EP4673950A1

EP4673950A1 - System and method for modelling molecular interactions

Info

Publication number: EP4673950A1
Application number: EP24707592.2A
Authority: EP
Inventors: Nikolaj Thomas ZINNER; Marco MAJLAND; Patrick ETTENHUBER; Stig Elkjær RASMUSSEN
Original assignee: Kvantify Aps
Current assignee: Kvantify Aps
Priority date: 2023-03-01
Filing date: 2024-03-01
Publication date: 2026-01-07
Also published as: WO2024180257A1

Abstract

Disclosed is a method for calculating an eigenstate of a Hamiltonian, comprising the steps of providing at least one classical computing unit and at least one quantum computing unit comprising a plurality of qubits, providing a plurality of physical properties of at least one molecule, initializing an initial state on the at least one quantum computing unit to model the properties of the provided physical properties of the at least one molecule, selecting a plurality of operators from a pool of operators using the at least one quantum computing unit performing measurement operations on the diagonal of the Hamiltonian to calculate the eigenstate of the Hamiltonian describing the at least one molecule, estimating an energy of the eigenstate of the Hamiltonian using the selected plurality of operators, and repeating the previous steps until a convergence criterion is met.

Description

System and method for modelling molecular interactions

Field of invention

The present disclosure relates to algorithms for finding, e.g., state energies of a molecular system, such as, e.g., application of one or more hybrid quantum-classical algorithms for finding state energies of a molecular system, such as, e.g., algorithms that when executed by a hybrid quantum-classical computing unit is able to predict and/or find state energies of a molecular system.

Background

ADAPT-VQE (Adaptive Variational Quantum Eigensolver) is a research area within the field of quantum computing. It is a hybrid quantum-classical algorithm used to find the energy eigenstates of a molecular system. This is important in the field of computational chemistry, as the ground state energy of a molecule is a key parameter for predicting its chemical properties. The goal of the ADAPT-VQE algorithm is to find the ground state energy and wave function of a molecule using a quantum computer. This is a computationally challenging problem, as the size of the Hilbert space of the molecule grows exponentially with the number of electrons and basis functions.

The ADAPT-VQE algorithm combines a classical optimization algorithm with a variational quantum circuit. The circuit is designed to prepare a quantum state that approximates the ground state of the molecular system, and the classical optimizer is used to adjust the parameters of the circuit to minimize the energy of the system.

One of the key advantages of ADAPT-VQE is its adaptivity. The algorithm can adjust the circuit to focus on specific regions of the energy landscape, which can improve its efficiency and accuracy. This makes ADAPT-VQE a promising approach for simulating molecular systems using quantum computers.

However, ADAPT-VQE can have a too high measurement overhead, which refers to the number of times a quantum circuit must be executed to obtain the necessary information to optimize the circuit parameters. This overhead can be a significant limitation in practical implementations of ADAPT-VQE, as it increases the computational cost and can lead to increased errors due to noise. One of the main sources of measurement overhead in ADAPT-VQE is the need to measure operators that corresponds to the gradient of an augmented energy functional where the current wave function has been extended by an operator from the operator pool. The steep scaling then results from the necessity of measuring this quantity for each operator from the operator pool, which becomes very time consuming for large molecules and basis sets, and may ultimately be impossible on current quantum computers.

Summary

Considering the prior art described above, it is a purpose of the present disclosure to provide improved algorithms for predicting and/or finding state energies of a molecular system; such as, e g, provide application of one or more improved hybrid quantum- classical algorithms for predicting and/or finding state energies of a molecular system; such as, e g, provide improved algorithms that when executed by a hybrid quantum- classical computer system predict and/or find state energies of a molecular system; such as ,e.g., to improve the ADAPT-VQE algorithm. To mitigate the measurement overhead, the presently disclosed algorithms may use Slater determinants populations obtained from diagonal measurements which may be ranked using, for example, perturbation theory to evaluate the importance of operators in the wavefunction. Moreover and/or thus, a difference between the presently disclosed approach and ADAPT-VQE is how the measure of importance for operators in the wavefunction is obtained.

In a first aspect, the disclosure provides a method for calculating and/or estimating an eigenstate of a Hamiltonian, the method comprising the steps of providing at least one classical computing unit and at least one quantum computing unit. Said quantum computing unit may comprise qubits of any platform that allows to control the qubit state with a set of universal gates, such as the publicly available superconducting qubit based quantum computers from Google or IBM. In addition, said quantum computing unit may comprise qubits that are controlled with a non-universal set of gates. Then, a plurality of physical properties of at least one molecule may be provided. An initial state may be initialized on the qubits on the at least one quantum computing unit to model the properties of the provided physical properties of the at least one molecule, a plurality of operators is selected from a pool of operators using the at least one quantum computing unit performing diagonal measurement operations to obtain population of Slater determinants which are used to calculate the eigenstate of the Hamiltonian, using a ranking based on, for example, perturbation theory, describing the at least one molecule. Finally, energy of the eigenstate of the Hamiltonian is estimated using the selected plurality of operators and the previous steps are repeated until a convergence criterion is met.

The step of “providing a plurality of physical properties of said at least one molecule” may be understood by the skilled person within present context as “providing data representing a plurality of physical properties of the at least one molecule to the classical computing unit and/or the quantum computing unit”. As such, the provided plurality of physical properties of the at least one molecule may be understood by the skilled person as data representing, e.g., a mass, type and/or number of atoms of the at least one molecule, spin of, e.g., at least one electron of the at least one molecule.

The step of “initializing an initial state on the at least one quantum computing unit to model the provided physical properties of the at least one molecule” may be understood by the skilled person within present context as “initializing an initial state on the at least one quantum computing unit to model at least part of the provided data representing a plurality of physical properties of the at least one molecule”. If one chooses a Hartree Fock ground state and an adaptive Ansatz (see below for examples of such), the step of “initialising an initial step on the at least one quantum computing unit” may comprise to simulate the initial state (first iteration) of the adaptive Ansatz, which may, e.g., be the ground state of the quantum computing unit, which may be (chosen as) a state of the quantum computing unit in which each qubit of the quantum computing unit is in its ground state.

To initialize an initial state on the quantum computing unit may be understood by a person skilled in the art as populating the states of the qubits of the quantum computing unit according to some predefined scheme at least in part based on at least part of the at least one molecule. As such, the quantum computer unit may comprise a set of qubits, wherein the qubits prior to initialization are all in a ground state. In this context, the step of initializing an initial state on the quantum computing unit may be understood as performing operations on the qubits of the quantum computing unit in order to selectively excite and/or flip (or not excite and/or flip) at least one, or each, of the qubits of the quantum computing unit according to some scheme. The scheme at least in part being based on the at least one molecule and/or at least in part being based on at least one physical parameter of the at least one molecule. Examples of how to initialize an initial state in a quantum computing unit is given in, e.g.,:

Philips, S.G.J., Mqdzik, M.T., Amitonov, S.V. etal. Universal control of a six-qubit quantum processor in silicon. Nature 609, 919-924 (2022).

://doi .org/10.1038/S41586-022-051 17-x and

Abobeih, M.H., Wang, Y., Randall, J. etal. Fault-tolerant operation of a logical qubit in a diamond quantum processor. Nature 606, 884-889 (2022). https://doi.Org/10.1038/S41586-022-04819-6.

The term “perform diagonal measurement operations on the eigenstate” may be understood by the skilled person in present context as “performing diagonal measurement operations on the diagonal eigenvectors of the current eigenstate”. This may be done, for instance, with the eigenstate of the at least one molecule being computed based at least in part on a reference state, for instance the Hartree Fock ground state, and based at least in part on performing operation iterations on the Hartree Fock ground state so as to simulate a (desired) state of the at least one molecule. Moreover, the eigenstate of the at least one molecule may be simulated and/or approximated by building an adaptive Ansatz, the adaptive Ansatz being computed at least in part by successively applying parameterized unitary operations to the reference state, see the Detailed description for an example of how such an adaptive Ansatz may be computed.

Accordingly, the step d. of “selecting a plurality of operators from a pool of operators using the at least one quantum computing unit performing diagonal measurement operations on the eigenstate to calculate the eigenstate of the Hamiltonian describing the at least one molecule” may be understood as:

- evaluating an importance value for at least one operator of the pool of operators of the Hamiltonian, such as, e.g., one or more, such as all, single and double excitation operators in the operator pool.

This step may also be performed on a classical computing unit. The step e. of estimating an eigenvalue, e.g., energy, of the eigenstate of the Hamiltonian using the at least one classical computing unit and the at least one quantum computing unit applying the selected plurality of operators may, in other words, be understood as one or more of the following steps:

- operating the operator(s) having the largest importance value to operate on the current eigen state of the quantum computing unit, to put the quantum computing unit in the next eigen state (next iteration of the adaptive Ansatz),

- perform a readout of the quantum computing unit, which may be done by, e.g., performing, e.g., parallel measurement operations on each of the qubits of the quantum computing unit,

- determine the population of the next eigen state, e.g., the coefficients of the, e.g., diagonal basis vectors, such as, e.g., Slater determinants in the next eigen state (next iteration of the adaptive Ansatz) at least in part on the basis of the readout of the quantum computing unit,

- generating population data representing at least in part the population of the next eigen state,

- the classical computing unit receiving the population data, and the classical computing unit determining an energy of the next eigen state at least in part on the basis of the population data.

Operating the operator(s) with the largest importance value to operate on the (initial) state of the quantum computing unit may be done, e.g., by performing a series of QUBIT manipulations such as, e.g., gate sweeping, spin flipping, optical excitation, wherein the specific QUBIT manipulations corresponding to operating the operator(s) may generally be hardware specific as understood by the person skilled in the art.

In above terminology, in the next iteration of (repeating) the steps of “selecting a plurality of operators from a pool of operators using the at least one quantum computing unit to perform diagonal measurement operations on the eigenstate to calculate the eigenstate of the Hamiltonian describing the at least one molecule”, and “estimating an energy of the eigenstate of the Hamiltonian using the at least one classical computing unit and the at least one quantum computing unit applying the selected plurality of operators”, the quantum computing unit will already be in the next eigen state, and the operator(s) with the largest importance value will thus operate on the next eigen state to put the quantum computing unit in the next next eigen state.

In some embodiments, the step of "the classical computing unit receiving the population data” may be omitted and the step of “determining an energy of the next eigen state at least in part on the basis of the population data” is performed at least in part by the quantum computing unit.

In some embodiments, the step of “estimating an energy of the eigenstate of the Hamiltonian using the at least one classical computing unit and the at least one quantum computing unit applying the selected plurality of operators” or simply “estimating an energy of the eigenstate of the Hamiltonian using the selected plurality of operators” is optional. In this respect, the inventors have found that it may be possible to define and/or use a different set of operators for computing the adaptive Ansatz not requiring estimation of an energy of the calculated and/or estimated eigenstate of the Hamiltonian describing the at least one molecule.

In some embodiments the convergence criterion in the step of “repeating steps d - e until a convergence criterion is met” is that a given repetition of steps d-e results in a change in energy obtained in step e from one iteration to the next below some predefined threshold. This may be particularly advantageous if it is desired to minimize the number of iterations, and thus computations on the quantum and/or classical computing unit. In some embodiments, the convergence criterion is that with repeated iterations of steps d-e, the change in energy obtained in step e. converges to zero. The result is a calculated and/or estimated eigenstate of the Hamiltonian describing the at least one molecule.

In summary, an eigenstate of a Hamiltonian describing at least one molecule may be calculated and/or estimated by performing the steps of claim 1 .

In a second aspect, the disclosure provides a computer program having instructions which, when executed by a computing device or computing system, cause the computing device or computing system to carry out the method of calculating an eigenstate of a Hamiltonian describing at least one molecule. In a third aspect, the disclosure provides use of one or more hybrid quantum-classical algorithms for finding state energies of a molecular system, such as, e g, one or more of the steps of the first aspect that when executed by a hybrid quantum-classical computing unit is able to predict and/or find state energies of a molecular system.

Relevant pools of operators might be obtained from publicly available libraries of operators for performing quantum molecular simulations, such as Qiskitm PySCF or OpenFermion. Pools of operators might also be produced accordingly to the particular characteristics of the molecule being simulated. The pool of operators may, e.g., comprise single and/or double and/or triple excitation operators.

Thus, it is possible to calculate eigenstates of Hamiltonians with an algorithm that requires only operators performing operations such as sampling population of Slater determinants of the wavefunction rather than expectation values of gradients. Sampling Slater determinants requires only the measurement of a single operator rather than measuring a quantity of operators scaling to the forth power required in the state of the art ADAPT-VQE algorithm. Hence, the number of operations to calculate the importance measures for operators in the pool of operators may be reduced, improving the performance of the disclosed algorithm over the state of the art algorithms.

One example of selecting a plurality of operators from a pool of operators is disclosed below. A way of reducing the number of operators by selecting a plurality of operators from a pool of operators is by defining an importance value, or, in other words, an importance metric. An example of such importance value - or importance metric - is given by:

Where S^<k) denotes the multi-set of Slater determinants given by where D_id denotes the frequency of each Slater determinant, denotes an adaptive Ansatz wavefunction, denotes the skew-Hermitian operators acting on the Hamiltonian H.

Instead of computing an energy gradient, the inventors realized that an advantageous selection process of operators may be derived by defining a heuristic gradient,, wherein the phases of the wavefunctions are omitted, and the real part of the wavefunction of a given molecular system is chosen to realize the equation of the importance value written above. The importance value estimates which operators may be included in the calculation of the energy eigenstates of a molecule. Having that importance value allows a user to perform a single measurement on a quantum computing unit, in order to determine whether an operator, or a plurality of operators, may be used or not to estimate the energy of a molecule. In contrast, ADAPT-VQE systems generally requires N⁵ measurements to reach the same conclusion as discussed in the background.

For the equation given above, if the importance value for a given operator converges to zero, then that operator may be neglected, while if the importance value for a given operator is above a threshold value, that operator may be selected. Thus, by performing such a process, it is possible to estimate the energy eigenstates of a molecule much faster compared to state of the art methods such as the ADAPT-VQE algorithm. As a result, in contrast to the ADAPT-VQE algorithm, the present disclosure requires less measurements -and possibly only a single measurement- in each algorithm iteration to determine the importance value, thereby efficiently reducing the measurement time needed for molecule calculations.

Description of the drawings

The invention will in the following be described in greater detail with reference to the accompanying drawings:

Fig. 1 a flow chart diagram of the steps performed by the herein disclosed calculation algorithm.

Fig. 2 shows a plot of the convergence of the full configuration interaction (FCI) calculation using a state of the art algorithm in dashed lines and the herein disclosed calculation algorithm in solid lines, wherein each line shows the calculation performed with a different number of samplings of expectation values and wherein the state vector solid line resembles the result for a number of samplings — > °°.

Fig. 3 A,B shows a plot of the FCI for the H4 molecule.

Fig. 4 shows importance value calculations as a function of different operators and number of iterations, for the ADAPT- VQE algorithm and the presently disclosed algorithm ran at different simulators and quantum computers.

Fig. 5 shows a diagram of the steps of the adaptive algorithm using a four-qubit Ansatz. Fig. 6 shows a plot of the FCI for the LiH molecule.

Detailed description

The present disclosure relates to an application of one or more hybrid quantum- classical algorithms for finding state energies of a molecular system. To start with, the Ansatz used for estimating the electronic wave function may be adaptively updated according to a metric designed to minimize the energy of the resulting wave function under the given Hamiltonian. In the present disclosure, this metric may be based on the population of Slater determinants in the wave function, ordered to determine the importance of the excitation evolution operators. An adaptive Ansatz is built by successively applying parameterized unitary operators to a reference state | t>o>- The reference state may be chosen as the Hartree Fock ground state. The adaptive Ansatz for the k-th iteration may be written as:

Where A^(k) is the set of operators in the Ansatz at the k-th iteration. The skew-Hermitian operators are given as is a fermionic excitation operator enumerated by p. In state of the art ADAPT VQE algorithms, the importance value used to evaluate operators is the energy gradient with respect to the parameter of the excitation evolution operator, wherein ^k+1) is the energy of the (k+1 )th iteration, described as

The above importance value of ADAPT VQE algorithms relies on measuring all the commutator operators [A_M, W]resu Iting to measurements in the order of N⁵ for a system with N qubits, in order to estimate whether an operator may be selected or discarded.

In contrast to ADAPT -VQE algorithms, as described in the following sections of the detailed description, the present disclosure may have as a starting point the energy gradient with respect to the parameter of the excitation evolution operator as written above, and then samples the population of Slater determinants in the Ansatz given above, to establish the importance value for the excitation evolution operators.

Such an importance value may require orders of magnitude less computation in the quantum computing unit compared to ADAPT -VQE algorithms to determine whether an operator may be used to estimate the energy of a molecular system. In an embodiment, the number of measurements performed to determine whether an operator may be used to estimate the energy of a molecular system is preferably less than N², more preferably less than N, most preferably less than 10, such as 1 measurement. The importance value used in the present disclosure is able to in principle compute all the S^(k) terms, to estimate the importance value of an operator and determine whether it should be selected or discarded. However, in an embodiment, it is possible that the quantum computer unit chooses a number of relevant S^(k) terms to compute, instead of computing all the terms. The phrase “relevant terms” refers to the terms that are necessary to determine whether an operator may be selected or not. Such a process may further increase the efficiency of the presently disclosed method.

Furthermore, regarding the selection process of an operator using the importance value, the process may vary on the size of the operator pool as well as on the complexity of the system. For example, for a simple molecule, it may be possible for the algorithm to select one operator from the operator pool on each iteration, without leading to a largely time-consuming process. However, for a more complicated molecular structure, a plurality of operators may be selected on each iteration, in order to speed up the process. It may be advantageous to speed up the process, as that may reduce the noise in the quantum computing unit, leading to more precise calculations.

In an embodiment, when an operator is used in the Ansatz, it may then be removed from the operator pool A in order to prevent repeating the same operator indefinitely in the Ansatz. However, it may be beneficial to repeat the same operators at a later stage in the adaptive iterations, in order to ensure full convergence of the algorithm. Such a process depends on the type of molecule that is to be computed, and it may also depend on other parameters such as the size and the chemical structure of the molecule. A threshold value may be used, in order to trigger the addition of all operations to the operator pool, following the formula max^ap) < e where is the importance value and e is the threshold.

In a preferred embodiment, the at least one quantum computing unit comprises a plurality of qubits. Said qubits might be defined in any of the qubit platforms that comprise a universal or non-universal set of gates that allow to initialize, control and measure each qubit and the interaction between them. The term of “initializing each qubit” may be understood by the skilled person within present context as “initializing an initial state on the at least one quantum computing unit to model the provided data representing a plurality of physical properties of the at least one molecule”.

The provided plurality of physical properties of the at least one molecule may be a wave function, atomic coordinates, spin and/or charge specification of each atom comprising said molecule. The step of “providing a plurality of physical properties of said at least one molecule” may be understood by the skilled person within present context as “providing data representing a plurality of physical properties of said at least one molecule to the classical computing unit and/or the quantum computing unit”. Other physical properties describing structural properties of the molecule and each atom, any quantum mechanical property and/or any property required to model the eigenstate of the Hamiltonian of the atoms comprising the molecule might be used as a provided physical property.

Additionally, the provided physical properties of the at least one molecule may comprise a collective spin and/or charge specification of said molecule. Other physical properties describing the collective properties of the molecule may be used as a provided physical property.

In some embodiments, the physical properties of one or more molecules comprises a reference state. Said reference state is for example a linear combination of slater determinants, a Hartree-Fock state or a Density Functional Theory determinant. The provided physical properties of one or more molecules might be at least one molecular integral. Moreover, the provided physical properties might be a combination of any of the previously mentioned properties.

The reference state and the at least one molecular integral may be obtained from the at least one classical computing unit via an ansatz calculation method such as Hartree- Fock. Other methods used in quantum chemistry for the estimation of ansatze electronic structure calculations may be used, such as for example density functional theory, coupled cluster theory, Moller-Plesset perturbation theory or random phase approximation.

In some embodiments, the provided physical properties of one or more molecules comprises at least one hyperparameter. Such hyperparameter may comprise a pool of operators provided to calculate the ansatz. The operators of the pool of operators may be assigned an importance value and later they may be ranked according to a higher importance value. Later, the ranked operators according to higher importance sequential order may be used to estimate the eigenstate energy of the initial wave function. The energy estimation is performed by minimizing the energy expectation, for example calculated by a Variational Quantum Eigensolver. Other methods to minimize the energy expectation of the wave function might be used for example quantum phase estimation.

Advantageously, the at least one hyperparameter may perform convergence criterion operations. These convergence criterion operations determine a difference in estimated eigenstate energy between iterations or a maximum number of iterations. When the calculated estimated eigenstate energy is smaller than a threshold value, the convergence criterion is met and the calculations stop as the energy is considered to be correctly estimated. The threshold value of the difference in estimated eigenstate energy between iterations is preferably smaller than 10^-3 Hartree, more preferably smaller than 10^-5 Hartree or even more preferably smaller than 10^-7 Hartree. In the field, an energy difference smaller than 10^-3 Hartree between iterations is considered chemical precision.

Similarly, when the maximum number of iterations exceed a threshold number the convergence criterion is met and the calculations stop as the energy may be considered to be correctly estimated. The threshold number of iterations is preferably bigger than 10³, more preferably bigger than 10⁴ or even more preferably bigger than 10⁵. The threshold number of iterations of the method at which the energy is considered to be correctly estimated is highly dependent on several factors such as the parameters of the molecule, the characteristics of the quantum computing unit and efficiency of the used computational tools. In some examples, a threshold number of iterations of 10² might be considered to be correctly estimated.

In some embodiments, the quantum computing unit comprising the plurality of qubits is initially configured to model the properties of a Hamiltonian describing the at least one molecule. This is achieved, for example, by initializing the qubits in the desired configuration by using a qubit initializing method such, as for example, adiabatic state preparation, quantum phase estimation Trotterization. In addition, the quantum computing unit comprising the plurality of qubits may initially be configured and/or initialized to model the properties of the one or more reference state of the at least one molecule.

Preferably, the initial configuration of the plurality of qubits is performed using a mapping scheme in the classical computing unit such as for example Jordan-Wigner or Bravyi-Kitaev mapping schemes.

The used plurality of operators are excitation operators, in some embodiments, of the at least one molecule, such as many-body molecular excitation operators or particlehole excitation operators. Then, the quantum computing unit may perform diagonal measurement operations to obtain population of Slater determinants, which may also be obtained through measuring the diagonal elements of the Hamiltonian. These measurements may be used to evaluate importance measures for the operators in the operator pool using, for example, Epstein-Nesbet perturbation theory operations or diagonal squared gradient norm calculation methods. In some embodiments, the at least one quantum computing unit performing measurement operations is configured to determine Slater determinant populations of the wavefunction to determine the importance of operators in the pool of operators. Later, the plurality of operators from the pool of operators are ordered based on an importance value which may be calculated using the population of Slater determinants and Epstein-Nesbet perturbation theory or the diagonal squared gradient norm. Such a technique holds significant advantages compared to the state of the art ADAPT-VQE algorithm, where the importance value is the gradient of the energy with respect to the parameter of the excitation evolution operator. That reflects a large number of measurements in the order of N⁵ for each operator, where N is the number of qubits, in order to determine whether said operator may be selected for estimating the energy of the wavefunction of a molecule. In contrast, determining Slater determinant populations to determine the importance value of operators may require only a single measurement per operator, reducing the time and cost of performing operator-selection measurements.

Figure 1 shows a flow chart diagram 100 of the steps performed by the herein disclosed method, providing at least one classical computing unit and at least one quantum computing unit comprising a plurality of qubits. The first step of calculating a molecular Hamiltonian with molecular integrals and a reference state (101) relates to providing at plurality of physical properties of a molecule, and initialize an initial state on the at least one quantum computing unit to model the physical properties of the molecule. The initial state may be encoded on a quantum computing unit (102), for example using commercial quantum computing units such as lonQ’s Harmony, or Rigetti. Then, a pool of operators may be defined (103), which may be used to iteratively construct the molecular wavefunction. In an embodiment, the pool of operators comprises skew-Hermitian operators. An importance value may be defined (104) by using a population of Slater determinants in the Ansatz, which is linked to performing diagonal measurement operations. As a result, performing such a process allows the selection of operators by conducting a single measurement, e.g., per iteration in k-space of the adaptive Ansatz, which may correspond to performing steps d-e, in comparison to N⁴ measurements that would be necessary when using an ADAPT-VQE algorithm. Therefore, a plurality of operators from the pool of operators may be selected using the at least one quantum computing unit (105), selecting operators based on the value of the importance value. For example, the importance value may be written as:

Then, based on the selection of operators, the at least one quantum computing unit may be used to perform diagonal measurement operations (106) on the eigenstate to calculate the eigenstate of the Hamiltonian describing the molecule (107). The energy of the eigenstate of the Hamiltonian may be estimated using at least one classical computing unit (108) and at least one quantum computing unit by applying the selected operators. Finally, the above steps are repeated until a convergence criterion is met.

Figure 5 shows a diagram of the steps of the adaptive algorithm using a four-qubit Ansatz which may be used in this example to find the ground state of dihydrogen. An initial state of the molecule is prepared (500) using a number of quantum logic gates. A number of operators may be selected using either the ADAPT VQE protocol (502) or the solution of the present disclosure (503). As discussed in the sections above, in contrast to the ADAPT-VQE algorithm, the present disclosure requires less measurements -and possibly only a single measurement- in each algorithm iteration to determine the importance value, thereby efficiently reducing the measurement time needed for molecule calculations. Then, the selected operators may be used to act on the Ansatz wavefunction (501 ). After selecting a number of operators from the pool of operators, the energy of the eigenstate of the Hamiltonian may be optimized (504), by performing a number of iterations K, with the aim of minimizing the eigenenergy of the Hamiltonian when operator(s) acts on it. After the convergence criterion is met following a number of iterations, the final optimized circuit is found (505).

Figure 2 shows a plot of the convergence (200) of a calculation comparison performed using the method herein disclosed and the state of the art algorithm, wherein the errors (201) of the calculations are depicted on the Y-axis, and the number of parameters used is depicted on the X-axis (202). The dashed lines correspond to a performed calculation using the state of the art algorithm ADAPT-VQE and the solid lines show a performed calculation using the herein disclosed calculation algorithm, using the state vector (203), 100 shots (204), 500 shots (205) or 1000 shots calculation (206). A larger number of shots may lead to a reduction of qubit noise. Each of the lines show the calculation performed with a different number of samplings of expectation values and wherein the state vector solid line (203) resembles the result for a number of samplings — > °°. It is observed that the full configuration interaction (FCI) is reduced faster (requiring a smaller number of parameters) using the herein disclosed method, in solid lines.

Moreover, figure 3A illustrates the errors (302) of the presently disclosed algorithm calculated at different quantum computers and simulations (300) as a function of parameters used (303) for the H₄ molecule. Another advantage of the present disclosure over state of the art full configuration interaction (FCI) calculations, is that the method of the present disclosure converges to chemical precision at 20 adaptive iterations for an exemplary system. In comparison, as seen in Fig. 3A, ADAPT-VQE (301) barely reaches the limit of chemical precision after 25 iterations. A state vector (306), a density matrix (307) and a random sampling (308) simulation is also shown. It is clear that the method of the present disclosure reaches an error level lower than the chemical precision (304) at approximately 20 parameters, while ADAPT-VQE algorithms require more than 25 parameters to barely reach an error level equivalent to the chemical precision. Figure 3B shows the errors of the algorithms as a function of number of CNOT gates (305).

Preferably, the used initial Hamiltonian for initializing the qubits and performing the previously disclosed operations is a stationary state such as a ground state or an excited state.

The obtained energies via the presently disclosed method may be used to determine the binding affinity between a drug and ligand target. The obtained information may be used for the development of novel drugs and to screen potential candidates with a probability of successfully bind a target biomolecule. Other applications in other fields than drug development requiring the binding or interaction of two or more molecules may benefit from the results obtained via the disclosed method.

The present disclosure further relates to a computer program having instructions which, when executed by a computing device or computing system, cause the computing device or computing system to carry out any embodiment of the presently disclosed method of calculating an eigenstate of a Hamiltonian describing at least one molecule. The computer program may be stored on any suitable type of storage media, such as non-transitory storage media.

The at least one classical computing unit and/or at least one quantum computing unit may further comprise peripheral components, such as one or more memories, which may be used for storing instructions that may be executed by any of the processors. The system may further comprise any of: internal and external network interfaces, input and/or output ports, a keyboard or mouse etc.

As would be understood by a person skilled in the art, a processing unit also may be a single processor in a multi-core/multiprocessor system. Both the computing hardware accelerator and the central processing unit may be connected to a data communication infrastructure.

The at least one classical computing unit and at least one quantum computing unit may include a memory, such as a random access memory (RAM) and/or a read-only memory (ROM), or any suitable type of memory. The system may further comprise a communication interface that allows software and/or data to be transferred between the system and external devices. Software and/or data transferred via the communications interface may be in any suitable form of electric, optical or RF signals. The communications interface may comprise, for example, a cable or a wireless interface.

Examples

Below, an example is shown of how the selection of a plurality of operators from a pool of operators may be performed.

In the below example, the presently disclosed algorithm is referred to as FAST-VQE. Figure 4 shows a number of measurements of the importance value as a function of various single and double excitation operators (400) and as a function of iterations of the algorithm (401). The dark points in the measurements (402) reflect a close to zero value of the importance value corresponding to an operator, while the light points (403) reflect a finite value of the importance value corresponding to an operator. The second plot shown (404) is the importance value for the ADAPT-VQE algorithm computed on a state vector simulator, while the other plots correspond to the FAST-VQE algorithm computed either on a simulator such as the SV simulator (405) or computed using a quantum computer such as Rigetti Aspem-M-3 (406) or lonQ Harmony (408). In these measurements, the importance value is plotted for the chosen operators (400) in the first 15 iterations (401 ) of the adaptive algorithm. The first two measurement plots (404,405) are noiseless, as they are computed on a simulator. The first noteworthy feature when comparing the noiseless simulations of FAST-VQE and ADAPT-VQE on state vector simulators is that the ranking of operators is a lot more consistent for each iteration using FAST-VQE; The Oth iteration (407) ranks the operators almost identically to the following iterations, with a few minor changes. On each row of the measurement plots, each operator is selected, usually the operator with the highest value, for example the left-most operator, and therefore it may be removed from the operator pool. Then, the following iteration may be performed, having the same operators apart from the one removed. Such a process may continue until the importance value max_M(a_M) gets below a threshold value e, where the pool of operators is refilled. Strikingly, the plots corresponding to the FAST-VQE algorithm on quantum computers (406) show that the FAST-VQE algorithm may run for 15 iterations before the importance value is below a given threshold, where the importance value is refilled. On the other hand, in ADAPT-VQE calculations (404) there seems to be no apparent order, as the pool of operators needs to be refilled after each iteration.

The above simulations and measurements show that the algorithm of the present disclosure works as intended, and it is clearly superior to the ADAPT-VQE algorithm, allowing the algorithm of present disclosure to successfully select the important operators for the calculation of the energy eigenstates of a molecule, while discarding the unimportant ones. Therefore, by requiring less measurements to select or discard an operator, the calculation of a molecule may be executed at a higher speed and at an increased accuracy. Importantly, the reduction of measurements required to estimate whether an operator may be selected or discarded has a direct effect on the performance of the at least one quantum computing unit, as the quantum computing unit has significant errors when computing for extended periods of time. Therefore, requiring just a single measurement may circumvent the measurement bottleneck in quantum computing units for estimating the energies of molecular structures. Similar to the calculations of FCI errors shown in Fig. 3, Fig. 6 compares FCI errors (600) for the ADAPT-VQE method (601 ) and the presently disclosed method executed in Rigetti Aspen-M-3 (602), as a function of number of parameters (603), for the LiH molecule with an interatomic distance of 1.5 A. A state vector (606) simulation and a density matrix (605) simulation is also depicted. In this example, each calculation is performed 10 times and the shaded areas shown the 95% confidence interval. It is clear that the presently disclosed method outperforms the state of the art ADAPT-VQE solution as the presently disclosed method reaches chemical precision (604) at around 15 parameters, while ADAPT-VQE does not reach chemical precision even at 40 parameters used.

Items

1 . A method for calculating and/or determining an eigenstate of a Hamiltonian, comprising the steps of a. providing at least one classical computing unit and at least one quantum computing unit comprising a plurality of qubits, b. providing a plurality of physical properties of at least one molecule, c. initializing an initial state on the at least one quantum computing unit to model the properties of the provided physical properties of the at least one molecule, d. selecting a plurality of operators from a pool of operators using the at least one quantum computing unit performing diagonal measurement operations on the eigenstate to calculate the eigenstate of the Hamiltonian describing the at least one molecule, e. estimating an energy of the eigenstate of the Hamiltonian using the selected plurality of operators, and f. repeating steps d - e until a convergence criterion is met.

2. The method according to item 1 , wherein the at least one quantum computing unit comprises a plurality of qubits.

3. The method according to any of the preceding items, wherein the physical properties of the at least one molecule is a wave function, atomic coordinates, spin or charge specification of each atom comprising said molecule.

4. The method according to any of the preceding items, wherein the physical properties of the at least one molecule is a spin or charge specification of said molecule.

5. The method according to any of the preceding items, wherein the physical property of one or more molecules is a reference state.

6. The method according to any of the preceding items, wherein the reference state is for example a linear combination of slater determinants, a Hartree-Fock state or a Density Functional Theory determinant. The method according to any of the preceding items, wherein the physical properties of one or more molecules is at least one molecular integral. The method according to any of items 6 - 7, wherein the reference state and the at least one molecular integral is obtained from the at least one classical computing unit via an ansatz calculation method such as Hartree-Fock. The method according to any of the preceding items, wherein the physical properties of one or more molecules is at least one hyperparameter. The method according to item 9, wherein the at least one hyperparameter comprises the pool of operators. The method according to any of items 9 - 10, wherein the operators of the pool of operators are ranked according to a higher importance value. The method according to any of items 9 - 11 , wherein the ranked operators according to higher importance sequential order estimate the eigenstate energy of the wave function. The method according to any of items 9 - 12, wherein the energy of the eigenstate is estimated by minimizing the energy expectation from the selected plurality of operators. The method according to any of items 9 - 13, wherein the minimization of energy expectation is performed by a Variational Quantum Eigensolver. The method according to any of the preceding items, wherein the at least one hyperparameter performs convergence criterion operations. The method according to any of the preceding items, wherein the convergence criterion is determined by a difference in estimated eigenstate energy between iterations or a maximum number of iterations. 17. The method according to item 16, wherein the difference in estimated eigenstate energy is smaller than a threshold value.

18. The method according to any of items 16 - 17, wherein the threshold value of the difference in estimated eigenstate energy is preferably smaller than 10^-3 Hartree, more preferably smaller than 10^-5 Hartree or even more preferably smaller than 10^-7 Hartree.

19. The method according to any of the preceding items, wherein the maximum number of iterations exceed a threshold number.

20. The method according to item 19, wherein the threshold number of iterations is preferably smaller than 10⁵, more preferably smaller than 10⁴ or even more preferably smaller than 10³.

21 . The method according to any of the preceding items, wherein at least one quantum computing unit comprising the plurality of qubits is initially configured to model the properties of a Hamiltonian describing the at least one molecule.

22. The method according to any of the preceding items, wherein the at least one quantum computing unit comprising the plurality of qubits is initially configured to model the properties of one or more reference state of the at least one molecule.

23. The method according to any of items 21 - 22, wherein the initial configuration of the plurality of qubits is performed using a mapping scheme in the classical computing unit.

24. The method according to item 23, wherein the mapping scheme is for example Jordan-Wigner or Bravyi-Kitaev.

25. The method according to any of the preceding items, wherein the plurality of operators are excitation operators of the at least one molecule, such as manybody molecular excitation operators or particle-hole excitation operators. 26. The method according to any of the preceding items, wherein the at least one quantum computing unit performing diagonal measurement operations on the diagonal of the Hamiltonian are perturbation operations.

27. The method according to item 26, wherein the performed perturbation operations are Epstein-Nesbet perturbation theory operations.

28. The method according to any of items 26 - 27, wherein the at least one quantum computing unit performing diagonal measurement operations is configured to perform Slater determinant populations of the plurality of operators from the pool of operators.

29. The method according to any of items 26 - 28, wherein the plurality of operators from the pool of operators are ordered based on an importance value calculated performing Epstein-Nesbet perturbation theory operations.

30. The method according to any of the preceding items, wherein the eigenstate of the Hamiltonian is a stationary state.

31 . The method according to any of the preceding items, wherein the stationary state is a ground state.

32. The method according to any of the preceding items, wherein the stationary state of the wave function is an excited state.

33. A system for calculating an eigenstate of a Hamiltonian comprising at least one classical computing unit, at least one quantum computing unit and a processing unit configured to perform the method according to any of the preceding items.

34. A computer program having instructions which, when executed by a computing device or computing system, cause the computing device or computing system to carry out the method of calculating an eigenstate of a Hamiltonian describing at least one molecule according to any one of items 1 - 33.

Claims

1 . A method for calculating and/or estimating an eigenstate of a Hamiltonian describing at least one molecule, comprising the steps of a. providing at least one classical computing unit, and at least one quantum computing unit comprising a plurality of qubits, b. providing a plurality of physical properties of said at least one molecule, c. initializing an initial state on the at least one quantum computing unit to model the provided physical properties of the at least one molecule, d. selecting a plurality of operators from a pool of operators using the at least one quantum computing unit to perform diagonal measurement operations on the eigenstate to calculate the eigenstate of the Hamiltonian describing the at least one molecule, e. estimating an energy of the eigenstate of the Hamiltonian using the at least one classical computing unit and the at least one quantum computing unit applying the selected plurality of operators, and f. repeating steps d - e until a convergence criterion is met.

2. The method according to claim 1 , wherein the physical properties of the at least one molecule is selected from the group of: a wave function, atomic coordinates, spin or charge specification of the molecule and/or each atom comprising said molecule.

3. The method according to any of the preceding claims, wherein the physical properties of the at least one molecule comprise a reference state such as for example a linear combination of slater determinants, a Hartree-Fock state or a Density Functional Theory determinant.

4. The method according to any of the preceding claims, wherein the physical properties of the at least one molecule is at least one molecular integral obtained from the at least one classical computing unit via an ansatz calculation method such as Hartree-Fock.

5. The method according to any of the preceding claims, wherein the physical properties of the at least one molecules is at least one hyperparameter comprising the pool of operators, such as many-body molecular excitation operators or particle-hole excitation operators.

6. The method according to claim 5, wherein the operators of the pool of operators are ranked according to a higher importance value estimated by a Variational Quantum Eigensolver minimizing the energy expectation from the selected plurality of operators.

7. The method according to any of claims 5 - 6, wherein the ranked operators according to higher importance sequential order estimate the eigenstate energy of the wave function.

8. The method according to any of the preceding claims, wherein the at least one hyperparameter performs the convergence criterion operations.

9. The method according to any of the preceding claims, wherein the convergence criterion is determined by a difference in estimated eigenstate energy between iterations preferably smaller than 10^-3 Hartree, more preferably smaller than 10^-5 Hartree or even more preferably smaller than 10^-7 Hartree, or a maximum number of iterations preferably smaller than 10⁵, more preferably smaller than

10⁴ or even more preferably smaller than 10³.

10. The method according to any of the preceding claims, wherein at least one quantum computing unit comprising the plurality of qubits is initially configured to model a reference state of a Hamiltonian describing the at least one molecule using a mapping scheme in the classical computing unit such as for example Jordan-Wigner or Bravyi-Kitaev.

11 . The method according to any of the preceding claims, wherein the at least one quantum computing unit performs measurement operations on the diagonal of the Hamiltonian, such as Epstein-Nesbet perturbation theory operations or diagonal squared gradient norm calculation methods.

12. The method according to any of claims 10 - 11 , wherein the at least one quantum computing unit performing measurement operations is configured to perform Slater determinant populations of the plurality of operators from the pool of operators.

13. The method according to any of claims 10 - 12, wherein the plurality of operators from the pool of operators are ordered based on an importance value calculated performing Epstein-Nesbet perturbation theory operations.

14. The method according to any of the preceding claims, wherein the eigenstate of the Hamiltonian is a stationary state, such as a ground state or an excited state.

15. A system for calculating an eigenstate of a Hamiltonian comprising at least one classical computing unit and at least one quantum computing unit comprising a plurality of qubits, the system configured to perform the method according to any of the preceding claims.