DESC:FORM 2

THE PATENTS ACT, 1970
(39 of 1970)
&
THE PATENT RULES, 2003

COMPLETE SPECIFICATION
(See Section 10 and Rule 13)

Title of invention:
METHOD AND SYSTEM FOR NESTED QUANTUM-CLASSICAL HYBRID MOLECULAR GEOMETRY OPTIMIZATION

Applicant:
Tata Consultancy Services Limited
A company Incorporated in India under the Companies Act, 1956
Having address:
Nirmal Building, 9th Floor,
Nariman Point, Mumbai 400021,
Maharashtra, India

The following specification particularly describes the invention and the manner in which it is to be performed.
CROSS-REFERENCE TO RELATED APPLICATIONS AND PRIORITY
The present application claims priority from Indian provisional patent application no. 202221061644, filed on October 28, 2022. The entire contents of the aforementioned application are incorporated herein by reference.

TECHNICAL FIELD
The disclosure herein generally relates to quantum computing, and, more particularly, to system and method for nested quantum-classical hybrid molecular geometry optimization.
BACKGROUND
As quantum hardware and algorithms continue to develop, various industry sectors, particularly the pharmaceutical and material design domains, are interested to see the utility and performance of quantum computation paradigm for applicability to specific problems. Variational quantum algorithms employ a quantum computer to prepare the wavefunction of the system under study and to measure the system’s energy as the expectation value of the target Hamiltonian. The subroutine of adjusting the parameters of the wave function to lower the measured expectation value is outsourced to a classical device. In fact, the promise held by quantum computers lies in representing the state of a system under study with fewer resources than their classical counterparts owing to the quantum mechanical effects of superposition, entanglement, and interference.
For example, matter or molecules are made up of electrons, protons and nuclei with a specific geometrical configuration. These geometries govern their equilibrium configuration and the reaction mechanisms they participate in, for synthesizing or discovering new molecules or materials. Calculating the most stable geometry or tracing the reaction pathway at desired precision and at scale is not tractable using classical Density Functional Theory (DFT) and Wave Function Theory (WFT) based methods on classical computers. Using a quantum computer/algorithm makes it feasible by mapping molecular particle states to qubits (encoding). However, current Noisy Intermediate-scale Quantum (NISQ) computers are limited in qubit counts and qubit coherence (determines length of the program) to run larger molecular simulations.

SUMMARY
Embodiments of the present disclosure present technological improvements as solutions to one or more of the above-mentioned technical problems recognized by the inventors in conventional systems. For example, in one embodiment, a method for nested quantum-classical hybrid molecular geometry optimization is provided. The method includes receiving, by one or more classical hardware processors of a system comprising the one or more classical hardware processors communicably coupled to a plurality of unentangled Quantum Processor Units (QPUs) via an interface, a plurality of initial nuclear coordinates associated with an initial geometry of a molecule, wherein the plurality of initial nuclear coordinates is initialized based on Hartree-Fock electronic structure method. Further, the method includes obtaining, by the one or more classical hardware processors and the plurality of unentangled QPUs, an equilibrium geometry of the molecule associated with all degrees of freedom by iteratively performing, until energy gradient associated with a plurality of current nuclear coordinates is equal to a predefined threshold, actions comprising: (i) generating, by the one or more classical hardware processors, a parameterized electronic Hamiltonian based on the plurality of current nuclear coordinates using an approximation technique, wherein the plurality of initial nuclear coordinates is assigned to the plurality of current nuclear coordinates (ii) generating (204B), by the one or more classical hardware processors, a Qubit Hamiltonian based on the parameterized electronic Hamiltonian using a Fermion to Qubit mapping technique (iii) generating (204C), by the plurality of unentangled QPUs, a trial state associated with the molecule using a quantum circuit comprising a plurality of tunable parameters (iv) computing, by the plurality of unentangled QPUs, an electronic energy associated with the molecule for the trial state based on the Qubit Hamiltonian using a Quantum Expectation Estimation (QEE) technique (v) computing, by the one or more classical hardware processors, a ground state energy for the plurality of current nuclear coordinates by optimizing the plurality of tunable parameters using a classical optimizer until one of a plurality of constraints is satisfied (vi) estimating, by the one or more classical hardware processors, and the plurality of unentangled QPUs, an electronic energy gradient associated with the plurality of current nuclear coordinates based on the ground state energy using a numerical differentiation technique; and (vii) updating, by the one or more classical hardware processors, the plurality of current nuclear coordinates based on the electronic energy gradient using a classical optimizer.
In another aspect, a system for nested quantum-classical hybrid molecular geometry optimization is provided. The system includes one or more classical hardware processors communicably coupled to a plurality of unentangled Quantum Processor Units (QPUs) via interfaces, wherein the one or more classical hardware processors comprises at least one memory storing programmed instructions; one or more Input /Output (I/O) interfaces; and one or more hardware processors operatively coupled to the at least one memory, wherein the one or more classical hardware processors (108) are configured by the programmed instructions to receive a plurality of initial nuclear coordinates associated with an initial geometry of a molecule, wherein the plurality of initial nuclear coordinates is initialized based on Hartree-Fock electronic structure method. Further, at least one of the one or more hardware processors (108) and the plurality of unentangled QPUs (126) are configured by the programmed instructions to: obtain an equilibrium geometry of the molecule associated with all degrees of freedom by iteratively performing, until energy gradient associated with a plurality of current nuclear coordinates is equal to a predefined threshold, actions comprising: (i) generating, by the one or more classical hardware processors, a parameterized electronic Hamiltonian based on the plurality of current nuclear coordinates using an approximation technique, wherein the plurality of initial nuclear coordinates are assigned to the plurality of current nuclear coordinates (ii) generating, by the one or more classical hardware processors, a Qubit Hamiltonian based on the parameterized electronic Hamiltonian using a Fermion to Qubit mapping technique (iii) generating, by the plurality of unentangled QPUs, a trial state associated with the molecule using a quantum circuit comprising a plurality of tunable parameters (iv) computing, by the plurality of unentangled QPUs, an electronic energy associated with the molecule for the trial state based on the Qubit Hamiltonian using a Quantum Expectation Estimation (QEE) technique (v) computing, the one or more classical hardware processors a ground state energy for the plurality of current nuclear coordinates by optimizing the plurality of tunable parameters using a classical optimizer until one of a plurality of constraints is satisfied (vi) estimating, by the one or more classical hardware processors, an electronic energy gradient associated with the plurality of current nuclear coordinates based on the ground state energy using a numerical differentiation technique and (vii) updating, by the one or more classical hardware processors, the plurality of current nuclear coordinates based on the electronic energy gradient using a classical optimizer.
In yet another aspect, a computer program product including a non-transitory computer-readable medium having embodied therein a computer program for nested quantum-classical hybrid molecular geometry optimization is provided. The computer readable program, when executed on a system comprising one or more classical hardware processors communicably coupled to a plurality of unentangled Quantum Processor Units (QPUs) via interfaces, causes the computing device to receive a plurality of initial nuclear coordinates associated with an initial geometry of a molecule, wherein the plurality of initial nuclear coordinates is initialized based on Hartree-Fock electronic structure method. Further, the computer readable program, when executed on the system comprising one or more classical hardware processors communicably coupled to a plurality of unentangled QPUs via interfaces, causes the computing device to: obtain an equilibrium geometry of the molecule associated with all degrees of freedom by iteratively performing, until energy gradient associated with a plurality of current nuclear coordinates is equal to a predefined threshold, actions comprising: (i) generating, by the one or more classical hardware processors, a parameterized electronic Hamiltonian based on the plurality of current nuclear coordinates using an approximation technique, wherein the plurality of initial nuclear coordinates are assigned to the plurality of current nuclear coordinates (ii) generating, by the one or more classical hardware processors, a Qubit Hamiltonian based on the parameterized electronic Hamiltonian using a Fermion to Qubit mapping technique (iii) generating, by the plurality of unentangled QPUs, a trial state associated with the molecule using a quantum circuit comprising a plurality of tunable parameters (iv) computing, by the plurality of unentangled QPUs, an electronic energy associated with the molecule for the trial state based on the Qubit Hamiltonian using a Quantum Expectation Estimation (QEE) technique (v) computing, the one or more classical hardware processors a ground state energy for the plurality of current nuclear coordinates by optimizing the plurality of tunable parameters using a classical optimizer until one of a plurality of constraints is satisfied (vi) estimating, by the one or more classical hardware processors, an electronic energy gradient associated with the plurality of current nuclear coordinates based on the ground state energy using a numerical differentiation technique and (vii) updating, by the one or more classical hardware processors, the plurality of current nuclear coordinates based on the electronic energy gradient using a classical optimizer.
It is to be understood that both the foregoing general description and the following detailed description are exemplary and explanatory only and are not restrictive of the invention, as claimed.
.

BRIEF DESCRIPTION OF THE DRAWINGS
The accompanying drawings, which are incorporated in and constitute a part of this disclosure, illustrate exemplary embodiments and, together with the description, serve to explain the disclosed principles:
FIG. 1 is a functional block diagram of a system for nested quantum-classical hybrid molecular geometry optimization, in accordance with some embodiments of the present disclosure.
FIG. 2 is an exemplary flow diagram illustrating a hybrid quantum-classical processor implemented method for nested quantum-classical hybrid molecular geometry optimization, implemented by the system of FIG. 1, in accordance with some embodiments of the present disclosure.
FIG. 3 illustrates Nucleophilic Substitution, second order (SN2) reaction pathway for CH3F + Cl- (complex of methyl fluoride and chloride ion) using Intrinsic Reaction Coordinates-Variational Quantum Eigen solver -Energy (IRC-VQE), as a function of various MO (Molecular Orbital) basis sets, in accordance with some embodiments of the present disclosure.
FIGS. 4A through 4C illustrates evolution of the internal molecular coordinates across the SN2 reaction pathway using IRC-VQE, in accordance with some embodiments of the present disclosure.
FIGS. 5A and 5B illustrates optimization of CH3Cl in different basis sets using the finite-difference based variational quantum approach, in accordance with some embodiments of the present disclosure.
FIG. 6A through 6D are simulation results showing the performance of the IRC-VQE tested over quantum hardware (IBM) and Quantum Assembly (QASM) language simulator, in accordance with some embodiments of the present disclosure.

DETAILED DESCRIPTION OF EMBODIMENTS
Exemplary embodiments are described with reference to the accompanying drawings. In the figures, the left-most digit(s) of a reference number identifies the figure in which the reference number first appears. Wherever convenient, the same reference numbers are used throughout the drawings to refer to the same or like parts. While examples and features of disclosed principles are described herein, modifications, adaptations, and other implementations are possible without departing from the scope of the disclosed embodiments. It is intended that the following detailed description be considered as exemplary only, with the true scope being indicated by the following embodiments described herein.
Variational quantum algorithms employ a quantum computer to prepare the wavefunction of the system under study and to measure the system’s energy as the expectation value of the target Hamiltonian. The subroutine of adjusting the parameters of the wavefunction to lower the measured expectation value is outsourced to a classical device. In fact, the promise held by quantum computers lies in representing the state of a system under study with fewer resources than their classical counterparts owing to the quantum mechanical effects of superposition, entanglement, and interference.
For example, matter or molecules are made up of electrons, protons and nuclei with a specific geometrical configuration. These geometries govern their equilibrium configuration and the reaction mechanisms they participate in, for synthesizing or discovering new molecules or materials. Calculating the most stable geometry or tracing the reaction pathway at desired precision and at scale is not tractable using classical Density Functional Theory (DFT) and Wave Function Theory (WFT) based methods on classical computers. Using a quantum computer/algorithm makes it feasible by mapping molecular particle states to qubits (encoding). However, current Noisy Intermediate-scale Quantum (NISQ) computers are limited in qubit counts and qubit coherence (determines length of the program) to run larger molecular simulations.
To overcome the limitations of the conventional approaches, the present disclosure provides a quantum-classical hybrid molecular geometry optimization. The present disclosure is a nested approach for geometry optimization of a linear or nom-linear molecules with all degrees of freedom. In this technique, the Hamiltonian of the molecular system for an initial set of nuclear coordinates associated with a molecule is obtained initially. It is further encoded into a qubit Hamiltonian. Further, a trial state parameterized by ? is optimized (by adjusting ?) over a quantum-classical hybrid architecture to obtain the ground state energy for the fixed initial set of nuclear coordinates. Then the fixed initial set of nuclear coordinates are being updated to move closer to the energy minimum on the Potential Energy Surface (PES). So, inside the inner loop, optimization of circuit parameters ? takes place, and in the outer loop the nuclear coordinates x_0 are updated. This hybrid approach used in the present disclosure overcomes the problems faced by classical computers and quantum computers individually for solving geometry optimization problem for all degrees of freedom. Further, the nested approach improves efficiency and accuracy of the geometry optimization problem for all degrees of freedom.
Referring now to the drawings, and more particularly to FIGS. 1 through 6D, where similar reference characters denote corresponding features consistently throughout the figures, there are shown preferred embodiments and these embodiments are described in the context of the following exemplary system and/or method.
FIG. 1 is a functional block diagram of a system 100 for nested quantum-classical hybrid molecular geometry optimization, in accordance with some embodiments of the present disclosure. The system 100 includes a classical computing system 102, a quantum computing system 104 and a communication interface 106.
The classical computing system 102 comprises classical hardware processors 108, at least one memory such as a memory 110, an I/O interface 118. The classical hardware processors 108, the memory 110, and the Input /Output (I/O) interface 118 may be coupled by a system bus such as a system bus 114 or a similar mechanism. In an embodiment, the classical hardware processors 108 can be one or more hardware processors. The classical hardware processors 108 and the hardware processors is interchangeably used throughout the document. Similarly, the classical computing system is a normal computing system.
The I/O interface 118 may include a variety of software and hardware interfaces, for example, a web interface, a graphical user interface, and the like., for example, interfaces for peripheral device(s), such as a keyboard, a mouse, an external memory, a printer and the like. Further, the I/O interface 118 may enable the system 100 to communicate with other devices, such as web servers, and external databases.
The I/O interface 118 can facilitate multiple communications within a wide variety of networks and protocol types, including wired networks, for example, local area network (LAN), cable, etc., and wireless networks, such as Wireless LAN (WLAN), cellular, or satellite. For the purpose, the I/O interface 118 may include one or more ports for connecting several computing systems with one another or to another server computer.
The one or more hardware processors 108 may be implemented as one or more microprocessors, microcomputers, microcontrollers, digital signal processors, central processing units, node machines, logic circuitries, and/or any devices that manipulate signals based on operational instructions. Among other capabilities, the one or more hardware processors 108 is configured to fetch and execute computer-readable instructions stored in the memory 110.
The memory 110 may include any computer-readable medium known in the art including, for example, volatile memory, such as static random access memory (SRAM) and dynamic random access memory (DRAM), and/or non-volatile memory, such as read only memory (ROM), erasable programmable ROM, flash memories, hard disks, optical disks, and magnetic tapes. In an embodiment, the memory 110 includes a plurality of modules 112. The memory 110 also includes a data repository (or repository) 116 for storing data processed, received, and generated by the plurality of modules 112.
The plurality of modules 112 include programs or coded instructions that supplement applications or functions performed by the system 100 for quantum-classical hybrid molecular geometry optimization. The plurality of modules 112, amongst other things, can include routines, programs, objects, components, and data structures, which performs particular tasks or implement particular abstract data types. The plurality of modules 112 may also be used as, signal processor(s), node machine(s), logic circuitries, and/or any other device or component that manipulates signals based on operational instructions. Further, the plurality of modules 112 can be used by hardware, by computer-readable instructions executed by the one or more hardware processors 108, or by a combination thereof. The plurality of modules 112 can include various sub-modules (not shown).
The data repository (or repository) 116 may include a plurality of abstracted piece of code for refinement and data that is processed, received, or generated as a result of the execution of the plurality of modules in the module(s) 112.
Although the data repository 116 is shown internal to the system 100, it will be noted that, in alternate embodiments, the data repository 116 can also be implemented external to the system 100, where the data repository 116 may be stored within a database (repository 116) communicatively coupled to the system 100. The data contained within such external database may be periodically updated. For example, new data may be added into the database (not shown in FIG. 1) and/or existing data may be modified and/or non-useful data may be deleted from the database. In one example, the data may be stored in an external system, such as a Lightweight Directory Access Protocol (LDAP) directory and a Relational Database Management System (RDBMS).
The example quantum computing system 104 shown in FIG. 1 includes a control system 122, a signal delivery system 124, a plurality of Quantum Processing Units (QPUs) 126 and a quantum memory 128. The plurality of QPUs is unentangled and hence alternatively called as the plurality of unentangled QPUs. The quantum computing system 104 may include additional or different features, and the components of a quantum computing system may operate as described with respect to FIG. 1 or in another manner.
The example quantum computing system 104 shown in FIG. 1 can perform quantum computational tasks (such as, for example, quantum simulations or other quantum computational tasks) by executing quantum algorithms. In some implementations, the quantum computing system 104 can perform quantum computation by storing and manipulating information within individual quantum states of a composite quantum system. For example, Qubits (i.e., Quantum bits) can be stored in and represented by an effective two-level sub-manifold of a quantum coherent physical system in the plurality of QPUs 126.
In an embodiment, the quantum computing system 104 can operate using gate-based models for quantum computing. For example, the Qubits can be initialized in an initial state, and a quantum logic circuit comprised of a series of quantum logic gates can be applied to transform the qubits and extract measurements representing the output of the quantum computation.
The example QPUs 126 shown in FIG. 1 may be implemented, for example, as a superconducting quantum integrated circuit that includes Qubit devices. The Qubit devices may be used to store and process quantum information, for example, by operating as ancilla Qubits, data Qubits or other types of Qubits in a quantum algorithm. Coupler devices in the superconducting quantum integrated circuit may be used to perform quantum logic operations on single qubits or conditional quantum logic operations on multiple qubits. In some instances, the conditional quantum logic can be performed in a manner that allows large-scale entanglement within the QPUs 126. Control signals may be delivered to the superconducting quantum integrated circuit, for example, to manipulate the quantum states of individual Qubits and the joint states of multiple Qubits. In some instances, information can be read from the superconducting quantum integrated circuit by measuring the quantum states of the qubit devices. The QPUs 126 may be implemented using another type of physical system.
The example QPUs 126, and in some cases all or part of the signal delivery system 124, can be maintained in a controlled cryogenic environment. The environment can be provided, for example, by shielding equipment, cryogenic equipment, and other types of environmental control systems. In some examples, the components in the QPUs 126 operate in a cryogenic temperature regime and are subject to very low electromagnetic and thermal noise. For example, magnetic shielding can be used to shield the system components from stray magnetic fields, optical shielding can be used to shield the system components from optical noise, thermal shielding and cryogenic equipment can be used to maintain the system components at controlled temperature, etc.
In the example shown in FIG. 1, the signal delivery system 124 provides communication between the control system 122 and the QPUs 126. For example, the signal delivery system 124 can receive control signals from the control system 122 and deliver the control signals to the QPUs 126. In some instances, the signal delivery system 124 performs preprocessing, signal conditioning, or other operations to the control signals before delivering them to the QPUs 126.
In an embodiment, the signal delivery system 124 includes connectors or other hardware elements that transfer signals between the QPUs 126 and the control system 122. For example, the connection hardware can include signal lines, signal processing hardware, filters, feedthrough devices (e.g., light-tight feedthroughs, etc.), and other types of components. In some implementations, the connection hardware can span multiple different temperature and noise regimes. For example, the connection hardware can include a series of temperature stages that decrease between a higher temperature regime (e.g., at the control system 122) and a lower temperature regime (e.g., at the QPUs 126).
In the example quantum computer system 104 shown in FIG. 1, the control system 122 controls operation of the QPUs 126. The example control system 122 may include data processors, signal generators, interface components and other types of systems or subsystems. Components of the example control system 122 may operate in a room temperature regime, an intermediate temperature regime, or both. For example, the control system 122 can be configured to operate at much higher temperatures and be subject to much higher levels of noise than are present in the environment of the QPUs 126.
In some embodiments, the control system 122 includes a classical computing system that executes software to compile instructions for the QPUs 126. For example, the control system 122 may decompose a quantum logic circuit or quantum computing program into discrete control operations or sets of control operations that can be executed by the hardware in the QPUs 126. In some examples, the control system 122 applies a quantum logic circuit by generating signals that cause the Qubit devices and other devices in the QPUs 126 to execute operations. For instance, the operations may correspond to single-Qubit gates, two-Qubit gates, Qubit measurements, etc. The control system 122 can generate control signals that are communicated to the QPUs 126 by the signal delivery system 124, and the devices in the QPUs 126 can execute the operations in response to the control signals.
In some other embodiments, the control system 122 includes one or more classical computers or classical computing components that produce a control sequence, for instance, based on a quantum computer program to be executed. For example, a classical processor may convert a quantum computer program to an instruction set for the native gate set or architecture of the QPUs 126. In some cases, the control system 122 includes a microwave signal source (e.g., an arbitrary waveform generator), a bias signal source (e.g., a direct current source) and other components that generate control signals to be delivered to the QPUs 126. The control signals may be generated based on a control sequence provided, for instance, by a classical processor in the control system 122. The example control system 122 may include conversion hardware that digitizes response signals received from the QPUs 126. The digitized response signals may be provided, for example, to a classical processor in the control system 122.
In some embodiments, the quantum computer system 104 includes multiple quantum information processors that operate as respective quantum processor units (QPU). In some cases, each QPU can operate independent of the others. For instance, the quantum computer system 104 may be configured to operate according to a distributed quantum computation model, or the quantum computer system 104 may utilize multiple QPUs in another manner. In some implementations, the quantum computer system 104 includes multiple control systems, and each QPU may be controlled by a dedicated control system. In some implementations, a single control system can control multiple QPUs; for instance, the control system 122 may include multiple domains that each control a respective QPU.
In some instances, the quantum computing system 104 uses multiple QPUs to execute multiple unentangled quantum computations (e.g., multiple VQE) that collectively simulate a single quantum mechanical system.
In an embodiment, the quantum memory 128 is a quantum-mechanical version of classical computer memory. The classical computer memory stores information as binary states and the quantum memory 128 stores a quantum state for later retrieval. These states hold useful computational information known as Qubits.
In an embodiment, the communication interface 106 which connects the classical computing system 102 and the quantum computing system 104 is a high speed digital interface.
FIG. 2 is an exemplary flow diagram illustrating a method 200 for nested quantum-classical hybrid molecular geometry optimization implemented by the system of FIG. 1 according to some embodiments of the present disclosure. In an embodiment, the system 100, comprising the one or more classical hardware processors communicably coupled to the plurality of unentangled QPUs 126 includes one or more data storage devices or the memory 110 operatively coupled to the classical hardware processors 108, (also referred to as one or more hardware processor(s) 108) and the quantum memory 128 operatively coupled to the plurality of unentangled QPUs and are configured to store instructions for execution of steps of the method 200 by the one or more hardware processors 108 and the plurality of unentangled QPUs 126. The steps of the method 200 of the present disclosure will now be explained with reference to the components or blocks of the system 100 as depicted in FIG. 1 and the steps of flow diagram as depicted in FIG. 2. The method 200 may be described in the general context of computer executable instructions. Generally, computer executable instructions can include routines, programs, objects, components, data structures, procedures, modules, functions, etc., that perform particular functions or implement particular abstract data types. The method 200 may also be practiced in a distributed computing environment where functions are performed by remote processing devices that are linked through a communication network. The order in which the method 200 is described is not intended to be construed as a limitation, and any number of the described method blocks can be combined in any order to implement the method 200, or an alternative method. Furthermore, the method 200 can be implemented in any suitable hardware, software, firmware, or combination thereof.
Now referring to FIG. 2, at step 202 of the method 200, the one or more classical hardware processors of the system comprising the one or more classical hardware processors communicably coupled to a plurality of unentangled Quantum Processor Units (QPUs) are configured by the programmed instructions to receive via respective interfaces a plurality of initial nuclear coordinates x_0 associated with an initial geometry of a molecule. In an embodiment, the molecule is a linear molecule. In another embodiment, the molecule is a non-linear molecule.
In an embodiment, the plurality of initial nuclear coordinates are initialized based on Hartree-Fock electronic structure method. The Hartree-Fock electronic structure method is used to ensure that the plurality of initial nuclear coordinates values is a good initial guess to the equilibrium geometry of the molecule by computing the full dimensional Hessian either analytically or numerically.
In another embodiment, the plurality of initial nuclear coordinate values are numerically verified by verifying that all eigenvalues of the Hessian are positive except for 6 (5 for linear molecules) zero eigenvalues that correspond to translational and rotational motion of the entire molecule. The geometry optimized at the classical Hartree-Fock (HF) level of theory often serves as a good initial geometry to begin with.
At step 204 of the method 200, the one or more classical hardware processors 108 and the plurality of unentangled QPUs 126 are configured by the programmed instructions to obtain an equilibrium geometry of the molecule associated with all degrees of freedom by iteratively performing steps 204A through 204G until energy gradient associated with the plurality of current nuclear coordinates is equal to a predefined threshold.
At step 204A of the method 200, the one or more classical hardware processors 108 are configured by the programmed instructions to generate a parameterized electronic Hamiltonian based on the plurality of current nuclear coordinates using an approximation technique, wherein the plurality of initial nuclear coordinates are assigned to the plurality of current nuclear coordinates. The parameterized electronic Hamiltonian is constructed to measure physical observables, such as energy, by evaluating the expectation value of the target Hamiltonian.
At step 204B of the method 200, the one or more classical hardware processors 108 are configured by the programmed instructions to generate a Qubit Hamiltonian based on the parameterized electronic Hamiltonian using a Fermion to Qubit mapping technique. Here, the electronic Hamiltonian is converted into a qubit Hamitonian by mapping the fermionic ladder operators to Pauli operators using a Fermion-to-Qubit mapping technique. Some of the exemplary qubit mapping techniques are Jordan-Wigner mapping, Bravyi-Kitaev mapping, Parity mapping, Bravyi-Kitaev Superfast mapping, etc. The mappings are such that the number of Qubits are proportional to the number of MO (Molecular Orbital) of the electronic structure problem. For example, if a state of a Fermionic system is |0001>, which indicates that there is one electron occupying the first orbital out of the four orbitals, then the corresponding Qubits state under parity mapping is |1111>.
A Qubit is known as the quantum bit, which is the smallest unit of quantum information, and can be considered equivalent to the bit in classical computing. Unlike bit, a Qubit can stay in a superposition of 0 and 1 states, hence a Qubit register can store exponential amount of information simultaneously, compared to classical register. For example, an N=3 bit classical register can store N=3 bit information, at a given point of time, whereas a N=3 Qubit quantum register can store 2N (23 = 8) bits of information simultaneously, each bitstring is 3-bit in length. Moreover, the Qubits can form quantum entanglement, which is a property of quantum mechanics, by which two Qubits’ states get correlated or entangled and can naturally store or manipulate electronic correlation.
In an embodiment, the electronic Hamiltonian is transformed to an equivalent sequence of Pauli operators acting on qubits of the quantum computer using an encoding as shown in equation [1]. Here, h_pq (x_0 ) and g_pqrs (x_0 ) are the one-body and two-body integrals computed efficiently using the classical HF method in the chosen molecular orbital basis, a_p^† and a_q are the fermionic creation and annihilation operators for orbitals p and q respectively.
H_ele (x_0 )=?_pq¦?h_pq (x_0 ) a_p^† a_q+1/2 ?_pqrs¦?g_pqrs (x_0 ) a_p^† a_q^† a_r a_s ??………………[1]
The qubit representation of the electronic Hamiltonian can be written as shown in equation [2]
H_q (x_0 )=?_j¦?h_j (x_0)P_j ?=?_j¦?h_j (x_0)?_i¦s_i^j ? ……………[2]
where h_j (x_0) are real coefficients inheriting the dependence on the nuclear coordinates x_0 and s_i^j?{I,X,Y,Z} is a single-qubit Pauli operator. The scaling of the number of s_i^j required to represent one fermionic ladder operator depends on the encoding technique used. The wavefunction of the chemical system being simulated is encoded into the quantum state of a set of qubits, which can then be conveniently manipulated using the available set of gates on the quantum computer.
At step 204C of the method 200, the plurality of unentangled QPUs 126 are configured by the programmed instructions to generate a trial state associated with the molecule using a quantum circuit comprising a plurality of tunable parameters. The quantum circuit includes layers of parameterized single-qubit rotation and two-qubit entangling gates that generates a trial state |?(?)>, where ? is a set of tunable parameters.
At step 204D of the method 200, the plurality of unentangled QPUs 126 are configured by the programmed instructions to compute an electronic energy associated with the molecule for the trial state based on the Qubit Hamiltonian using a Quantum Expectation Estimation (QEE) technique.
In an embodiment, the electronic energy of the system for the prepared trial state is calculated as the expectation value of the qubit Hamiltonian H_q (x_0 ) as given in equation [3]
E(?,x_0 )= ………………….[3]
At step 204E of the method 200, the one or more classical hardware processors 108 are configured by the programmed instructions to compute a ground state energy for the plurality of current nuclear coordinates by optimizing the plurality of tunable parameters using a classical optimizer until one of a plurality of constraints is satisfied. The plurality of constraints includes (i) if difference between the electronic energies of successive iterations is less than a predefined energy threshold and (ii) if an energy gradient associated with the plurality of tunable parameters is less than a predefined energy gradient threshold.
In an embodiment, an approximation to the ground state of the system is obtained by optimizing ? keeping x_0 fixed using classical gradient-free or gradient based optimization algorithms till the energy converges to a minimum, or gradient of energy with respect to circuit parameters becomes close to zero within a tolerable limit.
The energy gradient with respect to circuit parameters ?E(?,x_0 )/?? can be evaluated analytically using the parameter-shift rule. The energy obtained at this point corresponds to the ground state energy of the system for the specified set of nuclear coordinates x_0.
At step 204F of the method 200, the one or more classical hardware processors 108 and the plurality of unentangled QPUs 126 are configured by the programmed instructions to estimate an electronic energy gradient associated with the plurality of current nuclear coordinates based on the ground state energy using a numerical differentiation technique.
At step 204G of the method 200, the one or more classical hardware processors 108 are configured by the programmed instructions to update the plurality of current nuclear coordinates based on the electronic energy gradient using a classical optimizer.
In an embodiment, the plurality of current nuclear coordinates are updated either using a perturbative approach maintaining the symmetry of the molecule or using numerical optimization methods and the process repeats until the energy gradient with respect to the nuclear coordinates vanishes. The energy gradient with respect to nuclear coordinates is computed via the Hellman-Feynman theorem
?E(?,x)/(?x_i )= ……………………..[4]
where the derivatives (??H?_q (x))/(?x_i ) of the Hamiltonian can be evaluated analytically or numerically using finite-differences.
In an embodiment, an Atomic Simulation Environment (ASE) package is used for simulating the present disclosure. A molecule, ion, a lattice or any collection of atoms are defined using the Atoms class provided by the ASE package and all the related properties including atomic numbers, positions, masses, charges, magnetic moments, and velocities are stored in database. With the data stored in Atoms object, ASE enables the straightforward calculation of potential energies and forces of the atomic arrangement according to the underlying algorithm of the calculator. These quantities can then be used to perform many different simulations. The various simulations supported by ASE currently include molecular dynamics with different controls such as thermostats, geometry optimization using atomic forces and genetic algorithm, transition state optimization using the Nudged Elastic Band (NEB) method, global geometry optimization using basin hopping, vibrational mode analysis among many others. In addition, ASE provides a database module using which a user can conveniently store and retrieve the results of the simulations performed in a number of file formats (more than 65 file formats are being supported). Further, a plurality of open-source Deep Learning and Machine Learning (DL/ML) packages, such as Torch Anakin-me (TorchANI), are available to work on top of ASE. In such cases, ASE is used as an interface to communicate with different electronic structure codes to train their DL/ML models and possibly infer more accurate estimates of molecular properties.
In an embodiment, to enable quantum-classical hybrid simulation of molecules and materials in a seamless manner (which is relevant for the current NISQ era), there is a need to bring quantum and classical workflows under a unified framework. One can think of outsourcing the computationally expensive workloads of classical devices, such as determining near-exact ground state energies and forces, to quantum processors, thus gaining a polynomial, if not exponential, advantage in resource utilization. The present disclosure demonstrated this idea of bringing together quantum and classical workflows under a common framework by integrating a Variational Quantum Eigen Solver (VQE)-based energy and force calculator to atomic simulation environment (ASE) package. Such an interface can be written for any quantum algorithm capable of estimating ground state energy of molecules, i.e., a plurality of quantum algorithms like VQE, QPE, QITE, etc.
In an embodiment, the functionality of the present disclosure is demonstrated by optimizing the structure of water (H2O) molecule using the in-built local optimizers available in ASE. The steps undertaken for this simulation task are:
Instantiate the ASE Atoms class to represent a water molecule with initial geometry as given by classical HF level of theory.
Import the hybrid quantum energy and force calculator and configure it by passing the necessary parameters as a Python dictionary through a simple text file.
Attach the calculator to the Atoms object created in step i.
Import and create an instance of the local optimizer, say BFGS, from the module ase.optimize
Pass the Atoms object to the optimizer and run the optimizer until the force on all atoms goes below a predefined threshold (?10?^(-5) Hartree/Angstorms)
To illustrate the flexibility of the present disclosure, the energy and forces computation is performed in different basis sets. Here, the geometry optimization is performed in both 6-31G* and cc-pVTZ basis sets. The results are presented in Table I.
Table 1: The optimized geometries and minimum energies as obtained from the ASE interface to hybrid quantum calculator
MO Basis set Optimized rOH (Å) Optimized aHOH (degrees) Minimum Energy (Hartrees) Iterations
6-31G* 0.94767139 105.6029291 -76.009489361 7
cc-pVTZ 0.94065276 106.006883 -76.057836158 5

In an embodiment, the IRC-VQE applied to Conformational Isomer of Ethane (C2H6) is explained below. For C2H6, the H-C-C-H torsional angle is chosen as the reaction coordinate (IRC). For each torsion angle, the H atoms were further relaxed along the C-H bond to optimize the geometry and minimum energy was computed for each iteration.
In an embodiment, a finite-difference based approach has been constructed to illustrate scalability of the present disclosure, wherein starting from the reactant IRC, the reaction pathway can be traversed towards the products state via intermediate Transition State (TS) by stepping along the eigenmode directions using Newton-Raphson step. Following is the algorithmic workflow:
Considering mass-weighted cartesian coordinates x, starting with initial x_0 for the reactants’ coordinates, step change ?x is given by
?x= -H^(-1) g …………………[5]
where ?x=x-x_0 is the Newton-Raphson step taken on the PES from initial IRC/guess geometry x_0 and H is the Hessian evaluated at x_0 H is often used as Identity matrix).
Introducing a real Unitary U that diagonalizes the H we obtain
?x =-U?(U?^T H^(-1) U)?(U?^T g) ………….……………..[6]
=-?_(a,k)¦?U_(a,k) (??_a?^(-1) d_(a,k) ?)U?_(a,k)^T g_k ?
= ?_a¦?-g_a/?_a ? V^a
where g is the gradient vector evaluated at x_0 and V^a is the eigenvector with eigenvalue ?_a .
Since H is a symmetric matrix, its eigenvectors V^a,V^b,…etc., are orthogonal to each other. Thus, it could be stepped along the eigenmode directions that are uncoupled to each other.
For finding a minimum, it is essential to ensure that the gradient vanishes, and the Hessian is positive-definite, which corresponds to the step opposite to the gradient component g_a along the eigenmode direction V^a.
For locating a TS, it is essential to ensure that the gradient vanishes, and the Hessian has exactly one negative eigenvalue, which corresponds to the step uphill in energy along the eigenvector with negative eigenvalue.
Hence, the Newton-Raphson step (H^(-1) g) is sufficient for locating both the minima corresponding to reactants and products on the PES as well as the TS (transition state) which corresponds to the first order saddle point on the PES.
In an embodiment, Nucleophilic Substitution, second order (SN2) reaction pathway tracing is performed using the present disclosure. SN2 reaction is a bimolecular nucleophilic substitution reaction of an organic compound, in which an electrophile (electron-withdrawing group) gets substituted by a nucleophile (electron-pair donor) via a single transition state, where the bond-breaking and bond-forming occurs simultaneously. For e.g., in a SN2 reaction between fluoromethane (CH3F) and chloride ion (Cl-), the carbon (C) (with pentacoordinate) at the transition state pushes out the leaving F- group of CH3F to the opposite side to form CH3Cl. The reaction pathway tracing of such SN2 reaction is performed using the present disclosure. The plane of three hydrogen atoms are kept fixed at x=0 (YZ plane) and the relaxation of chlorine and fluorine atoms along the three-fold symmetry axis (x-axis) for each IRC of C atom is allowed. The optimized geometry of F- + CH3Cl complex was obtained, wherein the optimized geometry of one IRC (x-coordinate of C) becomes the initial geometry of the next IRC. To probe the influence of the MO basis sets, 6-31G*, cc-pVDZ and 6-311++G** are used. Further, the active space transformation CAS (2,2) has been applied followed by Parity mapping, so that the number of qubits required for quantum simulation is 2 with (UCCSD) ansatz circuit depth of 15.
In an embodiment, the simulation results depicted in FIG. 3 illustrates SN2 reaction pathway for CH3F + Cl- using IRC-VQE, as a function of various MO basis sets. Now referring to FIG. 3, the minimum expected ground state energy for CH3Cl + F- complex was only observed with 6-311++G** (306) MO basis sets, emphasizing further the importance of orbital functionals on the accuracy of computing. The TS was also not observed for 6-31G* (302) and cc-pvDZ (304) MO basis sets. It has been observed that the present disclosure has obtained the height of the energy barrier between CH3Cl + F- and the TS configuration to be 5.648 milli-Hartree, which is in very close agreement with the classically computed barrier height of approximately 5.322 milli-Hartree.
In an embodiment, FIGS. 4A through 4C demonstrates the evolution of each of the internal coordinates of the SN2 reaction across the reaction pathway on the PES. Now referring to FIG. 4A through 4C, it was observed that the variation in C-H bond length along the reaction path for the entire range of reaction coordinates is very small (˜0.08 Å) compared to the range of variations in C-F (˜1.2 Å) and C-Cl bond lengths (˜1.75 Å). The optimized C-H bond lengths showed symmetric behavior with respect to the plane of H atoms (x=0.0) along the reaction path. Near the TS, where the bonding between C, Cl and F atoms are not well-defined, the corresponding bond lengths showed an abrupt transition in the 6-311++G** (406) basis set whereas in the other basis sets they varied smoothly. This is consistent with the energy diagram as no TS was observed in these basis sets. As the C, F and Cl atoms are subjected to relax along the same line, and the C-H bond length variation is negligibly small compared to C’s movement on the x-axis, the H atoms take almost fixed positions in the plane, thus resulting in a linear evolution of F-C-H and Cl-C-H angles.
In an embodiment, to perform the transition state search and the vibrational frequency calculation from the geometrical data obtained by IRC-VQE, the present disclosure followed the approach described below:
Compute the non-mass weighted Hessian numerically or analytically at the optimized (initial/IRC-VQE transition state) geometry, check if the Hessian has 6 zero eigenvalues (5 for linear molecules), 3N-6 positive eigenvalues for minimum; 3N-7 positive and 1 negative eigenvalue for TS
Compute the mass-weighted Hessian from the non-mass weighted Hessian
H_(j,k)^' = H_(j,k)^ ??(m?_j m_k)?^(-0.5)………………..[7]
From the eigenvalues of the mass-weighted Hessian, compute vibrational frequencies
?^2 x_j=?_k¦?H_(j,k)^' x_k ?………………………[8]
Eigenvectors of the mass-weighted Hessian gives the normal modes of vibration
The transition state (first order saddle point) corresponds to the normalized eigenvector with negative eigenvalue, which gives the IRC direction.
Table II illustrates the eigenvalues & vibrational frequencies for SN2 reaction at transition state, wherein it was observed that one negative eigenvalue and the corresponding imaginary vibrational frequency (bold). The associated eigenvector was found to be parallel to the x-axis for C, Cl and F atoms. Also, it was observed that six zero eigenvalues corresponding to three translational and three rotational degrees of motion.
Table II: Eigenvalues & vibrational frequencies for SN2 reaction at the transition state
Eigenvalues
(Hartree/A2) Vibrational
Frequencies(cm-1) Eigenvalues
(Hartree/A2) Vibrational
Frequencies(cm-1)
1.566025 3354.2127 +0.0 j 0.057994 236.5486 +0.0 j
4.078312 3553.0428 +0.0 j 0.057997 236.5335 +0.0 j
4.078311 3553.0399 +0.0 j 6.75113e-05 6.8053 +0.0 j
-0.433398 0.0 + 473.378 j 6.50592e-05 4.8044 +0.0 j
0.431176 1271.7456 +0.0j 6.50105e-06 4.6091 +0.0 j
0.242063 1267.5429 +0.0 j -4.9589e-06 0.0 +1.8054 j
0.444211 1527.0498 +0.0 j -4.5771e-07 0.0 +0.54902 j
0.444214 1527.0357 +0.0 j -6.8326e-07 0.0 +0.66869 j
0.184989 1032.2589 +0.0 j
0.184990 1032.2526 +0.0 j
In an embodiment, to study the role played by different active spaces in geometry optimization over quantum processors, the case of CH3Cl molecule in the 6-311++G** higher-order basis was taken and different Active Spaces (AS) ranging from the minimum configuration of two active electrons in two molecular orbitals to six electrons in six orbitals was considered. The present disclosure primarily focused on the improvement in the calculated ground state energy and the amount of quantum resources required as a function of the size of AS. The results are presented in Table III. Given the performance analysis of various classical optimizers in the previous test case of H2O molecule, Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm is used for updating the nuclear coordinates of this molecule and Unitary variant coupled-cluster singles and doubles (UCCSD) ansatz is used to prepare the ground state wavefunction. The problem is viewed as a four-variable optimization problem by fixing the plane of the three hydrogen (H) atoms and confining the motion of carbon and chlorine atoms along the axis perpendicular to the plane of H atoms.
Table III: Influence of different active spaces on the computed minimum energy and equilibrium geometry
CAS (2,2) CAS (2,4) CAS (4,4) CAS (4,6) CAS (6,6)
Minimum Energy (Ha) -499.13262836
-499.1326289
-499.13262937
-499.1326423
-499.1326554

rCCl (Å) 1.78857 1.78857 1.788564 1.788626 1.788614
rCH (Å) 1.07819 1.07819 1.07819 1.07819 1.07819
aClCH (°) 108.237841 108.238035 108.238035 108.23728 108.23752
aHCH (°) 110.676142 110.675697 110.67573 110.67651 110.67638
Iterations 4 5 6 4 5
As evident from Table III, a significant improvement in the ground state energy for the specified set of active spaces is not observed. There is a systematic improvement in the simulation outcomes with increasing size of AS, however the optimal choice of AS depends highly on the system considered. In recent years, techniques have been proposed to automatically select the optimal AS for molecular systems. The scaling of the quantum resources in terms of the number of qubits, entangling gates, ansatz depth under Parity mapping with different choices of AS is presented in IV.
Table IV: Scaling of quantum resources with the size of the active space. CAS(n,m) represents an active space composed of n active electrons and m molecular orbitals
CAS(2,2) CAS(2,4) CAS(4,4) CAS(4,6) CAS(6,6) CAS(6,8) CAS(8,8) CAS(8,10)
No. of Qubits 2 6 6 10 10 14 14 18
No. of Parameters 3 15 26 92 117 315 360 804
Ansatz Depth 15 851 1647 9390 12248 43035 49529 134148
Total Gates 21 1527 2866 16794 21697 77763 89172 245004
No. of multi-qubit gates 4 560 1096 7080 9200 34448 39568 111416
In an embodiment, influence of choice of dataset associated with the present disclosure is analyzed as explained below: For example, the effect of basis sets in geometry optimization routines that use variational quantum algorithms to calculate molecular energy and forces are analyzed as illustrated in FIGS. 5A and 5B. Towards this goal, the molecular structure of CH3Cl molecule is optimized in increasingly larger basis sets of STO-6G (502), 6-31G* (504), 6-311++G** (506), and cc-pVDZ (508). The geometric parameters chosen to represent the structure are the four internal coordinates: C-Cl and C-H bond lengths, and H-C-Cl and H-C-H angles denoted as P1, P2, P3, P4, respectively. The initial estimates for these parameters were set to the optimized coordinates of the classical HF method in the respective basis. The active space was fixed to CAS (2,2) and all other essential arguments required for the quantum computation of energy and force were kept the same as that of the case of H2O molecule.
FIGS. 5A and 5B illustrates optimization of CH3Cl in different basis sets using the finite-difference based variational quantum approach. Here, FIG. 5A illustrates the correspondence between the geometric parameters P_x plotted on the x-axis and the internal coordinates of the molecule and FIG. 5B illustrates the difference between the optimized parameters and experimental values plotted for each basis set. The gray band 510 denotes error limit within ±0.05 Å of experimental values.
As shown in FIG. 5B, the deviation of C-Cl (P1) and C-H (P2) bond lengths from experimental equilibrium values are not significant in all the chosen basis sets. However, the error in the two bond angles (represented as P2 and P3 in FIG. 5A) becomes higher as the size of the basis set reduces, in particular, the optimized geometry in STO-6G basis set is significantly different from experiment. This could be understood from the general notion that bond stretching requires more energy than angle bending or torsional motion. One need to keep in mind that the choice of CAS (2,2) active space for all basis sets has brought in associated errors in the computation. As more realistic and higher-order basis sets are employed in the electronic structure calculation, more accurate estimates are obtained as observed previously in the one-dimensional problems related to lithium hydride (LiH) and hydrogen (H2).
The written description describes the subject matter herein to enable any person skilled in the art to make and use the embodiments. The scope of the subject matter embodiments is defined herein and may include other modifications that occur to those skilled in the art. Such other modifications are intended to be within the scope of the present disclosure if they have similar elements that do not differ from the literal language of the embodiments or if they include equivalent elements with insubstantial differences from the literal language of the embodiments described herein.
The embodiments of present disclosure herein addresses unresolved problem of Nested Quantum-Classical Hybrid Method and System for Molecular Geometry Optimization. The present disclosure Provides an efficient and accurate workflow to compute the minimum energy configuration of different molecular systems utilizing the inherent capabilities of both classical and quantum processors. The present disclosure proposed two methods useful for the NISQ (Noisy Intermediate Scale Quantum) era: 1) A nested, perturbative approach using VQE for optimizing only certain interatomic configurations (bond stretching, bending or torsion) for a symmetry-reduced, small molecule. This doesn't require the explicit calculation of energy gradients and improves the computational efficiency of the overall optimization process. 2) Quantum-Classical hybrid finite-difference-based approach: integrating VQE with a classical numerical optimizer towards a nested optimization can be implemented on current NISQ devices.
It is to be understood that the scope of the protection is extended to such a program and in addition to a computer-readable means having a message therein; such computer-readable storage means contain program-code means for implementation of one or more steps of the method, when the program runs on a server or mobile device or any suitable programmable device. The hardware device can be any kind of device which can be programmed including e.g., any kind of computer like a server or a personal computer, or the like, or any combination thereof. The device may also include means which could be e.g., hardware means like e.g., an application-specific integrated circuit (ASIC), a field-programmable gate array (FPGA), or a combination of hardware and software means, e.g., an ASIC and an FPGA, or at least one microprocessor and at least one memory with software processing components located therein. Thus, the means can include both hardware means and software means. The method embodiments described herein could be implemented in hardware and software. The device may also include software means. Alternatively, the embodiments may be implemented on different hardware devices, e.g., using a plurality of CPUs.
The embodiments herein can comprise hardware and software elements. The embodiments that are implemented in software include but are not limited to, firmware, resident software, microcode, etc. The functions performed by various components described herein may be implemented in other components or combinations of other components. For the purposes of this description, a computer-usable or computer readable medium can be any apparatus that can comprise, store, communicate, propagate, or transport the program for use by or in connection with the instruction execution system, apparatus, or device.
The illustrated steps are set out to explain the exemplary embodiments shown, and it should be anticipated that ongoing technological development will change the manner in which particular functions are performed. These examples are presented herein for purposes of illustration, and not limitation. Further, the boundaries of the functional building blocks have been arbitrarily defined herein for the convenience of the description. Alternative boundaries can be defined so long as the specified functions and relationships thereof are appropriately performed. Alternatives (including equivalents, extensions, variations, deviations, etc., of those described herein) will be apparent to persons skilled in the relevant art(s) based on the teachings contained herein. Such alternatives fall within the scope of the disclosed embodiments. Also, the words “comprising,” “having,” “containing,” and “including,” and other similar forms are intended to be equivalent in meaning and be open ended in that an item or items following any one of these words is not meant to be an exhaustive listing of such item or items, or meant to be limited to only the listed item or items. It must also be noted that as used herein, the singular forms “a,” “an,” and “the” include plural references unless the context clearly dictates otherwise.
Furthermore, one or more computer-readable storage media may be utilized in implementing embodiments consistent with the present disclosure. A computer-readable storage medium refers to any type of physical memory on which information or data readable by a processor may be stored. Thus, a computer-readable storage medium may store instructions for execution by one or more processors, including instructions for causing the processor(s) to perform steps or stages consistent with the embodiments described herein. The term “computer-readable medium” should be understood to include tangible items and exclude carrier waves and transient signals, i.e., be non-transitory. Examples include random access memory (RAM), read-only memory (ROM), volatile memory, nonvolatile memory, hard drives, CD ROMs, DVDs, flash drives, disks, and any other known physical storage media.
It is intended that the disclosure and examples be considered as exemplary only, with a true scope of disclosed embodiments being indicated by the following claims.

,CLAIMS:
1. A quantum simulation method (200) comprising:
receiving (202), by one or more classical hardware processors of a system comprising the one or more classical hardware processors communicably coupled to a plurality of unentangled Quantum Processor Units (QPUs) via an interface, a plurality of initial nuclear coordinates associated with an initial geometry of a molecule, wherein the plurality of initial nuclear coordinates is initialized based on Hartree-Fock electronic structure method; and
obtaining (204), by the one or more classical hardware processors and the plurality of unentangled QPUs, an equilibrium geometry of the molecule associated with all degrees of freedom by iteratively performing, until energy gradient associated with a plurality of current nuclear coordinates is equal to a predefined threshold, actions comprising:
generating (204A), by the one or more classical hardware processors, a parameterized electronic Hamiltonian based on the plurality of current nuclear coordinates using an approximation technique, wherein the plurality of initial nuclear coordinates is assigned to the plurality of current nuclear coordinates;
generating (204B), by the one or more classical hardware processors, a Qubit Hamiltonian based on the parameterized electronic Hamiltonian using a Fermion to Qubit mapping technique;
generating (204C), by the plurality of unentangled QPUs, a trial state associated with the molecule using a quantum circuit comprising a plurality of tunable parameters;
computing (204D), by the plurality of unentangled QPUs, an electronic energy associated with the molecule for the trial state based on the Qubit Hamiltonian using a Quantum Expectation Estimation (QEE) technique;
computing (204E), by the one or more classical hardware processors, a ground state energy for the plurality of current nuclear coordinates by optimizing the plurality of tunable parameters using a classical optimizer until one of a plurality of constraints is satisfied;
estimating (204F), by the one or more classical hardware processors, and the plurality of unentangled QPUs, an electronic energy gradient associated with the plurality of current nuclear coordinates based on the ground state energy using a numerical differentiation technique; and
updating (204G), by the one or more classical hardware processors, the plurality of current nuclear coordinates based on the electronic energy gradient using a classical optimizer.
2. The method as claimed in claim 1, wherein the molecule is one of a) a linear molecule and b) a non-linear molecule.
3. The method as claimed in claim 1, wherein the electronic energy represents the expectation value of the qubit Hamiltonian.
4. The method as claimed in claim 1, wherein the plurality of constraints comprises (i) checking whether difference between the electronic energies of successive iterations is less than a predefined energy threshold and (ii) checking whether an energy gradient associated with the plurality of tunable parameters is less than a predefined energy gradient threshold.
5. A system (100) comprising:
one or more classical hardware processors (108) communicably coupled to a plurality of unentangled QPUs (126) via interfaces, wherein the one or more classical hardware processors comprises at least one memory (110) storing programmed instructions; one or more Input /Output (I/O) interfaces (118); and one or more hardware processors (108) operatively coupled to the at least one memory (110), wherein
the one or more classical hardware processors (108) are configured by the programmed instructions to:
receive a plurality of initial nuclear coordinates associated with an initial geometry of a molecule, wherein the plurality of initial nuclear coordinates is initialized based on Hartree-Fock electronic structure method; and
at least one of the one or more hardware processors (108) and the plurality of unentangled QPUs (126) are configured by the programmed instructions to:
obtain an equilibrium geometry of the molecule associated with all degrees of freedom by iteratively performing, until energy gradient associated with a plurality of current nuclear coordinates is equal to a predefined threshold, actions comprising:
generating, by the one or more classical hardware processors, a parameterized electronic Hamiltonian based on the plurality of current nuclear coordinates using an approximation technique, wherein the plurality of initial nuclear coordinates are assigned to the plurality of current nuclear coordinates;
generating, by the one or more classical hardware processors, a Qubit Hamiltonian based on the parameterized electronic Hamiltonian using a Fermion to Qubit mapping technique;
generating, by the plurality of unentangled QPUs, a trial state associated with the molecule using a quantum circuit comprising a plurality of tunable parameters;
computing, by the plurality of unentangled QPUs, an electronic energy associated with the molecule for the trial state based on the Qubit Hamiltonian using a Quantum Expectation Estimation (QEE) technique;
computing, the one or more classical hardware processors a ground state energy for the plurality of current nuclear coordinates by optimizing the plurality of tunable parameters using a classical optimizer until one of a plurality of constraints is satisfied;
estimating, by the one or more classical hardware processors, an electronic energy gradient associated with the plurality of current nuclear coordinates based on the ground state energy using a numerical differentiation technique; and
updating, by the one or more classical hardware processors, the plurality of current nuclear coordinates based on the electronic energy gradient using a classical optimizer.
6. The system of claim 5, wherein the molecule is one of a) a linear molecule and b) a non-linear molecule.
7. The system of claim 5, wherein the electronic energy represents the expectation value of the qubit Hamiltonian.
8. The system of claim 5, wherein the plurality of constraints comprises (i) checking whether difference between the electronic energies of successive iterations is less than a predefined energy threshold and (ii) checking whether an energy gradient associated with the plurality of tunable parameters is less than a predefined energy gradient threshold.