Radio Frequency driven Josephson circuits provide a rich platform to engineer a variety of nonlinear Hamiltonians for superconducting quantum circuits. While Josephson junctions mediatestrong interactions between microwave photons, some particular types of interaction Hamiltonians can only be obtained through the application of microwave drives (pumps) at well-chosen frequencies. For various applications, it is important to increase the pump strength without introducing undesired couplings and interferences that limit the fidelity of the operations. In this Letter, we analyze these limitations through the theoretical study of the steady state behavior of the driven-dissipative systems. Our general analysis, based on the Floquet-Markov theory, indicates that the ubiquitous circuit consisting of a transmon coupled to a harmonic oscillator suffers from strong limitations in this regard. In accordance with a parallel experimental study, we find that above a fairly low critical pump power the transmon state escapes the Josephson potential confinement and is sent to a statistical mixture of free-particle like states. Next, we illustrate that by diluting the non-linearity of the Josephson junction through a parallel inductive shunt, the picture changes significantly and one achieves very large dynamic ranges in the pump power. This theoretical study provides the ground for drastic modifications in Josephson circuit designs to be used in parametric Hamiltonian engineering experiments.

Strong microwave drives, referred to as pumps, are widely applied to superconducting circuits incorporating Josephson junctions in order to induce couplings between electromagneticmodes. This offers a variety of applications, from quantum-limited amplification, to quantum state and manifold stabilization. These couplings scale with the pump power, therefore, seeking stronger couplings requires a detailed understanding of the behavior of such circuits in the presence of stronger pumps. In this work, we probe the dynamics of a transmon qubit in a 3D cavity, for various pump powers and frequencies. For all pump frequencies, we find a critical pump power above which the transmon is driven into highly excited states, beyond the first seven states which we individually resolve through cavity spectroscopy. This observation is compatible with our theory describing the escape of the transmon state out of its Josephson potential well, into states resembling those of a free particle which does not induce any non-linear couplings.

The remarkable discovery of Quantum Error Correction (QEC), which can overcome the errors experienced by a bit of quantum information (qubit), was a critical advance that gives hopefor eventually realizing practical quantum computers. In principle, a system that implements QEC can actually pass a „break-even“ point and preserve quantum information for longer than the lifetime of its constituent parts. Reaching the break-even point, however, has thus far remained an outstanding and challenging goal. Several previous works have demonstrated elements of QEC in NMR, ions, nitrogen vacancy (NV) centers, photons, and superconducting transmons. However, these works primarily illustrate the signatures or scaling properties of QEC codes rather than test the capacity of the system to extend the lifetime of quantum information over time. Here we demonstrate a QEC system that reaches the break-even point by suppressing the natural errors due to energy loss for a qubit logically encoded in superpositions of coherent states, or cat states of a superconducting resonator. Moreover, the experiment implements a full QEC protocol by using real-time feedback to encode, monitor naturally occurring errors, decode, and correct. As measured by full process tomography, the enhanced lifetime of the encoded information is 320 microseconds without any post-selection. This is 20 times greater than that of the system’s transmon, over twice as long as an uncorrected logical encoding, and 10% longer than the highest quality element of the system (the resonator’s 0, 1 Fock states). Our results illustrate the power of novel, hardware efficient qubit encodings over traditional QEC schemes. Furthermore, they advance the field of experimental error correction from confirming the basic concepts to exploring the metrics that drive system performance and the challenges in implementing a fault-tolerant system.

We show that a quantum-limited phase-preserving amplifier can act as a which-path information eraser when followed by detection of both quadratures. This beam splitter with gain implementsa continuous joint measurement on the signal sources. As an application, we propose heralded remote entanglement generation between two qubits coupled dispersively to separate cavities. Dissimilar qubit-cavity pairs can be made indistinguishable by simple engineering of the cavity driving fields providing experimental flexibility and the prospect for scalability. Additionally, we find an analytic solution for the stochastic master equation, a quantum filter, yielding a thorough physical understanding of the nonlinear measurement process leading to an entangled state of the qubits.

The `Schr“odinger’s cat‘ thought experiment highlights the counterintuitive facet of quantum theory that entanglement can exist between microscopic and macroscopicsystems, producing a superposition of distinguishable states like the fictitious cat that is both alive and dead. The hallmark of entanglement is the detection of strong correlations between systems, for example by the violation of Bell’s inequality. Using the CHSH variant of the Bell test, this violation has been observed with photons, atoms, solid state spins, and artificial atoms in superconducting circuits. For larger, more distinguishable states, the conflict between quantum predictions and our classical expectations is typically resolved due to the rapid onset of decoherence. To investigate this reconciliation, one can employ a superposition of coherent states in an oscillator, known as a cat state. In contrast to discrete systems, one can continuously vary the size of the prepared cat state and therefore its dependence on decoherence. Here we demonstrate and quantify entanglement between an artificial atom and a cat state in a cavity, which we call a `Bell-cat‘ state. We use a circuit QED architecture, high-fidelity measurements, and real-time feedback control to violate Bell’s inequality without post-selection or corrections for measurement inefficiencies. Furthermore, we investigate the influence of decoherence by continuously varying the size of created Bell-cat states and characterize the entangled system by joint Wigner tomography. These techniques provide a toolset for quantum information processing with entangled qubits and resonators. While recent results have demonstrated a high level of control of such systems, this experiment demonstrates that information can be extracted efficiently and with high fidelity, a crucial requirement for quantum computing with resonators.

Physical systems usually exhibit quantum behavior, such as superpositions and entanglement, only when they are sufficiently decoupled from a lossy environment. Paradoxically, a speciallyengineered interaction with the environment can become a resource for the generation and protection of quantum states. This notion can be generalized to the confinement of a system into a manifold of quantum states, consisting of all coherent superpositions of multiple stable steady states. We have experimentally confined the state of a harmonic oscillator to the quantum manifold spanned by two coherent states of opposite phases. In particular, we have observed a Schrodinger cat state spontaneously squeeze out of vacuum, before decaying into a classical mixture. This was accomplished by designing a superconducting microwave resonator whose coupling to a cold bath is dominated by photon pair exchange. This experiment opens new avenues in the fields of nonlinear quantum optics and quantum information, where systems with multi-dimensional steady state manifolds can be used as error corrected logical qubits.

While dissipation is widely considered as being harmful for quantum coherence, it can, when properly engineered, lead to the stabilization of non-trivial pure quantum states. We proposea scheme for continuous generation and stabilization of Schr\“{o}dinger cat states in a cavity using dissipation engineering. We first generate non-classical photon states with definite parity by means of a two-photon drive and dissipation, and then stabilize these transient states against single-photon decay. The single-photon stabilization is autonomous, and is implemented through a second engineered bath, which exploits the photon number dependent frequency-splitting due to Kerr interactions in the strongly dispersive regime of circuit QED. Starting with the Hamiltonian of the baths plus cavity, we derive an effective model of only the cavity photon states along with analytic expressions for relevant physical quantities, such as the stabilization rate. The deterministic generation of such cat states is one of the key ingredients in performing universal quantum computation.

We present a new hardware-efficient paradigm for universal quantum computation which is based on encoding, protecting and manipulating quantum information in a quantum harmonic oscillator.This proposal exploits multi-photon driven dissipative processes to encode quantum information in logical bases composed of Schr\“odinger cat states. More precisely, we consider two schemes. In a first scheme, a two-photon driven dissipative process is used to stabilize a logical qubit basis of two-component Schr\“odinger cat states. While such a scheme ensures a protection of the logical qubit against the photon dephasing errors, the prominent error channel of single-photon loss induces bit-flip type errors that cannot be corrected. Therefore, we consider a second scheme based on a four-photon driven dissipative process which leads to the choice of four-component Schr\“odinger cat states as the logical qubit. Such a logical qubit can be protected against single-photon loss by continuous photon number parity measurements. Next, applying some specific Hamiltonians, we provide a set of universal quantum gates on the encoded qubits of each of the two schemes. In particular, we illustrate how these operations can be rendered fault-tolerant with respect to various decoherence channels of participating quantum systems. Finally, we also propose experimental schemes based on quantum superconducting circuits and inspired by methods used in Josephson parametric amplification, which should allow to achieve these driven dissipative processes along with the Hamiltonians ensuring the universal operations in an efficient manner.

Photons are ideal carriers for quantum information as they can have a long
coherence time and can be transmitted over long distances. These properties are
a consequence of their weakinteractions within a nearly linear medium. To
create and manipulate nonclassical states of light, however, one requires a
strong, nonlinear interaction at the single photon level. One approach to
generate suitable interactions is to couple photons to atoms, as in the strong
coupling regime of cavity QED systems. In these systems, however, one only
indirectly controls the quantum state of the light by manipulating the atoms. A
direct photon-photon interaction occurs in so-called Kerr media, which
typically induce only weak nonlinearity at the cost of significant loss. So
far, it has not been possible to reach the single-photon Kerr regime, where the
interaction strength between individual photons exceeds the loss rate. Here,
using a 3D circuit QED architecture, we engineer an artificial Kerr medium
which enters this regime and allows the observation of new quantum effects. We
realize a Gedankenexperiment proposed by Yurke and Stoler, in which the
collapse and revival of a coherent state can be observed. This time evolution
is a consequence of the quantization of the light field in the cavity and the
nonlinear interaction between individual photons. During this evolution
non-classical superpositions of coherent states, i.e. multi-component
Schroedinger cat states, are formed. We visualize this evolution by measuring
the Husimi Q-function and confirm the non-classical properties of these
transient states by Wigner tomography. The single-photon Kerr effect could be
employed in QND measurement of photons, single photon generation, autonomous
quantum feedback schemes and quantum logic operations.