Full quantum mechanical simulation method for real systems

Abstract

A computer-implemented method for solving low-energy excitation spectra and their corresponding eigenstates is provided. The solved eigenstates include the ground state and single-fermion excited states. The method comprises: calculating the ground state and single-fermion excited states of a system and / or isolated particle subsystem; using quantum mechanical transition matrix elements between hybrid fermions and long-range Coulomb and exchange interactions to calculate the coupling between fermions for a given charge and spin density; calculating the system free energy as a function of molecular structural properties based on an energy spectrum that depends on the particle position and particle orientation, where the position is the charge center of each particle; and simulating the particle system by integrating the time evolution of structural properties using the full quantum mechanical time evolution of the quantum state from a given initial state.

Description

[0001] CROSS-REFERENCE TO RELATED APPLICATIONS

[0002] This application claims priority to U.S. Provisional Application No. 63 / 104,122, filed on October 22, 2020, the entire contents of which are incorporated herein by reference.

[0003] introduce

[0004] The present disclosure is applicable to the field of first-principles quantum mechanical calculations of material properties. This disclosure provides computational methods for predicting the structure and function of molecules, atoms, atomic nuclei, and other particles (including biomolecules in the environment), as well as their interactions, by applying a novel approach to solving the exact quantum mechanical eigenstates and eigenvalues ​​of many-body systems, with computational time that scales polynomially with the number of particles studied.

[0005] In all existing methods, the nuclei in the system are treated classically, using the Born-Oppenheimer approximation, and only the electrons are treated quantum mechanically. Therefore, in a sense, these methods are all semi-quantum mechanical. The present disclosure provides a new method that can treat all nuclei and electrons as quantum mechanical particles, which was previously considered completely unfeasible. The method can be applied in practice mainly because it can accurately solve the low-energy eigenstates of the entire system using polynomial time scaling. Summary of the Invention

[0006] According to an embodiment of the present disclosure, the present disclosure provides a computer-implemented method for solving a low-energy excitation spectrum. The low-energy excitation spectrum includes the ground state energy and the corresponding eigenstates of the physical properties of the particle system, including the ground state |Vac γ > and single fermion excited states The method includes: calculating the ground state and single fermion excited states of the system and / or isolated particle subsystem; using quantum mechanical transition matrix elements between hybrid fermions and long-range Coulomb and exchange interactions to calculate the coupling between fermions under given charge and spin density; calculating the system free energy as a function of molecular structural properties based on the solved system energy spectrum, given particle positions and particle directions, where the position of the particle is the charge center of each particle; and simulating the particle system by integrating the time evolution of structural properties using the time evolution of the quantum state of a given initial state.

[0007] According to an embodiment of the present disclosure, a computer-implemented method for simulating particle interactions is provided. The method includes receiving initial conditions for particles under simulation; calculating an initial isolated state for each particle based on the corresponding initial conditions; calculating an initial system state, which is the tensor product of the initial isolated state of each particle at the initial position of each corresponding particle; calculating projection coefficients based on the initial system state and eigenstates of the particle's total Hamiltonian; simulating the evolution of the particle's time-dependent total quantum state based on the projection coefficients; and obtaining an expected value of the particle based on the particle's time-dependent total quantum state. The computational time for simulating the evolution of the particle's time-dependent total quantum state scales polynomially with the number of particles. BRIEF DESCRIPTION OF THE DRAWINGS

[0008] Figure 1 A flow chart depicting a computational algorithm consistent with the disclosed embodiments.

[0009] Figure 2 A flow chart depicts an exemplary process for a computerized system consistent with the disclosed embodiments to perform a full quantum mechanical simulation of a real system.

[0010] illustrate Technical Field

[0011] This disclosure is applicable to the field of first-principles quantum mechanical calculations of material properties. The disclosed method, implemented on a computer or any suitable programmable electronic device, provides the advantage of reducing molecular dynamics simulation time through a novel algorithm. This novel algorithm treats all nuclei and electrons in a system as distinct quantum mechanical particles, modeling all dynamic responses of the system at high temperatures as fully quantum mechanical responses. Background Art

[0012] Simulating interacting atomic systems, such as large biomolecules in solvent, is crucial for drug design and the design of new materials. Direct molecular dynamics (MD) simulations, using ab initio quantum mechanical and molecular mechanics (QM / MM) methods, are extremely powerful for studying chemical reaction mechanisms in complex environments. However, current state-of-the-art computer-implemented methods capable of accurately diagonalizing the many-body Hamiltonian scale exponentially with the number of particles studied, making them prohibitively expensive and impractical for large systems. Semiempirical QM / MM methods can significantly reduce the computational cost of QM / MM calculations during MD simulations, but these methods suffer from lower accuracy and, more fundamentally, fail to capture off-diagonal quantum entanglement effects of fermions in the system, most notably proton entanglement, which is crucial for understanding biomolecular systems at room temperature. Since the discovery of quantum mechanics nearly a century ago, a comprehensive first-principles quantum mechanical description of any system under investigation has been the holy grail of scientists. For a real system, as long as we have a Hamiltonian that correctly captures all the important interactions between quantum particles (especially fermions), the ultimate task at hand is to solve the eigenstates and eigenvalues ​​of the Hamiltonian, especially the states close to the ground state, which is defined as the lowest energy state of the Hamiltonian. Once these eigenstates and eigenvalues ​​of the system are known, given its initial conditions, the physical properties of the system (including its dynamical evolution) are known, because the Schrödinger equation:

[0013]

[0014] Once the wave function is known, according to the provisions of quantum mechanics, any physical observable, when measured in the state |ψ(t)>, will be the expectation of the operator corresponding to the observable, that is, The Schrödinger equation can be integrated under given initial conditions and expanded into the system eigenstate |ψ i > is a linear combination of

[0015]

[0016] so

[0017]

[0018] Therefore, if the eigenstate |ψ i >The evolution of the system is known by solving the following eigenvalue problem.

[0019]

[0020] Furthermore, physical observables can be predicted / calculated from first principles. The above can be easily generalized to a collection of initial states, where the initial conditions are specified by a density matrix. The general many-body fermion Hamiltonian has the form

[0021]

[0022] in

[0023]

[0024] One of the main obstacles to achieving the ultimate feast of applying the theory to all problems of human concern is that in quantum mechanics, due to the two-body interaction term The non-quadratic terms embodied in the equations indicate that the number of degrees of freedom of state scales exponentially with the size of the system, and hence the size of the Hamiltonian matrix is ​​also thought to grow exponentially. Consequently, the consensus is to use various approximations with varying degrees of success to avoid this exponential curse, depending on the area of ​​interest. DETAILED DESCRIPTION

[0025] The present disclosure provides a solution to the above-mentioned problem by applying a new method in simulation to arrive at an accurate quantum mechanical solution for a many-body system, whose computation time scales polynomially with the number of particles under study. In some embodiments of the present disclosure, the present disclosure provides a realization of a local "vacuum" state at each local position, and the observable quantities that can be measured are due to the inter-position coupling of particle excitations from these local vacuum states. We assume that all elementary quantum particles are fermions (bosonic nuclei are and can be modeled as composite fermions). Once the low-energy spectrum of a system and the corresponding eigenstates are known, the physical properties of the system can be derived from first principles. For any fermionic system, we can decompose the full Hilbert space into the tensor product of two sub-Hilbert spaces where S is the single-orbital fermion system at spatial location x and E is the rest of the system. For each x, we have the following complete set of orthogonal local states at x, defined by the fermion creation operator

[0026]

[0027] We define the Bogolyubov vacuum at x as

[0028]

[0029] where α(x) and β(x) are c-numbers, and

[0030] |α(x)| 2 +|β(x)| 2 =1,2|β(x)|2 =n Vac (x), (9)

[0031] Among them, n Vac (x) is the particle density at x in a non-empty Bogoliubov vacuum. We further define the following Bogoliubov transformation operator

[0032]

[0033]

[0034]

[0035]

[0036] We further verify the following:

[0037]

[0038]

[0039] and the following anti-commutative relation

[0040]

[0041]

[0042] We noticed Create fermions from |Vac(x)> with fractional charge |e|(|α| 2 -β| 2 )=|e|(1-2|β| 2 ) and spin σ. This transformation effectively defines a general class of fermions, b-fermions, which become electrons when β = 0, positrons when α = 0, and positrons when |β| 2 =1 / 2, it becomes a Majorana fermion.

[0043] The self-consistent effective Hamiltonian theory asserts that for any interacting many-body fermion system, there exist two chiral symmetry-breaking effective Hamiltonians: and They are quadratic, each providing a set of eigenstates, such as equation (4), which, in addition, give the exact ground state and single-fermion excited states of the full Hamiltonian (5). Since it is known that the quadratic Hamiltonian corresponds to a polynomial scaling in terms of degrees of freedom, which is proportional to the number of sites x, rather than an exponential scaling, the present disclosure provides a technical breakthrough in the computational algorithm for the exact quantum mechanical many-body solution, avoiding the various errors of approximating the neglect of off-diagonal long-range processes caused by ignoring the off-diagonal pairing terms in the mathematical implementation of the continuous Bogoliubov transform. This technical breakthrough also allows computer simulation of the dynamic processes of large systems, rather than conducting resource-consuming and high-risk trial-and-error experiments in the laboratory.

[0044] The following is a detailed description of the self-consistent loop in our algorithm. The flowchart of the algorithm is as follows Figure 1 shown.

[0045] 1.A1: Establishing the parameters of the Hartree-Fock mean-field Hamiltonian

[0046] For a self-consistent Hartree-Fock calculation, the following quadratic Hamiltonian is established and solved

[0047]

[0048] That is, all E μ and v μσ,i All are found in the conclusions of Hartree-Fock.

[0049] 2.A2: Bogoliubov Transform

[0050] Whenever a Bogoliubov transform is introduced, the matrix size of the quadratic Hartree-Fock Hamiltonian doubles.

[0051]

[0052] in, is understood to be vector.

[0053] 3.A3: Splitting the Hamiltonian into Chiral Symmetry-Breaking Parts

[0054] We divide the above quadratic Hamiltonian into two parts,

[0055]

[0056] in

[0057]

[0058] and

[0059]

[0060] 4.A4: Solving for the Eigenstates of the Chiral Symmetry-Broken Hamiltonian

[0061] We solve the ground state of the chiral symmetry-broken Hamiltonian with charge conservation constraints. (Note that in the original Hartree-Fock calculation, there is no need to introduce a chemical potential because is a good quantum number.

[0062]

[0063] Note that the self-consistent calculation above appears to start with a fully spin-polarized Hartree-Fock solution, but the total charge does not satisfy the charge conservation condition. Due to the constraints, the chemical potential δ is introduced as a Lagrange multiplier.

[0064] 5.A5: Imposing a no-double-occupancy constraint

[0065] It is known that the ground state of the quadratic Hamiltonian has the form

[0066]

[0067] in

[0068] λ xσ，x′σ′ =-λ x′σ′，xσ

[0069] is antisymmetric and N γ is a normalization constant such that

[0070] <0 γ |0 γ >=1

[0071] Furthermore, the single-particle excited state is given by the diagonalization of the quadratic effective Hamiltonian

[0072]

[0073] for For each field pairing in , we can further perform additional Bogolyubov transformations. The additional Bogolyubov transformations will change the equation in Equation (19) to Therefore an extra loop is required.

[0074] 6.A6: Constructing a new Hartree-Fock Hamiltonian from the complete many-body Hamiltonian in a new chiral symmetry-breaking basis

[0075] Under the new fermion operators defined with the new chiral symmetry-breaking vacuum, the full many-body Hamiltonian can be reconstructed, discarding those terms where the annihilation operators outnumber the production operators. This can be done by directly replacing operator. Due to the non-double occupation restriction, the isotopic Hubbard U will not exist, i.e., it is renormalized. If the new basis converges, the self-consistent calculation is complete; otherwise, go to step A1.

[0076] According to an embodiment of the present disclosure, a computer-implemented method for solving a low-energy excitation spectrum is provided. The low-energy excitation spectrum includes the ground state energy and the corresponding eigenstates of the physical properties of the particle system, including the ground state |Vac γ > and single fermion excited states

[0077] The method involves computing the ground state and single fermion excited states of a system and / or isolated particle subsystems. The computational steps are described in the algorithm detailed in A1 to A6 and are described in Figure 1 Described in .

[0078] The method also involves calculating the coupling between fermions at a given charge and spin density using elements of the quantum mechanical transition matrix between hybrid fermions, as well as long-range Coulomb and exchange interactions. Hartree-Fock calculations can be used to update the coupling matrix elements whenever the single-particle ground state is updated due to a Bogoliubov transformation. Hybrid fermions here also refer to mixtures of particle and antiparticle / hole states.

[0079] The method also includes calculating the free energy of the system as a function of molecular structural properties based on the solved system energy spectrum, given the particle positions, which are the charge centers of each particle, and the particle orientations.

[0080] The method also includes simulating the particle system by integrating the time evolution of the structural properties using the time evolution of the quantum state given its initial state.

[0081] Here, the simulated particle system includes at least one of gas, liquid, nanodevice, biomolecule (eg, protein, RNA, and / or DNA), polymer, and small molecule.

[0082] In some embodiments, the method for solving the low-energy excitation spectrum may further include identifying a plurality of coherent off-diagonal long-range ordered quantum states at room temperature. Coherent quantum states are the qubit building blocks of quantum computers and quantum memory storage. Here, the coherent quantum state is represented by the antisymmetric off-diagonal matrix λ in equation (20) xσ，x′σ′ The singular value identifiers of .

[0083] In some embodiments, the simulated particle system may include one or more molecules used to design a new material. The method may also include inputting data corresponding to the designed material; after the simulation is complete, generating data related to the position, velocity, and energy of the one or more molecules; and estimating macroscopic properties of the one or more molecules and the effectiveness of the designed material.

[0084] In some embodiments, the method for solving the low-energy excitation spectrum can also include identifying energy transfer channels and / or frequencies during bond formation and / or breakage between particles. Here, the energy transfer channels are represented by the off-diagonal long-range pairing terms λ in Eq. (20) xσ，x′σ′ Logo.

[0085] In some embodiments, energy transfer channels and / or frequencies are identified for designing less invasive treatments. The method may also include targeting electrical signals in the identified channels and / or frequencies to enhance and / or prevent bond formation and / or breakage. Here, the energy transfer channels are represented by the off-diagonal long-range pairing terms λ in Eq. (20) xσ，x′σ′ In our design, whenever the energy transfer is disturbed by the target electrical signal, the corresponding off-diagonal long-range ordering will be affected.

[0086] In some embodiments, the simulated particle system may include one or more molecules for designing new drugs. The method may also include designing a drug for a drug target; selecting a new drug; inputting data corresponding to the selected new drug; after the simulation is completed, generating data related to the position, velocity, and energy of the molecules; estimating the macroscopic properties of the molecules and the effectiveness of the selected drug; and estimating the effectiveness of the selected drug in enhancing or hindering the formation of bonds between biological molecules (such as protein molecules).

[0087] Figure 2 A flowchart depicts an exemplary process for performing a full quantum mechanical simulation of a real system by a computerized system consistent with the disclosed embodiments. In these steps, whenever reference is made to the computation of states, they refer to the state embodied in the above algorithm. Figure 1 The calculation process is shown. The antisymmetric non-diagonal pairing matrix λ in the ground state (41) xσ，x′σ′ The singular values ​​of will also be the same as the energy transmission channels of the corresponding off-diagonal long-range ordered quantum states.

[0088] In step 102, the computerized system receives initial conditions for particles to be simulated. A particle can refer to a mathematical representation of a microscopic physical object, such as an atom, an atomic nucleus, a molecule, an electron, and / or other similar molecular, atomic, or subatomic object. The initial conditions of a particle can refer to parameters of the particle at an initial time. For example, the initial time can be the time at the start of the simulation. Examples of initial conditions can include the particle's initial mass, velocity, direction, position, energy level, quantum state, polarity, parity, and / or other physical properties.

[0089] In step 104, the computerized system calculates the initial particle state. In some embodiments, the computing system can calculate the initial particle state for each particle included in the simulation. The initial particle state can represent the physical state of the particle in isolation (without regard to other particles) in the simulation. The computerized system can calculate the initial particle state based on the initial conditions of the particles. For example, the initial state of each isolated particle can be calculated by obtaining an equilibrium state (or ground state) in which the first excited state has a large gap. In another example, the initial particle state can be calculated as a thermal distribution of each Boltzmann distribution.

[0090] In step 106, the computerized system calculates an initial system state. In some embodiments, the computerized system can calculate the initial system state by calculating the tensor product of the initial particle state of each particle at the initial position of each corresponding particle. The initial position of the particle can be the center of mass position of the corresponding particle at the beginning of the simulation. In some embodiments, the initial position can be included as part of the initial condition.

[0091] In step 108, the computerized system calculates projection coefficients. Because the initial system state calculated in step 106 is not an eigenstate of the overall Hamiltonian, the eigenstates of the overall Hamiltonian need to be obtained. In some embodiments, the computerized system may project the initial system state obtained in step 106 onto the eigenstates of the Hamiltonian to obtain the projection coefficients. For example, the computerized system may perform the projection by calculating the inner product of the quantum mechanical Hilbert space vectors corresponding to 1) the initial system state and 2) the eigenstates of the overall Hamiltonian, the solution of which is central to the present disclosure.

[0092] In step 110, the computerized system simulates the evolution of the time-dependent full quantum state of the particle. Based on the projection coefficients calculated in step 108, the complete Hamiltonian of the particle can be obtained. The computerized system can then derive the full quantum state of the particle at a given time. In some embodiments, the computerized system can use the time evolution of the quantum state given its initial state to perform a simulation of the evolution of the particle over a period of time.

[0093] In step 112, the computerized system obtains expected values ​​for the particles. Once the simulation results for the full quantum state of the particles are obtained, the computerized system can obtain expected values ​​for various parameters related to the physical state of the particles. Examples of expected values ​​can include mass, velocity, direction, position, energy level, and / or other physical properties of the particle or particle system.

[0094] It is important to point out that the size of the matrix is ​​proportional to the number of sites multiplied by the number / type of fermions in the system (i.e. N atoms ×N orbitals) scales linearly, rather than exponentially as in other fully quantum mechanical approaches.

[0095] The present disclosure achieves the technical advantage or effect of performing complex molecular dynamics simulations at extremely fast speeds while using a computer with conventional computing capabilities.

[0096] Another aspect of the present disclosure relates to an apparatus or system for simulating an atomic system.The computer comprises a data input device, a data output device, a processor and a computer program stored in a computer memory.

[0097] Applying the methods disclosed in this article to the field of quantum computers can help find candidate materials that exhibit long quantum coherence times at room temperature.

[0098] The method disclosed in this article can be applied in the field of materials design: by applying the above method to solve the complete eigenstates and eigenvalues ​​of the system under study, the material properties of interest can be deduced. Therefore, this method can test more material components, thereby increasing the possibility of finding useful materials with target properties.

[0099] The method disclosed in this article can be applied to the field of drug design: by applying the above method to solve the complete eigenstates and eigenvalues ​​of the molecular system under study, the properties of the molecules and the interactions between the molecules can be deduced. Therefore, this method can test a larger number of molecules, such as compounds, thereby increasing the possibility of finding useful compounds for treating diseases.

[0100] The methods disclosed herein may also be applied in the field of medical procedure design to test the effects of electromagnetic signals targeted to energy delivery pathways on biomolecular bond formation and / or breaking.

[0101] Other embodiments of the invention will be apparent to those skilled in the art from consideration of the specification and practice of the invention disclosed herein. It is intended that the specification and examples be considered as exemplary only, with the true scope and spirit of the invention being indicated by the following claims.

Claims

1. A computer-implemented method for solving low-energy excitation spectra, including ground state energies and single fermion excitation energies, as well as the corresponding eigenstates of the particle system, including ground state |Vac γ > and single fermion excited states The method comprises: calculating the ground state and single fermion excited state of the particle system; Calculate the coupling between fermions for a given charge and spin density using quantum mechanical transition matrix elements between hybrid fermions and long-range Coulomb and exchange interactions; Calculate the free energy of the system as a function of molecular structural properties based on the solved energy spectrum of the system, given the positions of the particles where the center of charge of each particle is located, and the particle orientations; and The particle system is simulated by integrating the time evolution of structural properties using the time evolution of the quantum state of a given initial state.

2. The computer-implemented method of claim 1, wherein: The simulated particle system includes at least one of nanodevices, biomolecules and small molecules.

3. The computer-implemented method of claim 1 , further comprising: Identification of multiple coherent off-diagonal long-range ordered quantum states at room temperature, where coherent quantum states are the building blocks of qubits stored in quantum computers and quantum memories.

4. The computer-implemented method of claim 1 , wherein: The simulated particle system includes one or more molecules for designing a new material, and the method further includes: Input data corresponding to the design materials; After the simulation is complete, data related to the position, velocity, and energy of one or more molecules is generated; and Estimate the macroscopic properties of one or more molecules and the effectiveness of designed materials.

5. The computer-implemented method of claim 1 , further comprising: Identify energy transfer channels and / or frequencies during bond formation and / or breakage between particles.

6. The computer-implemented method of claim 5, wherein identifying the energy delivery channels and / or frequencies is used to design a less invasive approach to treatment, the method further comprising: Electromagnetic signals in identified channels and / or frequencies are targeted to enhance and / or inhibit bond formation and / or breaking.

7. The computer-implemented method of claim 1 , wherein: The simulated particle system includes one or more molecules for designing new drugs, and the method further includes: Select new drugs; Enter data corresponding to the selected new drugs and biomolecules; After the simulation is completed, data related to the position, velocity, and energy of the molecules are generated; Evaluate the macroscopic properties of new drug molecules and the effectiveness of selected drugs; and Estimation of the effectiveness of selected drugs in enhancing or hindering the formation of bonds between molecules of biomolecules.

8. The computer-implemented method of claim 1 , wherein: The particles include at least one of atoms, atomic nuclei and molecules.

9. The computer-implemented method of claim 1 , wherein: At least one of atoms, atomic nuclei, and molecules is considered to be a quantum mechanical particle.

10. The computer-implemented method of claim 1, wherein the initial isolated state of each particle is an equilibrium state.

11. The computer-implemented method of claim 1 , wherein calculating a ground state and a single-fermion excited state of the particle system comprises: Establish the parameters of the Hartree-Fock mean-field Hamiltonian and solve the Hamiltonian; Perform Bogolliubov transformation on the Hamiltonian; Split the Hamiltonian into its chiral symmetry-broken parts; Solve the Hamiltonian to obtain the eigenstates of the chiral symmetry-breaking Hamiltonian; impose non-dual occupancy restrictions; and In the new chiral symmetry-breaking basis, a new Hartree-Fock Hamiltonian is constructed from the complete many-body Hamiltonian.

12. The computer-implemented method of claim 1, wherein when the particle system comprises a particle subsystem, the method comprises calculating a ground state and single fermion excited states of the particle subsystem in isolation.

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