FORM 2
THE PATENTS ACT, 1970
(39 of 1970)
&
THE PATENT RULES, 2003
COMPLETE SPECIFICATION
(See Section 10 and Rule 13)
Title of invention:
IMPLEMENTATION OF DENSITY FUNCTIONAL THEORY ON QUANTUM COMPUTING THROUGH META GENERALIZED GRADIENT APPROXIMATION
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.
2
TECHNICAL FIELD
[001]
The disclosure herein generally relates to quantum computing, and, more particularly, to a method and system for implementation of density functional theory on quantum computing through meta generalized gradient approximation.
BACKGROUND 5
[002]
As quantum hardware and algorithms continue to develop, various industry sectors, particularly the pharmaceutical and material design domains, are applying quantum computation paradigm to their specific problems. For example, many chemical compounds are synthesized and predicted for practical use. Current software packages calculate the energetics and other physical properties of 10 physical/chemical systems using a quantum mechanical (QM) electronic self-consistent field approach called the Kohn-Sham Density functional theory (KS-DFT). However, a computational time complexity bottleneck of the DFT approach has remained a cubic function of the number of electronic orbitals. This is also known as the cubic scaling wall bottleneck in DFT. Most industrially relevant 15 chemical systems in Material sciences industry are bulk materials. One supercell made from a collection of neighboring unit cells involves around thousands of atoms which corresponds to several thousands of electronic orbitals even in the least correlated basis. Currently these chemical systems are treated approximately within the QM/Molecular Mechanical (MM) approach, where only a subsystem with 20 strong quantum correlations is simulated using DFT and the rest is treated only via the MM technique. Identifying these subsystems requires doing prior Molecular mechanical and or Force field analysis and adds to additional overhead cost. The gap in applying this DFT technology to the complete system arises from the cubic scaling wall bottleneck i.e. for a 50000 electronic orbital system (corresponding to 25 a supercell of the bulk material) computing even one KS-DFT step would require 0.001 secs on supercomputer FUGAKU having 𝑅𝑚𝑎𝑥 of 442 PETA Flops.
[003]
For all practical purposes, the DFT code should converge within 100 self-consistency steps therefore to simulate one large supercell on FUGAKU would require 0.1 seconds. For real world simulations one would require screening across 30
3
several lakhs of molecular configurations in a supercell and that would require more
than one day. The output electronic density computed from DFT needs to be in turn passed into Force-Field or Molecular mechanical modelling suites that then recomputes the geometric positioning of atoms and the DFT energies needs to be recomputed. Altogether that implies that for slightly larger system such calculations 5 can enter several days of calculations even on the largest of the supercomputers. Similarly the chemical systems in Pharma industry that correspond to Protein drug or protein-protein systems involve more than thousands of atoms which corresponds to a several thousands of electronic orbitals. These chemical systems are treated approximately within the QM/Molecular Mechanical (MM) approach, 10 only a subsystem is treated with strong quantum correlations and is simulated using DFT and the rest is treated only via the MM technique. Drug molecules in the pharma industry have more than 500 Molecular weight, screening across different drug molecules enters across several days of efforts. If time computational complexity of DFT for general chemical systems is reduced even nominally from 15 cubic to quadratic or linear, then several days of calculations can be reduced to a few days or within a day. This would enable faster and more screening in a short duration of time, handling larger QM regions and reduce product discovery time drastically.
[004]
In conventional KS DFT, the computational complexity scales 20 cubically to system size as a consequence of delocalized nature of wave functions which are the eigen solutions of the Kohn–Sham single particle Hamiltonian. For scaling the DFT calculations to large systems, there have been attempts to develop a technique which scales linearly with system size. One such technique is Order-N Electronic Total Energy Package (ONETEP) which uses a basis of non-orthogonal 25 generalized Wannier functions (NGWFs) expressed in terms of periodic cardinal sine (psinc) functions, which are in turn equivalent to a basis of plane-waves. ONETEP, therefore combines benefits of linear scaling with a level of accuracy and variational bounds comparable to traditional cubic-scaling plane-wave approaches. However, ONETEP is primarily designed for periodic systems restricting its 30 application to certain materials and structures. Further, in the current scenario, there
4
have
been efforts to speed up the DFT calculations. The conventions method for speeding up the DFT calculations include Exascale computing using central processing unit (CPU), general processing unit (GPU), message passing interface (MPI), Multiprocessing, and/or the like. However, these have memory and processing limitations. There have been attempts made to implement DFT on a 5 combination of classical and quantum processors. One such approach implements DFT on a classical processor and optimizes the DFT results on a quantum processor. However, the complex calculations are still performed on the classical processor thereby bottlenecks in DFT calculations are still challenging to be overcome. 10
SUMMARY
[005]
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 implementation of density functional theory on quantum 15 computing through meta generalized gradient approximation is provided. The method includes receiving, via the one or more classical hardware processors, a chemical compound whose one or more properties to be extracted, wherein the chemical compound is at least one of (i) a molecule, and (ii) a solid; obtaining, via the one or more classical hardware processors, a plurality of atomic coordinates of 20 each of a plurality of atoms present in the chemical compound; determining, via the one or more classical hardware processors, (i) a plurality of electron integrals, (ii) a core Hamiltonian matrix, and (iii) a collocation matrix, from the plurality of atomic coordinates of each of the plurality of atoms present in the chemical compound, wherein the collocation matrix is a rectangular matrix of dimension 𝑁𝑔 25 and 𝑁𝑎𝑜, and wherein 𝑁𝑔 represents a number of real space grid points, 𝑁𝑎𝑜 is a number of basis functions; determining, via the plurality of unentangled QPUs, an initial density matrix of the chemical compound from the core Hamiltonian matrix, wherein the initial density matrix is constructed using (i) a single particles unitary rotation matrix which is obtained by diagonalizing the core Hamiltonian matrix, 30 and (ii) electron occupancies; iteratively updating, the initial density matrix until a
5
convergence criteria is satisfied to obtain a final density matrix of the chemical
compound, wherein iteratively updating the initial density matrix at each iteration comprises: (a) computing a direct (𝐽) matrix from the initial density matrix; (b) determining a correlation exchange matrix based on the initial density matrix, the collocation matrix, a Gradient of collocation matrix, and a Hessian of the 5 collocation matrix for meta Generalized Gradient Approximation (meta GGA) by: (i) encoding the collocation matrix on (a) control qubits composed of a first set of qubits and a second set of qubits, and (b) a plurality of ancilla qubits in a second quantum circuit, to obtain a first quantum circuit block, wherein the first set of qubits is associated with the number of points on the numerical grid, the second set 10 of qubits is associated with the number of atomic orbitals, and the plurality of ancilla qubits are used to store numerical entries of the collocation matrix and the initial density matrix; (ii) encoding a gradient of the collocation matrix on the quantum circuit to obtain a gradient to obtain a first gradient quantum circuit block; (iii) encoding a hessian of the collocation matrix on the quantum circuit to obtain a 15 first hessian quantum circuit block; (iv)composing the first quantum circuit block with a second quantum circuit block that encodes an intermediate density matrix on (a) control qubits composed of the second set of qubits and (b) an ancilla qubit in the quantum circuit, to construct a third quantum circuit block; (v) encoding an electronic density on the quantum circuit by performing a sequential matrix 20 multiplication of the collocation matrix, the density matrix, and the collocation matrix to obtain a fourth quantum circuit block; (vi) composing the third quantum circuit block with the first gradient quantum circuit block that encoded the gradient of the collocation matrix, to construct a second gradient quantum circuit block that encodes a gradient of the electronic density of the chemical compound; (vii) 25 composing the first hessian quantum circuit block that encodes the hessian of the collocation matrix with the first quantum circuit block and the second quantum circuit block that encodes the initial density matrix sequentially, to construct a second hessian quantum circuit block that encodes a first component of hessian of the electronic density; (viii) composing the first hessian quantum circuit block that 30 encodes the hessian of the collocation matrix with the second quantum circuit block
6
that encodes the initial density matrix and the first quantum circuit block that
encoded the collocation matrix sequentially, to construct a third hessian quantum circuit block that encodes a second component of hessian of the electronic density; (ix) composing the first hessian quantum circuit block that encodes the hessian of the collocation matrix with the second quantum circuit block that encodes the initial 5 density matrix and the first gradient quantum circuit block encodes the gradient of the collocation matrix, to construct a fourth hessian quantum circuit block that encodes a third component of hessian of the electronic density of the chemical compound; (x) computing the hessian of the electronic density of the chemical compound by adding the first, second and third component of hessian of the 10 electronic density of the chemical compound using the plurality of ancilla qubits; (xi) encoding the Hessian of the electronic density of the chemical compound on (a) control qubits composed of a first set of qubits and a second set of qubits, and (b) the plurality of ancilla qubits, to obtain a fifth hessian quantum circuit block; (xii) putting the fourth quantum circuit block with a fifth quantum circuit block for 15 a phase angle shifted reflector, the second gradient quantum circuit block that encodes the gradient of the electronic density, and the fifth hessian quantum circuit block that encodes the hessian of the electronic density into a quantum signal processing sequence, to create a sixth quantum circuit block that encodes a Chebyshev polynomial approximation of a nonlinear function of the electronic 20 density, wherein the nonlinear function is a derivative of the electronic energy density with respect to the electronic density; (xiii) composing the sixth quantum circuit block with the first quantum circuit block, to create a seventh quantum circuit block that encodes a 𝑍 matrix; and (xiv) composing the seventh quantum circuit block with the first quantum circuit block, to create an eight quantum circuit 25 block that encodes the correlation exchange matrix; (c) determining a Fock (𝐹) matrix by adding the 𝐽 matrix and the correlation exchange matrix; (d) performing a qubitized diagonalization of the 𝐹 matrix to obtain an intermediate density matrix; and (e) repeating the steps of computing the direct (𝐽) matrix till performing the qubitized diagonalization of the 𝐹 matrix until the convergence criteria is satisfied 30 to obtain the final density matrix of the chemical compound, wherein the
7
intermediate density matrix is considered as the initial density matrix
until the convergence criteria is satisfied; and utilizing, via the one or more classical hardware processors, the final density matrix of the chemical compound to extract the one or more properties of the chemical compound.
[006]
In another aspect, a system for implementation of density functional 5 theory on quantum computing through meta generalized gradient approximation is provided. The system comprises one or more classical hardware processors (108) and a plurality of unentangled quantum processors (122), wherein the one or more classical hardware processors (108) are communicably coupled to the plurality of unentangled quantum processors (122) by respective interfaces, wherein the one or 10 more classical hardware processors are communicably coupled to at least one memory (110) storing programmed instructions; one or more Input /Output (I/O) interfaces (116); and the plurality of unentangled quantum processors (122) are operatively coupled to the at least one quantum memory (124), wherein the one or more classical hardware processors (108) and the plurality of unentangled quantum 15 processors (122) are configured by the programmed instructions to receive a chemical compound whose one or more properties to be extracted, wherein the chemical compound is at least one of (i) a molecule, and (ii) a solid; obtain a plurality of atomic coordinates of each of a plurality of atoms present in the chemical compound; determine (i) a plurality of electron integrals, (ii) a core 20 Hamiltonian matrix, and (iii) a collocation matrix, from the plurality of atomic coordinates of each of the plurality of atoms present in the chemical compound, wherein the collocation matrix is a rectangular matrix of dimension 𝑁𝑔 and 𝑁𝑎𝑜, and wherein 𝑁𝑔 represents a number of real space grid points, 𝑁𝑎𝑜 is a number of basis functions; determine an initial density matrix of the chemical compound from 25 the core Hamiltonian matrix, wherein the initial density matrix is constructed using (i) a single particles unitary rotation matrix which is obtained by diagonalizing the core Hamiltonian matrix, and (ii) electron occupancies; iteratively updating, the initial density matrix until a convergence criteria is satisfied to obtain a final density matrix of the chemical compound, wherein iteratively updating the initial density 30 matrix at each iteration comprises: (a) computing a direct (𝐽) matrix from the initial
8
density matrix; (b) determin
ing a correlation exchange matrix based on the initial density matrix, the collocation matrix, a Gradient of collocation matrix, and a Hessian of the collocation matrix for meta Generalized Gradient Approximation (meta GGA) by: (i) encoding the collocation matrix on (a) control qubits composed of a first set of qubits and a second set of qubits, and (b) a plurality of ancilla qubits 5 in a second quantum circuit, to obtain a first quantum circuit block, wherein the first set of qubits is associated with the number of points on the numerical grid, the second set of qubits is associated with the number of atomic orbitals, and the plurality of ancilla qubits are used to store numerical entries of the collocation matrix and the initial density matrix; (ii) encoding a gradient of the collocation 10 matrix on the quantum circuit to obtain a gradient to obtain a first gradient quantum circuit block; (iii) encoding a hessian of the collocation matrix on the quantum circuit to obtain a first hessian quantum circuit block; (iv)composing the first quantum circuit block with a second quantum circuit block that encodes an intermediate density matrix on (a) control qubits composed of the second set of 15 qubits and (b) an ancilla qubit in the quantum circuit, to construct a third quantum circuit block; (v) encoding an electronic density on the quantum circuit by performing a sequential matrix multiplication of the collocation matrix, the density matrix, and the collocation matrix to obtain a fourth quantum circuit block; (vi) composing the third quantum circuit block with the first gradient quantum circuit 20 block that encoded the gradient of the collocation matrix, to construct a second gradient quantum circuit block that encodes a gradient of the electronic density of the chemical compound; (vii) composing the first hessian quantum circuit block that encodes the hessian of the collocation matrix with the first quantum circuit block and the second quantum circuit block that encodes the initial density matrix 25 sequentially, to construct a second hessian quantum circuit block that encodes a first component of hessian of the electronic density; (viii) composing the first hessian quantum circuit block that encodes the hessian of the collocation matrix with the second quantum circuit block that encodes the initial density matrix and the first quantum circuit block that encoded the collocation matrix sequentially, to 30 construct a third hessian quantum circuit block that encodes a second component
9
of hessian of the electronic density; (ix) composing the first hessian quantum circuit
block that encodes the hessian of the collocation matrix with the second quantum circuit block that encodes the initial density matrix and the first gradient quantum circuit block encodes the gradient of the collocation matrix, to construct a fourth hessian quantum circuit block that encodes a third component of hessian of the 5 electronic density of the chemical compound; (x) computing the hessian of the electronic density of the chemical compound by adding the first, second and third component of hessian of the electronic density of the chemical compound using the plurality of ancilla qubits; (xi) encoding the Hessian of the electronic density of the chemical compound on (a) control qubits composed of a first set of qubits and a 10 second set of qubits, and (b) the plurality of ancilla qubits, to obtain a fifth hessian quantum circuit block; (xii) putting the fourth quantum circuit block with a fifth quantum circuit block for a phase angle shifted reflector, the second gradient quantum circuit block that encodes the gradient of the electronic density, and the fifth hessian quantum circuit block that encodes the hessian of the electronic density 15 into a quantum signal processing sequence, to create a sixth quantum circuit block that encodes a Chebyshev polynomial approximation of a nonlinear function of the electronic density, wherein the nonlinear function is a derivative of the electronic energy density with respect to the electronic density; (xiii) composing the sixth quantum circuit block with the first quantum circuit block, to create a seventh 20 quantum circuit block that encodes a 𝑍 matrix; and (xiv) composing the seventh quantum circuit block with the first quantum circuit block, to create an eight quantum circuit block that encodes the correlation exchange matrix; (c) determining a Fock (𝐹) matrix by adding the 𝐽 matrix and the correlation exchange matrix; (d) performing a qubitized diagonalization of the 𝐹 matrix to obtain an intermediate 25 density matrix; and (e) repeating the steps of computing the direct (𝐽) matrix till performing the qubitized diagonalization of the 𝐹 matrix until the convergence criteria is satisfied to obtain the final density matrix of the chemical compound, wherein the intermediate density matrix is considered as the initial density matrix until the convergence criteria is satisfied; and utilize, the final density matrix of the 30
10
chemical compound to extract the one or more properties of the chemical
compound.
[007]
In yet another aspect, a computer program product including a non-transitory computer-readable medium having embodied therein a computer program implementation of density functional theory on quantum computing 5 through meta generalized gradient approximation 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 chemical compound whose one or more properties to be extracted, wherein the 10 chemical compound is at least one of (i) a molecule, and (ii) a solid; obtain a plurality of atomic coordinates of each of a plurality of atoms present in the chemical compound; determine (i) a plurality of electron integrals, (ii) a core Hamiltonian matrix, and (iii) a collocation matrix, from the plurality of atomic coordinates of each of the plurality of atoms present in the chemical compound, 15 wherein the collocation matrix is a rectangular matrix of dimension 𝑁𝑔 and 𝑁𝑎𝑜, and wherein 𝑁𝑔 represents a number of real space grid points, 𝑁𝑎𝑜 is a number of basis functions; determine an initial density matrix of the chemical compound from the core Hamiltonian matrix, wherein the initial density matrix is constructed using (i) a single particles unitary rotation matrix which is obtained by diagonalizing the 20 core Hamiltonian matrix, and (ii) electron occupancies; iteratively updating, the initial density matrix until a convergence criteria is satisfied to obtain a final density matrix of the chemical compound, wherein iteratively updating the initial density matrix at each iteration comprises: (a) computing a direct (𝐽) matrix from the initial density matrix; (b) determining a correlation exchange matrix based on the initial 25 density matrix, the collocation matrix, a Gradient of collocation matrix, and a Hessian of the collocation matrix for meta Generalized Gradient Approximation (meta GGA) by: (i) encoding the collocation matrix on (a) control qubits composed of a first set of qubits and a second set of qubits, and (b) a plurality of ancilla qubits in a second quantum circuit, to obtain a first quantum circuit block, wherein the 30 first set of qubits is associated with the number of points on the numerical grid, the
11
second set of qubits is associated with the number of atomic orbitals, and the
plurality of ancilla qubits are used to store numerical entries of the collocation matrix and the initial density matrix; (ii) encoding a gradient of the collocation matrix on the quantum circuit to obtain a gradient to obtain a first gradient quantum circuit block; (iii) encoding a hessian of the collocation matrix on the quantum 5 circuit to obtain a first hessian quantum circuit block; (iv)composing the first quantum circuit block with a second quantum circuit block that encodes an intermediate density matrix on (a) control qubits composed of the second set of qubits and (b) an ancilla qubit in the quantum circuit, to construct a third quantum circuit block; (v) encoding an electronic density on the quantum circuit by 10 performing a sequential matrix multiplication of the collocation matrix, the density matrix, and the collocation matrix to obtain a fourth quantum circuit block; (vi) composing the third quantum circuit block with the first gradient quantum circuit block that encoded the gradient of the collocation matrix, to construct a second gradient quantum circuit block that encodes a gradient of the electronic density of 15 the chemical compound; (vii) composing the first hessian quantum circuit block that encodes the hessian of the collocation matrix with the first quantum circuit block and the second quantum circuit block that encodes the initial density matrix sequentially, to construct a second hessian quantum circuit block that encodes a first component of hessian of the electronic density; (viii) composing the first 20 hessian quantum circuit block that encodes the hessian of the collocation matrix with the second quantum circuit block that encodes the initial density matrix and the first quantum circuit block that encoded the collocation matrix sequentially, to construct a third hessian quantum circuit block that encodes a second component of hessian of the electronic density; (ix) composing the first hessian quantum circuit 25 block that encodes the hessian of the collocation matrix with the second quantum circuit block that encodes the initial density matrix and the first gradient quantum circuit block encodes the gradient of the collocation matrix, to construct a fourth hessian quantum circuit block that encodes a third component of hessian of the electronic density of the chemical compound; (x) computing the hessian of the 30 electronic density of the chemical compound by adding the first, second and third
12
component of hessian of the electronic density of the chemical compound using the
plurality of ancilla qubits; (xi) encoding the Hessian of the electronic density of the chemical compound on (a) control qubits composed of a first set of qubits and a second set of qubits, and (b) the plurality of ancilla qubits, to obtain a fifth hessian quantum circuit block; (xii) putting the fourth quantum circuit block with a fifth 5 quantum circuit block for a phase angle shifted reflector, the second gradient quantum circuit block that encodes the gradient of the electronic density, and the fifth hessian quantum circuit block that encodes the hessian of the electronic density into a quantum signal processing sequence, to create a sixth quantum circuit block that encodes a Chebyshev polynomial approximation of a nonlinear function of the 10 electronic density, wherein the nonlinear function is a derivative of the electronic energy density with respect to the electronic density; (xiii) composing the sixth quantum circuit block with the first quantum circuit block, to create a seventh quantum circuit block that encodes a 𝑍 matrix; and (xiv) composing the seventh quantum circuit block with the first quantum circuit block, to create an eight 15 quantum circuit block that encodes the correlation exchange matrix; (c) determining a Fock (𝐹) matrix by adding the 𝐽 matrix and the correlation exchange matrix; (d) performing a qubitized diagonalization of the 𝐹 matrix to obtain an intermediate density matrix; and (e) repeating the steps of computing the direct (𝐽) matrix till performing the qubitized diagonalization of the 𝐹 matrix until the convergence 20 criteria is satisfied to obtain the final density matrix of the chemical compound, wherein the intermediate density matrix is considered as the initial density matrix until the convergence criteria is satisfied; and utilize, the final density matrix of the chemical compound to extract the one or more properties of the chemical compound. 25
[008]
In an embodiment of the present disclosure, the convergence criteria is satisfied when a norm of a difference between the intermediate density matrix at a current iteration and the initial density matrix at a previous iteration is lesser than a predefined threshold.
[009]
In an embodiment of the present disclosure, the (i) control qubits 30 composed of the first set of qubits and the second set of qubits, and (ii) the plurality
13
of ancilla qubits, to obtain the first quantum circuit block, is mathematically
represented as: 𝑙𝑜𝑔 𝑁𝑔+ 𝑙𝑜𝑔 𝑁𝑎𝑜+2 𝑞𝑢𝑏𝑖𝑡𝑠, where 𝑙𝑜𝑔 𝑁𝑔 represents the first set of qubits, 𝑙𝑜𝑔 𝑁𝑎𝑜 represents the second set of qubits, and 2 qubits represent the plurality of ancilla qubits.
[010]
In an embodiment of the present disclosure, the hessian of the 5 collocation matrix is obtained by encoding the collocation matrix for a set of real space grid points (+/−𝑑𝑒𝑙𝑡𝑎,0,0),(0,+/−𝑑𝑒𝑙𝑡𝑎,0),(0,0,+/−𝑑𝑒𝑙𝑡𝑎),(+/𝑑𝑒𝑙𝑡𝑎,0,+/−𝑑𝑒𝑙𝑡𝑎),(+/𝑑𝑒𝑙𝑡𝑎,+/−𝑑𝑒𝑙𝑡𝑎,0),(0,+/−𝑑𝑒𝑙𝑡𝑎,+/−𝑑𝑒𝑙𝑡𝑎) on (a) control qubits composed of the first set of qubits and the second set of qubits, and (b) the plurality of ancilla qubits to obtain the first hessian quantum circuit block. 10
[011]
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 15
[012]
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:
[013]
FIG. 1 is a functional block diagram of a system for implementation of density functional theory on quantum computing through meta generalized 20 gradient approximation, in accordance with some embodiments of the present disclosure.
[014]
FIGS. 2A through 2E, collectively referred as FIG. 2 is an exemplary flow diagram illustrating for implementation of density functional theory on quantum computing through meta generalized gradient approximation, 25 implemented by the system of FIG. 1, in accordance with some embodiments of the present disclosure.
[015]
FIG. 3 illustrates an alternative representation of the flow diagram of FIG. 2, in accordance with some embodiments of the present disclosure.
14
[016]
FIG. 4 illustrates a quantum circuit used to compute a direct matrix in the method illustrated in FIG. 2, in accordance with some embodiments of the present disclosure.
[017]
FIG. 5 illustrates a first quantum circuit block that encodes a collocation matrix in the method illustrated in FIG. 2, in accordance with some 5 embodiments of the present disclosure.
[018]
FIGS. 6A through 6F illustrates quantum circuit blocks that encode different components of the Hessian of the collocation matrix in the method illustrated in FIG. 2, in accordance with some embodiments of the present disclosure. 10
[019]
FIG. 7 illustrates a quantum circuit comprising the second hessian quantum circuit block, the third hessian quantum circuit block, and the fourth hessian quantum circuit block that encode the first, second and third components of the hessian of electronic density respectively in the method illustrated in FIG. 2, in accordance with some embodiments of the present disclosure. 15
[020]
FIG. 8 illustrates a fifth quantum circuit block that encodes Hessian of the electronic density in the method illustrated in FIG. 2, in accordance with some embodiments of the present disclosure.
[021]
FIG. 9 illustrates the eight quantum circuit block that encodes the correlation exchange matrix in the method illustrated in FIG. 2, in accordance with 20 some embodiments of the present disclosure.
DETAILED DESCRIPTION OF EMBODIMENTS
[022] Exemplary embodiments are described with reference to the accompanying drawings. In the figures, the left-most digit(s) of a reference number 25 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. 30
15
[023]
As quantum hardware and algorithms continue to develop, various industry sectors, particularly the pharmaceutical and material design domains, are applying quantum computation paradigm to their specific problems. For example, many chemical compounds are synthesized and predicted for practical use. Current software packages calculate the energetics and other physical properties of 5 physical/chemical systems using a quantum mechanical (QM) electronic self-consistent field approach called the Kohn-Sham Density functional theory (KS-DFT). However, the computational time complexity bottleneck of the DFT approach has remained a cubic function of the number of electronic orbitals. This is also known as the cubic scaling wall bottleneck in DFT. Further, conventional 10 methods implement the DFT simulations on classical processors such as CPU, GPU etc. which is time consuming. Some state of the art techniques implement DFT on a combination of classical and quantum processors. However, the complex calculations are still performed on a classical processor which doesn’t overcome the bottlenecks in DFT calculations. 15
[024]
The present disclosure addresses the unresolved problem of computational time complexity bottleneck of the DFT in the conventional methods by providing a system and method for implementation of density functional theory on quantum computing through meta generalized gradient approximation. In the present disclosure, diagonalization problem is mapped to a sequence of quadratic 20 least squares problem and solved as a quantum linear systems problem using implementation of density functional theory on quantum computing through meta generalized gradient approximation. More specifically, the present disclosure provides a quantum circuit for computing meta generalized gradient approximation (meta GGA) density functional computations efficiently on a quantum processor. 25 Initially, atomic coordinates of each atom in a chemical compound whose one or more properties must be extracted is received. Electron integrals, a core Hamiltonian, and a collocation matrix is computed from the atomic coordinates. The core Hamiltonian is diagonalized to obtain a density matrix of the chemical compound which is further updated iteratively until a convergence criteria is 30 satisfied, using the meta GGA. At each iteration, a direct matrix is computed from
16
the density matrix, a correlation exchange matrix is computed from the direct matrix
and the collocation matrix, a Fock matrix is computed by adding the direct matrix and the correlation exchange matrix and the Fock matrix is diagonalized to obtain updated density matrix which is used in subsequent iteration. This is repeated until norm of a difference between the updated density matrix at a current iteration and 5 the density matrix at a previous iteration is lesser than a predefined threshold to obtain a final density matrix of the chemical compound which can be used to extract the one or more properties of the chemical compound.
[025]
Referring now to the drawings, and more particularly to FIGS. 1 through 5. where similar reference characters denote corresponding features 10 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.
[026]
FIG. 1 is a functional block diagram of a system 100 for implementing density functional theory on quantum processors through local 15 density approximation for simulating chemical compounds, 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.
[027]
The classical computing system 102 comprises one or more classical 20 hardware processors 108, at least one memory such as a memory 110, and an I/O interface 116. The one or more classical hardware processors 108, the memory 110, and the Input /Output (I/O) interface 116 may be coupled by a system bus such as a system bus 112 or a similar mechanism. In an embodiment, the one or more classical hardware processors 108 can be one or more hardware processors. The 25 one or more classical hardware processors and the hardware processors are interchangeably used throughout the document. Similarly, the classical computing system 102 is a normal computing system.
[028]
The I/O interface 116 may include a variety of software and hardware interfaces, for example, a web interface, a graphical user interface, and 30 the like., for example, interfaces for peripheral device(s), such as a keyboard, a
17
mouse, an external memory, a printer and the like. Further, the I/O interface 116
may enable the system 100 to communicate with other devices, such as web servers, and external databases. The I/O interface 116 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 5 networks, such as Wireless LAN (WLAN), cellular, or satellite. For the purpose, the I/O interface 116 may include one or more ports for connecting several computing systems with one another or to another server computer.
[029]
The one or more hardware processors 108 may be implemented as one or more microprocessors, microcomputers, microcontrollers, digital signal 10 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.
[030]
The memory 110 may include any computer-readable medium 15 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 data repository 114. The data repository 20 (or repository) 114 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 method illustrated in FIGS. 2 and 3. Although the data repository 114 is shown internal to the system 100, it should be noted that, in alternate embodiments, the data repository 114 can also be implemented external to the 25 system 100, where the data repository 114 may be stored within a database (repository 114) 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, 30 the data may be stored in an external system, such as a Lightweight Directory
18
Access Protocol (LDAP) directory and a Relational Database Management System
(RDBMS).
[031]
The quantum computing system 104 shown in FIG. 1 includes a control system 118, a signal delivery system 120, a plurality of quantum processors (alternatively referred as Quantum Processing Units (QPUs)) 122, and a quantum 5 memory 124. The plurality of quantum processors 122 are unentangled and hence alternatively referred as the plurality of unentangled quantum processors 122. The quantum computing system 104 may include additional or different features, and the components of a quantum computing system 104 may operate as described with respect to FIG. 1 or in alternative manner. 10
[032]
In an embodiment, the quantum computing system 104 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 15 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 unentangled quantum processors 122. 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 20 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.
[033]
The plurality of unentangled quantum processors 122 shown in FIG. 1 may be implemented, for example, as a superconducting quantum integrated 25 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 30 qubits. In some instances, the conditional quantum logic can be performed in a
19
manner that allows large
-scale entanglement within the plurality of unentangled quantum processors 122. 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 5 measuring the quantum states of the qubit devices. The plurality of unentangled quantum processors 122 may be implemented using another type of physical system.
[034]
The plurality of unentangled quantum processors 122, and in some cases all or part of the signal delivery system 120, can be maintained in a controlled 10 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 plurality of unentangled quantum processors 122 operate in a cryogenic temperature regime and are subject to very low electromagnetic and thermal noise. For example, magnetic shielding 15 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.
[035]
In an embodiment, the signal delivery system 120 provides 20 communication between the control system 118 and the plurality of unentangled quantum processors 122. For example, the signal delivery system 120 can receive control signals from the control system 118 and deliver the control signals to the plurality of unentangled quantum processors 122. In some instances, the signal delivery system 120 performs preprocessing, signal conditioning, or other 25 operations to the control signals before delivering them to the plurality of unentangled quantum processors 122. In an embodiment, the signal delivery system 120 includes connectors or other hardware elements that transfer signals between the plurality of unentangled quantum processors 122 and the control system 118. For example, the connection hardware can include signal lines, signal processing 30 hardware, filters, feedthrough devices (e.g., light-tight feedthroughs, etc.), and
20
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 118) and a lower temperature regime (e.g., at the plurality of unentangled quantum processors 122). 5
[036]
In an embodiment, the control system 118 controls operation of the plurality of unentangled quantum processors 122. For example, the control system 118 may include data processors, signal generators, interface components and other types of systems or subsystems. Components of the example control system 118 may operate in a room temperature regime, an intermediate temperature 10 regime, or both. For example, the control system 118 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 plurality of unentangled quantum processors 122.
[037]
In some embodiments, the control system 118 includes a classical computing system that executes software to compile instructions for the plurality 15 of unentangled quantum processors 122. For example, the control system 118 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 plurality of unentangled quantum processors 122. In some examples, the control system 118 applies a quantum logic circuit by generating signals that cause 20 the Qubit devices and other devices in the plurality of unentangled quantum processors 122 to execute operations. For instance, the operations may correspond to single-Qubit gates, two-Qubit gates, Qubit measurements, etc. The control system 118 can generate control signals that are communicated to the plurality of unentangled quantum processors 122 by the signal delivery system 120, and the 25 devices in the plurality of unentangled quantum processors 122 can execute the operations in response to the control signals.
[038]
In some other embodiments, the control system 118 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 30 example, a classical processor may convert a quantum computer program to an
21
instruction set for the native gate set or architecture of the plurality of unentangled
quantum processors 122. In some cases, the control system 118 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 plurality of unentangled quantum processors 122. The 5 control signals may be generated based on a control sequence provided, for instance, by a classical processor in the control system 118. The example control system 118 may include conversion hardware that digitizes response signals received from the plurality of unentangled quantum processors 122. The digitized response signals may be provided, for example, to a classical processor in the 10 control system 118.
[039]
In some embodiments, the quantum computer system 104 includes multiple quantum information processors that operate as respective unentangled quantum processors 122. In some cases, each quantum processor can operate independently. For instance, the quantum computer system 104 may be configured 15 to operate according to a distributed quantum computation model, or the quantum computer system 104 may utilize multiple quantum processors in another manner. In some implementations, the quantum computer system 104 includes multiple control systems, and each quantum processor may be controlled by a dedicated control system. In some implementations, a single control system can control 20 multiple quantum processors. For instance, the control system 118 may include multiple domains that each control a respective quantum processor. In some instances, the quantum computing system 104 uses multiple quantum processors to execute multiple unentangled quantum computations (e.g., multiple Variational Quantum Eigen solver (VQE)) that collectively simulate a single quantum 25 mechanical system.
[040] In an embodiment, the quantum memory 124 is a quantum-mechanical version of classical computer memory 110. The classical computer memory 100 stores information such as binary states and the quantum memory 124 stores a quantum state for later retrieval. These states hold useful computational 30 information known as Qubits. In an embodiment, the communication interface 106
22
which connects the classical computing system 102 and the quantum computing
system 104 is a high speed digital interface.
[041]
FIGS. 2A through 2E, collectively referred as FIG. 2 is an exemplary flow diagram illustrating a method 200 for implementation of density functional theory on quantum computing through meta generalized gradient approximation, 5 implemented by the system of FIG. 1, in accordance with some embodiments of the present disclosure. In an embodiment, the system 100 is 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 quantum processors 122. FIG. 3 illustrates an alternative representation of the flow diagram of FIG. 2, in accordance 10 with some embodiments of the present disclosure.
[042]
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 FIGS. 2 and 3. The method 200 may be described in the general context of computer executable instructions. 15 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 20 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. 25
[043]
Now referring to FIG. 2, at step 202 of the method 200, one or more classical hardware processors are configured to receive a chemical compound whose one or more properties are to be extracted. The chemical compound is at least one of (i) a molecule, and (ii) a solid (202). In the context of the present disclosure, the expression ‘solid’ refers to a periodic arrangements of atoms such 30 as a crystalline form of a molecule. For example, in pharma industry, the chemical
23
compound maybe
a drug lead and its conformations whose physical properties must be analyzed to determine feasibility of synthesizing the drug from the drug lead. The one or more properties of drug lead includes one or more physical and one or more chemical properties such as highest occupied molecular orbital and lowest unoccupied molecular (HOMO-LUMO) gap in solvent medium, acid-base 5 dissociation constant (𝑝𝐾𝑎) of a drug, dipole moment, partial charges, and/or the like.
[044]
Further, at step 204 of the method 200, the one or more classical hardware processors are configured to obtain a plurality of atomic coordinates of each of a plurality of atoms present in the chemical compound. For example, for 10 drug paracetamol, a list of the plurality of atomic coordinates (𝑋,𝑌,𝑍) with a corresponding atomic symbol is obtained as: (2.8660 -2.5950 0.0000 𝑂), ( 4.5981 1.4050 0.0000 𝑂), (2.8660 1.4050 0.0000 𝑁), (2.8660 0.4050 0.0000 𝐶), (1.4631 0.2150 0.0000 𝐻), and/or the like.
[045]
Further, at step 206 of the method 200, the one or more classical 15 hardware processors are configured to determine (i) a plurality of electron integrals, (ii) a core Hamiltonian matrix, and (iii) a collocation matrix, from the plurality of atomic coordinates of each of the plurality of atoms present in the chemical compound. The collocation matrix is a rectangular matrix of dimension 𝑁𝑔 and 𝑁𝑎𝑜, where 𝑁𝑔 represents a number of real space grid points, 𝑁𝑎𝑜 is a number of basis 20 functions. The collocation matrix comprises a plurality of basis functions of a plurality of atomic orbitals and a plurality of points on numerical grid. Each of the plurality of basis functions is a Gaussian wave function centered around the plurality of atomic coordinates. Thus, the plurality of basis functions are represented as Gaussians functions as a function of the plurality atomic coordinates 25 of the plurality of atoms and an angular and spin momentum. The collocation matrix is mathematically represented using equation (1) below:
Φ=( ϕ1(𝑥1,𝑦1,𝑧1),………ϕ1(𝑥𝑁𝑔,𝑦𝑁𝑔,𝑧𝑁𝑔)⋮⋮ϕ𝑁𝑎𝑜(𝑥1,𝑦1,𝑧1),………ϕ𝑁𝑎𝑜(𝑥𝑁𝑔,𝑦𝑁𝑔,𝑧𝑁𝑔)) (1)
24
The plurality of basis functions are used to compute the plurality of electron integrals. The plurality of electron integrals include one-electron integrals and two-electron integrals which are further bifurcated as three centered and two centered electron integrals. Thus, the plurality of electron integrals include (a) 3 center 2 electron integrals, b) 2 center 2 electron integrals and c) 2 center 1 electron 5 integrals. The 3 center 2 electron integrals and 2 center 2 electron integrals describe a Coulomb repulsion integral that are computed in atomic orbital basis and can be represented in either spherical or cartesian coordinates. The 2 center 1 electron integrals describes an overlap between the basis states. The plurality of electron integrals are determined using tools such as LIBCINT, Python-based Simulations 10 of Chemistry Framework (PySCF), NorthWest computational Chemistry (NWchem), and/or the like. The core Hamiltonian matrix which is a 𝑁𝑏× 𝑁𝑏 matrix is determined using the Kohn-Sham (KS) hybrid Density Functional Theory (DFT) process which is initiated from a set of 𝑁𝑏 basis functions 𝔹 = {𝜙𝜇(𝑟)}𝜇=1𝑁𝑏 to compute the core Hamiltonian matrix ℎ. ℎ constitutes free 15 particle energy of electrons and one-body nuclear potential arising from electron-nucleus interactions, both these terms are independent of electronic density. Therefore ℎ needs to be computed while initiating a self-consistent field (SCF) procedure.
[046]
Further, at step 208 of the method 200, the one or more classical 20 hardware processors and the plurality of unentangled QPUs are configured to determine an initial density matrix of the chemical compound from the core Hamiltonian matrix. The initial density matrix is constructed using (i) a single particles unitary rotation matrix which is obtained by diagonalizing the core Hamiltonian matrix, and (ii) electron occupancies (e.g., refer Indian patent 25 application IN 20232106141 titled ‘Method And System For Reducing Time Complexity Of Density Functional Theory Calculations With Qubitized Diagonalization’).
[047]
Once the density matrix is determined, at step 210 of the method 200, the one or more classical hardware processors and the plurality of unentangled 30 QPUs are configured to iteratively update the initial density matrix until a
25
convergence criteria is satisfied to obtain a final density matrix of the chemical
compound. The convergence criteria is satisfied when a norm of a difference between an intermediate density matrix at a current iteration and the initial density matrix at a previous iteration is lesser than a predefined threshold. For iteratively updating the initial density matrix at each iteration, steps 210a to 210e are 5 performed at each iteration to update the initial density matrix. In an embodiment, the step 210a is better understood by way of following description as exemplary explanation.
[048]
In the Kohn-Sham (KS) hybrid Density Functional Theory (DFT) process, the direct (𝐽) matrix and an exact-exchange (𝐾) matrix are electronic 10 density dependent terms which are computed from an Electronic Repulsion Integral (ERI) tensor. Further, in the Kohn-Sham (KS) hybrid Density Functional Theory (DFT) process, a correlation exchange potential or a correlation exchange matrix (𝑉𝑥𝑐) is computed from an electronic density, its gradient and/or a Kinetic energy density. Together 𝐽, 𝐾 and 𝑉𝑥𝑐 make a hybrid Kohn-Sham Fock Matrix functional 15 𝐹 as given by equation (2)
F[D]=h+J[D]−𝑐K[D]+V𝑥𝑐[D] (2)
In equation (2), 𝐷 is a one-particle density matrix discretized in a basis of {𝜙𝜇} and 𝑐 ∈ ℝ+ is a modulation for the exact-exchange term. The terms 𝐽, 𝐾 can be computed from an atomic orbital (AO)-integral approach using the ERI tensor 20 represented as ⟨ij|kl⟩ and 𝐷 as shown below in equations (3) and (4), respectively.
Jij=Σ⟨ij|kl⟩Dkl𝑘𝑙 (3) Kij=Σ⟨ik|jl⟩Dkl𝑘𝑙 (4)
In equations (3) and (4), the ERI tensor ⟨ij|kl⟩ is obtained by integrating space of basis functions in ℝ3 according to equation (5) shown below: 25
⟨ij|kl⟩=∫𝑑3r𝑑3r′𝜙𝑖(r)𝜙𝑗(r)1|r−r′|𝜙𝑘(r′)𝜙𝑙(r′). (5)
[049]
The correlation-exchange potential 𝑉𝑥𝑐 in the discretized basis {𝜙𝑖} is computed from equation (6) shown below, where the basis functions are defined according to equation (7).
26
𝑉𝑖𝑗𝑥𝑐=∫𝑑3r𝜙𝑖(r)δE𝑥𝑐[𝜌(r)]δ𝜌(r)𝜙𝑗(r)𝑅3. (6)
𝜙𝑎(r)=(𝑥−𝐴𝑥)𝑎𝑥(𝒴−𝐴𝒴)𝑎𝒴(𝑧−𝐴𝑧)𝑎𝑧exp(−𝜗(𝑥−𝐴𝑥)2)exp(−𝜗(𝒴− 𝐴𝒴)2)exp(−𝜗(𝑧−𝐴𝑧)2) (7)
In equation 6, 𝐴𝑥, 𝐴𝒴, 𝐴𝑧 are nuclear coordinate locations of the Gaussian functions, 5 𝑎 = (𝑎𝑥,𝑎𝑦,𝑎𝑧) are the integer cartesian quanta and 𝜗 is the cartesian Gaussian exponent. In equation (6), E𝑥𝑐 is an exchange energy functional evaluated for the electronic density 𝑟 and its form depends on a DFT method implemented using a plurality of classes of Density Functional Theory (DFT) protocols. The plurality of classes of DFT protocols comprise local density approximation (LDA), generalized 10 gradient approximation (GGA), meta GGA, hybrid DFT and double hybrid DFT. In the context of the present disclosure, the DFT protocol used is meta Generalized Gradient Approximation (meta GGA). The ERI tensor is further represented using a Cholesky decomposition representation as given by equation (8) below.
⟨ij|kl⟩=Σ𝐿𝑖𝑗𝑃𝑃𝐿𝑘𝑙𝑃 (8) 15
𝐿𝑃 represents Cholesky matrices that can be obtained from a Density-Fitting (DF) or a resolution of identity (RI) method as in equation (9).
𝐿𝑖𝑗𝑃=1√𝓌𝑃Σ𝑢𝑄𝑃⟨ij|𝑄⟩.𝑄 (9)
In equation (9), ⟨ij|𝑄⟩ are 3-center 2-electron integrals represented in terms of the auxiliary basis functions {𝜒𝑃(𝑟)}𝑃=1𝑁𝑎𝑢𝑥, and 𝑢𝑄𝑃, 𝓌𝑃 are obtained from the spectral 20 decomposition of 2-center 2-electron integrals 𝑉𝑃𝑄=Σ𝑢𝑃𝑅𝓌𝑅𝑢𝑄𝑅∗𝑅 represented in the auxiliary basis functions. The 3-center 2-electron integrals are given by equation (10) and the two center 𝑉𝑃𝑄= ⟨P|𝑄⟩ represents two center two electron integrals which are given by equation (11).
⟨ij|P⟩=∫𝑑3r𝑑3r′𝜙𝑖(r)𝜙𝑗(r)𝛬𝑃(r′)|r−r′| (10) 25
⟨P|Q⟩=∫𝑑3r𝛬𝑃(r)𝛬𝑄(r)|r−r′| (11)
The 𝐽 and 𝐾 matrix elements are computed using the RI approach as in equation (12).
27
𝐽𝑖𝑗=Σ𝐿𝑖𝑗𝑃𝑃Σ𝐿𝑘𝑙𝑃𝑘𝑙𝐷𝑘𝑙,𝐾𝑖𝑗=Σ𝐿𝑖𝑙𝑃𝑃𝑙Σ𝐿𝑗𝑘𝑃𝑘𝐷𝑘𝑙 (12)
When the KS-DFT is classically computed, the complexity of computing 𝐽 and 𝐾 is 𝑂(𝑝𝑁2) and this complexity resides in computing 𝐾. Computing ℎ and 𝑉𝑥𝑐 has lower complexity of 𝑂(𝑁2).
[050]
For executing any of the plurality of classes of Density Functional 5 Theory (DFT) protocols, next step is to solve generalized eigenvalue problem for the Fock matrix accounting for an overlap between basis functions 𝑆𝑖𝑗= ⟨𝜙𝑖|𝜙𝑗⟩ and an intermediate density matrix 𝐷(𝑙) of a current iteration as given by equation (13).
𝐹[𝐷(𝑙)]𝐶(𝑙+1)=𝑆𝐶(𝑙+1)𝐸̂(𝑙+1) (13) 10
In equation (13), the one particle density matrix or intermediate density matrix 𝐷(𝑙)=𝐶(𝑙)𝑊𝐶(𝑙)†, where an occupancy matrix 𝑊𝑖𝑗= 𝜃(𝜇 − 𝐸𝑖)𝛿𝑖𝑗 fills up 𝑁𝑒/2 lowest energy levels below a chemical potential 𝜇.
[051]
In the present disclosure, at step 210a, a direct matrix (alternatively referred to as 𝐽 matrix or Coulomb matrix) is computed from the initial density 15 matrix using a quantum circuit as illustrated in FIG. 4. The quantum circuit comprises a first set of qubits, a second set of qubits and a plurality of ancilla qubits. The first set of qubits is associated with number of auxiliary basis functions in density fitting approximation of the plurality electron integrals (more particularly the 2 centered 2 electron integral), and the second set of qubits is associated with 20 number of basis functions. In FIG. 4, ‘𝑐’ represents the first set of qubits, ‘𝑟1, 𝑟2’ represent the second set of qubits and ‘𝑎1, 𝑎2’ represent the plurality of ancilla qubits. In an embodiment, the quantum circuit is mathematically represented as: 𝑙𝑜𝑔 𝑁𝑎𝑢𝑥+2𝑙𝑜𝑔 𝑁𝑎𝑜+2 qubits. Initially, the initial density matrix is encoded (alternatively referred as loaded or block encoded) on the second set of qubits in 25 the quantum circuit to form a first quantum circuit component (represented by block 402 in FIG. 4). As shown in block 402, the control is loaded on 𝑟2 and data elements of the initial density matrix are loaded on ancilla qubit 𝑎2. The density matrix is loaded using its Eigen decomposition representation that involves: i) NaoC2 Givens rotations on 𝑙𝑜𝑔 𝑁𝑎𝑜 qubits, ii) Then block encoding a diagonal matrix of 1’s in 30
28
initial
𝑁𝑒/2 diagonal blocks and zeros in the rest using 2𝑙𝑜𝑔 𝑁𝑎𝑜 qubits, iii) Another set of NaoC2 Givens rotation with conjugate angles on 𝑁𝑒/2 diagonals. This is mathematically represented by equation (14).
𝑂(𝐷)=𝐼2⊗𝑚⊗[(𝐶⊗𝐼2⊗𝑛⊗𝐼2𝑎1)(Σ|𝑘,𝑘⟩⟨𝑘,𝑘|⊗(𝜃(𝜇𝑘−𝐸𝑘)𝐼𝑎1+(1−𝜃(𝜇−𝐸𝑘))𝑌𝑎1)(𝐶†⊗𝐼2⊗𝑛⊗𝐼2𝑎1)]⊗𝐼𝑎2 ….. (14) 5
The readouts from the encoding of the initial density matrix are obtained by equation (15).
⟨𝑃,𝑖,0,0,0|𝑂(𝐷)|𝑃,𝑗,0,0,0⟩=Σ𝐶𝑖𝑘𝑁𝑒/2𝑘=1𝐶𝑘𝑗∗=𝐷𝑖𝑗, where 𝑁𝑒=2Σθ(μ−𝑁𝑘=1𝐸𝑘) …. (15)
[052]
Once the initial density matrix is encoded, a Cholesky tensor is 10 encoded on the first quantum circuit to form a first Cholesky circuit component (represented by block 404 in FIG. 4). The Cholesky tensor is obtained from one or more electron integrals among the plurality of electron integrals, particularly, 3-center 2-electron and 2-center 2-electron integrals. As shown in the block 404, the controls are encoded on the 𝑙𝑜𝑔 𝑁𝑎𝑢𝑥 + 2𝑙𝑜𝑔 𝑁𝑎𝑜 qubits and the data is loaded on 15 the ancilla qubits. This is mathematically represented by equation 15.
𝑉𝐿=Σ[|𝑐,𝑟1,𝑟2,0⟩⟨𝑐,𝑟1,𝑟2,0|⊗𝐼2+|𝑐,𝑟1,𝑟2,1⟩⟨𝑐,𝑟,1|⊗𝑁𝑎𝑢𝑥−1,𝑁𝑎𝑜−1𝑐=0,𝑟1=0,𝑟2=0 (𝐿𝑟1,𝑟2𝑐′𝐼+𝑖√1−(𝐿𝑟1,𝑟2𝑐′)2𝑌)] ….. (15)
[053]
Further, the first quantum circuit component is composed with the first Cholesky circuit component and a diffusion operator to create a second 20 quantum circuit component that processes the density matrix to generate an intermediate state vector as illustrated by block 406. The diffusion operator 𝑅 is given by equation 16.
𝑅=Σ[2|𝑘,+,0⟩⟨𝑙,+,0|𝑁𝑎𝑢𝑥⊗𝐼2⊗2n −𝐼]𝑘𝑙 ….. (16)
[054]
Next, transpose of the Cholesky tensor is encoded on the first 25 quantum circuit to form a second Cholesky circuit component as illustrated by block 408. It is mathematically represented by equation 17.
29
𝑉′𝐿=Σ[|𝑐,𝑟1,𝑟2,0⟩⟨𝑐,𝑟1,𝑟2,0|⊗ (𝐿𝑟1,𝑟2𝑐′𝐼+𝑁𝑎𝑢𝑥−1,𝑁𝑎𝑜−1𝑐=0,𝑟1=0,𝑟2=0𝑖√1−(𝐿𝑟1,𝑟2𝑐′)2𝑌)+|𝑐,𝑟1,𝑟2,1⟩⟨𝑐,𝑟,1|⊗𝐼2] …. (17)
[055]
Further, the second quantum circuit component is composed with the second Cholesky circuit component (to form block 410) for processing the intermediate state vector to obtain a plurality of states at the second set of qubits in 5 the first quantum circuit. At step 210a6, sequences of bitstrings are read from the second set of qubits (by block 412) to obtain the direct matrix. This step is mathematically represented by equation 18.
⟨0,𝑖,0,0,0|𝐻⊗𝑚𝑉𝐿′𝑅𝑉𝐿𝑂(𝐷)|0,𝑗,0,1,0⟩=Σ𝐿𝑖𝑗𝑎𝐿𝑘𝑙𝑎𝐷𝑘𝑙=𝐽𝑖𝑗 ….. (18)
[056]
Once the direct (J) matrix is obtained, at step 210b, a correlation 10 exchange matrix is determined based on the initial density matrix, the collocation matrix, a Gradient of collocation matrix, and a Hessian of the collocation matrix for meta Generalized Gradient Approximation (meta GGA). The correlation exchange matrix is computed using a second quantum circuit. The second quantum circuit comprises (a) control qubits composed of a first set of qubits and a second set of 15 qubits, and (b) a plurality of ancilla qubits. The first set of qubits is associated with the number of points on the numerical grid, the second set of qubits is associated with the number of atomic orbitals, and the plurality of ancilla qubits are used to store numerical entries of the collocation matrix and the initial density matrix. In an embodiment, the second quantum circuit is mathematically represented as: 20 𝑙𝑜𝑔 𝑁𝑔 + 2𝑙𝑜𝑔 𝑁𝑎𝑜+2 qubits. Here, 𝑙𝑜𝑔 𝑁𝑔 represents the first set of qubits, 𝑙𝑜𝑔 𝑁𝑎𝑜 represents the second set of qubits, and 2 qubits represent the plurality of ancilla qubits.
[057]
Initially, at step 210b1, the collocation matrix is encoded on (a) control qubits composed of a first set of qubits and a second set of qubits, and (b) a 25 plurality of ancilla qubits in the second quantum circuit to obtain a first quantum circuit block a shown in FIG. 5. FIG. 5 illustrates a first quantum circuit block that encodes a collocation matrix in the method illustrated in FIG. 2, in accordance with some embodiments of the present disclosure. In an embodiment, a quantum circuit
30
block could be an independently functioning quantum circuit or a part of the
quantum circuit. In an embodiment, a set of Hadamard gates and a set of multi-controlled RY gates are used to encode the collocation matrix. As shown in FIG. 5, each entry of the collocation matrix is being encoded on 𝑎2 ancilla qubit with controls on 𝑁𝑎𝑜 and 𝑁𝑔 sets of qubits. 5
[058]
At step 210b2, a gradient of the collocation matrix is computed and encoded on the quantum circuit to obtain a first gradient quantum circuit. The first gradient quantum circuit is obtained by block encoding a plurality of sub-collocation matrices (six sub-collocation matrices in the present disclosure) obtained from the collocation matrix for positions (𝑥+/− 𝑑𝑒𝑙𝑡𝑎 𝑥,𝑦,𝑧), (𝑥,𝑦+10 /−𝑑𝑒𝑙𝑡𝑎 𝑦,𝑧), (𝑥,𝑦,𝑧+/−𝑑𝑒𝑙𝑡𝑎 𝑧) using three additional ancilla qubits. Further, the plurality of sub-collocation matrices are sandwiched across the set of Hadamard gates to block encode finite differences of each of the plurality of sub-collocation matrices for three directions including a X-direction, a Y-direction, and a Z-direction. 15
[059]
At step 210b3, a hessian of the collocation matrix on the quantum circuit is computed and encoded to obtain a first hessian quantum circuit block. The hessian of the collocation matrix is obtained by encoding the collocation matrix for a set of real space grid points (+/−𝑑𝑒𝑙𝑡𝑎,0,0),(0,+/−𝑑𝑒𝑙𝑡𝑎,0),(0,0,+/−𝑑𝑒𝑙𝑡𝑎),(+/𝑑𝑒𝑙𝑡𝑎,0,+/−𝑑𝑒𝑙𝑡𝑎),(+/𝑑𝑒𝑙𝑡𝑎,+/−𝑑𝑒𝑙𝑡𝑎,0),(0,+/−𝑑𝑒𝑙𝑡𝑎,+/20 −𝑑𝑒𝑙𝑡𝑎) on (a) control qubits composed of the first set of qubits and the second set of qubits, and (b) the plurality of ancilla qubits to obtain the third quantum circuit block. In other words, a plurality of collocation matrices are obtained from the collocation matrix with shifted coordinates corresponding to 15 basis functions which are mathematically represented by equation 19 25
Φ±Δ𝑥,±Δ𝑦=( 𝜙1(𝑥1±Δ𝑥,𝑦1±Δ𝑦,𝑧1),………𝜙1(𝑥𝑁𝑔±Δ𝑥,𝑦𝑁𝑔±Δ𝑦,𝑧𝑁𝑔)⋮⋮𝜙𝑁𝑎𝑜(𝑥1±Δ𝑥,𝑦1±Δ𝑦,𝑧1),………𝜙𝑁𝑎𝑜(𝑥𝑁𝑔±Δ𝑥,𝑦𝑁𝑔±Δ𝑦,𝑧𝑁𝑔))
31
Φ±Δ𝑥,±Δ𝑧=( 𝜙1(𝑥1±Δ𝑥,𝑦1𝑦,𝑧1±Δ𝑧),………𝜙𝑁𝑎𝑜(𝑥𝑁𝑔±Δ𝑥,𝑦𝑁𝑔,𝑧𝑁𝑔±Δ𝑧)⋮⋮𝜙1(𝑥1±Δ𝑥,𝑦1,𝑧1±Δ𝑧),………𝜙𝑁𝑎𝑜(𝑥𝑁𝑔±Δ𝑥,𝑦𝑁𝑔,𝑧𝑁𝑔±Δ𝑧))
Φ±Δ𝑦,±Δ𝑧=( 𝜙1(𝑥1,𝑦1±Δ𝑦,𝑧1±Δ𝑧),………𝜙𝑁𝑎𝑜(𝑥𝑁𝑔,𝑦𝑁𝑔±Δ𝑦,𝑧𝑁𝑔±Δ𝑧)⋮⋮𝜙1(𝑥1,𝑦1±Δ𝑦,𝑧1±Δ𝑧),………𝜙𝑁𝑎𝑜(𝑥𝑁𝑔,𝑦𝑁𝑔±Δ𝑥,𝑧𝑁𝑔±Δ𝑧)) Φ±Δ𝑥,±Δ𝑦5 =( 𝜙1(𝑥1±Δ𝑥,𝑦1±Δ𝑦,𝑧1),………𝜙1(𝑥𝑁𝑔±Δ𝑥,𝑦𝑁𝑔±Δ𝑦,𝑧𝑁𝑔)⋮⋮𝜙𝑁𝑎𝑜(𝑥1±Δ𝑥,𝑦1±Δ𝑦,𝑧1),………𝜙𝑁𝑎𝑜(𝑥𝑁𝑔±Δ𝑥,𝑦𝑁𝑔±Δ𝑦,𝑧𝑁𝑔)) Φ±2Δ𝑥=( 𝜙1(𝑥1±2Δ𝑥,𝑦1𝑦,𝑧1),………𝜙𝑁𝑎𝑜(𝑥𝑁𝑔±2Δ𝑥,𝑦𝑁𝑔,𝑧𝑁𝑔)⋮⋮𝜙1(𝑥1±2Δ𝑥,𝑦1),………𝜙𝑁𝑎𝑜(𝑥𝑁𝑔±2Δ𝑥,𝑦𝑁𝑔,𝑧𝑁𝑔)) Φ±2Δ𝑦=( 𝜙1(𝑥1,𝑦1±2Δ𝑦,𝑧1±Δ𝑧),………𝜙𝑁𝑎𝑜(𝑥𝑁𝑔,𝑦𝑁𝑔±2Δ𝑦,𝑧𝑁𝑔)⋮⋮𝜙1(𝑥1,𝑦1±2Δ𝑦,𝑧1±Δ𝑧),………𝜙𝑁𝑎𝑜(𝑥𝑁𝑔,𝑦𝑁𝑔±2Δ𝑦,𝑧𝑁𝑔))
Φ±2Δ𝑧=10 ( 𝜙1(𝑥1,𝑦1,𝑧1±2Δ𝑧),………𝜙𝑁𝑎𝑜(𝑥𝑁𝑔,𝑦𝑁𝑔,𝑧𝑁𝑔)⋮⋮𝜙1(𝑥1,𝑦1±2Δ𝑦,𝑧1±Δ𝑧),………𝜙𝑁𝑎𝑜(𝑥𝑁𝑔,𝑦𝑁𝑔±2Δ𝑦,𝑧𝑁𝑔))
(19)
The plurality of collocation matrices for different position shift are: (𝑥,𝑦,𝑧),(𝑥+/−𝑑𝑒𝑙𝑡𝑎 𝑥,𝑦,𝑧),,(𝑥,𝑦+/−𝑑𝑒𝑙𝑡𝑎 𝑦,𝑧),(𝑥,𝑦,𝑧+/−𝑑𝑒𝑙𝑡𝑎 𝑧),(𝑥+/−𝑑𝑒𝑙𝑡𝑎 𝑥,𝑦+/−𝑑𝑒𝑙𝑡𝑎 𝑦,𝑧),(𝑥+/−𝑑𝑒𝑙𝑡𝑎 𝑥,𝑦,𝑧+/𝑑𝑒𝑙𝑡𝑎 𝑧),(𝑥,𝑦+/15 −𝑑𝑒𝑙𝑡𝑎 𝑦,𝑧+/−𝑑𝑒𝑙𝑡𝑎 𝑧). Altogether 25 collocation matrices are block encoded with 5 qubits. And then, finite differences are performed to compute different
32
components of the Hessian of the collocation matrix as shown in FIGS. 6A through 6F. The finite difference formulas for different components of the Hessian of the collocation matrix is given by, 𝜕𝑗2Φ=Φ2Δ𝑗+Φ−2Δ𝑗−2Φ 𝜕𝑥𝑦2Φ=(ΦΔ𝑥,Δ𝑦−ΦΔ𝑥,−Δ𝑦)−(Φ−Δ𝑥,Δ𝑦−Φ−Δ𝑥,−Δ𝑦) 5 𝜕𝑦𝑧2Φ=(ΦΔ𝑦,Δ𝑧−ΦΔ𝑦,−Δ𝑧)−(Φ−Δ𝑦,Δ𝑧−Φ−Δ𝑦,−Δ𝑧)
𝜕𝑧𝑥2Φ=(ΦΔ𝑧,Δ𝑥−ΦΔ𝑧,−Δ𝑥)−(Φ−Δ𝑧,Δ𝑥−Φ−Δ𝑧,−Δ𝑥) (20)
As shown in equation (20), six independent components of the Hessian of the collocation matrix are obtained. The above 19 matrices are encoded on a third 10 quantum circuit and their differences corresponding to the finite difference formulas are computed. As a first step, the third quantum circuit mathematically expressed as 𝑛𝑔 +𝑛𝑎𝑜+ 2 + 2 qubits is used, where 𝑛𝑔= 𝑙𝑜𝑔 𝑁𝑔 , 𝑛𝑎𝑜= 𝑙𝑜𝑔 𝑁𝑎𝑜 arranged as |⟩𝑁𝑎𝑜|⟩𝑁𝑔|⟩1|⟩𝑎1 |⟩𝑎2 to compute gradients of the density matrix as shown in equation (21). 15 𝑉∂𝑗2Φ=𝐼2⊗(𝑛𝑎𝑜+𝑛𝑔)𝐻⊗2𝐼2⊗2𝐻Σ(|𝑐,𝑟,0,0,0⟩⟨𝑐,𝑟,0,0,0|⊗[Φ+Δ𝑗,𝑐,𝑟𝐼𝑟,𝑐+𝑖√1−Φ+Δ𝑗,𝑐,𝑟2𝑌] +|𝑐,𝑟,0,0,1⟩⟨𝑐,𝑟,0,0,1|⊗𝐼2+|𝑐,𝑟,0,1,0⟩⟨𝑐,𝑟,0,1,0|⊗[Φ−Δ𝑗,𝑐,𝑟𝐼+𝑖√1−Φ−Δ𝑗,𝑐,𝑟2𝑌]+|𝑐,𝑟,0,1,1⟩⟨𝑐,𝑟,0,1,1|⊗𝐼2+|𝑐,𝑟,1,0,0⟩⟨𝑐,𝑟,1,0,0|⊗[Φ𝑐,𝑟𝐼+𝑖√1−Φ𝑐,𝑟2𝑌]20 +|𝑐,𝑟,1,0,1⟩⟨𝑐,𝑟,1,0,1|⊗𝐼2+|𝑐,𝑟,1,1⟩⟨𝑐,𝑟,1,1|⊗𝐼2⊗2)𝐼2⊗(𝑛𝑎𝑜+𝑛𝑔)𝐻⊗2𝐼2⊗2𝐻
33
𝑉∂𝑖𝑗2Φ=𝐼2⊗(𝑛𝑎𝑜+𝑛𝑔)𝐻⊗2𝐼2⊗2𝐻Σ(|𝑐,𝑟,0,0,0⟩⟨𝑐,𝑟,0,0,0|⊗[Φ+Δ𝑖,+Δ𝑗,𝑐,𝑟𝐼𝑟,𝑐+𝑖√1−Φ+Δ𝑖,+Δ𝑗,𝑐,𝑟2𝑌]+|𝑐,𝑟,0,0,1⟩⟨𝑐,𝑟,0,0,1|⊗𝐼2+|𝑐,𝑟,0,1,0⟩⟨𝑐,𝑟,0,1,0|⊗[Φ−Δ𝑖,+Δ𝑗,𝑐,𝑟𝐼+𝑖√1−Φ−Δ𝑖,+Δ𝑗,𝑐,𝑟2𝑌]+|𝑐,𝑟,0,1,1⟩⟨𝑐,𝑟,0,1,1|⊗𝐼25 +|𝑐,𝑟,1,0,0⟩⟨𝑐,𝑟,1,0,0|⊗[Φ+Δ𝑖,−Δ𝑗,𝑐,𝑟𝐼+𝑖√1−Φ+Δ𝑖,−Δ𝑗,𝑐,𝑟2𝑌]+|𝑐,𝑟,1,0,1⟩⟨𝑐,𝑟,1,0,1|⊗𝐼2+|𝑐,𝑟,1,1,0⟩⟨𝑐,𝑟,1,1,0|⊗[Φ−Δ𝑖,−Δ𝑗,𝑐,𝑟𝐼+𝑖√1−Φ−Δ𝑖,−Δ𝑗,𝑐,𝑟2𝑌]+|𝑐,𝑟,1,1,1⟩⟨𝑐,𝑟,1,1,1|⊗𝐼2)𝐼2⊗(𝑛𝑎𝑜+𝑛𝑔)𝐻⊗2𝐼2⊗2𝐻
10 (21)
[060]
At step 210b4, the first quantum circuit block is composed with a second quantum circuit block to construct a third quantum circuit block. The second quantum circuit block encodes the initial density matrix on (a) control qubits composed of the second set of qubits and (b) an ancilla qubit in the quantum circuit. 15 The initial density matrix is loaded using its Eigen decomposition representation that involves: i) NaoC2 Givens rotations on 𝑙𝑜𝑔 𝑁𝑎𝑜 qubits, ii) Then block encoding a diagonal matrix of 1’s in the initial 𝑁𝑒/2 diagonal blocks and zeros in the rest using 2𝑙𝑜𝑔 𝑁𝑎𝑜 qubits, iii) Another set of NaoC2 Givens rotation with conjugate angles on 𝑁𝑒/2 diagonals. This is mathematically represented by equation 13. The 20 third quantum circuit block encodes a matrix multiplication of the collocation matrix and the initial density matrix.
[061]
At step 210b5, an electronic density is encoded on the quantum circuit by performing a sequential matrix multiplication of the collocation matrix, the density matrix, and the collocation matrix to obtain a fourth quantum circuit 25 block. After obtaining (i) the first quantum circuit block for encoding the collocation matrix, and (ii) the second quantum circuit block for encoding the initial density matrix, another collocation matrix is block-encoded for matrix multiplication using the fourth quantum circuit block. The matrix multiplication
34
encodes the data for electronic density on log Ng qubits. This is a state preparation
oracle for loading the electronic density.
[062]
At step 210b6, the third quantum circuit block that encodes the matrix multiplication of the collocation matrix and the initial density matrix is composed with the first gradient quantum circuit block that encoded the gradient of 5 the collocation matrix, to construct a second gradient quantum circuit block that encodes a gradient of the electronic density of the chemical compound. This is obtained when another set of three gradient matrices which are obtained by computing the gradient of the collocation matrix are block encoded using two ancilla qubits. The matrix multiplication in the second gradient quantum circuit 10 block encodes the data for the gradient of electronic density on 𝑙𝑜𝑔 𝑁𝑔+2 qubits and the gradient of the collocation. This is the state preparation oracle for loading the gradient of the electronic density.
[063]
At step 210b7, the first hessian quantum circuit block that encodes the hessian of the collocation matrix is composed sequentially with the first 15 quantum circuit block and the second quantum circuit block that encodes the initial density matrix to construct a second hessian quantum circuit block that encodes a first component of hessian of the electronic density. The first component of the hessian of the electronic density is obtained by a sequential matrix multiplication of the hessian of the collocation matrix, the collocation matrix, and the initial 20 density matrix. Similarly, at steps 210b8, the first hessian quantum circuit block that encodes the hessian of the collocation matrix is composed sequentially with the second quantum circuit block that encodes the initial density matrix and the first quantum circuit block that encoded the collocation matrix to construct a third hessian quantum circuit block that encodes a second component of hessian of the 25 electronic density. The second component of the hessian of the electronic density is obtained by a sequential matrix multiplication of the hessian of the collocation matrix, the initial density matrix, and the collocation matrix.
[064]
Further, at step 210b9, a fourth hessian quantum circuit block is constructed that encodes a third component of hessian of the electronic density of 30 the chemical compound by sequentially composing the first hessian quantum circuit
35
block that encodes the hessian of the collocation matrix with the second quantum
circuit block that encodes the initial density matrix and the first gradient quantum circuit block encodes the gradient of the collocation matrix. The third component of the hessian of the electronic density is obtained by a sequential matrix multiplication of the hessian of the collocation matrix, the initial density matrix, 5 and the gradient of the collocation matrix. FIG. 7 illustrates a quantum circuit comprising the second hessian quantum circuit block, the third hessian quantum circuit block, and the fourth hessian quantum circuit block that encode the first, second and third components of the hessian of electronic density respectively in the method illustrated in FIG. 2, in accordance with some embodiments of the present 10 disclosure.
[065]
Further, at step 210b10, the hessian of the electronic density of the chemical compound is computed by adding the first, second and third component of hessian of the electronic density of the chemical compound using the plurality of ancilla qubits. This step is mathematically represented by equation (22). 15
𝜕𝑖𝑗2(𝜌(𝑟1),…𝜌(𝑟𝑁𝑔))𝑇=𝜕𝑖Φ𝑇𝐷𝜕𝑗Φ+𝜕𝑖𝑗2Φ𝑇𝐷Φ+Φ𝑇𝐷𝜕𝑖𝑗2Φ (22)
[066]
Furthermore, at step 210b11, the Hessian of the electronic density of the chemical compound is encoded on (a) control qubits composed of a first set of qubits and a second set of qubits, and (b) the plurality of ancilla qubits to obtain a fifth hessian quantum circuit block. FIG. 8 illustrates a fifth quantum circuit block 20 that encodes Hessian of the electronic density in the method illustrated in FIG. 2, in accordance with some embodiments of the present disclosure. The fifth hessian quantum circuit block is mathematically represented by equation (23).
∂𝑖𝑗2ρ(𝑥,𝑦,𝑧)=⟨0,0,0,0,𝑐,0,0|𝐻⊗2(|0,0⟩⟨0,0|⊗𝑉∂𝑖Φ𝑅1𝑉𝐷𝑉∂𝑗Φ+|0,1⟩⟨0,1|⊗𝑉∂𝑖𝑗2Φ𝑅1𝑉𝐷𝑉Φ+|1,0⟩⟨1,0|⊗𝑉Φ𝑅1𝑉𝐷𝑉𝜕𝑖𝑗2Φ+25 |11⟩⟨11|⊗𝐼2⊗(𝑛𝑔+𝑛𝑎𝑜+2+2))𝐻⊗2|0,0,0,0,𝑐,0,0⟩ (23)
[067]
At step 210b12, the fourth quantum circuit block is put with a fifth quantum circuit block for a phase angle shifted reflector, the second gradient
36
quantum circuit block that encodes the gradient of the electronic density, and the
fifth hessian quantum circuit block that encodes the hessian of the electronic density into a quantum signal processing sequence. The quantum signal processing sequence create a sixth quantum circuit block that encodes a Chebyshev polynomial approximation of a nonlinear function of the electronic density. The nonlinear 5 function is a derivative of the electronic energy density with respect to the electronic density. This means a quantum signal processing sequence is constructed to encode a meta-GGA density functional.
[068]
At step 210b13, the sixth quantum circuit block is composed with the first quantum circuit block, to create a seventh quantum circuit block that 10 encodes a 𝑍 matrix. This step is mathematically represented by equation (24)
𝑉𝑍=Π[(𝐻⊗ 𝑛𝑎𝑜⊗ 𝛪⊗(𝑛𝑔+2))𝑉Φ𝑅1𝑉𝐷𝑅1𝑉Φ((1+𝑒−𝑖ϕ𝑘)|0,𝑖,0,0⟩⟨ 0,𝑖,0,0|𝑁𝑘=1−1)] Π[(𝐻⊗ 𝑛𝑎𝑜⊗ 𝛪⊗(𝑛𝑔+2))𝑉∇Φ𝑅1𝑉𝐷𝑅1𝑉Φ((1+𝑒−𝑖ϕ𝑘)|0,𝑖,0,0⟩⟨ 0,𝑖,0,0|−1)]𝑁𝑘=1 15 Π[(𝐻⊗ 𝑛𝑎𝑜⊗ 𝛪⊗(𝑛𝑔+2))𝑉∇Φ𝑅1𝑉𝐷𝑅1𝑉∇Φ((1+𝑒−𝑖ϕ𝑘)|0,𝑖,0,0⟩⟨ 0,𝑖,0,0|−1)]𝑁𝑘=1 Π[(𝐻⊗ 𝑛𝑎𝑜⊗ 𝛪⊗(𝑛𝑔+2))𝑉∇2Φ𝑅1𝑉𝐷𝑅1𝑉Φ((1+𝑒−𝑖ϕ𝑘)|0,𝑖,0,0⟩⟨ 0,𝑖,0,0|−1)]𝑁𝑘=1 Π[(𝐻⊗ 𝑛𝑎𝑜⊗ 𝛪⊗(𝑛𝑔+2))𝑉Φ𝑅1𝑉𝐷𝑅1𝑉∇2Φ((1+𝑒−𝑖ϕ𝑘)|0,𝑖,0,0⟩⟨ 0,𝑖,0,0|−1)]𝑁𝑘=1 20 (𝐻⊗𝑛𝑎𝑜⊗𝐼⊗(𝑛𝑔+2))𝑉Φ𝐻⊗𝑛𝑔 (24)
25
[069]
Further, at step 210b14, the seventh quantum circuit block is composed with the first quantum circuit block, to create an eight quantum circuit block that encodes the correlation exchange matrix. FIG. 9 illustrates the eight quantum circuit block that encodes the correlation exchange matrix in the method
37
illustrated in FIG. 2, in accordance with some embodiments of the present
disclosure. As shown in FIG. 9, In order to compute the Hessian contribution to the exchange potential, a gradient of energy density needs to be computed with respect to the electronic density. The energy density is a nonlinear function of the electronic density, its gradient and its Hessian. A quantum signal processing sequence is 5 generated using the block encoding oracles of the electronic density, its 3 components of the gradient and 6 components of the Hessian to obtain the electronic energy density and its derivative with respect to its density. This is then sandwiched via the first quantum circuit block that encodes the Collocation matrix to create the correlation exchange matrix, 𝑉𝑥𝑐. This step is mathematically represented by 10 equation (25).
𝑉𝑖𝑗𝑥𝑐=⟨𝑖,0,0,0|𝐻⊗𝑛𝑔𝑉Φ𝑉𝑍|𝑗,0,0,0⟩=ΣΦ𝑖𝑟𝑟𝑑𝐸(𝜌,∇𝜌,∇2𝜌)𝑑𝜌(𝑟)Φ𝑗𝑟 (25)
[070]
Once the correlation exchange matrix is determined, at step 210c, a Fock matrix is computed by adding the direct matrix and the correlation exchange matrix. At step 210d, qubitized diagonalization of the Fock matrix is performed to 15 obtain an intermediate density matrix. At step 210e, the steps 210a through 210d are repeated until the convergence criteria is satisfied by considering the intermediate density matrix as the initial density matrix to obtain the final density matrix of the chemical compound.
[071]
Finally, at step 212, the one or more classical hardware processors 20 are configured to utilize the final density matrix of the chemical compound to extract the one or more properties of the chemical compound. The final density matrix enables computation of properties such as the HOMO-LUMO gap from energies of the highest occupied Kohn Sham orbital and lowest unoccupied orbital, the charge distribution from integrating the electronic density in regions around the 25 different atoms where the electronic density is obtained from the initial density matrix, a dipole moment from an expectation value of perturbation electric field operator with respect to the final density matrix.
USE CASE EXAMPLES
38
[072]
Example 1: For drug screening, an accurate potential energy surface of a drug molecule is obtained from the method 200 disclosed herein. Then, gradients are studied from the PES and the forces associated with the displacement of the ligand molecules in protein environment are calculated. This will enable more accurate assessment of ligand conformations and protein docking faces. Further 5 dynamical response functions can be calculated which helps better determination of the binding.
[073]
Example 2: For selecting cathode materials for rechargeable batteries, accurate optimized geometry of the cathode molecules in electrolyte environment, and associated ground state energy are calculated from method 200. 10 This will then be used to compute more accurate energy enthalpy differences for the redox reactions. In turn the energy enthalpy differences are used to calculate high open cell voltages(OCV) (equilibrium voltage). A higher OCV leads to a higher cut-off for the recharging voltage. Cathode materials with different co-doped transition metal oxides (like Co, Ni, Mn) are screened using high OCV as the 15 deciding factor. As there are large number of properties to optimize such as number of battery recharging cycles, charging percentage, time to recharge, weight of battery material. The number of battery materials are more than a million, thus faster quantum chemistry calculation is required to screen them based on Density functional theory. 20
[074]
Example 3: For drug conformational search and drug design, the molecular descriptors, conformations, solvation energy is obtained from the method 200 disclosed herein. Then from the conformational energies and the descriptors the subset of drug molecule structure and conformations with one or more target properties are prepared via high throughput experimentation. 25
[075]
Example 4: Screening corrosion inhibitor molecules for aluminum sheets where efficacy is required to be enhanced, low cost of inhibitor molecules and availability is maintained, and adverse medical effects are ensured.
EXPERIMENTAL RESULTS 30
39
[076]
The meta generalized gradient approximation (meta GGA) functional computation was performed on classical processor using conventional methods and quantum processor using the method 200. The classical complexity is measured in terms of space and time complexity. Similarly, the quantum complexity is measured in terms of number of qubits and gate complexity. The 5 results are recorded in table 1.
Table 1
Quantity
Classical Complexity
Quantum Complexity
Space
Time
No.of Qubits
Gate complexity
𝑉𝑥𝑐[ρ,∇ρ,∇2ρ](meta−GGA)
𝑂(𝑁2𝑁𝑔)
𝑂(2𝑁2𝑁𝑔)
log𝑁+log𝑁𝑔
16(log𝑁+log𝑁𝑔)
For the meta GGA density functional computation, the correlation exchange potential is a function of the electronic density, gradient of the electronic density 10 and the Hessian of the electronic density. For computing the electronic density, the number of qubits needed to encode the electronic density is given by 𝑙𝑜𝑔 𝑁 +𝑙𝑜𝑔 𝑁𝑔, where 𝑁𝑔 is number of points on numerical grid and 𝑁 is number of basis functions. Further, 3 additional ancilla qubits are needed to encode different components of the Hessian of the electronic density. Gate complexity is associated 15 with a diffusion operator that enables a multiplication between the Collocation matrix and the Hessian of the collocation matrix and the initial density matrix. The gate complexity scales logarithmically with the number of basis functions. Classically, the space complexity to store the integrals is 𝑁2𝑁𝑔 and the time complexity is 𝑁2𝑁𝑔. 20
[077] 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 25
40
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.
[078] 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 5 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 10 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 15 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.
[079] The embodiments herein can comprise hardware and software elements. The embodiments that are implemented in software include but are not 20 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 25 connection with the instruction execution system, apparatus, or device.
[080] 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. 30 Further, the boundaries of the functional building blocks have been arbitrarily
41
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 5 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 10 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.
[081] 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 15 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 20 carrier waves and transient signals, i.e., be non-transitory. Examples include random access memory (RAM), read-only memory (ROM), volatile memory, non-volatile memory, hard drives, CD ROMs, DVDs, flash drives, disks, and any other known physical storage media.
[082] It is intended that the disclosure and examples be considered as 25 exemplary only, with a true scope of disclosed embodiments being indicated by the following claims.
42
We Claim:
1.
A 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 5 interfaces, wherein the method comprising:
receiving, via the one or more classical hardware processors, a chemical compound whose one or more properties to be extracted, wherein the chemical compound is at least one of (i) a molecule, and (ii) a solid (202); 10
obtaining, via the one or more classical hardware processors, a plurality of atomic coordinates of each of a plurality of atoms present in the chemical compound (204);
determining, via the one or more classical hardware processors, (i) a plurality of electron integrals, (ii) a core Hamiltonian matrix, and (iii) a 15 collocation matrix, from the plurality of atomic coordinates of each of the plurality of atoms present in the chemical compound, wherein the collocation matrix is a rectangular matrix of dimension 𝑁𝑔 and 𝑁𝑎𝑜, and wherein 𝑁𝑔 represents a number of real space grid points, 𝑁𝑎𝑜 is a number of basis functions (206); 20
determining, via the plurality of unentangled QPUs, an initial density matrix of the chemical compound from the core Hamiltonian matrix, wherein the initial density matrix is constructed using (i) a single particles unitary rotation matrix which is obtained by diagonalizing the core Hamiltonian matrix, and (ii) electron occupancies (208); 25
iteratively updating, via the plurality of unentangled QPUs, the initial density matrix until a convergence criteria is satisfied to obtain a final density matrix of the chemical compound (210), wherein iteratively updating the initial density matrix at each iteration comprises:
(a)
computing a direct (J) matrix from the initial density matrix 30 (210a);
43
(b)
determining a correlation exchange matrix based on the initial density matrix, the collocation matrix, a Gradient of collocation matrix, and a Hessian of the collocation matrix for meta Generalized Gradient Approximation (meta GGA) (210b) by:
(i)
encoding the collocation matrix on (a) control qubits 5 composed of a first set of qubits and a second set of qubits, and (b) a plurality of ancilla qubits in a second quantum circuit, to obtain a first quantum circuit block, wherein the first set of qubits is associated with the number of points on the numerical grid, the second set of qubits is associated with 10 the number of atomic orbitals, and the plurality of ancilla qubits are used to store numerical entries of the collocation matrix and the initial density matrix (210b1);
(ii)
encoding a gradient of the collocation matrix on the quantum circuit to obtain a gradient to obtain a first gradient 15 quantum circuit block;
(iii)
encoding a hessian of the collocation matrix on the quantum circuit to obtain a first hessian quantum circuit block (210b2);
(iv)
composing the first quantum circuit block with a 20 second quantum circuit block that encodes an intermediate density matrix on (a) control qubits composed of the second set of qubits and (b) an ancilla qubit in the quantum circuit, to construct a third quantum circuit block (210b3);
(v)
encoding an electronic density on the quantum circuit 25 by performing a sequential matrix multiplication of the collocation matrix, the density matrix, and the collocation matrix to obtain a fourth quantum circuit block (210b4);
(vi)
composing the third quantum circuit block with the first gradient quantum circuit block that encoded the gradient 30 of the collocation matrix, to construct a second gradient
44
quantum circuit block that encodes a gradient of the
electronic density of the chemical compound (210b5);
(vii)
composing the first hessian quantum circuit block that encodes the hessian of the collocation matrix with the first quantum circuit block and the second quantum circuit 5 block that encodes the initial density matrix sequentially, to construct a second hessian quantum circuit block that encodes a first component of hessian of the electronic density (210b6);
(viii)
composing the first hessian quantum circuit block 10 that encodes the hessian of the collocation matrix with the second quantum circuit block that encodes the initial density matrix and the first quantum circuit block that encoded the collocation matrix sequentially, to construct a third hessian quantum circuit block that encodes a second component of 15 hessian of the electronic density (210b7);
(ix)
composing the first hessian quantum circuit block that encodes the hessian of the collocation matrix with the second quantum circuit block that encodes the initial density matrix and the first gradient quantum circuit block encodes 20 the gradient of the collocation matrix, to construct a fourth hessian quantum circuit block that encodes a third component of hessian of the electronic density of the chemical compound (210b8);
(x)
computing the hessian of the electronic density of the 25 chemical compound by adding the first, second and third component of hessian of the electronic density of the chemical compound using the plurality of ancilla qubits (210b9);
(xi)
encoding the Hessian of the electronic density of the 30 chemical compound on (a) control qubits composed of a first
45
set of qubits and a second set of qubits, and (b) the plurality
of ancilla qubits, to obtain a fifth hessian quantum circuit block (210b10);
(xii)
putting the fourth quantum circuit block with a fifth quantum circuit block for a phase angle shifted reflector, the 5 second gradient quantum circuit block that encodes the gradient of the electronic density, and the fifth hessian quantum circuit block that encodes the hessian of the electronic density into a quantum signal processing sequence, to create a sixth quantum circuit block that 10 encodes a Chebyshev polynomial approximation of a nonlinear function of the electronic density, wherein the nonlinear function is a derivative of the electronic energy density with respect to the electronic density (210b11);
(xiii)
composing the sixth quantum circuit block with the 15 first quantum circuit block, to create a seventh quantum circuit block that encodes a 𝑍 matrix (210b12); and
(xiv)
composing the seventh quantum circuit block with the first quantum circuit block, to create an eight quantum circuit block that encodes the correlation exchange matrix 20 (210b13 );
(c)
determining a Fock (𝐹) matrix by adding the 𝐽 matrix and the correlation exchange matrix (210c);
(d)
performing a qubitized diagonalization of the 𝐹 matrix to obtain an intermediate density matrix (210d); and
(e)
repeating the steps of computing the direct (𝐽) matrix till performing the qubitized diagonalization of the 𝐹 matrix until the convergence criteria is satisfied to obtain the final density matrix of the chemical compound, wherein the intermediate density matrix is considered as the initial density matrix until the convergence criteria 30 is satisfied (210e); and

utilizing, via the one or more classical hardware processors, the final density matrix of the chemical compound to extract the one or more properties of the chemical compound (212).
2.
The method as claimed in claim 1, wherein the convergence criteria is 5 satisfied when a norm of a difference between the intermediate density matrix at a current iteration and the initial density matrix at a previous iteration is lesser than a predefined threshold.
3.
The method as claimed in claim 1, wherein the (i) control qubits composed 10 of the first set of qubits and the second set of qubits, and (ii) the plurality of ancilla qubits, to obtain the first quantum circuit block, is mathematically represented as: 𝑙𝑜𝑔 𝑁𝑔+ 𝑙𝑜𝑔 𝑁𝑎𝑜+2 𝑞𝑢𝑏𝑖𝑡𝑠, where 𝑙𝑜𝑔 𝑁𝑔 represents the first set of qubits, 𝑙𝑜𝑔 𝑁𝑎𝑜 represents the second set of qubits, and 2 qubits represent the plurality of ancilla qubits.
4.
The method as claimed in claim 1, wherein the hessian of the collocation matrix is obtained by encoding the collocation matrix for a set of real space grid points (+/−𝑑𝑒𝑙𝑡𝑎,0,0),(0,+/−𝑑𝑒𝑙𝑡𝑎,0),(0,0,+/−𝑑𝑒𝑙𝑡𝑎),(+/𝑑𝑒𝑙𝑡𝑎,0,+/−𝑑𝑒𝑙𝑡𝑎),(+/𝑑𝑒𝑙𝑡𝑎,+/−𝑑𝑒𝑙𝑡𝑎,0),(0,+/−𝑑𝑒𝑙𝑡𝑎,+/20 −𝑑𝑒𝑙𝑡𝑎) on (a) control qubits composed of the first set of qubits and the second set of qubits, and (b) the plurality of ancilla qubits to obtain the first hessian quantum circuit block.
5.
A system (100) comprising:
one or more classical hardware processors (108) and a plurality of unentangled quantum processors (122), wherein the one or more classical hardware processors (108) are communicably coupled to the plurality of unentangled quantum processors (122) by respective interfaces, wherein the one or more classical hardware processors are communicably coupled to at 30 least one memory (110) storing programmed instructions; one or more Input

/Output (I/O) interfaces (116); and the plurality of unentangled quantum processors (122) are operatively coupled to the at least one quantum memory (124), wherein the one or more classical hardware processors (108) and the plurality of unentangled quantum processors (122) are configured by the programmed instructions to:
receive a chemical compound whose one or more properties to be extracted, wherein the chemical compound is at least one of (i) a molecule, and (ii) a solid (202);
obtain a plurality of atomic coordinates of each of a plurality of atoms present in the chemical compound (204);
determine (i) a plurality of electron integrals, (ii) a core Hamiltonian matrix, and (iii) a collocation matrix, from the plurality of atomic coordinates of each of the plurality of atoms present in the chemical compound, wherein the collocation matrix is a rectangular matrix of dimension 𝑁𝑔 and 𝑁𝑎𝑜, and wherein 𝑁𝑔 represents a number of real space 15 grid points, 𝑁𝑎𝑜 is a number of basis functions (206);
determine an initial density matrix of the chemical compound from the core Hamiltonian matrix, wherein the initial density matrix is constructed using (i) a single particles unitary rotation matrix which is obtained by diagonalizing the core Hamiltonian matrix, and (ii) electron 20 occupancies (208);
iteratively update the initial density matrix until a convergence criteria is satisfied to obtain a final density matrix of the chemical compound (210), wherein iteratively updating the initial density matrix at each iteration comprises:
(a)
computing a direct (𝐽) matrix from the initial density matrix (210a);
(b)
determining a correlation exchange matrix based on the initial density matrix, the collocation matrix, a Gradient of collocation matrix, and a Hessian of the collocation matrix for meta Generalized Gradient Approximation (meta GGA) (210b) by:

(i)
encoding the collocation matrix on (a) control qubits composed of a first set of qubits and a second set of qubits, and (b) a plurality of ancilla qubits in a second quantum circuit, to obtain a first quantum circuit block, wherein the first set of qubits is associated with the number of points on 5 the numerical grid, the second set of qubits is associated with the number of atomic orbitals, and the plurality of ancilla qubits are used to store numerical entries of the collocation matrix and the initial density matrix (210b1);
(ii)
encoding a gradient of the collocation matrix on the 10 quantum circuit to obtain a gradient to obtain a first gradient quantum circuit block(210b2);
(iii)
encoding a hessian of the collocation matrix on the quantum circuit to obtain a first hessian quantum circuit block (210b3);
(iv)
composing the first quantum circuit block with a second quantum circuit block that encodes an intermediate density matrix on (a) control qubits composed of the second set of qubits and (b) an ancilla qubit in the quantum circuit, to construct a third quantum circuit block (210b4);
(v)
encoding an electronic density on the quantum circuit by performing a sequential matrix multiplication of the collocation matrix, the density matrix, and the collocation matrix to obtain a fourth quantum circuit block (210b5);
(vi)
composing the third quantum circuit block with the 25 first gradient quantum circuit block that encoded the gradient of the collocation matrix, to construct a second gradient quantum circuit block that encodes a gradient of the electronic density of the chemical compound (210b6);
(vii)
composing the first hessian quantum circuit block 30 that encodes the hessian of the collocation matrix with the

first quantum circuit block and the second quantum circuit
block that encodes the initial density matrix sequentially, to construct a second hessian quantum circuit block that encodes a first component of hessian of the electronic density (210b7);
(viii)
composing the first hessian quantum circuit block that encodes the hessian of the collocation matrix with the second quantum circuit block that encodes the initial density matrix and the first quantum circuit block that encoded the collocation matrix sequentially, to construct a third hessian quantum circuit block that encodes a second component of hessian of the electronic density (210b8);
(ix)
composing the first hessian quantum circuit block that encodes the hessian of the collocation matrix with the second quantum circuit block that encodes the initial density 15 matrix and the first gradient quantum circuit block encodes the gradient of the collocation matrix, to construct a fourth hessian quantum circuit block that encodes a third component of hessian of the electronic density of the chemical compound (210b9);
(x)
computing the hessian of the electronic density of the chemical compound by adding the first, second and third component of hessian of the electronic density of the chemical compound using the plurality of ancilla qubits (210b10);
(xi)
encoding the Hessian of the electronic density of the chemical compound on (a) control qubits composed of a first set of qubits and a second set of qubits, and (b) the plurality of ancilla qubits, to obtain a fifth hessian quantum circuit block (210b11);

(xii)
putting the fourth quantum circuit block with a fifth quantum circuit block for a phase angle shifted reflector, the second gradient quantum circuit block that encodes the gradient of the electronic density, and the fifth hessian quantum circuit block that encodes the hessian of the electronic density into a quantum signal processing sequence, to create a sixth quantum circuit block that encodes a Chebyshev polynomial approximation of a nonlinear function of the electronic density, wherein the nonlinear function is a derivative of the electronic energy density with respect to the electronic density (210b12);
(xiii)
composing the sixth quantum circuit block with the first quantum circuit block, to create a seventh quantum circuit block that encodes a Z matrix (210b13); and
(xiv)
composing the seventh quantum circuit block with the first quantum circuit block, to create an eight quantum circuit block that encodes the correlation exchange matrix (210b14);
(c)
determining a Fock (𝐹) matrix by adding the 𝐽 matrix and the correlation exchange matrix (210c);
(d)
performing a qubitized diagonalization of the 𝐹 matrix to obtain an intermediate density matrix (210d); and
(e)
repeating the steps of computing the direct (𝐽) matrix till performing the qubitized diagonalization of the 𝐹 matrix until the convergence criteria is satisfied to obtain the final density matrix of the chemical compound, wherein the intermediate density matrix is considered as the initial density matrix until the convergence criteria is satisfied (210e); and
utilize the final density matrix of the chemical compound to extract the one or more properties of the chemical compound (212).

6.
The system as claimed in claim 5, wherein the convergence criteria is satisfied when a norm of a difference between the intermediate density matrix at a current iteration and the initial density matrix at a previous iteration is lesser than a predefined threshold.
7.
The system as claimed in claim 5, wherein the (i) control qubits composed of the first set of qubits and the second set of qubits, and (ii) the plurality of ancilla qubits, to obtain the first quantum circuit block, is mathematically represented as: 𝑙𝑜𝑔 𝑁𝑔+ 𝑙𝑜𝑔 𝑁𝑎𝑜+2 𝑞𝑢𝑏𝑖𝑡𝑠, where 𝑙𝑜𝑔 𝑁𝑔 represents the first set of qubits, 𝑙𝑜𝑔 𝑁𝑎𝑜 represents the second set of qubits, and 2 qubits represent the plurality of ancilla qubits.
8.
The system as claimed in claim 5, wherein the hessian of the collocation matrix is obtained by encoding the collocation matrix for a set of real space grid points (+/−𝑑𝑒𝑙𝑡𝑎,0,0),(0,+/−𝑑𝑒𝑙𝑡𝑎,0),(0,0,+/−𝑑𝑒𝑙𝑡𝑎),(+/𝑑𝑒𝑙𝑡𝑎,0,+/−𝑑𝑒𝑙𝑡𝑎),(+/𝑑𝑒𝑙𝑡𝑎,+/−𝑑𝑒𝑙𝑡𝑎,0),(0,+/−𝑑𝑒𝑙𝑡𝑎,+/−𝑑𝑒𝑙𝑡𝑎) on (a) control qubits composed of the first set of qubits and the second set of qubits, and (b) the plurality of ancilla qubits to obtain the first hessian quantum circuit block.