WO2023232175A1 - Procédé de détermination des énergies de l'état fondamental d'un système physique au moyen d'un ordinateur quantique - Google Patents

Procédé de détermination des énergies de l'état fondamental d'un système physique au moyen d'un ordinateur quantique Download PDF

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Publication number
WO2023232175A1
WO2023232175A1 PCT/DE2023/100263 DE2023100263W WO2023232175A1 WO 2023232175 A1 WO2023232175 A1 WO 2023232175A1 DE 2023100263 W DE2023100263 W DE 2023100263W WO 2023232175 A1 WO2023232175 A1 WO 2023232175A1
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energy
functional
expected value
ground state
observable
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PCT/DE2023/100263
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German (de)
English (en)
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Nicolas Vogt
Florian Eich
Peter SCHMITTECKERT
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HQS Quantum Simulations GmbH
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
    • G06N10/00—Quantum computing, i.e. information processing based on quantum-mechanical phenomena
    • G06N10/60—Quantum algorithms, e.g. based on quantum optimisation, quantum Fourier or Hadamard transforms
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
    • G06N10/00—Quantum computing, i.e. information processing based on quantum-mechanical phenomena
    • G06N10/20—Models of quantum computing, e.g. quantum circuits or universal quantum computers
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
    • G06N10/00—Quantum computing, i.e. information processing based on quantum-mechanical phenomena
    • G06N10/40—Physical realisations or architectures of quantum processors or components for manipulating qubits, e.g. qubit coupling or qubit control

Definitions

  • the invention relates to a method for determining ground state energies of a physical system using a quantum computer.
  • a typical use case for quantum computers is determining the ground state energy of the Hamiltonian of a quantum mechanical system.
  • the two most common methods for estimating ground state energy on quantum computers are variational approaches and quantum phase estimation (QPE).
  • VQE Variational Quantum Eigensolver
  • test states can be generally written as, where 9 is the vector of classical parameters, which U ⁇ di) are unitary transformations performed on the quantum computer and depend on 6 and is an initial state that can be efficiently initialized on the quantum computer.
  • NISQ Noisesy-Intermediate-Scale Quantum and refers to logic gate-based quantum computers.
  • Dissipation processes and other errors can change the available space of test states, so that a good approximation of the ground state is not found.
  • An example is the loss of electrons during simulation of a particle number-preserving electron system on a quantum computer with dissipation.
  • QPE Quantum Phase Estimation
  • a quantum Fourier transform is applied to the anzilla qubits. With a probability that depends on the initial state of the m system qubits, after measuring the anzilla qubits one obtains an stood in which the m system qubits contain the ground state. In this case, measuring the ancestral qubits results in a binary representation of the phase Eoto, from which the ground state energy Eo is obtained.
  • DFT density functional theory
  • the ground state energy is determined by minimization of an energy functional.
  • the DFT is a Legendre-transformed description of the quantum mechanical problem, in which a change from the local potentials to the local densities is made as a parameter of the system. The energy is no longer viewed as a functional of the local potentials, but as a functional of the local densities.
  • the variational parameters are not external quantities that characterize a class of wave functions, for example, but rather expected values of the quantum mechanical problem itself, the electron density in the case of DFT. Since not all contributions to the energy functional are explicitly known, they must be approximated.
  • DFT can be generalized to, among other things, reduce the contributions to be approximated in the energy functional.
  • electron density spin magnetization, electron and spin current, kinetic energy density, stress tensor, double occupancy, etc. can be considered as additional variational parameters.
  • the term functional theories is used to describe methods that describe physical systems through the extremization of a functional that depends on intrinsic system quantities or combinations of intrinsic and external quantities.
  • the object of the invention is, among other things, the determination of the ground state energy of a physical system using a quantum computer using a particularly small number of measurements on the quantum computer.
  • the invention is a method for determining ground state energies of a physical system using a quantum computer.
  • the method is characterized in that a test state is prepared for the physical system on the quantum computer, at least one expected value is determined together with commutating partial amounts of the at least one expected value of the test state and at least one expected value of an observable, using the at least one specific expected value of the observable the ground state energies of the test states are determined by generating a functional of the expected value of the test state as a function of the expected values of the observables for the physical system from the at least one specific expected value of the observables and the at least one expected value of the test state and by applying an energy minimization to the functional the ground state energy is determined.
  • the energy and expected values of (local) observables obtained in the context of a quantum simulation are related to one another in the sense of a functional theory.
  • a functional is constructed from the measurement data, the minimization of which provides a new approximate value for the ground state energy.
  • the external variational parameters are replaced by system-internal expected values. Examples of such observables are listed and described in more detail below.
  • the constructed functional is minimized.
  • the selection of the expected values represents an adjustment screw that allows the invention to be adapted to the specific problem. Essentially, the selection of observables influences the complexity of the functional to be approximated and the so-called N-representability conditions, which guarantee that the expected value of an observable can be realized by a quantum state ⁇ ip).
  • a starting state I 'init is first initialized on a qubit register of the quantum computer.
  • To determine the ground state energy of a quantum mechanical system one ideally chooses a state that already has a large overlap with the real ground state, but can be initialized efficiently with a few gates on the quantum computer.
  • An example is the ground state of an interacting electron system described within the framework of a so-called mean field approximation, which can be obtained from functional theories.
  • the Ground states of square mean-field Hamiltonians can be efficiently produced on the quantum computer (see, for example, Jiang, Z., Sung, KJ, Kechedzhi, K., Smelyanskiy, VN & Boixo, S. Quantum Algorithms to Simulate Many-Body Physics of Correlated Fermions. Phys. Rev. Applied 9, 044036 (Apr. 2018).
  • the invention presented has several advantages compared to usual variational approaches, which result from the combination of variational quantum computer algorithms with methods from functional theory.
  • VQE Variational approaches such as VQE are limited to the Fock space of possible test states that can be reached by the unitary transformations Ui ß ) from the starting state ⁇ ipintt).
  • the presented invention is not strictly limited to the Fock space of the test states and can in principle also find the ground state energy if the ground state does not correspond to any of the possible test states.
  • states close to the ground state must be sampled for a precise determination of the ground state energy, this only applies to ground states with small deviations from possible test states. For example, extrapolation is not possible if the true ground state and the possible test states have completely different symmetries.
  • each state of the quantum computer is a valid state of the simulated spin system and can be used to determine the ground state.
  • the main advantage of the invention over usual variational approaches is the faster convergence of the method, i.e. with a small number of measurements on the quantum computer.
  • Typical variational approaches require a large portion of the measurements for the final optimization steps, which only minimally improve the approximated ground state energy.
  • Black box optimizers in particular require many steps at the end of the optimization to rule out the existence of a deeper local minimum.
  • both gradient-based and black-box optimizers require multiple runs with different starting parameters to have a high probability of obtaining a good result.
  • VHA the unitary transformations in the preparation of the test state are chosen so that they correspond to the time evolution of an electron system under a gradual switching on of the interaction between electrons.
  • the classic parameters 9 correspond to pseudo time steps in the time development.
  • the classic optimizer should then find an optimized quasi-adiabatic time evolution that transfers the system from the non-interacting to the interacting ground state. This shows why it is not easy for the optimizer to find a clear optimum for parameters 9.
  • an adiabatic time evolution can have any number of revolutions If you take paths through the Fock or Hilbert space, as long as it is adiabatic enough and the initial Hamiltonian corresponds to the non-interacting system and the final Hamiltonian corresponds to the interacting system, you will get the interacting ground state. Analogously, there can in principle be many parameter sets 6 that transform the initial state into the correct basic state or a very similar state.
  • the invention uses physical observables to minimize the energy functional.
  • the constructed energy functional can provide a better approximation for the ground state energy through extrapolation, and on the other hand, the gradients of the energy functional with respect to the observables in a feedback mechanism can accelerate the optimization of the classical parameters.
  • stagnation in the minimization can be detected at an early stage and unnecessary measurements of the quantum register can be avoided.
  • the Hamilton operator H of the physical system under investigation is rewritten into a sum of products of Pauli operators for all qubits of the quantum register of the test state, where ( i ,X i , Y i ,Z i ') are the identity and Pauli operators, respectively, acting on the (/)th qubit.
  • (N) is the number of qubits in the quantum register used to store the basis states of the simulated system.
  • the expected values of the individual products of Pauli operators can be easily measured by applying a maximum of one-qubit rotation for each qubit and multiple projective measurements of the qubits in the Z-basis.
  • the expected value of the energy of the system in the test state is then determined by measurements from the quantum computer. For this purpose, “compatible” (commutating) partial energy contributions are determined simultaneously in order to optimize the number of measurements required.
  • the expected value of the energy in the test state is determined from the totality of all - generally non-commuting - partial contributions.
  • the central idea of the invention is not to use the classic parameters 9 of the test states for minimizing the energy - and thus the approximation of the ground state energy - but rather the measured expected values of the observables in relation to the test state, for example the local densities of the system . At the same time, an optimization of the measured observables can be carried out.
  • a functional of the energy can be generated from the measured observables and the energy of a large number of test states generated by the quantum computer depending on the expected values of the observables for the system under consideration.
  • the local kinetic expectation values (ct a cj a + hc ) are measured to determine the total kinetic energy, and the local densities are measured to obtain the contribution of the chemical potential to the total energy.
  • the local magnetization density and/or magnetization current can be used to describe spin systems.
  • One-particle reduced density matrix Local densities, kinetic energy and general expectation values of the one-particle reduced density matrix, see for example Gilbert, T. L. Hohenberg-Kohn theorem for nonlocal external potentials. Phys. Rev. B 12, 21 11 (1975).
  • Pair hopping plays an important role in the description of high-temperature superconductors.
  • abnormal 1 RDM expected values (1 RDM one-particle reduced density matrix) are of central importance.
  • Entanglement entropies are based on the partitioning of the system into local environments. They can be used to characterize the spread of information in the system.
  • the minimization breaks down into terms that are explicitly given by the expectation values of the observables, represented in the aforementioned equation by Ei Pi * h t , and a remainder, defined in the aforementioned equation as the infimum of the expectation value of H w over all quantum states , which provide the required expected values ⁇ .
  • H w is the Hamilton operator without terms that explicitly link to the observable
  • a multidimensional fit function can be created from the energies and the complete set of measured observables, for example with the help of machine learning techniques, the global minimum of which can be determined.
  • the fit function for determining the minimum itself can be represented, for example, by a neural network that was trained on the measured data.
  • the resulting energy functional can be minimized using classical optimization methods.
  • a physically motivated functional can be created for a specific system to be examined, also using already known approximations for specific models in the context of DFT. These functionals have parameters that can be approximately determined using the measured data. Finally, the minimum energy and thus the ground state energy can be calculated from the functionals with the fitted parameters.
  • the classic parameters 9 can be functionally fed back into the optimization in order to accelerate it.

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Abstract

L'invention concerne un procédé de détermination des énergies de l'état fondamental d'un système physique au moyen d'un ordinateur quantique, un état d'essai pour le système physique étant préparé sur l'ordinateur quantique. Outre les deux procédés les plus courants pour déterminer l'énergie de l'état fondamental sur l'ordinateur quantique, c'est-à-dire l'approche variationnelle et l'estimation de phase quantique (QPE), le procédé selon l'invention propose de déterminer des énergies de l'état fondamental d'un système physique au moyen d'un ordinateur quantique en déterminant au moins une valeur attendue d'une énergie par rapport à l'état d'essai et au moins une valeur attendue d'un observable, l'au moins une valeur attendue déterminée de l'observable étant utilisée pour déterminer les énergies de l'état fondamental de l'état d'essai en utilisant ladite valeur attendue déterminée de l'observable et ladite valeur attendue de l'énergie par rapport à l'état d'essai pour générer une fonctionnelle pour l'énergie sur la base des valeurs attendues de l'observable pour le système physique et déterminer l'énergie de l'état fondamental en appliquant une minimisation d'énergie à la fonctionnelle.
PCT/DE2023/100263 2022-05-31 2023-04-05 Procédé de détermination des énergies de l'état fondamental d'un système physique au moyen d'un ordinateur quantique WO2023232175A1 (fr)

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DE102022113761.9 2022-05-31
DE102022113761.9A DE102022113761A1 (de) 2022-05-31 2022-05-31 Verfahren zur Bestimmung von Grundzustandsenergien eines physikalischen Systems mittels eines Quantencomputers

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CN117690534A (zh) * 2024-01-25 2024-03-12 合肥综合性国家科学中心人工智能研究院(安徽省人工智能实验室) 一种量子材料超导性判断方法

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Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN117690534A (zh) * 2024-01-25 2024-03-12 合肥综合性国家科学中心人工智能研究院(安徽省人工智能实验室) 一种量子材料超导性判断方法
CN117690534B (zh) * 2024-01-25 2024-04-19 合肥综合性国家科学中心人工智能研究院(安徽省人工智能实验室) 一种量子材料超导性判断方法

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