Variational study of fermionic and bosonic systems with non-Gaussian states: Theory and applications

Abstract

We present a new variational method for investigating the ground state and out of equilibrium dynamics of quantum many-body bosonic and fermionic systems. Our approach is based on constructing variational wavefunctions which extend Gaussian states by including generalized canonical transformations between the fields. The key advantage of such states compared to simple Gaussian states is presence of non-factorizable correlations and the possibility of describing states with strong entanglement between particles. In contrast to the commonly used canonical transformations, such as the polaron or Lang-Firsov transformations, we allow parameters of the transformations to be time dependent, which extends their regions of applicability. We derive equations of motion for the parameters characterizing the states both in real and imaginary time using the differential structure of the variational manifold. The ground state can be found by following the imaginary time evolution until it converges to a steady state. Collective excitations in the system can be obtained by linearizing the real-time equations of motion in the vicinity of the imaginary time steady-state solution. Our formalism allows us not only to determine the energy spectrum of quasiparticles and their lifetime, but to obtain the complete spectral functions and to explore far out of equilibrium dynamics such as coherent evolution following a quantum quench. We illustrate and benchmark this framework with several examples: a single polaron in the Holstein and Su-Schrieer-Heeger models, non-equilibrium dynamics in the spin-boson and Kondo models, the superconducting to charge density wave phase transitions in the Holstein model. arXiv:1707.05902v1 [quant-ph]

Theory of variational quantum simulation

“…We further elaborate on the implementation with quantum circuits in Sec. 6. We conclude our work and discuss future directions in Sec.…”

“…for both static problems in finding the ground state and ground state energy, and dynamical problems in simulating the real and imaginary time evolution of quantum states [4][5][6][12][13][14][15][16]. In modern science, the variational method has become a versatile tool for simulating various problems when the target system state can be well modelled classically.…”

Gaussian time-dependent variational principle for the Bose-Hubbard model

“…Projected Hamiltonian. Finally, there is a well-known alternative [25][26][27] to compute excitation spectra from a tangent plane based on the projected Hamiltonian. Instead of linearizing the equations of motion, we can directly take the tangent plane as variational ansatz for eigenstates by projecting the full Hamiltonian onto it, i.e., H P = P |ψ Ĥ P |ψ , and then computing its spectrum.…”

Efficient variational approach to dynamics of a spatially extended bosonic Kondo model

“…(12) we explicitly include a phase factor θ, which is necessary to obtain the absorption spectrum of the system as detailed later. The time-evolution equation can be obtained from the timedependent variational principle [91][92][93]. Specifically, we project the exact real-time evolution of the environmental state (in the transformed frame),…”

“…and define the 2N b ×2N b matrix Ω including the environmental parity operator by [91] Ω ≡ P envb †b GS…”