Quantum simulators are attractive as a means to study many-body quantum systems that are not amenable to classical numerical treatment. A versatile framework for quantum simulationis offered by superconducting circuits. In this perspective, we discuss how superconducting circuits allow the engineering of a wide variety of interactions, which in turn allows the simulation of a wide variety of model Hamiltonians. In particular we focus on strong photon-photon interactions mediated by nonlinear elements. This includes on-site, nearest-neighbour and four-body interactions in lattice models, allowing the implementation of extended Bose-Hubbard models and the toric code. We discuss not only the present state in analogue quantum simulation, but also future perspectives of superconducting quantum simulation that open up when concatenating quantum gates in emerging quantum computing platforms.

We derive a theory for the generation of arbitrary spin-spin interactions in superconducting circuits via periodic time modulation of the individual qubits or the qubit-qubit interactions.The modulation frequencies in our approach are in the microwave or radio frequency regime so that the required fields can be generated with standard generators. Among others, our approach is suitable for generating spin lattices that exhibit quantum spin liquid behavior such as Kitaev’s honeycomb model.

Topological order is now being established as a central criterion for characterizing and classifying ground states of condensed matter systems and complements categorizations basedon symmetries. Fractional quantum Hall systems and quantum spin liquids are receiving substantial interest because of their intriguing quantum correlations, their exotic excitations and prospects for protecting stored quantum information against errors. Here we show that the Hamiltonian of the central model of this class of systems, the Toric Code, can be directly implemented as an analog quantum simulator in lattices of superconducting circuits. The four-body interactions, which lie at its heart, are in our concept realized via Superconducting Quantum Interference Devices (SQUIDs) that are driven by a suitably oscillating flux bias. All physical qubits and coupling SQUIDs can be individually controlled with high precision. Topologically ordered states can be prepared via an adiabatic ramp of the stabilizer interactions. Strings of qubit operators, including the stabilizers and correlations along non-contractible loops, can be read out via a capacitive coupling to read-out resonators. Moreover, the available single qubit operations allow to create and propagate elementary excitations of the Toric Code and to verify their fractional statistics. The architecture we propose allows to implement a large variety of many-body interactions and thus provides a versatile analog quantum simulator for topological order and lattice gauge theories.

We propose a hybrid system with quantum mechanical three-body interactions between photons, phonons, and qubit excitations. These interactions take place in a circuit quantum electrodynamicalarchitecture with a superconducting microwave resonator coupled to a transmon qubit whose shunt capacitance is free to mechanically oscillate. We show that this system design features a three-mode polariton–mechanical mode and a nonlinear transmon–mechanical mode interaction in the strong coupling regime. Together with the strong resonator–transmon interaction, these properties provide intriguing opportunities for manipulations of this hybrid quantum system. We show, in particular, the feasibility of cooling the mechanical motion down to its ground state and preparing various nonclassical states including mechanical Fock and cat states and hybrid tripartite entangled states.

We study the properties of an array of QED-cavities coupled by nonlinear elements in the presence of photon leakage and driven by a coherent source. The main effect of the nonlinearcouplings is to provide an effective cross-Kerr interaction between nearest-neighbor cavities. Additionally correlated photon hopping between neighboring cavities arises. We provide a detailed mean-field analysis of the steady-state phase diagram as a function of the system parameters, the leakage and the external driving, and show the emergence of a number of different quantum phases. A photon crystal associated to a spatial modulation of the photon blockade appears. The steady state can also display oscillating behavior and bi-stability. In some regions the crystalline ordering may coexist with the oscillating behavior. Furthermore we study the effect of short-range quantum fluctuations by employing a cluster mean-field analysis. Focusing on the corrections to the photon crystal boundaries, we show that, apart for some quantitative differences, the cluster mean field supports the findings of the simple single-site analysis. In the last part of the paper we concentrate on the possibility to build up the class of arrays introduced here, by means of superconducting circuits of existing technology. We consider a realistic choice of the parameters for this specific implementation and discuss some properties of the steady-state phase diagram.

We consider a superconducting coplanar waveguide resonator where the central conductor is interrupted by a series of uniformly spaced Josephson junctions. The device forms an extendedmedium that is optically nonlinear on the single photon level with normal modes that inherit the full nonlinearity of the junctions but are nonetheless accessible via the resonator ports. For specific plasma frequencies of the junctions a set of normal modes clusters in a narrow band and eventually become entirely degenerate. Upon increasing the intensity of a red detuned drive on these modes, we observe a sharp and synchronized switching from low occupation quantum states to high occupation classical fields, accompanied by a pronounced jump from low to high output intensity.

Arrays of circuit cavities offer fascinating perspectives for exploring quantum many-body systems in a driven dissipative regime where excitation losses are continuously compensatedby coherent input drives. Here we investigate a system consisting of three transmission line resonators, where the two outer ones are driven by coherent input sources and the central resonator interacts with a superconducting qubit. Whereas a low excitation number regime of such a device has been considered previously with a numerical integration, we here specifically address the high excitation density regime. This is of particular interest as intra cavity fields might undergo a transition from low excitation number quantum fields to high amplitude classical fields when increasing the input drives. We present analytical approximations to these regimes in the form of two methods. The first method is a Bogoliubov expansion in quantum fluctuations which can be understood as an approximation for weak nonlinearities. As the second method we introduce a combination of mean-field decoupling for the photon tunneling with an exact approach to a driven Kerr nonlinearity which can be understood as an approximation for low tunneling rates.

We introduce and study the properties of an array of QED cavities coupled by
non-linear elements, in the presence of photon leakage and driven by a coherent
source. The non-linearcouplings lead to photon hopping and to nearest-neighbor
Kerr terms. By tuning the system parameters, the steady state of the array can
exhibit a photon crystal associated to a periodic modulation of the photon
blockade. In some cases the crystalline ordering may coexist with phase
synchronisation. The class of cavity arrays we consider can be built with
superconducting circuits of existing technology.

We introduce a circuit quantum electrodynamical setup for a quantum single
photon transistor. In our approach single photons propagate in two open
transmission lines that are coupledvia two interacting transmon qubits. The
interaction is such that photons are not exchanged between the two transmission
lines but a photon in one line can completely block respectively enable the
propagation of photons in the other line. High on-off ratios can be achieved
for feasible experimental parameters. Our approach is inherently scalable as
all photon pulses can have the same pulse shape and carrier frequency such that
output signals of one transistor can be input signals for a consecutive
transistor.

We present the Josephson junction intersected superconducting transmission
line resonator. In contrast to the Josephson parametric amplifier, Josephson
bifurcation amplifier and Josephsonparametric converter we consider the regime
of few microwave photons. We review the derivation of eigenmode frequencies and
zero point fluctuations of the nonlinear transmission line resonator and the
derivation of the eigenmode Kerr nonlinearities. Remarkably these
nonlinearities can reach values comparable to Transmon qubits rendering the
device ideal for accessing the strongly correlated regime. This is particularly
interesting for investigation of quantum many-body dynamics of interacting
particles under the influence of drive and dissipation. We provide current
profiles for the device modes and investigate the coupling between resonators
in a network of nonlinear transmission line resonators.