One of the key applications of quantum information is simulating nature. Fermions are ubiquitous in nature, appearing in condensed matter systems, chemistry and high energy physics. However, universally simulating their interactions is arguably one of the largest challenges, because of the difficulties arising from anticommutativity. Here we use digital methods to construct the required arbitrary interactions, and perform quantum simulation of up to four fermionic modes with a superconducting quantum circuit. We employ in excess of 300 quantum logic gates, and reach fidelities that are consistent with a simple model of uncorrelated errors. The presented approach is in principle scalable to a larger number of modes, and arbitrary spatial dimensions.

We propose the digital quantum simulation of a minimal AdS/CFT model in controllable quantum platforms. We consider the Sachdev-Ye-Kitaev model describing interacting Majorana fermions with randomly distributed all-to-all couplings, encoding nonlocal fermionic operators onto qubits to efficiently implement their dynamics via digital techniques. Moreover, we also give a method for probing non-equilibrium dynamics and the scrambling of information. Finally, our approach serves as a protocol for reproducing a simplified low-dimensional model of quantum gravity in advanced quantum platforms as trapped ions and superconducting circuits.Holographic duality [1] posits the equivalence, subject to certain conditions, of quantum gravity and ordinary quantum field theories. The most celebrated such correspondence is conjectured to exist between N = 4 supersymmetric YangMills theory in four dimensions and type IIB string theory on AdS 5 × S 5 . Such dualities offer the exciting prospect of probing quantum gravity effects by studying the well-defined equivalent quantum field theory. Nevertheless, this is still a hard problem because the semiclassical gravity regime is located at strong coupling and for a large number of local degrees of freedom N 1. Furthermore, a fully nonperturbative understanding of the dual field theory is likely necessary in order to resolve the most puzzling aspects of quantum black holes, such as the famous information loss paradox [2]. We may therefore opt for studying the dual field theory on the lattice, by rewriting the problem in terms of a quantum many-body system suitable for simulation on a classical computer [3,4]. Even this powerful technique faces important challenges and limitations, such as the sign problem [5], and the inapplicability of Euclidean lattice methods for intrinsically Lorentzian physics. It is precisely the latter kind of problem one needs to understand in order to describe black hole formation [6] and evaporation.It is essential to develop alternative avenues of dealing with strongly coupled quantum many-body systems; both for their own sake, as well as with an eye on quantum gravity. As pointed out originally by Feynman [7], quantum systems themselves are vastly more computationally efficient at solving many-body Hamiltonians than classical computer simulations. With the recent advent of quantum technologies [8][9][10][11][12], it is then natural to consider multiqubit systems that encode a dual gravity theory via quantum simulation. Currently, four-dimensional gauge theories such as the aforementioned N = 4 theory appear out of reach (see, however, [13] for work on QCD in this context). Instead, we start by looking elsewhere for simpler models which nevertheless have a holographic interpretation.In this Letter, we propose the digital quantum simulation of the simplest known AdS/CFT duality, namely the SachdevYe-Kitaev (SYK) model [14][15][16]. We consider different variants of the model, two in terms of Majorana fermions, and two with complex fermions. We then propose dig...

We report on ultrastrong coupling between a superconducting flux qubit and a resonant mode of a system comprised of two superconducting coplanar stripline resonators coupled galvanically to the qubit. With a coupling strength as high as 17.5% of the mode frequency, exceeding that of previous circuit quantum electrodynamics experiments, we observe a pronounced Bloch-Siegert shift. The spectroscopic response of our multimode system reveals a clear breakdown of the Jaynes-Cummings approximation. In contrast to earlier experiments, the high coupling strength is achieved without making use of an additional inductance provided by a Josephson junction.

We propose an analog-digital quantum simulation of fermion-fermion scattering mediated by a continuum of bosonic modes within a circuit quantum electrodynamics scenario. This quantum technology naturally provides strong coupling of superconducting qubits with a continuum of electromagnetic modes in an open transmission line. In this way, we propose qubits to efficiently simulate fermionic modes via digital techniques, while we consider the continuum complexity of an open transmission line to simulate the continuum complexity of bosonic modes in quantum field theories. Therefore, we believe that the complexity-simulating-complexity concept should become a leading paradigm in any effort towards scalable quantum simulations. In general, the numerical simulations of QFTs are computationally hard, with the processing time growing exponentially with the system size. Nevertheless, a quantum simulator [3][4][5] could provide an efficient way to emulate these theories [6][7][8][9][10][11][12][13][14] in polynomial time. For instance, the remarkable developments in superconducting circuits and circuit quantum electrodynamics (QED) [15][16][17][18][19][20][21][22][23], specifically concerning their improvements in controllability and scalability [24,25], make them suitable candidates for developing a quantum simulator [26].

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]