EP-4736081-A1 - A QUANTUM COMPUTING SYSTEM AND A METHOD FOR USING SUCH A QUANTUM COMPUTING SYSTEM

Abstract

A computer-implemented method is disclosed herein which comprises: providing a Hamiltonian representation of a physical system; using a first quantum circuit to block encode the Hamiltonian into a matrix; and using a second quantum circuit to perform a Quantum Singular Value Transformation to invert the matrix. The inverted matrix is used to provide the Green's function corresponding to the physical system for determining one or more physical properties of the physical system.

Inventors

• FITZPATRICK, Nathan
• RALLI, Alexis

Assignees

• Quantinuum Ltd

Dates

Publication Date

20260506

Application Date

20240719

Claims (1)

1. Claims 1. A computer-implemented method comprises: providing a Hamiltonian representation of a physical system; using a first quantum circuit to block encode the Hamiltonian into a matrix M; and using a second quantum circuit to perform a Quantum Singular Value Transformation (QSVT) to invert the matrix, wherein the inverted matrix is used to provide a Green’s function corresponding to the physical system for determining one or more physical properties of the physical system. 2. A computer-implemented method according to claim 1, wherein the first quantum circuit implements a double-factorisation of an operator of the Hamiltonian and a mid-circuit measurement to determine the values of the matrix M. 3. The computer-implemented method of claim 1 or claim 2, wherein the Green’s function is expressed in matrix form (i, j) and comprises an advanced component according to equation 2a and a retarded component according to equation 2b.: 4. The computer-implemented method of claim 3, wherein z = ω + iδ is a complex frequency in which w >> δ. 5. The computer-implemented method of claim 2, 3 or 4, wherein the block encoding is performed using a linear combination of unitaries (LCU), optionally wherein the LCU is given as a linear combination of Pauli operators. 6. The computer-implemented method of any of claims 2 to 5, wherein the block encoding is performed with respect to the following complex-shifted Hamiltonians: 7. The computer-implemented method of claim 6, where the QSVT is configured to approximately invert the singular values of the block encoded matrix using the block encodings of ^ (^) and ^ (^) to allow separate calculation of the inverse parts of equations 2a and 2b to reduce the cir depth. 8. The computer-implemented method of any preceding claim, wherein inverting the matrix includes generating quantum signal-processing angles to implement a desired inverse function that will be applied to the singular values of the block encoded matrix. 9. The computer-implemented method of claim 8, wherein the desired inverse function is a polynomial approximation of f(x) = 1/x. 10. The computer-implemented method of claim 9, wherein the polynomial expression is defined in the range x = [−1, −1/κ] ∪ [1/κ, 1]. 11. The computer-implemented method of any of claims 8 to 10, further comprising selecting k to be large enough such that there is no singular value in the range (-1/k) – (+1/k). 12. The computer implemented method of any of claims 8 to 11, further comprising constructing the second quantum circuit to implement the QSVT algorithm using the matrix M and the quantum signal-processing angles. 13. The computer-implemented method of any preceding claim, wherein the Hamiltonian is not unitary but the matrix M is unitary. 14. The computer-implemented method of any preceding claim, wherein the inverted matrix provides matrix elements for calculating the single-particle, many-body Green’s function corresponding to the physical system. 15. The method of any preceding claim, further comprising performing a Hadamard test to evaluate the real and complex parts of each entry in the Green’s function. 16. The method of any preceding claim, further comprising using the Green’s function to calculate one or more physical properties of the physical system, optionally wherein the physical properties include at least one of excitation and ionization energies, ground-state energies, transition matrix elements, absorption coefficients, dynamical polarizabilities, and elastic and inelastic electron cross-sections. 17. The method of any preceding claim, wherein the Green’s function is determined for a two- site, single-impurity Anderson model, optionally for approximating the Mott insulator phase transition. 18. The method of any preceding claim, wherein the method is implemented at least partly on a quantum computing system. 19. A computer system configured to: provide a Hamiltonian representation of a physical system; use a first quantum circuit to block encode the Hamiltonian into a matrix M; use a second quantum circuit to perform a Quantum Singular Value Transformation (QSVT) to invert the matrix, wherein the inverted matrix is configured to provide matrix elements for calculating a Green’s function corresponding to the physical system, for determining one or more physical properties of the physical system. 20. A computer system according to claim 19, wherein the first quantum circuit implements a double-factorisation of an operator of the Hamiltonian and a mid-circuit measurement to determine the values of the matrix M.

Description

A QUANTUM COMPUTING SYSTEM AND A METHOD FOR USING SUCH A QUANTUM COMPUTING SYSTEM Technical field This patent specification relates to quantum computing systems and methods for using such quantum computing systems to help determine physical properties of physical systems. Background Quantum computers are inherently suitable for determining properties of complex physical systems, such as modelling the energy states of molecules, because they exploit quantum phenomena. Unlike classical digital computers in which the basic unit of computation, the bit, has in one of two discrete binary states (1 or 0), quantum computers utilise qubits which may exist in a superposition of many different computational states. This allows a quantum computer to investigate many different states in parallel. However, quantum computers are vulnerable to noise which may lead to decoherence between the different quantum states. This noise problem becomes more significant as the number of qubits increases for more complex computational processing. There is interest in developing procedures and implementations of quantum computers for determining the physical (e.g. spectroscopic) properties of physical systems such as atoms and molecules. Summary Provided are computer-implemented methods and computer systems for use when analysing physical properties of quantum systems, such as optical and electronic properties and behaviours including quantum state changes. The described methods and systems are useful, for example, when simulating a molecular system - which may be an individual molecule or a solid material. The invention is defined in the appended claims. A computer-implemented method is disclosed herein which comprises: providing a Hamiltonian representation of a physical system; using a first quantum circuit to block encode the Hamiltonian into a matrix; and using a second quantum circuit to perform a Quantum Singular Value Transformation to invert the matrix. The inverted matrix is then used to provide a Green’s function corresponding to the physical system, for determining one or more physical properties of the physical system. For example, the inverted matrix may provide matrix elements for calculating a single-particle, many-body Green’s function. The determined properties could be ground state energies or excitation and ionization energies, for example. The input representation of the physical system is preferably provided as a computer-readable expression including one or more operators representing the dynamic response behaviour of the physical quantum system when manipulated by the operators, such as defining a relationship between an external perturbation or internal interaction and the system’s response. For example, a Hamiltonian representation of the energy of a physical quantum system may be provided as an input to a quantum computation system for implementing the invention to calculate the energy of an excited state. For example, the Hamiltonian or other input representation may be provided as an input from a separate computer system or from a storage device. A ground state or other reference state of the quantum system may also be an input. The method of the invention can be applied to different physical systems using input representations of those physical systems that include different excitation (and annihilation or ‘de-excitation’) operators relating to the dynamic response behaviour of the respective physical system. The system and method disclosed herein may also utilise conventional computing systems, for example as an adjunct to a quantum computing system or as a simulation environment for a quantum computing system. These methods may be implemented in a hybrid classical-quantum computer system, including a classical (digital) computing apparatus that generates one or more measurable quantum circuits to block encode the Hamiltonian into a matrix and to perform the Quantum Singular Value Transformation (QSVT) to invert the matrix (that is used to calculate the Single-Particle Green’s Function) and a quantum computing apparatus that executes the quantum circuits on its qubits or qudits and outputs measurement results indicating changes to the quantum states of the physical quantum system. Allocation of appropriate tasks to each of a classical binary digital computer and a coupled quantum computer allows exploitation of the respective capabilities of each. This can provide significant advantages when the quantum circuits are executed on resource-constrained noisy intermediate-scale quantum (NISQ) computers, if using the method as described below. The methods and systems described below enable analysis of the properties and behaviours of real- world physical quantum systems, using computer-readable expressions to describe those properties and behaviours, and provides a computationally acceptable mitigation of the errors that result from an approximation. This method enables a reduction in the number of single-