Shadow hamiltonian simulation using a quantum computer

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.

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.

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.