Solving strongly correlated electron models on a quantum computer

Abstract

One of the main applications of future quantum computers will be the simulation of quantum models. While the evolution of a quantum state under a Hamiltonian is straightforward (if sometimes expensive), using quantum computers to determine the ground state phase diagram of a quantum model and the properties of its phases is more involved. Using the Hubbard model as a prototypical example, we here show all the steps necessary to determine its phase diagram and ground state properties on a quantum computer. In particular, we discuss strategies for efficiently determining and preparing the ground state of the Hubbard model starting from various mean-field states with broken symmetry. We present an efficient procedure to prepare arbitrary Slater determinants as initial states and present the complete set of quantum circuits needed to evolve from these to the ground state of the Hubbard model. We show that, using efficient nesting of the various terms each time step in the evolution can be performed with just $\mathcal{O}(N)$ gates and $\mathcal{O}(\log N)$ circuit depth. We give explicit circuits to measure arbitrary local observables and static and dynamic correlation functions, both in the time and frequency domain. We further present efficient non-destructive approaches to measurement that avoid the need to re-prepare the ground state after each measurement and that quadratically reduce the measurement error.Comment: 25 pages, 14 figure

Control of Fano resonances and slow light using Bose-Einstein condensates in a nanocavity

“…Transition from Mott insulator state to superfluidity of atoms [78] in an optical lattice coupled to a vibrating mirror has been analyzed, as an example of a strongly interacting quantum system subject to the optomechanical interaction [79]. Recently, a comprehensive strategy for using quantum computers to solve models of strongly correlated electrons, using the Hubbard model as a prototypical example has been reported [80]. More recently, interferometric phase detection controlled by Fano resonances and manipulation of slow light propagation have been reported in the x-ray regime [25,81].…”

Hybrid Quantum-Classical Approach to Correlated Materials

“…Because of the constraint of unitary evolution on a quantum computer, we can measure the Green's function only in real time (or real frequencies [18]). For t ≥ 0, the particle and hole Green's functions in real time are defined as…”

“…Possible choices for the initial Hamiltonian could be either the atomic limit of turning off all hopping terms, such that the ground state becomes a simple product state of occupied and unoccupied spin orbitals, or turning off interactions, such that the initial state is a Slater determinant that can be efficiently prepared using techniques discussed in Ref. [18]. At the end of the adiabatic process, QPE can be used to measure the energy of the state and collapse the wave function into an eigenstate jΨi of the Hamiltonian.…”

“…One approach is to approximate the material with idealized model Hamiltonians, such as the Hubbard model [19], which have a sufficiently small number of interaction terms that they can easily be studied on quantum computers [18]. While capturing qualitative phenomena, such simple models do not offer a quantitative description of real materials.…”

“…In contrast, it has been shown that quantum computers can in principle solve certain electronic structure problems [8,9] in polynomial time. Recent advances towards building a small quantum computer [10][11][12] have led to increasing interest in what a small quantum computer could realistically simulate, and it has been shown that the simulation of small molecules [13][14][15][16][17] and simplified model Hamiltonians [18] is within reach. However, in part because of the multiplicity of relevant interaction terms, the scaling of the currently known algorithms is not benign enough to allow naive direct simulations of complex correlated materials, for which thousands of electrons would have to be considered.…”