dynamic correlation treatment, in which the CAS wavefunction is used as a fixed starting point is described. The CAS wavefunction is used as a reference state that is not varied anymore. The following methods account for dynamic correlation of the “inactive space” and among “active space” and “inactive space” in a further computational step utilizing the principles of the invention with respect to the distribution of the sub-problems to be computed. In the following, four exemplary embodiments are presented. The first embodiment leverages a NISQ-QPU that is not entangled with the FT- QPU or higher-fidelity NISQ-QPU used for the CAS wavefunction calculation and thus this is compatible with the unentangled quantum computing system described with respect to Fig. 9. A further embodiment leverages a NISQ-QPU that is entangled with the FT-QPU or higher-fidelity NISQ-QPU used for the CAS wavefunction calculation and thus this is compatible with the entangled quantum computing system described with respect to Fig. 9. The remaining two embodiments are focused on post-processing on a CPU. Fig. 14 shows an overview of the embodiments that will be discussed in the following. The following sections treat the CAS method as separating between predominantly static correlation within the “active space” on the one hand, and dynamic correlation among electrons in all electronic orbitals, both active and inactive, on the other hand. However, the methods described below can also be applied in an embedding context. For example, the CASCI method can be used to calculate both the static and dynamic correlation of electrons in electronic orbitals located on a fragment of a molecular system or material, and a dynamic correlation method can be used to calculate the interaction between the CASCI fragment and the surrounding electronic orbitals. A fragment refers to a spatially connected part of the molecular system comprising the respective electron orbitals associated with this part of the molecular system and is independent of the active and inactive space subproblems. For example, a fragment can be associated with spatially connected atoms of the molecular system and comprise the electron orbitals of these atoms. A separation between electronic orbitals inside and outside of the CASCI fragment can be achieved through orbital localization methods, which can be aided by other techniques such as natural orbitals, natural orbitals for orbital pairs or subsets, orbital-specific virtual orbitals, and domains of projected atomic orbitals or other functions. A molecule or material can also contain multiple fragments that are treated at CASCI level with approximate interaction, for example at mean-field level, between the fragments. Afterwards, a correlated interaction is calculated using the dynamic correlation methods described below.

[0134] In the following a method referring to a NISQ unitary dynamic correlation (NISQ-UDC) approach leveraging the unentangled quantum computing system is described. For relevant molecules, such as transition metal compounds, the number of electronic orbitals that are assigned to the “active space” defining an “active space” sub-problem is typically significantly smaller than the number of electronic orbitals remaining in the “inactive space” defining the “inactive space” sub-problem. As discussed above, FT-QPU(s) and / or higher- fidelity NISQ-QPU(s) with only a small number of available qubits can be utilized together with larger NISQ-QPU(s), i.e. NISQ-QPU(s) with a larger number of qubits but a lower fidelity. Such a hardware setting is ideally suited to combine a CAS calculation, wherein a FT-QPU or a higher-fidelity NISQ-QPU is used to solve the “active space” sub-problem at a high level of accuracy, with an approximate dynamic correlation method for the “inactive space” sub-problem and for accounting for dynamic correlation within the “inactive space” sub-problem and between “active space” and “inactive space” sub-problems utilizing a NISQ-QPU. Based on the preceding CAS calculation determining the CAS wavefunction as described above, in a further computational step a unitary dynamic correlation (UDC) ansatz can be applied in combination with the variational quantum eigensolver (VQE) or the variational Hamiltonian ansatz (VHA) to approximately account for dynamic correlation utilizing a NISQ-QPU. The “inactive space” part can further be solved utilizing known methods. In many cases the solution of the “inactive space” part can be trivial. Moreover, the solution can also be generated already taking a solution of the “active space” part into account, for example, adapting the electron orbitals of the “inactive space” part based on a solution of the “active space” part. This solution of the “inactive space” part can then be integrated into the CAS wavefunction as part of the interaction representation.

[0135] In more detail the method can be represented by a parameterized unitary operator U(9 that acts both on the “inactive space” and “active space” parts of the previously determined CAS wavefunction: with 9 being a set of parameters that are optimized within a variational procedure, e.g. VQE or VHA. One specific example is that U(0) = eT~T^ is a unitary coupled cluster (UCC) operator. The method is referred to as an “internally contracted multi reference unitary coupled cluster method” (ic-MR-UCC). For an introductory overview on MR-CC methods, see the article “Perspective: Multireference coupled cluster theories of dynamical electron correlation, F. A. Evangelista, The Journal of Chemical Physics: Vol 149, No 3” incorporated herein by reference.

[0136] The cluster operator citations from arbitrary electronic orbitals p, r to arbitrary electronic orbitals q, s to account for dynamic correlation. Usually, the expansion only includes the singles and doubles term as explicitly shown above leading to the UCCSD operator and thus the ic-MR-UCCSD method. However, the method can also be systematically extended to higher-order excitations like triple, quadruple, etc. excitations leading to ic-MR-UCCSDT, ic-MR-UCCSDTQ, etc. variants.qandpare electronic creation and annihilation operators, respectively. Restrictions can be applied to the indices pq in the singles term, the indices pqrs in the doubles term, or the corresponding indices in higher-order terms. Examples for possible restrictions on the indices include the following. Whenever both indices p, q in tpq q prefer to inactive electronic orbitals, the term can be restricted to include only contributions withoccupied inactive electronic orbitals i and unoccupied inactive electronic orbitals a. Whenever all indices p, q, r, s in pqrs tpq qsr prefer to inactive electronic orbitals, the term can be restricted to include only contributions tjab ttajb^b^j^iwi^oc" cupied inactive electronic orbitals i, j and unoccupied inactive electronic orbitals a, b. No terms 'Ztuttududtortuvw ttuvw^u^w^v^w with all indices t, u, v, w referring exclusively to active electronic orbitals can be included. Terms including at least one creation operator and at least one annihilation operator in the occupied inactive space can be excluded. Terms with at least one creation and at least one annihilation operator in the unoccupied inactive space can be excluded. Examples for such excluded terms include inactive occupied electronic orbitals i, j, inactive unoccupied electronic orbitals a, b and arbitrary electronic orbitals p, q.

[0137] The method can then be performed as described in the following. In a first step the CAS method and process described above using a FT-QPU or higher-fidelity NISQ-QPU and the quantum phase estimation (QPE) algorithm is applied to determine the exact or a nearexact solution of the “active space” sub-problem in form of the CAS wavefunction. In an embodiment, only the “active space” sub-problem is solved on the FT-QPU or higher-fidelity NISQ-QPU which means that typically as many logical qubits as there are electronic orbitals in the “active space” are used to represent the solution of the “active space” sub-problem. Depending on the algorithm additional qubits can be required to implement the algorithm, e.g., ancilla qubits for the QPE algorithm. Alternatively a NISQ-QPU (and, e.g., the variational quantum eigensolver, VQE) to solve the “active space” sub-problem can be used in this step. Alternatively, it is also possible to run the CAS (or alternatively RAS or GAS) method entirely on a CPU, and proceed with the step described below, e.g. the dynamic correlation treatment on a NISQ-QPU. This way, for calculating the CAS wavefunction a quantum computing system could be used that comprises quantum computers with the same or a low fidelity.

[0138] In the second step the Cl coefficients can be determined as described above. The CAS wavefunction or the most relevant Cl coefficients that were read out, can be stored on the CPU and kept fixed throughout all of the following steps. In the third step the dynamic correlation of the “inactive space” and among “active space” and “inactive space” can be taken into account utilizing a NISQ-QPU by executing again parts of the process depicted in Fig. 10 as described in the following. First a representation of the CAS wavefunction is prepared on a quantum computer with a suitable fidelity, for example, a NISQ-QPU. For example, the entire CAS wavefunction, and not only the “active space” solution, can be reconstructed according to Eq. (1), for example, using the most relevant Cl coefficients as described above, on the NISQ-QPU via where the summation over is restricted to the n most important basis states, e.g. Slater determinants, as described above, e.g. the ones where is largerthan a predetermined threshold. This typically requires as many qubits as there are electronic orbitals in the molecule, for example, electronic orbitals in the “active space” plus electronic orbitals in the “inactive space”. Depending on the algorithm used to prepare the CAS wavefunction on the respective quantum computer additional quits can be required for the preparation step, e.g., ancilla qubits to prepare sums of states. In a next step a UDC operator U(&) can be applied to generate a trial state according to Eq. (5) with 0 being a set of variational parameters that will be adjusted on the CPU later within a VQE or VHA procedure. One example of such a unitary operator is the unitary coupled cluster (UCC) operator described above, where tpqand tpqrsare the sets of variational parameters. In the first iteration step, parameters can be either chosen randomly or by means of other computationally inexpensive traditional methods, e.g. based on perturbation theory, that are run on a CPU in advance. Next, the electronic creation and annihilation operators can be encoded in qubit operators, namely Pauli operators, e.g. by means of the Jordan-Wigner or Bravyi-Kitaev transformation. The encoded UDC operator, U(&), is then translated into quantum gates that are applied to the previously reconstructed CAS wavefunction on the quantum computer to prepare a parameterized trial state according to Eq. (5). Strategies like the FSIM network algorithm using low-rank decomposition orthe CZ algorithm can be used. The one- and two-particle reduced density matrices can then be measured on the quantum computer. The measured quantities allow for the evaluation of the energy of the trial state I^NISQ-UDC ) with respect to the molecular Hamiltonian H on a classical computer, e.g. a CPU, according to:

[0139] If the energy and the variational parameters are not converged yet, the iteration continues by updating the set of variational parameters such that the energy E is minimized in a variational procedure on the classical computer. If the energy and the variational parameters are converged, the iteration proceeds with the final step. The final energy and further properties of the solution can then be provided to the user. Also, further post-processing on a classical computer, e.g. a CPU, can be performed including application to real-world problems related to chemical products like molecules, solids, materials, etc.

[0140] As mentioned above multireference dynamic correlation approaches have an advantage over the CAS method as they account for dynamic correlation across the “active space” and “inactive space”. In particular, the ic-MR-UCC method provides a more systematic approach to compute observable quantities compared to MC-PDFT, as it does not rely on the choice of a pair density functional as MC-PDFT. Additionally, the accuracy can be increased systematically by going from ic-MR-UCCSD to ic-MR-UCCSDT to ic-MR- UCCSDTQ etc. Traditional internally contracted multireference methods for CPUs, such as second-order complete active space perturbation theory (CASPT2), NEVPT2, ic-MRCI or ic-MRCC, require computation of three-particle, four-particle or even higher-order reduced density matrices. Needing to compute these reduced density matrices in calculations on CPUs, or to measure these reduced density matrices in calculations on QPUs, renders such methods extremely expensive or entirely unfeasible when used in combination with large “active spaces”. As there is no need to determine reduced density matrices of high order for the above described ic-MR-UCC method, it permits larger electronic-structure problems, e.g. larger molecules, to be simulated with higher accuracy than previously possible. Another specific advantage of ic-MR-UCC over perturbative methods such as CASPT2 and NEVPT2 is its increased accuracy, analogous to the well-documented advantages of CCSD and UCCSD over second-order M0ller-Plesset perturbation theory (MP2) and to the advantages of traditional multireference coupled cluster over perturbative multireference methods. Unlike in MRCI, where the accuracy of results deteriorates with an increasing number of atoms due to lack of size consistency and size extensivity, the quality of ic-MR-UCC results is retained.

[0141] In the following a FT-NISQ unitary dynamic correlation (FT-NISQ-UDC) method utilizing an entangled quantum computing system is described, wherein the FT-QPU can also be replaced by a higher-fidelity NISQ-QPU. As an alternative to the exemplary method described above it is also possible to utilize an entangled quantum computing system to account for dynamic correlation. As a result of a successfully converged CAS calculation using a respective quantum computing system as described above a highly accurate solution, for example, highly accurate eigenfunctions, | ^active )FT-QPU. of the “active space” sub-problem, for instance, “active space” Hamiltonian, is available on the respective quantum computer, preferably, a FT-QPU or higher-fidelity NISQ-QPU. Furthermore, all resulting electronic orbitals can be stored on a classical computer, e.g. a CPU. The latter information can be used to represent the electronic orbitals defining the “inactive space” sub-problem on a NISQ-QPU with a respective fidelity using a state that can easily be prepared on the NISQ- QPU, |‘I’inactive>NisQ- QPU.e9- by initializing a qubit in state 1 if the inactive electronic orbital is occupied and in state 0 if the inactive electronic orbital is unoccupied according to the Jordan-Wigner transformation. It is noted that, preferably, the FT-, higher-fidelity NISQ- and / or the NISQ-QPU are located on the same chip to facilitate entanglement in a later step. However, using a quantum bus such as optical fibers and photons, spatially separated QPUs can be entangled as well.

[0142] To approximately account for dynamic correlation within the “inactive space” sub-problem and between “active space” and “inactive space” sub-problems by using an entangled quantum computing system a parameterized unitary dynamic correlation (UDC) operator, t / ent(0). that entangles quantum computers with different fidelities can be applied according to: with 0 being a set of variational parameters within a variational procedure, e.g. within the variational quantum eigensolver (VQE) or a variational Hamiltonian ansatz (VHA). One specific example is that Uent(0) = eT~Ttis a unitary coupled cluster (UCC) operator. Such a choice would also result in an ic-MR-UCC method, wherein the difference to the analogous method described above with respect to the unentangles quantum computing system is that here the entangled quantum computing system is used. Further details on the operator, possible truncations and possible restrictions on indices are already described above and can also be applied in this embodiment. By describing electronic excitations / transitions between electronic orbitals, the coupled cluster operator T accounts for dynamic correlation. Those electronic excitations / transitions that involve at least one active and at least one inactive electronic orbital will create entanglement between the respective quantum computers, e.g. a term with p referring to an active electronic orbital that has been mapped onto a FT-QPU or a higher-fidelity NISQ-QPU and q, r, s referring to occupied or unoccupied inactive electronic orbitals that have been mapped onto the NISQ-QPU will create entanglement between qubits located on two different QPUs.

[0143] The method can comprise the following steps. In a first step the CAS method and process as described above is performed using, for example, a FT-QPU or higher-fidelity NISQ- QPU and, e.g., a quantum phase estimation algorithm (QPE) to determine the exact or a near-exact solution of the “active space” sub-problem. Note, that only the “active space” sub-problem is solved on the FT-QPU or higher-fidelity NISQ-QPU which means that typically as many logical qubits as there are electronic orbitals in the “active space” sub-problem are used to represent the solution of the “active space” sub-problem. Depending on the algorithm additional qubits can be required to implement the algorithm, e.g., ancilla qubits forthe QPE algorithm. Afterthe CAS procedure is finished the electronic orbitals can be stored on a CPU. In the next step the dynamic correlation of the “inactive space” and between “active space” and “inactive space” is accounted for by using an entangled quantum computing system in combination with a variational approach similar to the general procedure exemplarily depicted in Fig. 11. For example, the solution of the “active space” sub-problem can be prepared on a higher-fidelity NISQ-QPU or FT-QPU resulting in ltt’active>FT-QPu . e g- by applying the quantum phase estimation (QPE) algorithm using the electronic orbitals that are stored on the CPU. If a CASSCF has been used, since the optimized electronic orbitals are stored, it is not necessary to run the entire iterative CASSCF procedure again but a preparation of the solution of the “active space” sub-problem is sufficient, i.e. a single CASCI calculation using the optimized and stored electronic orbitals. Simultaneously, the electronic orbitals that are stored on the CPU can be utilized to prepare a wavefunction describing the “inactive space” sub-problem on a NISQ-QPU resulting in I 'i’inactive^NisQ-QPu - This can require as many qubits on the NISQ-QPU as there are electronic orbitals in the “inactive space”. In summary, the entire CAS wavefunction is prepared on the QPUs of the quantum computing system. A UDC operator Uent(0) that entangles the respective quantum computers on which the sub-problems, here wavefunctions of the “active space” and “inactive space”, are prepared, is applied to generate a trial state according to Eq. (8) with 0 being a set of variational parameters that is adjusted on the CPU within, e.g., a VQE procedure. One example of such a UDC operator is a unitary coupled cluster (UCC) operator described above. In the first iteration step, the variational parameters can be either chosen randomly or by means of other computationally inexpensive traditional methods, e.g. based on perturbation theory, that are run on a CPU in advance. The one- and two-particle reduced density matrices can then be measured on the QPUs as result of the computation and the energy of the trial state |T'FT-NISQ_UDC) with respect to the molecular Hamiltonian can be evaluated on the CPU in analogy to Eq. (7). If the energy and the variational parameters are not converged yet, the method performes a respective iteration of the above steps after updating the set of variational parameters on the CPU such that the energy is minimized, e.g. in a variational procedure. If the energy and the variational parameters are converged, the method can proceed with the final step. Preferably the electronic orbitals that are stored on the CPU as a result of the CAS calculation are kept fixed during the procedure, i.e. the CAS wavefunction that serves as a reference for the dynamic correlation treatment is not varied any more. However, it is also possible to adjust the electronic orbitals within optional self-consistent field (SCF) steps. The final energy and further properties of the solution can then be provided to the user. Also, further post-processing on the CPU can be performed including application to real-world problems related to chemical products like molecules, solids, materials, etc. The multireference dynamic correlation approach described above exhibits very similar advantages over other methods as described already with respect to the NISQ-UDC method above. Furthermore, in comparison to the NISQ-UDC method that leverages the unentangled quantum computing system there are additional advantages of the FT-NISQ-UDC method. In comparison to the NISQ-UDC method, the FT-NISQ-UDC method described above neither relies on the measurement of Cl coefficients nor on the approximate reconstruction of the CAS wavefunction on a different hardware device. This brings the advantage that the unitary operator accounting for dynamic correlation is applied to the exact CAS wavefunction and not an approximately reconstructed one. Such an approach may help to eliminate problems that can result from an approximate reconstruction of the CAS wavefunction, such as intruder states. Furthermore, whereas in case of the unentangled quantum computing system the entire CAS wavefunction comprising both “active space” and “inactive space” is represented on a NISQ-QPU, in the method described above it is sufficient to only represent the wavefunction of the “inactive space” on the NISQ-QPU resulting in a reduced requirement with respect to the number of qubits on the NISQ-QPU.

[0144] In the following some preferred applications of the above described embodiments are provided. As a result of the methods described above using a quantum computer, total energies and properties of the electronic structure of a chemical product, e.g., molecular and, optionally, periodic materials can be calculated and provided to a user. Using those computational results it is possible to predict relevant quantities for real-world applications such as technical application properties of chemical products, e.g., molecules. From those, recommendations can be derived to discover new materials and chemicals, improve chemical processes, tailor molecules, solids and materials to desired properties and make research activities more efficient by reducing the number of required expensive lab and production trials.

[0145] An important example for a technical application property that can be determined is the chemical reactivity, wherein the prediction of thermodynamic and kinetic quantities of chemical reactions relies on the calculation of free enthalpies such as reaction free enthalpies and free enthalpies of activation, respectively. For example, the reaction free enthalpy is indicative of whether a chemical reaction can occur in principle or not and the free enthalpy of activation is indicative of the rate, i.e. speed, of a chemical reaction. Among the most important and difficult to calculate contributions to the respectively required free enthalpies are energy differences between reactants, products and the transition state, e.g. the highest point in energy along the reaction pathway from reactants to products. For example, the reaction energy, which is one of the main contributions to chemical thermodynamics, is obtained by subtracting the sum of the total energies of all reactants from the sum of the total energies of all products. In analogy, the activation energy, which is one of the main contributions to chemical kinetics, is obtained by subtracting the sum of the total energies of all educts from the total energy of the transition state. The total energies of respective products, reactants and transition states can advantageously be calculated utilizing the methods of the invention as described above. The calculation of the free enthalpies of all potential reaction pathways, i.e., taking into account all possible transition states, intermediates and products, ultimately allows to predict the outcome if molecules are brought to reaction by identifying the one or more energetically most favorable reaction pathways. A highly accurate calculation of all species within a chemical reactive network, i.e. reactants, products, transition states and intermediates, and their total energies is indispensable for a reliable prediction of thermodynamic and kinetic quantities of chemical reactions, such as reaction free enthalpies and free enthalpies of activation; this highly accurate calculation is one requirement for the computational design of new chemical products and improvement of industrial chemical production processes as well as for other technical application areas like the understanding and based on that the suppression of the degradation of chemical products, the prediction of the microstructure of polymeric chemical products and thus the computational fine-tuning of technical application properties of chemical products.

[0146] Further examples for technical application properties that can be determined are spectra and spectroscopic properties, e.g., referring to electronically excited states of electronic structure problems. For example, calculating the differences in total energies between different electronic states, e.g., between the ground state and one or more electronically excited states, e.g., with particular spin multiplicities or electronic configurations that are different from the ground state, allow the prediction of spectra and spectroscopic properties of chemical products, which are relevant to understand the influence of radiation on a chemical product, e.g. in photovoltaics and photochemical synthesis as well as degradation processes. The calculation of electronically excited states is furthermore a prerequisite for the computational design of, e.g., dyes or photoinitiators as well as more complex assemblies like organic electronic materials.

[0147] Another example for technical application properties that can be determined with the above described methods are molecular properties beyond energies, for example, electrostatic multipole moments, hyperfine couplings, electrical fields and their gradients relevant for Mbssbauer spectroscopy, diamagnetic shieldings relevant for nuclear magnetic resonance (NMR) spectroscopy, etc. Generally, the calculation of physico-chemical properties is important for molecular structure and property elucidation. Moreover, the above described invention is particularly advantageous for the solving of problems in molecular systems that exhibit strong static correlation. Nevertheless, the invention is not limited to solving statically correlated electronic structure problems and can be applied to all electronic structure problems. Statically correlated molecular systems usually comprise parts that exhibit a complex electronic structure which makes high-accuracy calculations often prohibitively expensive on classical computers already for small problem sizes. Examples for molecular systems that exhibit strong static correlation commonly include metal-organic compounds that contain transition metals such as iron, nickel, rhodium, palladium, etc., or contain lanthanides and actinides. Thus, the examples for molecular systems that exhibit strong static correlation include statically correlated homonuclear or oligonuclear center in weakly correlated environments, wherein the term “nuclear” refers to transition metal, lanthanide or actinide ions or atoms therein plus optionally single ions or single atoms of the ligands. Such transition metal compounds are of key importance for the design of new catalysts, chelating agents, homogenously catalyzed fine chemicals, enzymes, etc. Furthermore, static correlation also occurs outside of transition metal chemistry, e.g., during chemical reactions of organic molecules in bond-breaking and / or -forming situations, e.g. in certain transition states, and also occurs in certain common main-group- element molecules like ozone.

[0148] Specific industry-relevant chemical products to which the invention can be applied are, for example, catalysts. Catalysts play a crucial role in enabling or accelerating chemical reactions under mild conditions, e.g., mild temperatures and mild pressures, by interacting with the transition state and lowering its energy thus reducing the activation energy and increasing the rate of the chemical reaction. Today, catalysts are often used that contain 4d or 5d transition metals like rhodium and palladium, which are highly expensive, like the rhodium- based Wilkinson catalyst for hydroformylation. The goal of many technical objectives is to substitute those catalysts by catalysts that are cheaperto produce containing, e.g., cheaper 3d transition metals cobalt or iron. The above described invention can be particularly advantageous in the area of homogenous catalysis, e.g., for calculating oxidation reactions, reduction reactions, hydrogenations, carbonylations, etc. in large-scale chemical production processes such as the chemical production process of polymers but also in the synthesis of homogenously catalyzed basic chemicals. For example, utilizing the above described invention activation, energies for a predetermined catalytic cycle including undesired side reactions can be calculated for a set of predetermined catalysts before their synthesis in the laboratory. As described above, those activation energies contribute to the calculation of chemical kinetics, in particular chemical reaction rates. In particular, determined chemical reaction rates of the potential catalyst can be compared with target chemical reaction rates and, based on the comparison, either i) a respective potential catalyst can be determined as target catalyst, or ii) a new potential catalyst can be provided and the determination of the reaction rates according to the invention can be repeated. Thus, only those catalysts for which the chemical reaction rate is larger than a predetermined threshold and for which no major undesired side reactions are predicted are finally selected for synthesis and further investigation in the laboratory, for instance, by providing control data that causes a respective synthesis of the respective catalyst.

[0149] Other specific industry-relevant chemical products to which the invention can be applied are chelating agents. The development of chelating agents, which are tailored towards certain metal ions, can be enhanced by the calculation of complex formation constants. A complex formation constant is a thermodynamic quantity that indicates how thermodynamically stable the resulting complex of the metal ion and chelating agent is. A reliable computational prediction typically requires a highly accurate calculation of total energies of the respective ions and molecules which is particularly challenging for the case of transitionmetal ions. Additionally, a reliable prediction of the selectivity of chelating agents is challenging, since both its experimental determination and its computational prediction are difficult. Selectivity is a means that describes how strongly a particular chelating agent preferably binds to a specific metal ion compared to other metal ions, which can be derived from respective complex formation constants. A very well-known example for a chelating agent is ethylenediaminetetraacetic acid (EDTA) which can be used, e.g., for the solubilisation of a Fe3+ion. The goal of many technical objectives is to design new chelating agents with a predetermined selectivity that exhibit favorable properties like biodegradability or are less hazardous to water organisms. Chelating agents are used in a large variety of technical applications, e.g., to suppress the undesired influence of metal ions in washing and cleaning processes. Furthermore, chelating agents are also used in mining for selective extraction of metals. For example, using the above described invention reaction energies for a predetermined chelating agent can be calculated that ultimately contribute to the calculation of complex formation constants and thus the selectivity with respect to different transition metal ions. These calculations are done, e.g., for a set of predetermined chelating agents before their synthesis in the laboratory. In particular, determined selectivities of the potential chelating agent can be compared with target selectivities and, based on the comparison, either i) a respective potential chelating agent can be determined as target chelating agent, or ii) a new potential chelating agent can be provided and the determination of the selectivities according to the invention can be repeated. Thus, only those chelating agents that fulfill certain predetermined criteria / technical application properties, e.g., with respect to selectivity, are finally selected for synthesis and further investigation in the laboratory, for instance, by providing control data that causes a respective synthesis of the respective chelating agent. Further specific industry-relevant chemical products to which the invention can be applied include (bio-)macromolecular systems as well as large biomolecules with active centers, e.g., enzymes like peptidases and esterases.

[0150] More detailed examples how the above described properties can be calculated are provided in the following. One example refers to the application to spectroscopy shown schematically and exemplarily in Fig. 15. Spectroscopy is a non-invasive way of experimentally studying a system, either in comparison with other systems or in different environments and / or under different physico-chemical conditions to elucidate the system’s molecular structure, properties and chemical reactivity. Different experimental spectroscopic techniques spanning different ranges of the electromagnetic spectrum and their combination can lead to a more comprehensive picture of investigated systems such as molecules and materials. However, the growing sophistication of these experimental techniques makes it increasingly complex to interpret spectroscopic results without the help of computational chemistry. For example, oftentimes the experimental results do not allow to directly deduce the desired information, e.g., in some cases from a measured spectrum the molecular structure cannot be directly deduced. However, the experimental results can be compared to computational results to obtain the desired information, e.g. the measured spectrum can be compared to a variety of computed spectra assuming different molecular structures to determine the molecular structure for which the measured and computed spectra agree best.

[0151] In the following a determining of technical application properties related to spectroscopy from the above described solution of the electronic structure problem is described. In UV / Vis spectroscopy, excited-state properties obtained as solution to the electronic structure problem can be used. In particular, energies of electronically excited states, i.e. energies of states with predefined spin multiplicities or electronic orbital configurations that are different from the ground state, and the energy of the respective electronic ground state, for instance, both obtained by utilizing a gate-based quantum computer, can be used. Additionally and optionally, vibrational contributions, such as Franck-Condon profiles obtained by utilizing, e.g., a boson sampling photonic quantum computer as well as other contributions, such as line widths obtained via approximate calculations on a classical computer or as input from a user, can be used. As output, technical application properties related to spectroscopy can be calculated that can be directly related to experimentally accessible properties and thus support solving the real-world chemistry or material problem. For instance, electronic absorption spectra can be computed, that directly refer to, e.g., experimentally accessible UV / Vis spectra.

[0152] UV / Vis spectroscopy uses light in the visible and adjacent ranges, wherein the absorption or reflectance of the light in the visible range by a chemical product or material directly affects the perceived color of that chemical product or material. Thus, computed electronic absorption spectra can support, for instance, the design of new dyes. Another example refers to photoinitiators, which are molecules that create reactive species, such as free radicals, when exposed to radiation in the UV or visible range. The reactive species can then initiate, for instance, polymerization to create a polymer. In this context, the difference in computed energies between the energetically lowest electronically excited state and the electronic ground state directly relates to the required wavelength of a laser irradiating the photoinitiator to initiate this polymerization process. Further examples of real-world chemistry and material problems refer to photovoltaics and photochemical synthesis.

[0153] Furthermore, electronic emission spectra can be computed, that directly refer to, e.g., experimentally accessible fluorescence spectra. Electronic emission spectra are complementary to electronic absorption spectra, in that the former deal with transitions of electrons from an excited state to the ground state induced by emission of photons while the latter deals with transitions of electrons from the ground state to excited states induced by absorption of photons. For example, as an important technical application property, the difference in computed energies between an electronically excited state and the electronic ground state of a molecule or material is directly related to the color of the emitted light. For example, using this workflow, the color of the emitted light of a new potential organic light emitting diode material can be computed before actually synthesizing the material.

[0154] In infrared (IR) spectroscopy, the derivatives of the total ground state energy w.r.t to nuclear positions obtained as described above can be used. In particular, the second derivates are used, which can be either calculated entirely analytically, entirely numerically, or by using the analytically calculated first derivatives to calculate the second derivates numerically in a subsequent step. The calculated spectra, which are also referred to as vibrational spectra within the rigid rotor / harmonic oscillator approximation as they refer to molecular vibrations, can be directly related to experimentally accessible IR spectra. IR spectroscopy can be used to characterize new chemical and / or materials or to identify and verify known and unknown samples. Furthermore, vibronic spectra can be computed, that take into account simultaneous changes in electronic and vibrational energy levels of a chemical product or material due to the absorption of emission of a photon of the appropriate energy. Vibronic spectroscopy may provide information, such as bond-length, on electronic excited states of a molecule.

[0155] To facilitate the comparison of the above mentioned computed spectra with respective experimentally determined spectra, the calculated spectra are typically visualized by plotting calculated transition intensities or related quantities, such as absorbance, against the respectively calculated transition energies or related quantities, such as transition wavelengths. Further details on computational spectroscopy prediction in general, are described in the article “Computational molecular spectroscopy”, Vincenzo Barone et al., Nature Reviews Methods Primers 1 , 38 (2021).

[0156] A preferred example for a technical application property that can be determined is the chemical reactivity, wherein the prediction of thermodynamic and kinetic quantities of a single chemical reaction such as

[0157] TS nAA + nBB -> ncC + nvD relies on the calculation of reaction free enthalpies and free enthalpies of activation, respectively. For example, the reaction free enthalpy is indicative of whether a chemical reaction can occur in principle or not and the free enthalpy of activation is indicative of the rate, i.e. speed, of a chemical reaction.

[0158] The reaction free enthalpy, AGr, can be calculated by subtracting the sum of the free enthalpies of all reactants - A and B in the example above - from the sum of the free enthalpies of all products - C and D in the example above according to

[0159] AGr= ncGSoi (C) + ^D^SOI CD)_[^A^soi (^) +nB ^soi (^)] wherein the free enthalpy in solution, Gsol, of each individual chemical species is weighted by its respective stochiometric weight n. Analogously, the free enthalpy of activation, AG*, can be obtained by subtracting the sum of the stochiometrically weighted free enthalpies of all reactants from the free enthalpy of the transition state - TS in the example above - which is the highest point in free enthalpy along the reaction pathway from reactants to products, according to AG* - Gsol(TS) - [nAGsolG4) + nBGsol(B)]

[0160] Since chemical reactions typically happen in solution, e.g. in water, and not in the gas phase, all free enthalpies are preferably calculated considering the influence of a solvent. However, also a calculation omitting the influence of the solvent can be a reasonable approximation for many application cases. The following example takes the solvent into account, but can also be applied to approximations without taking the solvent into account by removing the solvent components accordingly. The free enthalpy in solution, Gsol, of an individual chemical species, e.g. a reactant A or B, a transition state TS of product C or D, can be obtained as the sum of the free enthalpy in the gas phase, Ggas, and the free enthalpy of solvation, AGsolvation, according to csol = cugas i ' ^ Ar^solvation

[0161] It is noted that sometimes the free enthalpy in solution is also named Gibbs free energy in solution. The free enthalpy in the gas phase is obtained at a predetermined temperature T according to

[0162] Ggas— E + ZPE — RT • ln(qtrans' Qrot ' Qvib) wherein E is the electronic energy. The high-level energy can be generated utilizing the method described above, for instance, utilizing a quantum computer. It is noted, that this energy can - depending on the electronic state of interest - both refer to the electronic ground state and an electronically excited state. ZPE is the vibrational zero-point energy,and Qtrans ■ Qrot ■ Qvib are the translational / rotational / vibrational partition functions, respectively. Both the ZPE and gvibcan be calculated from a vibrational spectrum, which can be generated, for instance, using a spectroscopy workflow as described above, in particular, the IR spectroscopy workflow, utilizing a quantum computer as described above. However, these quantities can also be obtained utilizing a classical computer in combination with generally less accurate methods such as density functional theory (DFT). R is the Avogadro gas constant.

[0163] The free enthalpy of solvation, AGsolvation, of a molecule in a predefined solvent can be obtained via solvation models such as the conductor like screening model for real solvents (COSMO-RS) utilizing a classical computer. Within the COSMO-RS approach, in a first step, the electron density, which is indicative of the distribution of the electric charge in the molecule, obtained utilizing a quantum computer as described above can be used to calculate the screening charge density o on the surface of the molecule. In a second step, this information can be used to calculate the chemical potential p of the molecule in the predetermined liquid solvent or mixture. The resulting chemical potential p is then used to calculate the free enthalpy of solvation. Further information on the COSMO-RS approach can be found in the book “COSMO-RS: From Quantum Chemistry to fluid phase thermodynamics and drug design”, Andreas Klamt, Elsevier (2005). It is noted that by including the influence of the solvent on the technical application properties, such as the reaction free enthalpies and free enthalpies of activation, implicitly via the COSMO-RS approach rather than straightforwardly calculating the technical application properties in solution by constructing a supersystem that explicitly includes solvent molecules, the computational cost can be significantly reduced.

[0164] The calculation of the reaction free enthalpies and free enthalpies of activation in solution for all potential reaction pathways, i.e., for multiple chemical reactions taking into account all possible transition states, intermediates and products, ultimately allows to determine the outcome if predefined molecules, i.e., reactants, are mixed in a test tube or vessel by identifying the one or more energetically most favorable reaction pathways. In particular, the highly accurate energy computation for all species of a chemical reactive network, i.e. reactants, products, transition states and intermediates, is indispensable for reliable determinations and thus for the design of new chemical products and materials and the improvement of industrial chemical processes as well as for other technical application areas like the understanding and, based on that the suppression of the degradation of chemical products, the determination of the microstructure of polymeric chemical products and thus the computational fine-tuning of technical application properties of chemical products.

[0165] Specific industry-relevant chemical products to which the chemical reactivity, i.e. reaction rate, workflow can preferably be applied are, for example, catalysts, as shown in Fig. 16. Catalysts play a crucial role in enabling or accelerating chemical reactions under mild conditions, e.g., mild temperatures and mild pressures, by interacting with the transition state and lowering its energy thus reducing the free enthalpy of activation and increasing the rate of the chemical reaction. Today, catalysts are often used that contain 4d or 5d transition metals like rhodium and palladium, which are highly expensive, like the rhodium-based Wilkinson catalyst for hydroformylation. The goal of many technical objectives is to substitute those catalysts by catalysts that are cheaper to produce containing, e.g., cheaper 3d transition metals like cobalt or iron. The chemical reactivity workflow can be particularly advantageous in the area of homogenous catalysis, e.g., for calculating oxidation reactions, reduction reactions, hydrogenations, carbonylations, etc. in large-scale chemical production processes such as the chemical production process of polymers but also in the synthesis of homogenously catalyzed fine chemicals. For example, utilizing the chemical reactivity workflow free enthalpies of activation for a predetermined catalytic cycle including undesired side reactions can be calculated for a set of predetermined catalysts before their synthesis in the laboratory. As described above, those free enthalpies of activation contribute to the calculation of chemical kinetics, in particular chemical reaction rates. In particular, determined chemical reaction rates of the potential catalyst can be compared with target chemical reaction rates and, based on the comparison, either i) a respective potential catalyst can be determined as target catalyst, or ii) a new potential catalyst can be provided and the determination of the reaction rates according to the invention can be repeated. Thus, only those catalysts for which the chemical reaction rate is larger than a predetermined threshold and for which no major undesired side reactions are predicted are finally selected for synthesis and further investigation in the laboratory, for instance, by providing control data that causes a respective synthesis of the respective catalyst. Moreover, a respective reaction process can also be controlled such that the catalyst is utilized in the reaction, for example, a feet to a reactor can be controlled accordingly.

[0166] Other specific industry-relevant chemical products to which the chemical reactivity workflow can be applied are chelating agents as shown in Fig. 17. The development of chelating agents, which are tailored towards certain metal ions, can be enhanced by the calculation of complex formation constants. A complex formation constant is a thermodynamic quantity that indicates how thermodynamically stable the resulting complex of the metal ion and chelating agent is. It can be calculated using the reaction free enthalpy for the reaction of chelating agent and metal ion to form the respective chelate complex in solution. A reliable computational determination is preferably based on an accurate calculation of the reaction free enthalpy which is particularly challenging for reactions of transitional metal ions. Additionally, a reliable determination of the selectivity of chelating agents is challenging, since both its experimental determination and its computational determination are difficult. However, utilizing a quantum computer as described above allows for such accurate solutions. Selectivity is a means that describes how strongly a particular chelating agent preferably binds to a specific metal ion compared to other metal ions, which can be derived from respective complex formation constants. A very well-known example for a chelating agent is ethylenediaminetetraacetic acid (EDTA) which can be used, e.g., for the solubilisation of a Fe3+ion. The goal of many technical objectives is to design new chelating agents with a predetermined selectivity that exhibit favorable properties like biodegradability or are less hazardous to water organisms. Chelating agents are used in a large variety of technical applications, e.g., to suppress the undesired influence of metal ions in washing and cleaning processes. Furthermore, chelating agents are also used in mining for selective extraction of metals. For example, using the chemical reactivity workflow described above reaction free enthalpies for a predetermined chelating agent can be calculated to obtain respective complex formation constants and thus the selectivity with respect to different transition metal ions. These calculations can be done, e.g., for a set of predetermined chelating agents before their synthesis in the laboratory. In particular, determined selectivities of the potential chelating agent can be compared with target selectivities and, based on the comparison, either i) a respective potential chelating agent can be determined as target chelating agent, or ii) a new potential chelating agent can be provided and the determination of the selectivities according to the invention can be repeated. Thus, only those chelating agents that fulfill certain predetermined criteria / technical application properties, e.g., with respect to selectivity, are finally selected for synthesis and further investigation in the laboratory, for instance, by providing control data that causes a respective synthesis of the respective chelating agent. Moreover, a respective reaction process can also be controlled such that the chelating agent is utilized in the reaction, for example, a feed to a reactor can be controlled accordingly.

[0167] Additionally, further technical application properties such as activity coefficients, solubility, partition coefficients and vapor pressure can be calculated using a slightly amended workflow as described above based on the COSMO-RS method. The equilibrium vapor pressure, for example, is also an experimentally accessible quantity and defined as the pressure exerted by a vapor in thermodynamic equilibrium with its condensed phases (liquid or solid) at a given temperature in a closed environment and is an indication of a liquid’s or solid’s evaporation rate. For example, the computation of the vapor pressure is important, e.g., to quantify the volatility of a hazardous or toxic chemical product at a given temperature before or instead of carrying out real-world experiments. In case the potentially hazardous or toxic chemical product is volatile, i.e., has a high vapor pressure, which would result in a high concentration of the chemical product in the breathing zone, additional safety measures can be established. Further information on generally determining the aforementioned technical application properties can be found in the article “Predicting accurate absolute binding energies in aqueous solution: thermodynamic considerations for electronic structure methods”, Jan H. Jensen, Phys. Chem. Chem. Phys. 17, 12441 (2015).

[0168] A further exemplary embodiment refers to quantitative structure-activity relationship (QSAR) or quantitative structure-property relationship (QSPR). QSAR or QSPR models are regression or classification models that relate a set of predictor variables, also called “descriptors”, to one or more output variables, e.g., the potency of a response variable in case of a regression model, in particular a partial least squares regression model, or to a categorical value in case of a classification model. Furthermore, neural networks can also be used as model.

[0169] In the context of this invention the predictor variables can refer to a set of solutions of electronic structure problem of electronic structure systems, e.g., molecules, wherein at least one of the electronic structure properties that is generated based on the solution of the electronic structure problem is calculated utilizing a quantum computer. However, the predictor variables can also refer to a set of technical application properties of real-world chemistry or material problems that are obtained by processing the electronic structure properties as described above. Furthermore, a combination with predictor variables obtained from other sources like chemoinformatic processing of structural information or experimental physico-chemical properties is possible as well. The predictor variables can be electronic, geometric, structural or physico-chemical properties and / or molecular descriptors and thus refer to quantities that are obtainable by one of the above described approaches or similar approaches, whereas the output variables refer to technical application properties of real-world chemistry or material problems, such as a biological activity or a chemical property of a molecule, that are not directly accessible by the above described or similar approaches.

[0170] The QSAR / QSPR model mathematically summarizes a supposed relationship between the selected set of predictor variables and the output variables. After the construction of the model, it is carefully validated with regard to robustness, prediction performance and its applicability domain. After successful validation, the QSAR / QSPR model can then finally be used to determine the output variables, e.g., technical application properties, of new real-world chemistry or material problems for a set of predictor variables that are obtained as described above utilizing a quantum computer. Since the quality of the prediction significantly depends on the accuracy of the provided predictor variables the utilization of a quantum computer is expected to be advantageous. Furthermore, the quality of the prediction also depends on other factors such as the selection of appropriate predictor variables, the used QSAR / QSPR model as well as its validation.

[0171] A specific technical application property that can be calculated using the above described QSAR / QSPR workflow is, for example, the biological activity of a chemical product such as a drug, toxicant or environmental pollutant using respective computed electronic structure properties or above described technical application properties as predictor variables. Biological activity can be expressed quantitatively as the concentration of a chemical product required to give a certain biological response including desirable therapeutic effects and undesirable side effects. For example, the QSAR / QSPR workflow can be used for a computational toxicological assessment of a new chemical product.

[0172] Other variations to the disclosed embodiments can be understood and effected by those skilled in the art in practicing the claimed invention, from a study of the drawings, the disclosure, and the appended claims.

[0173] For the processes and methods disclosed herein, the operations performed in the processes and methods may be implemented in differing order. Furthermore, the outlined operations are only provided as examples, and some of the operations may be optional, combined into fewer steps and operations, supplemented with further operations, or expanded into additional operations without detracting from the essence of the disclosed embodiments.

[0174] In the claims, the word "comprising" does not exclude other elements or steps, and the indefinite article "a" or "an" does not exclude a plurality.

[0175] A single unit or device may fulfill the functions of several items recited in the claims. The mere fact that certain measures are recited in mutually different dependent claims does not indicate that a combination of these measures cannot be used to advantage.

[0176] Procedures like the providing of the problem description, deriving sub-problems, generating control signals, etc. performed by one or several units or devices can be performed by any other number of units or devices. These procedures can be implemented as program code means of a computer program and / or as dedicated hardware.

[0177] A computer program product may be stored / distributed on a suitable medium, such as an optical storage medium or a solid-state medium, supplied together with or as part of other hardware, but may also be distributed in otherforms, such as via the Internet or otherwired or wireless telecommunication systems.

[0178] Any units described herein may be processing units that are part of a classical computing system. Processing units may include a general-purpose processor and may also include a field programmable gate array (FPGA), an application specific integrated circuit (ASIC), or any other specialized circuit. Any memory may be a physical system memory, which may be volatile, non-volatile, or some combination of the two. The term “memory” may include any computer-readable storage media such as a non-volatile mass storage. If the computing system is distributed, the processing and / or memory capability may be distributed as well. The computing system may include multiple structures as “executable components”. The term “executable component” is a structure well understood in the field of computing as being a structure that can be software, hardware, or a combination thereof. For instance, when implemented in software, one of ordinary skill in the art would understand that the structure of an executable component may include software objects, routines, methods, and so forth, that may be executed on the computing system. This may include both an executable component in the heap of a computing system, or on computer- readable storage media. The structure of the executable component may exist on a computer-readable medium such that, when interpreted by one or more processors of a computing system, e.g., by a processor thread, the computing system is caused to perform a function. Such structure may be computer readable directly by the processors, for instance, as is the case if the executable component were binary, or it may be structured to be interpretable and / or compiled, for instance, whether in a single stage or in multiple stages, so as to generate such binary that is directly interpretable by the processors. In other instances, structures may be hard coded or hard wired logic gates, that are implemented exclusively or near-exclusively in hardware, such as within a field programmable gate array (FPGA), an application specific integrated circuit (ASIC), or any other specialized circuit. Accordingly, the term “executable component” is a term for a structure that is well understood by those of ordinary skill in the art of computing, whether implemented in software, hardware, or a combination. Any embodiments herein are described with reference to acts that are performed by one or more processing units of the computing system. If such acts are implemented in software, one or more processors direct the operation of the computing system in response to having executed computer-executable instructions that constitute an executable component. Computing system may also contain communication channels that allow the computing system to communicate with other computing systems over, for example, network. A “network” is defined as one or more data links that enable the transport of electronic data between computing systems and / or modules and / or other electronic devices. When information is transferred or provided over a network or another communications connection, for example, either hardwired, wireless, or a combination of hardwired or wireless, to a computing system, the computing system properly views the connection as a transmission medium. Transmission media can include a network and / or data links which can be used to carry desired program code means in the form of computer-executable instructions or data structures and which can be accessed by a general-purpose or specialpurpose computing system or combinations. While not all computing systems require a user interface, in some embodiments, the computing system includes a user interface system for use in interfacing with a user. User interfaces act as input or output mechanism to users for instance via displays. Those skilled in the art will appreciate that at least parts of the invention may be practiced in network computing environments with many types of computing system configurations, including, personal computers, desktop computers, laptop computers, message processors, hand-held devices, multi-processor systems, microprocessor-based or programmable consumer electronics, network PCs, minicomputers, mainframe computers, mobile telephones, PDAs, pagers, routers, switches, datacenters, wearables, such as glasses, and the like. The invention may also be practiced in distributed system environments where local and remote computing system, which are linked, for example, either by hardwired data links, wireless data links, or by a combination of hardwired and wireless data links, through a network, both perform tasks. In a distributed system environment, program modules may be located in both local and remote memory storage devices.

[0179] Those skilled in the art will also appreciate that at least parts of the invention may be practiced in a cloud computing environment. Cloud computing environments may be distributed, although this is not required. When distributed, cloud computing environments may be distributed internationally within an organization and / or have components possessed across multiple organizations. In this description and the following claims, “cloud computing” is defined as a model for enabling on-demand network access to a shared pool of configurable computing resources, e.g., networks, servers, storage, applications, and services. The definition of “cloud computing” is not limited to any of the other numerous advantages that can be obtained from such a model when deployed. The computing systems of the figures include various components or functional blocks that may implement the various embodiments disclosed herein as explained. The various components or functional blocks may be implemented on a local computing system or may be implemented on a distributed computing system that includes elements resident in the cloud or that implement aspects of cloud computing. The various components or functional blocks may be implemented as software, hardware, or a combination of software and hardware. The computing systems shown in the figures may include more or less than the components illustrated in the figures and some of the components may be combined as circumstances warrant.

[0180] Any reference signs in the claims should not be construed as limiting the scope.

[0181] The invention refers to an apparatus for providing control signals for controlling a quantum computing system. The quantum computing system comprises at least two quantum computers with different fidelities. A problem providing unit provides a problem description indicative of a problem. A sub-problem deriving unit derives sub-problems from the provided problem description based on the fidelities of the at least two quantum computers. A control signal generation unit generates control signals for controlling the quantum computing system to cause a quantum computation of the derived sub-problems on the at least two quantum computers, wherein the sub-problems are distributed to the at least two quantum computers based on the fidelity of the respective at least two quantum computers.

Claims

Claims:1 . An apparatus for providing control signals for controlling a quantum computing system for performing a quantum computation of a problem, in particular, associated with a chemical product, wherein the quantum computing system comprises at least two quantum computers with different fidelities, wherein the apparatus comprises: a problem providing unit for providing a problem description indicative of the problem, a sub-problem deriving unit for deriving sub-problems from the provided problem description based on the fidelities of the at least two quantum computers, and a control signal generation unit for generating control signals for controlling the quantum computing system to cause a quantum computation of the derived sub-problems on the at least two quantum computers, wherein the sub-problems are distributed to the at least two quantum computers based on the fidelity of the respective at least two quantum computers.

2. The apparatus according to claim 1 , wherein at least two of the quantum computers are configured to be entangled and wherein the control signals are generated for further controlling the entanglement between the at least two quantum computers of the quantum computers of the quantum computing system during a parallel quantum computation of the sub-problems to distribute and share information between the at least two quantum computers during the quantum computation, wherein the entanglement of the two quantum computers is defined by at least one quantum element of each of the quantum computers being entangled.

3. The apparatus according to any of claims 1 and 2, wherein the deriving of the subproblems from the problem description based on the fidelities of the at least two quantum computers comprises identifying a sub-problem that requires a higher solution accuracy than another sub-problem, and wherein the control signals are generated such that the sub-problem requiring a higher solution accuracy is computed on a quantum computer with a higher fidelity and wherein the other sub-problem is computed on a quantum computer with a lower fidelity.

4. The apparatus according to any of the preceding claims, wherein the deriving of subproblems is further based on the number of logical quantum elements provided by the respective at least two quantum computers.

5. The apparatus according to any of the preceding claims, wherein the problem is an electronic structure problem associated with a property of a chemical product and the subproblems are derived such that a sub-problem is defined on an active space and another sub-problem is defined on an inactive space, and wherein the sub-problem defined on the active space is derived such that it is computable on a quantum computer with a higher fidelity and the sub-problem defined on the inactive space is computable on a quantum computer with a lower fidelity, wherein the control signals are generated such that the respective sub-problems are calculated on the respective quantum computers.

6. The apparatus according to any of the preceding claims, wherein the quantum computing system comprises at least one fault-tolerant quantum computer and one noisy inter- mediate-scale quantum computer, wherein the fault-tolerant quantum computer has a higher fidelity than the noisy intermediate-scale quantum computer.

7. A quantum computer system comprising: a controlling unit for receiving control signals generated by an apparatus according to claim 1 , and at least two quantum computers with different fidelities, wherein a quantum computation of a sub-problem, in particular, associated with a chemical product, is performed on the at least two quantum computers based on the received control signals based on the different fidelities of the at least two quantum computers.

8. A system according to claim 7, wherein at least two quantum computers of the quantum computers are configured for an entangled quantum computation, wherein the entanglement between the at least two quantum computers is controlled based on the control signals, and wherein the entanglement of the two quantum computers is defined by at least one quantum element of each of the quantum computers being entangled.

9. A system according to any of claims 7 and 8, wherein at least one of the at least two quantum computers is a fault-tolerant quantum computer and at least one of the at least two quantum computers is a noisy intermediate-scale quantum computer.

10. A problem solving apparatus for solving a problem, in particular, associated with a chemical product, wherein the apparatus comprises: a receiving unit for receiving quantum computation results of sub-problems performed by at least two quantum computers of a quantum computing system in accordance with control signals generated by an apparatus according to claim 1 , and a problem solving unit configured for determining a solution of the problem based on the results of the quantum computations of the sub-problems.

11. A system for generating a solution of a problem, in particular, associated with a chemical product, wherein the system comprises: an apparatus according to claim 1 , a quantum computer system according to claim 7, and a problem solving apparatus according to claim 10.

12. A computer implemented method for providing control signals for controlling a quantum computing system for performing a quantum computation of a problem, in particular, associated with a chemical product, wherein the quantum computing system comprises at least two quantum computers with different fidelities, wherein the method comprises: providing a problem description indicative of the problem, deriving sub-problems from the provided problem description based on the fidelities of the at least two quantum computers, and generating control signals for controlling the quantum computing system to cause a quantum computation of the derived sub-problems on the at least two quantum computers, wherein the sub-problems are distributed to the at least two quantum computers based on the fidelity of the respective at least two quantum computers.

13. A problem solving method for solving a problem, in particular, associated with a chemical product, wherein the method comprises:receiving quantum computation results of sub-problems performed by at least two quantum computers of a quantum computing system in accordance with control signals generated by a method according to claim 12, and determining a solution of the problem based on the results of the quantum computa- tions of the sub-problems.

14. A computer program product for providing control signals for controlling a quantum computing system for performing a quantum computation of a problem, wherein the computer program product comprises program code means for causing the apparatus according to any of claims 1 to 6to execute the method according to claim 12.

15. Use of the apparatus according to any of claims 1 to 6 for determining a technical application property of a chemical product based on a solution of an electronic structure problem generated utilizing control signals of the apparatus.

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