FORM 2
THE PATENTS ACT, 1970
(39 of 1970)
&
THE PATENT RULES, 2003
COMPLETE SPECIFICATION (See Section 10 and Rule 13)
Title of invention:
SIMULATING MOLECULAR REACTION PATHWAY USING
INTRINSIC REACTION COORDINATE DRIVEN VARIATIONAL
QUANTUM EIGENSOLVER
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
Preamble to the description
The following specification particularly describes the invention and the manner in which it is to be performed.

TECHNICAL FIELD [001] The disclosure herein generally relates to the field of quantum computing and, more particularly, to a method and system for simulating molecular reaction pathway using Intrinsic Reaction Coordinate driven Variational Quantum Eigen solver (IRC-VQE).
BACKGROUND
[002] Simulating accurately the molecular ground state energy surface or the ground state energy derivative/gradient thereon is a computationally complex problem that requires understanding of the electronic structure of the molecule. In electronic structure problem, accurate modeling of underlying quantum-mechanical electronic exchange-correlation is required, which produces exponential cost of computation over classical computer, that grows exponentially with the problem size (the number of electrons present in the molecule).
[003] The conventional methods address the electronic structure problem on Noisy Intermediate-Scale Quantum (NISQ) hardware and quantum simulators. In parallel, some other scientific developments are spawning towards quantum error correction codes and fault-tolerant quantum computers for noise-free quantum simulations. Further, several quantum algorithms and variational quantum algorithms or methodologies have also been proposed to model the electronic structure problem. However, accurate simulation of molecular energy surface of a multidimensional molecule and thereby obtaining the reaction pathway is a challenging task.
SUMMARY
[004] 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 simulating molecular reaction pathway using IRC-VQE is provided. The method includes receiving, by one or more hardware processors, a plurality of molecular parameters pertaining to a multidimensional molecule.

Further, the method includes generating, by the one or more classical hardware processors, an active space based on the plurality of molecular parameters, using an active space transformation technique, wherein the active space comprises a plurality of core orbitals and a plurality of active orbitals. Furthermore, the method includes generating, by the one or more classical hardware processors, a Fermionic Hamiltonian corresponding to the plurality of active orbitals in an effective potential of the core orbitals using an electronic Hamiltonian constructor. Furthermore, the method includes obtaining, by the one or more classical hardware processors, a Qubit Hamiltonian corresponding to the plurality of active orbitals by mapping the Fermionic Hamiltonian to a corresponding Qubit using a Fermionic Hamiltonian-to-Qubit mapping technique. Furthermore, the method includes obtaining, by the one or more classical hardware processors, a reduced Qubit Hamiltonian by reducing the Qubit Hamiltonian using a Qubit reduction technique. Finally, the method includes obtaining, by the one or more classical hardware processors and the plurality of unentangled QPUs, a reaction pathway associated with the multidimensional molecule based on the reduced qubit Hamiltonian using an Intrinsic Reaction Coordinates driven Variational Quantum Eigen solver (IRC-VQE).
[005] In another aspect, a system for simulating molecular reaction pathway using IRC-VQE is provided. The system includes one or more classical hardware processors and a plurality of unentangled Quantum Processor Units (QPUs), wherein the one or more classical hardware processors are communicably coupled to the plurality of unentangled QPUs by respective 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 hardware processors are configured by the programmed instructions to receive a plurality of molecular parameters pertaining to a multidimensional molecule. Further, the one or more hardware processors are configured by the programmed instructions to generate an active space based on the plurality of molecular parameters, using an active space transformation technique,

wherein the active space comprises a plurality of core orbitals and a plurality of active orbitals. Furthermore, the one or more hardware processors are configured by the programmed instructions to generate a Fermionic Hamiltonian corresponding to the plurality of active orbitals in an effective potential of the core orbitals using an electronic Hamiltonian constructor. Furthermore, the one or more hardware processors are configured by the programmed instructions to obtain a Qubit Hamiltonian corresponding to the plurality of active orbitals by mapping the Fermionic Hamiltonian to a corresponding Qubit using a Fermionic Hamiltonian-to-Qubit mapping technique. Furthermore, the one or more hardware processors are configured by the programmed instructions to obtain a reduced Qubit Hamiltonian by reducing the Qubit Hamiltonian using a Qubit reduction technique. Finally, the one or more hardware processors are configured by the programmed instructions to obtain a reaction pathway associated with the multidimensional molecule based on the reduced qubit Hamiltonian using an Intrinsic Reaction Coordinates driven Variational Quantum Eigen solver (IRC-VQE).
[006] In yet another aspect, a computer program product including a non-transitory computer-readable medium having embodied therein a computer program for simulating molecular reaction pathway using IRC-VQE is provided. The computer readable program, when executed on a computing device, causes the computing device to receive a plurality of molecular parameters pertaining to a multidimensional molecule. Further, computer readable program, when executed on a computing device, causes the computing device to generate an active space based on the plurality of molecular parameters, using an active space transformation technique, wherein the active space comprises a plurality of core orbitals and a plurality of active orbitals. Furthermore, the computer readable program, when executed on a computing device, causes the computing device to generate a Fermionic Hamiltonian corresponding to the plurality of active orbitals in an effective potential of the core orbitals using an electronic Hamiltonian constructor. Furthermore, the computer readable program, when executed on a computing device, causes the computing device to obtain a Qubit Hamiltonian corresponding to the plurality of active orbitals by mapping the Fermionic Hamiltonian to a

corresponding Qubit using a Fermionic Hamiltonian-to-Qubit mapping technique. Furthermore, the computer readable program, when executed on a computing device, causes the computing device to obtain a reduced Qubit Hamiltonian by reducing the Qubit Hamiltonian using a Qubit reduction technique. Finally, the computer readable program, when executed on a computing device, causes the computing device to obtain a reaction pathway associated with the multidimensional molecule based on the reduced qubit Hamiltonian using an Intrinsic Reaction Coordinates driven Variational Quantum Eigen solver (IRC-VQE).
[007] 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
[008] 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:
[009] FIG. 1 is a functional block diagram of a system for simulating molecular reaction pathway using IRC-VQE, in accordance with some embodiments of the present disclosure.
[0010] FIG. 2 is an exemplary flow diagram illustrating a hybrid quantum-classical processor implemented method for simulating molecular reaction pathway using IRC-VQE, implemented by the system of FIG. 1, in accordance with some embodiments of the present disclosure.
[0011] FIG. 3 is an overall functional architecture for the hybrid quantum-classical processor implemented method for simulating molecular reaction pathway using IRC-VQE implemented by the system of FIG. 1, in accordance with some embodiments of the present disclosure.
[0012] FIGS. 4A to 4H illustrates experimental results for the hybrid quantum-classical processor implemented method for simulating molecular

reaction pathway using IRC-VQE implemented by the system of FIG. 1, in accordance with some embodiments of the present disclosure.
DETAILED DESCRIPTION OF EMBODIMENTS
[0013] 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 spirit and scope of the disclosed embodiments.
[0014] Conventional methods on Noisy Intermediate-Scale Quantum (NISQ) hardware and simulators are not capable of providing an accurate computation of the reaction pathway of a multidimensional molecule.
[0015] Embodiments herein provide a method and system for a quantum-classical hybrid solver, called the Intrinsic Reaction Coordinate (IRC) driven Variational Quantum Eigen solver (IRC-VQE), which traces accurately the reaction dynamics and reaction pathway of a molecular system. In particular, the present disclosure provides a molecular system that shows intramolecular dynamics. In one of the embodiments, such molecules are called fluxional molecules. Initially, the system receives a plurality of molecular parameters pertaining to a multidimensional molecule. Further, the system generates an active space based on the plurality of molecular parameters using an active space transformation technique. The active space includes a plurality of core orbitals and a plurality of active orbitals. Post generating the active space, the system generates an electronic energy Hamiltonian corresponding to the plurality of active orbitals in an effective potential of the electrons occupying the core orbitals using an electronic Hamiltonian constructor. Further, the system obtains a Qubit Hamiltonian corresponding to the plurality of active orbitals using a Fermion-to-Qubit mapping technique. Further, the system obtains a reduced Qubit Hamiltonian using a qubit

reduction technique. Finally, the system obtains a reaction pathway associated with the multidimensional molecule based on the reduced qubit Hamiltonian using the IRC-VQE.
[0016] Referring now to the drawings, and more particularly to FIGS. 1 through 4H, 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.
[0017] FIG. 1 is a functional block diagram of a system for simulating molecular reaction pathway using IRC-VQE, 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.
[0018] 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 and the hardware processors is interchangeably used throughout the document. Similarly, the classical computing system is a normal computing system.
[0019] 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.
[0020] 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.
[0021] 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.
[0022] 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.
[0023] The plurality of modules 112 include programs or coded instructions that supplement applications or functions performed by the system 100 for dynamic context based authentication factor recommendation. 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). In an embodiment, the plurality of modules 112 includes an active space generation module (shown in FIG. 3), a Fermionic Hamiltonian computation module (shown in FIG. 3), a Qubits mapping module (shown in FIG. 3), a Qubits reduction module (shown in FIG. 3) and an IRC-VQE module (shown in FIG. 3). FIG. 3 is an overall

functional architecture for the processor implemented method for simulating molecular reaction pathway using IRC-VQE implemented by the system of FIG. 1, in accordance with some embodiments of the present disclosure.
[0024] 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.
[0025] 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).
[0026] 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 are 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.
[0027] 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.
[0028] 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.
[0029] 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.
[0030] 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.

[0031] 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.
[0032] 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).
[0033] 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.
[0034] 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.
[0035] 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.
[0036] 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.

[0037] 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.
[0038] 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.
[0039] 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.
[0040] FIG. 2 is an exemplary flow diagram illustrating a method 200 for simulating molecular reaction pathway using IRC-VQE implemented by the system of FIG. 1 according to some embodiments of the present disclosure. In an embodiment, the system 100 includes one or more data storage devices or the memory 110 operatively coupled to the one or more hardware processor(s) 108 and is configured to store instructions for execution of steps of the method 200 by the one or more hardware processors 108. 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 FIG. 4 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.

[0041] At step 202 of the method 200, the one or more hardware processors 108 are configured by the programmed instructions to receive the plurality of molecular parameters. The plurality of molecular parameters includes a plurality of molecular IRC (an initial IRC), a plurality of molecular bond lengths, a plurality of molecular bond angles, a molecular charge and a spin multiplicity value.
[0042] For example, the molecular parameters of an ammonia (NH3) molecule for C3v configuration is given in Table I.
Table I

Initial IRC molecular molecular Molecular molecular spin
bond bond orbital charge multiplicity
lengths angles basis value
C3v N-H = 1.01 N-H bond ccPVTZ 0 1
configuration angstroms angle =
106.70
[0043] IRC are mass-weighted, molecular nuclear coordinates that a molecular system takes, while moving down the product and reactant valleys with zero kinetic energy from a transition state. It can also be defined as the mass-weighted steepest descent path on the potential energy surface (PES) of a molecular system, starting primarily from the transition state to the product state or equilibrium state, as a path of chemical reaction.
[0044] At step 204 of the method 200, the active space generation module 302 executed by the one or more hardware processors 108 is configured by the programmed instructions to generate the active space based on the plurality of molecular parameters using the active space transformation technique. The active space includes a plurality of core orbitals and a plurality of active orbitals. The plurality of active orbitals includes a plurality of valence orbitals and a plurality of virtual orbitals. For example, the core orbitals hold two electrons, the valence orbitals are partially occupied by electrons and the virtual orbitals holds no electrons.

[0045] In an embodiment, the active space generation module 302 separates the full electronic structure problem into an active space, including active electrons and active orbitals, and its environment, which includes the remaining electrons and orbitals called the inactive orbitals. In one of the embodiments, orbitals near the Fermi-level are usually taken as active orbitals. Only the active orbitals are mapped to Qubits and given a quantum computing treatment whereas the rest is treated with an efficient classical algorithm like Density Functional Theory (DFT) or Hartree-Fock (HF). The effective Hamiltonian incorporates a mean field potential generated by the inactive electrons, hence replaces the mapping of inactive orbitals to Qubits. The active state generation significantly reduces the number of Qubits necessary to calculate the energy surface, making the entire calculation efficient while keeping a good level of accuracy, can go to higher order MO basis.
[0046] At step 206 of the method 200, the Fermionic Hamiltonian construction module (304) executed by the one or more hardware processors 108 is configured by the programmed instructions to generate the Fermionic Hamiltonian corresponding to each of the plurality of active orbitals using the electronic Hamiltonian constructor. The electronic energy Hamiltonian and the Fermionic Hamiltonian can be used interchangeably.
[0047] In an embodiment, the Fermionic Hamiltonian constructor includes non-relativistic time independent Schrodinger equation, to construct the electronic structure problem and electronic energy (Hamiltonian) operator. For example, the Fermionic Hamiltonian under Born-Oppenheimer approximation (of stationary nuclei) is as given in equation (1). In another embodiment, the electronic Hamiltonian constructor includes a Second Quantization framework, which constructs the Hamiltonian operator in second quantization formulation as given in equation (2).
…………………….(1)
Now referring to equation (1), R I and ZI are position and atomic number of Ith nucleus and ri is the position of ith electron.


Now referring to equation (2), hpq is one body integral which represents the kinetic energy of electrons and electron-nuclei Coulomb interaction, hpqrs is a two body integral representing the electron-electron Coulomb repulsion, ap+ and aq are Fermionic creation and annihilation operators respectively.
[0048] In an embodiment, the electronic Hamiltonian constructor is represented as second quantization operator including Fermionic creation and annihilation operators. The Fermionic creation and annihilation operators create or annihilate respectively the occupation of an electron in one of the electronic orbitals (MO) under the constraint of preserving the total number of electrons in the molecular system (under non-relativistic limit).
[0049] At step 208 of the method 200, the Qubits mapping module 306 executed by the one or more hardware processors 108 is configured by the programmed instructions to obtain the Qubit Hamiltonian corresponding to each of the plurality of active orbitals by mapping the Fermionic Hamiltonian to a corresponding Qubit 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>.
[0050] 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.
[0051] At step 210 of the method 200, the Qubits reduction module 308 executed by the one or more hardware processors 108 is configured by the programmed instructions to obtain the reduced Qubit Hamiltonian using the Qubit reduction technique. For example, the Qubit reduction techniques comprise Z2 symmetry technique and parity reduction technique. For example, the Qubits pertaining to the ammonia molecule before and after Qubit reduction is 4 and 2 respectively.
[0052] At step 212 of the method 200, the IRC-VQE module 308 executed by the one or more hardware processors 108 and the plurality of unentangled QPUs 126 is configured by the programmed instructions to obtain a reaction pathway associated with the multidimensional molecule based on the reduced qubit Hamiltonian using the IRC-VQE. For example, the reaction pathway of ammonia molecule is shown in FIG. 4C.
[0053] In an embodiment, the method of computing the molecular ground state energy associated with the multidimensional molecule based on the reduced Qubit Hamiltonian using the IRC-VQE includes the following method steps. At first, an initial IRC associated with the multidimensional molecule, the reduced Qubit Hamiltonian associated with the multidimensional molecule and a parameterized quantum circuit are received by the IRC-VQE module. The initial IRC is assigned to a current IRC variable. Further, the reaction pathway associated with the multidimensional molecule is obtained by iteratively performing the following steps until obtaining an equilibrium energy pertaining to one of, a reactant and a product pertaining to the multidimensional molecule. The iterative steps include: (i) identification of a plurality of perturbations associated with the multidimensional molecule based on the current IRC using a perturbations identification technique by a perturbation identification module 312, (ii) obtaining a plurality of ground state energies by computing a ground state energy corresponding to each of the plurality of perturbations based on reduced Qubit

Hamiltonian using VQE by the ground state energy computation module 314, (iii) identifying the equilibrium energy based on the plurality of ground state energies by the equilibrium energy selection module 316, wherein the minimum ground state energy from the plurality of ground state energies is identified as the equilibrium energy, and (iv) obtaining the IRC corresponding to the identified equilibrium energy and updating the current IRC by the IRC identification and updation module 318.
[0054] In an embodiment, the quantum circuit corresponding is simulated using a quantum processor based hardware-efficient circuit or chemistry-inspired circuit. For e.g., hardware-efficient circuit (Two Local, EfficientSU2 (SU(2) corresponds to Special Unitary 2x2 transformation or equivalent unitary gates such as Pauli rotation gates) for running on real quantum hardware and chemistry-inspired (Unitary Coupled Cluster Single and Double (UCCSD), Parallel Unitary Coupled Cluster Single and Double (PUCCSD), etc.) for quantum simulators. The chemistry-inspired circuits are more accurate. The simulated quantum circuit includes a plurality of parameterized quantum gates.
[0055] In an embodiment, obtaining the reaction pathway of the ammonia molecule (NH3) is explained as follows. The IRC coordinates associated with the Nitrogen (N) atom of the ammonia molecule is obtained initially. The three Hydrogen atoms (H) are relaxed by perturbing around their mean position. In an embodiment, a plurality of constraints are applied during perturbation and the plurality of constraints includes (i) All three N-H bond lengths should be same for each of the plurality of perturbations, and (ii) All three H-N-H bond angles should be same for each of the plurality of perturbations. Further, the ground state energy is computed for each of the plurality of perturbations using VQE. Further, a minimum ground state energy (where the energy gradient vanishes) is identified as the equilibrium energy and corresponding IRC of three Hydrogen atoms as the mean position of the three Hydrogen (H) atoms for the next iteration. The IRC corresponding to the identified equilibrium energy is obtained and swapped with the initial IRC. This iteration is performed until obtaining an equilibrium energy pertaining to one of, a reactant and a product of the ammonia molecule.

[0056] In an embodiment, the method of obtaining the plurality of ground state energies by computing a ground state energy corresponding to each of the plurality of perturbations based on the reduced Qubit Hamiltonian using VQE is explained below. Initially, the method receives an initial state, a plurality of parameters and a reduced Qubit Hamiltonian. The plurality of parameters are assigned with initial values. Further, the method generates a quantum trial state using the parameterized quantum circuit. Further a plurality of expectation values are computed by applying the reduced Qubit Hamiltonian on the quantum trial state. The above steps are executed by the plurality of unentangled QPUs. Further, the method computes the trial state energy of the quantum trial state from the expectation values based on the reduced qubit Hamiltonian by the one or more classical hardware processors. Finally, the method optimizes the plurality of parameters such that the trial state energy is minimized using an optimization technique. This optimization is performed, by the one or more classical hardware processors.
[0057] In an embodiment, the repository 320 of the FIG. 3 includes a Molecular Orbital (MO) Basis Set data repository, a molecular geometry repository, a PES repository, a thermodynamic properties data repository, and a reaction pathway repository.
[0058] In an embodiment, the Molecular Orbital (MO) Basis Set Data Repository comprises the minimal basis sets of Gaussian orbitals, viz. STO-kG (k = 3, 6, etc.), which approximates each STO (Slater type Orbital) with k GTOs (Gaussian type Orbital) functions in the least squares. (For e.g., in STO-3G basis, each STO is represented by a linear combination of three Gaussian functions, often their exponents are constrained to be the same for all orbitals belonging to the same shell.). It also comprises split-valence basis set with double zeta functions on the valence orbital, some of which are 3-21G or 6-31G or 6-31G* (also called the Pople basis set). Here, for X-YZG, X denotes the number of primitive Gaussians (G) for each core atomic orbital basis function, whereas Y and Z indicate two sets of primitive Gaussian basis functions (G) for each of the valence orbitals. The basis sets also comprise ‘correlation-consistent’ basis sets, termed as cc-pVnZ

(correlation-consistent polarized Valence n Zeta) where n = D, T, Q for double, triple or quadrupole zeta functions. Computational Chemistry employs molecular orbital (MO) or a linear combination of atomic orbitals as basis sets, which comprises a set of algebraic functions to describe the spatial distribution of electronic wave function, as well as their spin configuration. Designing of or selecting a molecular orbital basis set is a complex problem, as the flexibility of the basis set must inherently accommodate all the correlated electronic orbitals and the spatial and spin representation accurately for computation yet be computationally efficient. A larger MO basis set often provides a more accurate potential energy surface (PES) simulation of a molecule, though with significant increase in computational resources.
[0059] In an embodiment, the PES repository includes data of a molecule as a function of at least one of the molecular parameters. The potential energy surface data includes quantum hardware or quantum simulator measurement data, quantum trial state data, qubit mapping data, qubit reduction related data, quantum circuit data and parameters therein, classical hardware optimizer iteration counts data, time of execution data, noise model data of the Qubits, error mitigation data of the Qubits, wherein the molecular parameter data comprises molecular intrinsic reaction coordinate, molecular bond lengths, bond angles, torsional, rotational and reflectional degrees of freedom, total charge, spin multiplicity, and the like.
[0060] In an embodiment, thermodynamic properties data repository includes molecular property data comprising molecular specific heat, molecular vibrational energy levels, splitting of molecular vibrational energy levels or energy gaps, data corresponding to the quantum tunneling, other properties corresponding to molecular dipole moment and the like.
[0061] In an embodiment, the reaction pathway repository includes data corresponding to the reaction path data comprising the molecular ground state energy of the transition molecular state, reactant and product molecular states, intra¬molecular transitional states such as molecular inversion states, reflection states and the like.

[0062] In an embodiment, the molecular geometry repository comprises equilibrium molecular geometry data, data comprising molecular orbitals basis sets, molecular point group data, molecular space group data, and the like.
[0063] In an embodiment, the preset disclosure is experimented for the ammonia molecule which is a 3D molecule. Similarly, the present disclosure is applicable to any kind of molecules with one or more dimensions.
[0064] In an embodiment, the present disclosure is experimented as follows: IBM quantum simulators, viz., State vector and QASM (Quantum Assembly language) with number of shots as 1024 have been used for the VQE simulation. The present disclosure considered (CAS (2,2)) (Complete Active Space) transformation, by choosing only two electrons and two molecular orbitals (corresponding to four spin orbitals) near the Fermi-level for quantum simulation. Further Parity mapping is performed to reduce the qubit counts by two and used ccPVTZ(correlation-consistent Polarized Valence Triple Zeta) MO basis set for higher accuracy. For ansatz, the UCCSD ansatz is constructed with 3 variational parameters having ansatz circuit depth of 23. Further, SLSQP (Sequential Least SQuares Programming) is used ad optimizer with maximum iteration counts of 500 for the state vector simulator and L_BFGS_B (Limited-memory BFGS Bound) classical optimizers with maximum iteration of 1000 for QASM simulator.
[0065] FIGS. 4A to 4H illustrates experimental results for the hybrid quantum-classical processor implemented method for simulating molecular reaction pathway using IRC-VQE implemented by the system of FIG. 1, in accordance with some embodiments of the present disclosure.
[0066] For example, FIG. 4A illustrates the simulation of the equilibrium ground state energy of NH3 molecule, as a function of the perturbation of reaction coordinates of Hydrogen atoms for fixed intrinsic reaction coordinates of Nitrogen (N) atom at D3h configuration. The vanishing of the energy gradient as a function of the perturbation of Hydrogen atom reaction coordinates occurs at a ground state energy of -56.2098 Hartree.
[0067] For example, FIG. 4B illustrates the simulation of the equilibrium ground state energy of NH3 molecule, as a function of the perturbation of reaction

coordinates of Hydrogen atoms for fixed intrinsic reaction coordinates of Nitrogen (N) atom at C3v configuration. The vanishing of the energy gradient as a function of the perturbation of Hydrogen atom reaction coordinates occurs at a ground state energy of -56.2182 Hartree.
[0068] For example, FIG. 4C illustrates the NH3 potential energy surface corresponding to its inversion reaction pathway. The symmetric double-minima potential energy in the potential energy surface corresponds to direct and inverted C3v configuration of NH3. The computed N-H bond length is 1.00312 angstroms, and the H-N-H bond angle is 106.437 degrees. The potential energy barrier height separating the two minima has been found to be 8 milli Hartree, with a barrier width of 0.7632 angstroms.
[0069] For example, FIG. 4D illustrates the performance of the IRC-VQE for C3v configuration of NH3, over quantum hardware using a hardware efficient quantum trial state, which resulted the accurate ground state energy in less than 30 iterations.
[0070] For example, FIG. 4E illustrates the performance of the IRC-VQE for D3h configuration of NH3, over quantum hardware using a hardware efficient quantum trial state, which resulted the accurate ground state energy in less than 20 iterations. For example, FIG. 4F and FIG. 4G illustrate two hardware runs by perturbing the three hydrogen atoms around their equilibrium coordinates for fixed reaction coordinate of Nitrogen atom, to measure the equilibrium energy and configuration, with two different perturbation range of Hydrogen atom reaction coordinates. The simulations produced similar equilibrium energy and configuration, as generated by the earlier hardware runs.
[0071] For example, FIG. 4H illustrates the working of the IRC-VQE quantum algorithm on another fluxional molecular system, which is PH3 molecule. To probe accurately the inversion pathway on the potential energy surface (PES) of PH3, the perturbation ranges have been carefully constructed for the Hydrogen atoms, for different fixed Phosphorous reaction coordinates. The entire range of Phosphorous reaction coordinates have been divided in 7 segments, and the perturbation ranges have been selected from -0.26 angstrom to 0.26 angstrom with

a step-size of 0.02 angstrom. By this, one ensures that the equilibrium ground state energy of PH3 for each z-coordinate of Phosphorous could be simulated accurately. The inversion pathway of PH3 has been traced accurately, and the corresponding potential barrier height and width obtained from the simulation are 55.7 milli Hartree and 1.5434 angstrom, respectively.
[0072] Now comparing the PES of PH3 (FIG. 4H) with NH3 (FIG. 4C), the potential barrier height as well as the barrier width observed for PH3 is much larger than that of NH3. It indicates that the phenomena of quantum tunneling is not observed in PH3.
[0073] 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 by the claims 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 claims if they have similar elements that do not differ from the literal language of the claims or if they include equivalent elements with insubstantial differences from the literal language of the claims.
[0074] The embodiments of present disclosure herein address the unresolved problem of obtaining the reaction pathway associated with the multidimensional molecule using IRC-VQE. The present disclosure is a quantum-classical hybrid solver, which traces accurately the reaction dynamics and reaction pathway of a molecular system. In particular, the present disclosure provides a molecular system that shows intramolecular dynamics.
[0075] 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 modules 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, GPUs and edge computing devices.
[0076] 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 modules described herein may be implemented in other modules or combinations of other modules. 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 and spirit 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 and in the appended claims, 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. 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.
[0077] 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.

WE CLAIM:
1. A quantum simulation method (200) performed by a system comprising one or more classical hardware processors and a plurality of unentangled Quantum Processor Units (QPUs), wherein the one or more classical hardware processors are communicably coupled to the plurality of unentangled QPUs by respective interfaces, wherein the quantum simulation method comprising:
receiving (202), by one or more classical hardware processors, a plurality of molecular parameters pertaining to a multidimensional molecule;
generating (204), by the one or more classical hardware processors, an active space based on the plurality of molecular parameters, using an active space transformation technique, wherein the active space comprises a plurality of core orbitals and a plurality of active orbitals;
generating (206), by the one or more classical hardware processors, a Fermionic Hamiltonian corresponding to the plurality of active orbitals in an effective potential of the core orbitals using an electronic Hamiltonian constructor;
obtaining (208), by the one or more classical hardware processors, a Qubit Hamiltonian corresponding to the plurality of active orbitals by mapping the Fermionic Hamiltonian to a corresponding Qubit using a Fermionic Hamiltonian-to-Qubit mapping technique;
obtaining (210), by the one or more classical hardware processors, a reduced Qubit Hamiltonian by reducing the Qubit Hamiltonian using a Qubit reduction technique; and
obtaining (212), by the one or more classical hardware processors and the plurality of unentangled QPUs, a reaction pathway associated with the multidimensional molecule based on the reduced qubit Hamiltonian using an Intrinsic Reaction Coordinates driven Variational Quantum Eigen solver (IRC-VQE).

2. The method as claimed in claim 1, wherein the method of obtaining the reaction pathway associated with the multidimensional molecule based on the reduced Qubit Hamiltonian using the IRC-VQE comprises:
receiving, by the plurality of unentangled QPUs from the one or more classical hardware processors via the respective interfaces, an initial IRC associated with the multidimensional molecule, the reduced Qubit Hamiltonian associated with the multidimensional molecule, and a parameterized quantum circuit, wherein the initial IRC is assigned to a current IRC variable; and
obtaining the reaction pathway associated with the multidimensional molecule by iteratively performing until obtaining an equilibrium energy pertaining to one of a reactant and a product pertaining to the multidimensional molecule:
identifying, by the plurality of unentangled QPUs, a plurality of perturbations associated with the multidimensional molecule based on the current IRC using a perturbations identification technique;
obtaining, by the plurality of unentangled QPUs, a plurality of ground state energies by computing the ground state energy corresponding to each of the plurality of perturbations based on the reduced Qubit Hamiltonian using the VQE;
identifying, by the one or more classical hardware processors, the equilibrium energy based on the plurality of ground state energies, wherein a minimum ground state energy from the plurality of ground state energies is identified as the equilibrium energy; and
obtaining, by the one or more classical hardware processors, the IRC corresponding to the identified equilibrium energy, wherein the current IRC is swapped with the obtained IRC.

3. The method as claimed in claim 2, wherein the parameterized quantum circuit comprises a plurality of parameterized gates.
4. The method as claimed in claim 1, wherein the plurality of molecular parameters comprises a plurality of molecular IRC, a plurality of molecular bond lengths, a plurality of molecular bond angles, a molecular charge, and a spin multiplicity value.
5. The method as claimed in claim 1, wherein the plurality of active orbitals comprises a plurality of valence orbitals and a plurality of virtual orbitals.
6. A system (100) comprising:
one or more classical hardware processors (108) and a plurality of unentangled Quantum Processor Units (QPUs) (126), wherein the one or more classical hardware processors (108) are communicably coupled to the plurality of unentangled QPUs (126) by respective 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 hardware processors (108) are configured by the programmed instructions to:
receive a plurality of molecular parameters pertaining to a multidimensional molecule;
generate an active space based on the plurality of molecular parameters, using an active space transformation technique, wherein the active space comprises a plurality of core orbitals and a plurality of active orbitals;
generate a Fermionic Hamiltonian corresponding to the plurality of active orbitals in an effective potential of the core orbitals using an electronic Hamiltonian constructor;
obtain a Qubit Hamiltonian corresponding to the plurality of active orbitals using a Fermionic Hamiltonian-to-Qubit mapping technique;

obtain a reduced Qubit Hamiltonian by reducing the Qubit Hamiltonian using a Qubit reduction technique; and
obtain, a reaction pathway associated with the multidimensional molecule based on the reduced qubit Hamiltonian using an Intrinsic Reaction Coordinates driven Variational Quantum Eigen solver (IRC-VQE).
7. The system of claim 6, wherein the method of obtaining the reaction pathway associated with the multidimensional molecule based on the reduced Qubit Hamiltonian using the IRC-VQE comprises:
receiving, by the plurality of unentangled QPUs from the one or more classical hardware processors via the respective interfaces, an initial IRC associated with the multidimensional molecule, the reduced Qubit Hamiltonian associated with the multidimensional molecule, and a parameterized quantum circuit, wherein the initial IRC is assigned to a current IRC variable; and
obtaining the reaction pathway associated with the multidimensional molecule by iteratively performing until obtaining an equilibrium energy pertaining to one of a reactant and a product pertaining to the multidimensional molecule:
identifying, by the plurality of unentangled QPUs, a plurality of perturbations associated with the multidimensional molecule based on the current IRC using a perturbations identification technique;
obtaining, by the plurality of unentangled QPUs, a plurality of ground state energies by computing the ground state energy corresponding to each of the plurality of perturbations based on the reduced Qubit Hamiltonian using the VQE;
identifying, by the one or more classical hardware processors, the equilibrium energy based on the plurality of

ground state energies, wherein a minimum ground state energy from the plurality of ground state energies is identified as the equilibrium energy; and
obtaining, by the one or more classical hardware processors, the IRC corresponding to the identified equilibrium energy, wherein the current IRC is swapped with the obtained IRC.
8. The system of claim 7, wherein the parameterized quantum circuit comprises a plurality of parameterized gates.
9. The system of claim 6, wherein the plurality of molecular parameters comprises a plurality of molecular IRC, a plurality of molecular bond lengths, a plurality of molecular bond angles, a molecular charge, and a spin multiplicity value.
10. The system of claim 6, wherein the plurality of active orbitals comprises a plurality of valence orbitals and a plurality of virtual orbitals.