Simulating quantum mechanics is known to be a difficult computational problem, especially when dealing with large systems. However, this difficulty may be overcome by using some controllable quantum system to study another less controllable or accessible quantum system, i.e., quantum simulation. Quantum simulation promises to have applications in the study of many problems in, e.g., condensed-matter physics, high-energy physics, atomic physics, quantum chemistry and cosmology. Quantum simulation could be implemented using quantum computers, but also with simpler, analog devices that would require less control, and therefore, would be easier to construct. A number of quantum systems such as neutral atoms, ions, polar molecules, electrons in semiconductors, superconducting circuits, nuclear spins and photons have been proposed as quantum simulators. This review outlines the main theoretical and experimental aspects of quantum simulation and emphasizes some of the challenges and promises of this fast-growing field.

Hybrid quantum circuits combine two or more physical systems, with the goal of harnessing the advantages and strengths of the different systems in order to better explore new phenomena and potentially bring about novel quantum technologies. This article presents a brief overview of the progress achieved so far in the field of hybrid circuits involving atoms, spins and solid-state devices (including superconducting and nanomechanical systems). How these circuits combine elements from atomic physics, quantum optics, condensed matter physics, and nanoscience is discussed, and different possible approaches for integrating various systems into a single circuit are presented. In particular, hybrid quantum circuits can be fabricated on a chip, facilitating their future scalability, which is crucial for building future quantum technologies, including quantum detectors, simulators, and computers.

A transition between energy levels at an avoided crossing is known as a Landau-Zener transition. When a two-level system (TLS) is subject to periodic driving with sufficiently large amplitude, a sequence of transitions occurs. The phase accumulated between transitions (commonly known as the Stückelberg phase) may result in constructive or destructive interference. Accordingly, the physical observables of the system exhibit periodic dependence on the various system parameters. This phenomenon is often referred to as Landau-Zener-Stückelberg (LZS) interferometry. Phenomena related to LZS interferometry occur in a variety of physical systems. In particular, recent experiments on LZS interferometry in superconducting TLSs (qubits) have demonstrated the potential for using this kind of interferometry as an effective tool for obtaining the parameters characterizing the TLS as well as its interaction with the control fields and with the environment. Furthermore, strong driving could allow for fast and reliable control of the quantum system. Here we review recent experimental results on LZS interferometry, and we present related theory. 3 B. Superconducting qubits 3 C. Fano and Fabry-Perot interferometry using superconducting qubits 4 D. Hamiltonian and bases 4

The interaction between an atom and the electromagnetic field inside a cavity 1-6 has played a crucial role in developing our understanding of light-matter interaction, and is central to various quantum technologies, including lasers and many quantum computing architectures. Superconducting qubits 7,8 have allowed the realization of strong 9,10 and ultrastrong 11-13 coupling between artificial atoms and cavities. If the coupling strength g becomes as large as the atomic and cavity frequencies (∆ and ω o , respectively), the energy eigenstates including the ground state are predicted to be highly entangled 14 . There has been an ongoing debate 15-17 over whether it is fundamentally possible to realize this regime in realistic physical systems. By inductively coupling a flux qubit and an LC oscillator via Josephson junctions, we have realized circuits with g/ω o ranging from 0.72 to 1.34 and g/∆ 1. Using spectroscopy measurements, we have observed unconventional transition spectra that are characteristic of this new regime. Our results provide a basis for ground-state-based entangled pair generation and open a new direction of research on strongly correlated light-matter states in circuit quantum electrodynamics.We begin by describing the Hamiltonian of each component in the qubit-oscillator circuit, which comprises a superconducting flux qubit and an LC oscillator inductively coupled to each other by sharing a tunable inductance L c , as shown in the circuit diagram in Fig. 1a.The Hamiltonian of the flux qubit can be written in the basis of two states with persistent currents flowing in opposite directions around the qubit loop 18 , |L q and |R q , as H q = − (∆σ x + εσ z )/2, where ∆ and ε = 2I p 0 (n φq − n φq0 ) are the tunnel splitting and the energy bias between |L q and |R q , I p is the maximum persistent current, and σ x, z are Pauli matrices. Here, n φq is the normalized flux bias through the qubit loop in units of the superconducting flux quantum, 0 = h/2e, and n φq0 = 0.5 + k q , where k q is the integer that minimizes |n φq − n φq0 |. The macroscopic nature of the persistent-current states enables strong coupling to other circuit elements. Another important feature of the flux qubit is its strong anharmonicity: the two lowest energy levels are well isolated from the higher levels.The Hamiltonian of the LC oscillator can be written asC is the resonance frequency, L 0 is the inductance of the superconducting lead, L qc ( L c ) is the inductance across the qubit and coupler (see Supplementary Section 2), C is the capacitance, andâ (â † ) is the oscillator's annihilation (creation) operator. Figure 1b shows a laser microscope image of the lumped-element LC oscillator, where L 0 is designed to be as small as possible to maximize the zeropoint fluctuations in the currentand hence achieve strong coupling to the flux qubit, while C is adjusted so as to achieve a desired value of ω o . The freedom of choosing L 0 for large I zpf is one of the advantages of lumped-element LC oscillators over coplanar-waveguide ...

Remarkable progress towards realizing quantum computation has been achieved using natural and artificial atoms as qubits. This article presents a brief overview of the current status of different types of qubits. On the one hand, natural atoms (such as neutral atoms and ions) have long coherence times, and could be stored in large arrays, providing ideal "quantum memories". On the other hand, artificial atoms (such as superconducting circuits or semiconductor quantum dots) have the advantage of custom-designed features and could be used as "quantum processing units". Natural and artificial atoms can be coupled with each other and can also be interfaced with photons for long-distance communications. Hybrid devices made of natural/artificial atoms and photons may provide the next-generation design for quantum computers.

We consider a system composed of a two-level system (i.e., a qubit) and a harmonic oscillator in the ultrastrongcoupling regime, where the coupling strength is comparable to the qubit and oscillator energy scales. Special emphasis is placed on the possibility of preparing nonclassical states in this system. These nonclassical states include squeezed states, Schrödinger-cat states, and entangled states. We start by comparing the predictions of a number of analytical methods that can be used to describe the system under different assumptions, thus analyzing the properties of the system in various parameter regimes. We then examine the ground state of the system and analyze its nonclassical properties. We finally discuss some questions related to the possible experimental observation of the nonclassical states and the effect of decoherence.

This paper starts with a brief review of the topic of strong and weak pre-and post-selected (PPS) quantum measurements, as well as weak values, and afterwards presents original work. In particular, we develop a nonperturbative theory of weak PPS measurements of an arbitrary system with an arbitrary meter, for arbitrary initial states of the system and the meter. New and simple analytical formulas are obtained for the average and the distribution of the meter pointer variable. These formulas hold to all orders in the weak value. In the case of a mixed preselected state, in addition to the standard weak value, an associated weak value is required to describe weak PPS measurements. In the linear regime, the theory provides the generalized Aharonov-Albert-Vaidman formula. Moreover, we reveal two new regimes of weak PPS measurements: the strongly-nonlinear regime and the inverted region (the regime with a very large weak value), where the system-dependent contribution to the pointer deflection decreases with increasing the measurement strength. The optimal conditions for weak PPS measurements are obtained in the strongly-nonlinear regime, where the magnitude of the average pointer deflection is equal or close to the maximum. This maximum is independent of the measurement strength, being typically of the order of the pointer uncertainty. In the optimal regime, the small parameter of the theory is comparable to the overlap of the pre-and post-selected states. We show that the amplification coefficient in the weak PPS measurements is generally a product of two qualitatively different factors. The effects of the free system and meter Hamiltonians are discussed. We also estimate the size of the ensemble required for a measurement and identify optimal and efficient meters for weak measurements. Exact solutions are obtained for a certain class of the measured observables. These solutions are used for numerical calculations, the results of which agree with the theory. Moreover, the theory is extended to allow for a completely general postselection measurement. We also discuss time-symmetry properties of PPS measurements of any strength and the relation between PPS and standard (not post-selected) measurements.

A flux qubit can have a relatively long decoherence time at the degeneracy point, but away from this point the decoherence time is greatly reduced by dephasing. This limits the practical applications of flux qubits. Here we propose a new qubit design modified from the commonly used flux qubit by introducing an additional capacitor shunted in parallel to the smaller Josephson junction (JJ) in the loop. Our results show that the effects of noise can be considerably suppressed, particularly away from the degeneracy point, by both reducing the coupling energy of the JJ and increasing the shunt capacitance. This shunt capacitance provides a novel way to improve the qubit.Superconducting quantum circuits based on Josephson junctions (JJs) are promising candidates of qubits for scalable quantum computing (see, e.g., [1]). Like other types of superconducting qubits, flux qubits have been shown to have quantum coherent properties (see, e.g., [2, 3, 4, 5, 6, 7, 8]). A recent experiment [7] showed that this qubit has a long decoherence time T 2 (∼ 120 ns) at the degeneracy point; this T 2 can become as long as ∼ 4 µs by means of spin-echo techniques. However, even slightly away from the degeneracy point, the decoherence time is drastically reduced. This sensitivity to flux bias considerably limits the applications both for flux qubits for quantum computing, and also when performing quantum-optics and atomic-physics experiments on microelectronic chips with the qubit as an artificial atom.Typically, JJ circuits have two energy scales: the charging energy E c of the JJ, and the Josephson coupling energy E J of the junction. Ordinarily, a flux qubit works in the phase regime with E J /E c ≫ 1, where its decoherence is dominated by flux fluctuations. For the widely used three-junction flux qubit design [2,3,4,5,6], in addition to two identical JJs with coupling energy E J and charging energy E c , a third JJ, which has an area smaller by a factor α ∼ 0.7, is employed to properly adjust the qubit spectrum. Charge fluctuations can affect the decoherence of this flux qubit via the smaller junction.Here we search for an improved design for flux qubits. We show that reducing the ratio E J /E c suppresses the effects of flux noise, although charge noise becomes increasingly important. Reducing α further suppresses the effects of flux noise and considerably improves the decoherence properties away from the degeneracy point. As the effect of flux noise has been largely suppressed, charge noise would now be the dominant source of decoherence. It mainly comes from the charge fluctuations on the two islands separated by the smaller JJ and affects the qubit mainly through relaxation. We thus propose an improved flux qubit by introducing a large capacitor that shunts in parallel to the smaller JJ. This shunt capacitance suppresses the effects of the dominant charge noise in the two islands separated by the smaller JJ by reducing the charging energy. Our results reveal that using a larger shunt capacitor allows reducing both E J /E c and α ...