EP-4736082-A2 - SHADOW HAMILTONIAN SIMULATION USING A QUANTUM COMPUTER

EP4736082A2EP 4736082 A2EP4736082 A2EP 4736082A2EP-4736082-A2

Abstract

Methods, systems, and apparatus for quantum simulation of a quantum system. In one aspect, a method includes, for an observable generated from a set of observables, wherein a commutator of each observable in the set of observables with the first Hamiltonian is equal to a combination of observables in the set of observables: encoding, by a quantum computer, a vector of coefficients of a time-dependent representation of the observable in a quantum state of a register of qubits; simulating, by the quantum computer, time evolution of the quantum state under a second Hamiltonian to obtain an evolved quantum state, wherein the second Hamiltonian comprises a matrix of complex weights in the linear combination of observables; measuring, by the quantum computer, the evolved quantum state; and post-processing, by a classical processor, obtained measurement results to obtain an expectation value of the observable.

Inventors

O'BRIEN, THOMAS EUGENE
SOMMA, Rolando Diego
BABBUSH, Ryan

Assignees

Google LLC

Dates

Publication Date: 20260506
Application Date: 20240724

Claims (1)

CLAIMS 1. A method for quantum simulation of a quantum system characterized by a first Hamiltonian, the method comprising: for an observable generated from a set of observables, wherein a commutator of each observable in the set of observables with the first Hamiltonian is equal to a linear combination of observables in the set of observables: encoding, by a quantum computer, a vector of coefficients of a time-dependent representation of the observable in a quantum state of a register of qubits included in the quantum computer; simulating, by the quantum computer, time evolution of the quantum state under a second Hamiltonian to obtain an evolved quantum state, wherein the second Hamiltonian comprises a matrix of complex weights in the linear combination of observables; measuring, by the quantum computer, the evolved quantum state to obtain measurement results; and post-processing, by a classical processor, the measurement results to obtain an expectation value of the observable. 2. The method of claim 1, wherein: the observables in the set of observables generate a Lie algebra; the first Hamiltonian comprises a second linear combination of observables in the set of observables; and the second Hamiltonian comprises a complex valued matrix of real weights in the second linear combination of observables and structure factors of the Lie algebra. 3. The method of claim 1, wherein: the first Hamiltonian is an element of a Lie algebra; the set of observables define a real vector space; and the real vector space is invariant to conjugation of operators that are dependent on the Lie algebra. 4. The method of claim 1, wherein the first Hamiltonian has a larger dimension than the second Hamiltonian. 5. The method of claim 1, wherein the vector of coefficients has a dimension ^^ equal to a number of observables included in the set of observables; and the register of qubits comprises at least ⌈ ^^ ^^ ^^^ ^^^ ⌉ qubits. 6. The method of claim 1, wherein encoding the vector of coefficients of the time- dependent representation of the observable in the quantum state of the register of qubits included in the quantum computer comprises preparing the register of qubits in an initial quantum state that is proportional to the vector of coefficients at an initial time. 7. The method of claim 6, wherein preparing the register of qubits in the initial quantum state costs poly(n), where n represents the number of qubits in the register. 8. The method of claim 6, wherein amplitudes of the initial quantum state are proportional to an expectation value of the observable with respect to an initial state of the quantum system. 9. The method of claim 1, wherein simulating time evolution of the quantum state under the second Hamiltonian to obtain the evolved quantum state comprises performing a simulation in space equal to the dimension of the vector of coefficients. 10. The method of claim 1, wherein amplitudes of the evolved quantum state are proportional to a time-dependent expectation value of the observable with respect to an initial state of the quantum system. 11. The method of claim 1,wherein simulating time evolution of the quantum state under the second Hamiltonian for a time t comprises complexity poly(n,t), where n represents the number of qubits in the system. 12. The method of claim 1, wherein the quantum system comprises a spin system. 13. The method of claim 12, wherein: the first Hamiltonian comprises an Ising model in a transverse field; observables included in the first Hamiltonian generate a Lie algebra of a special unitary group of degree N, wherein N represents a number of spin orbitals in the quantum system; and either the encoded vector of coefficients of the time-dependent representation of the observable is proportional to expectation values of the observables included in the first Hamiltonian, the observables comprising ^ ^^ ^ ^^ ^ , ^^ ^ ^^ ^ , ^^ ^ ^ where ^^ ^ represents a Pauli X operator applied to spin j, ^^ ^ represents a Pauli Y operator applied to spin j, and ^^ ^ represents a Pauli Z operator applied to spin j; or the encoded vector of coefficients of the time-dependent representation of the observable is proportional to expectation values of the observables included in the first Hamiltonian, the observables comprising ^ ^^ ^ ^^ ^ ^^ ^ ᇲ ^^ ^ ᇲ , ^^ ^ ^^ ^ ^^ ^ ᇲ ^^ ^ ᇲ , ^^ ^ ^^ ^ ^^ ^ ᇲ , ^^ ^ ^^ ^ ^^ ^ ᇲ ^^ ^ ᇲ , ^^ ^ ^^ ^ ^^ ^ ᇲ , ^^ ^ ^^ ^ ^. comprises a free fermion system on a lattice. 15. The method of claim 14, wherein: the first Hamiltonian comprises a linear combination of products of fermionic creation and annihilation operators; and the set of observables generate a Lie algebra of a unitary group of degree N, wherein N represents a number of lattice sites in the lattice. 16. The method of claim 15, wherein the quantum simulation simulates dynamics of the first Hamiltonian within a one-excitation manifold and wherein each observable in the set of observables is equal to a sum of an annihilation operator for a respective spin and a creation operator for the respective spin. 17. The method of claim 1, wherein: the quantum simulation simulates Landau-Lifshits dynamics and the observables in the set of observables generate a Lie algebra of a unitary group of degree N, wherein N represents system size; the quantum simulation simulates the dynamics of a fermionic system and the observables in the set of observables generate a Lie algebra of a special unitary group of degree 2N; or the quantum simulation simulates the dynamics of a bosonic system and the observables in the set of observables generate a Lie algebra of a sympletic group of degree 2N. 18. The method of claim 1, wherein the observable generated from the set of observables comprises an element of the set of observables or a product of elements of the set of observables. 19. A system comprising: a quantum computer; and a classical computer coupled to the quantum computer, the classical computer comprising: one or more data processing apparatuses; and non-transitory computer readable storage media in data communication with the one or more data processing apparatuses and storing instructions executable by the data processing apparatuses; wherein the system is configured to perform operations for quantum simulation of a quantum system characterized by a first Hamiltonian, the operations comprising: for an observable generated from a set of observables, wherein a commutator of each observable in the set of observables with the first Hamiltonian is equal to a linear combination of observables in the set of observables: encoding, by a quantum computer, a vector of coefficients of a time-dependent representation of the observable in a quantum state of a register of qubits included in the quantum computer; simulating, by the quantum computer, time evolution of the quantum state under a second Hamiltonian to obtain an evolved quantum state, wherein the second Hamiltonian comprises a matrix of complex weights in the linear combination of observables; measuring, by the quantum computer, the evolved quantum state to obtain measurement results; and post-processing, by a classical processor, the measurement results to obtain an expectation value of the observable. 20. The system of claim 19, wherein the quantum computer comprises a fault tolerant quantum computer. 21. A method for spectroscopy of a quantum system characterized by a first Hamiltonian, the method comprising: for a set of observables, wherein a commutator of each observable in the set of observables with the first Hamiltonian is equal to a linear combination of observables in the set of observables: mapping, by a classical processor, the first Hamiltonian to a second Hamiltonian with lower dimension than the first Hamiltonian, comprising generating a matrix wherein elements of the matrix correspond to respective complex weights in the linear combinations of observables; performing, by a quantum computer, spectroscopy on the second Hamiltonian to obtain spectroscopy data; and processing, by the classical processor, the spectroscopy data to determine spectral and response properties of the quantum system. 22. The method of claim 21, wherein: the observables in the set of observables generate a Lie algebra; the first Hamiltonian comprises a second linear combination of observables in the set of observables; and the second Hamiltonian comprises a complex valued matrix of real weights in the second linear combination of observables and structure factors of the Lie algebra. 23. The method of claim 21, wherein: the first Hamiltonian is an element of a Lie algebra; the set of observables define a real vector space; and the real vector space is invariant to conjugation of operators that are dependent on the Lie algebra. 24. The method of claim 21, wherein performing spectroscopy on the second Hamiltonian comprises implementing a quantum Kernel Polynomial method. 25. The method of claim 24, further comprising using block encodings of the second Hamiltonian. 26. The method of claim 21, wherein the quantum system comprises a free fermion system on a lattice. 27. The method of claim 26, wherein: the first Hamiltonian comprises a linear combination of products of fermionic creation and annihilation operators; and the set of observables generate a Lie algebra of a unitary group of degree N, wherein N represents a number of lattice sites in the lattice. 28. The method of claim 21, wherein the quantum system comprises a spin system. 29. A system comprising: a quantum computer; and a classical computer coupled to the quantum computer, the classical computer comprising: one or more data processing apparatuses; and non-transitory computer readable storage media in data communication with the one or more data processing apparatuses and storing instructions executable by the data processing apparatuses; wherein the system is configured to perform operations for spectroscopy of a quantum system characterized by a first Hamiltonian, the operations comprising: for a set of observables, wherein a commutator of each observable in the set of observables with the first Hamiltonian is equal to a linear combination of observables in the set of observables: mapping, by a classical processor, the first Hamiltonian to a second Hamiltonian with lower dimension than the first Hamiltonian, comprising generating a matrix wherein elements of the matrix correspond to respective complex weights in the linear combinations of observables; performing, by a quantum computer, spectroscopy on the second Hamiltonian to obtain spectroscopy data; and processing, by the classical processor, the spectroscopy data to determine spectral and response properties of the quantum system. 30. The system of claim 9, wherein the quantum computer comprises a fault tolerant quantum computer.

Description

SHADOW HAMILTONIAN SIMULATION USING A QUANTUM COMPUTER BACKGROUND This disclosure relates to quantum computing and quantum simulation. SUMMARY This disclosure describes exponentially-improved simulations of quantum systems using a quantum computer. In general, one innovative aspect of the subject matter described in this specification can be implemented in a method for quantum simulation of a quantum system characterized by a first Hamiltonian, the method comprising: for an observable generated from a set of observables, wherein a commutator of each observable in the set of observables with the first Hamiltonian is equal to a linear combination of observables in the set of observables: encoding, by a quantum computer, a vector of coefficients of a time-dependent representation of the observable in a quantum state of a register of qubits included in the quantum computer; simulating, by the quantum computer, time evolution of the quantum state under a second Hamiltonian to obtain an evolved quantum state, wherein the second Hamiltonian comprises a matrix of complex weights in the linear combination of observables; measuring, by the quantum computer, the evolved quantum state to obtain measurement results; and post- processing, by a classical processor, the measurement results to obtain an expectation value of the observable. Other implementations of these aspects include corresponding computer systems, apparatus, and computer programs recorded on one or more computer storage devices, each configured to perform the actions of the methods. A system of one or more classical and quantum computers can be configured to perform particular operations or actions by virtue of having software, firmware, hardware, or a combination thereof installed on the system that in operation causes or cause the system to perform the actions. One or more computer programs can be configured to perform particular operations or actions by virtue of including instructions that, when executed by data processing apparatus, cause the apparatus to perform the actions. The foregoing and other implementations can each optionally include one or more of the following features, alone or in combination. In some implementations the observables in the set of observables generate a Lie algebra; the first Hamiltonian comprises a second linear combination of observables in the set of observables; and the second Hamiltonian comprises a complex valued matrix of real weights in the second linear combination of observables and structure factors of the Lie algebra. In some implementations the first Hamiltonian is an element of a Lie algebra; the set of observables define a real vector space; and the real vector space is invariant to conjugation of operators that are dependent on the Lie algebra. In some implementations the first Hamiltonian has a larger dimension than the second Hamiltonian. In some implementations the vector of coefficients has a dimension ^^ equal to a number of observables included in the set of observables; and the register of qubits comprises at least ⌈ ^^ ^^ ^^^ ^^^ ⌉ qubits. In some implementations encoding the vector of coefficients of the time-dependent representation of the observable in the quantum state of the register of qubits included in the quantum computer comprises preparing the register of qubits in an initial quantum state that is proportional to the vector of coefficients at an initial time. In some implementations preparing the register of qubits in the initial quantum state costs poly(n), where n represents the number of qubits in the register. In some implementations amplitudes of the initial quantum state are proportional to an expectation value of the observable with respect to an initial state of the quantum system. In some implementations simulating time evolution of the quantum state under the second Hamiltonian to obtain the evolved quantum state comprises performing a simulation in space equal to the dimension of the vector of coefficients. In some implementations amplitudes of the evolved quantum state are proportional to a time-dependent expectation value of the observable with respect to an initial state of the quantum system. In some implementations simulating time evolution of the quantum state under the second Hamiltonian for a time t comprises complexity poly(n,t), where n represents the number of qubits in the system. In some implementations the quantum system comprises a spin system. In some implementations the first Hamiltonian comprises an Ising model in a transverse field; observables included in the first Hamiltonian generate a Lie algebra of a special unitary group of degree N, wherein N represents a number of spin orbitals in the quantum system; and either the encoded vector of coefficients of the time-dependent representation of the observable is proportional to expectation values of the observables included in the first Hamiltonian, the observables comprising ^ ^^^ ^^^, ^^^ ^^^ , ^^^^ wher