Variational quantum eigensolver for dynamic correlation functions

Dynamical Self-energy Mapping (DSEM) for Creation of Sparse Hamiltonians Suitable for Quantum Computing

“…For details, see Sections and 2.2. The fictitious Hamiltonian given by eq and obtained by a perturbative method is passed to a nonperturbative solver (or a quantum machine). Evaluate Green’s function G (ω) ( F̃ , ṽ ) using a high-level solver such as exact diagonalization, quantum Monte Carlo, or one of the truncated configuration interaction (CI) schemes. In the case of quantum execution of this step, Green’s function evaluation can be done based on one of the algorithms described in refs − . Evaluate the self-energy Σ­(ω) ( F̃ , ṽ ) using the Dyson equation. Employ the self-energy evaluated in HLP1 to calculate the new Green’s function G (ω) according to Note that by writing Σ­(ω) ( F̃ , ṽ ), we explicitly denote that this self-energy came from the solution of the fictitious problem solved in HLP1 . Find chemical potential μ yielding a proper number of electrons. Evaluate a new density matrix γ from Green’s function obtained in CP4 and a new Fock matrix. Evaluate one-body electronic energy as Using new Green’s function and self-energy evaluate two-body energy according to Using Green’s function defined in LLP4 , it is possible to re-evaluate the self-energy in the LLP part of the algorithm and find a new set of effective integrals and continue iterating until electronic energies stop to change. …”

“…Evaluate Green’s function G (ω) ( F̃ , ṽ ) using a high-level solver such as exact diagonalization, quantum Monte Carlo, or one of the truncated configuration interaction (CI) schemes. In the case of quantum execution of this step, Green’s function evaluation can be done based on one of the algorithms described in refs − .…”

The Variational Quantum Eigensolver: a review of methods and best practices

“…The main promise of HEA is that it can be flexibly tailored to the specific native gate set of the device used, while at the same time being highly expressive. Its versatility and ease of construction resulted in it being widely used for numerous small-scale quantum experiments [128,227,[394][395][396][397]. 4 As above…”

“…However there exist other VQE approaches to directly target these correlation functions in either the (imaginary) time or frequency domains [580,601,605]. These include VQE-based variational approaches to directly solve the linear equations for the response of a system at a given frequency [606], which can be cast as a modified cost function, with similarly parameterized VQE ansatz [397,607,608]. While these approaches can describe the correlation functions to high accuracy over the whole energy range without restricting to a low-energy subspace, their challenge arises chiefly from the substantially more difficult optimization problem for the ansatz, originating from the larger condition number of the cost function, as well as the necessity for Hadamard tests to compute the transition amplitudes between the excited (or response) and ground states (for more details, see Appendix.…”

Probabilistic imaginary-time evolution by using forward and backward real-time evolution with a single ancilla: first-quantized eigensolver of quantum chemistry for ground states

“…Liu et al [17] analyzed their PITE circuit for the Grover's search algorithm [18,19], where they introduced one additional qubit per a pair of qubits for the nonunitary operation on the pair. The measurement-based generation of desired states is also used for other purposes such as linear equations [20], the Green's functions [21,22], linear-response functions [23], and ionization energies. [24] It is noted that those three kinds of ITE summarized above are not necessarily exclusive to each other.…”