Increasing representation accuracy of quantum simulations without additional quantum resources

A method for simulating a quantum system, the method comprising:

• obtaining a simulation output from a quantum simulation of the quantum system, wherein the quantum simulation comprises a quantum simulation in active space;
• and
• adjusting, by classical computation, the simulation output using multiple single particle rotations in full space to obtain an estimated energy of the quantum system, comprising solving a nonlinear optimization problem, wherein the nonlinear optimization problem comprises:
• an objective function comprising an expected value of the energy of the quantum system, and
• one or more constraints that specify unitary rotations of quantum system orbitals.

An apparatus comprising:

• quantum hardware;
• and
• one or more classical processors;
• wherein the apparatus is configured to perform operations comprising:
• selecting a first set of basis functions for the simulation, wherein the first set of basis functions comprises i) an active set of orbitals, and ii) a virtual set of orbitals;
• defining a set of expansion operators for the simulation, wherein expansion operators in the set of expansion operators approximate fermionic excitations in an active space spanned by the active set of orbitals and a virtual space spanned by the virtual set of orbitals;
• performing multiple quantum computations to determine a matrix representation of the Hamiltonian in a second set of basis functions, wherein each basis function in the second set of basis functions comprises a respective expansion operator applied to a wavefunction prepared in the active space, and wherein determining the matrix representation of the Hamiltonian comprises, for each matrix element:
• determining whether the matrix element comprises operators that act on the virtual space or operators that act on the active space only,
• in response to determining that the matrix element comprises operators that act on the virtual space, performing a classical computation to contract the matrix element to a matrix element comprising operators that act on the active space only, and
• measuring the operators that act on the active space only to determine a value for the matrix element;
• computing, using the determined matrix representation of the Hamiltonian in the second set of basis functions, eigenvalues and eigenvectors of the Hamiltonian;
• and
• determining properties of the physical system using the computed eigenvalues and eigenvectors.