We study a proof-of-principle example of the recently proposed hybrid quantum-classical simulation of strongly correlated fermion models in the thermodynamic limit. In a "two-site" dynamical mean-field theory (DMFT) approach we reduce the Hubbard model to an effective impurity model subject to self-consistency conditions. The resulting minimal two-site representation of the non-linear hybrid setup involves four qubits implementing the impurity problem, plus an ancilla qubit on which all measurements are performed. We outline a possible implementation with superconducting circuits feasible with near-future technology. IntroductionUsing highly controllable quantum devices to study other quantum systems, i.e., quantum simulation [1,2,3,4], offers a means to tackle strongly correlated fermion models that are intractable on classical computers. This is vital for understanding complex quantum materials [5] with strong electronic correlations that exhibit a plethora of exciting physical phenomena of immediate technological interest. Examples of such effects include the Mott metal-insulator transition [6,7], colossal magnetoresistance [8], and high-temperature superconductivity [9,10].Classical numerical methods have limited ability to study even significantly simplified toy models of strongly correlated fermions. For instance, exact diagonalization faces exponential scaling with the system size, while quantum Monte Carlo methods [11,12] are often crippled by the infamous fermionic sign problem [13]. Tensor network methods [14,15,16,17,18] are powerful in one spatial dimension where they track strong correlations accurately. However, in higher dimensional systems, correlations tend to grow more quickly with system size, making these methods computationally challenging.Another well-established approach to the study of strongly correlated fermionic lattice systems is dynamical mean-field theory (DMFT) [19]. It reduces the complexity of the original problem, e.g., the Hubbard model [20] in the thermodynamic limit, by mapping it onto a simpler impurity problem that is subject to a selfconsistency condition relating its properties to those of the original model. Since an impurity problem is local, the mapping corresponds to neglecting spatial fluctuations. In the limit of infinite spatial dimensions this mapping is exact, but for arXiv:1606.04839v1 [quant-ph]

We propose a non-linear, hybrid quantum-classical scheme for simulating non-equilibrium dynamics of strongly correlated fermions described by the Hubbard model in a Bethe lattice in the thermodynamic limit. Our scheme implements non-equilibrium dynamical mean field theory (DMFT) and uses a digital quantum simulator to solve a quantum impurity problem whose parameters are iterated to self-consistency via a classically computed feedback loop where quantum gate errors can be partly accounted for. We analyse the performance of the scheme in an example case.

Combining the Josephson effect with magnetism, or spin dependence in general, creates novel physical phenomena. The spin-asymmetric Josephson effect is a predicted phenomenon where a spin-dependent potential applied across a Josephson junction induces a spin-polarized Josephson current. Here, we propose an approach to observe the spin-asymmetric Josephson effect with spin-dependent superlattices, realizable, e.g., in ultracold atomic gases. We show that observing this effect is feasible by studying numerically the quantum dynamics of the system in one dimension. Furthermore, we show that the enhancement, or tunability, of the critical supercurrent in ferromagnetic Josephson junctions [F. S. Bergeret, A. F. Volkov, and K. B. Efetov, Phys. Rev. Lett. 86, 3140 (2001)] can be explained by the spin-asymmetric Josephson effect.

The spin-asymmetric Josephson effect is a proposed quantum-coherent tunneling phenomenon where Cooperpaired fermionic spin-1 2 particles, which are subjected to spin-dependent potentials across a Josephson junction, undergo frequency-synchronized alternating-current Josephson oscillations with spin-dependent amplitudes. Here, in line with present-day techniques in ultracold Fermi gas setups, we consider the regime of small Josephson oscillations and show that the Josephson plasma oscillation amplitude becomes spin dependent in the presence of spin-dependent potentials, while the Josephson plasma frequency is the same for both spin components. Detecting these spin-dependent Josephson plasma oscillations provides a possible means to establish the yet-unobserved spin-asymmetric Josephson effect with ultracold Fermi gases using existing experimental tools.