Quantum error correction (QEC) can overcome the errors experienced by qubits and is therefore an essential component of a future quantum computer. To implement QEC, a qubit is redundantly encoded in a higher-dimensional space using quantum states with carefully tailored symmetry properties. Projective measurements of these parity-type observables provide error syndrome information, with which errors can be corrected via simple operations. The 'break-even' point of QEC--at which the lifetime of a qubit exceeds the lifetime of the constituents of the system--has so far remained out of reach. Although previous works have demonstrated elements of QEC, they primarily illustrate the signatures or scaling properties of QEC codes rather than test the capacity of the system to preserve a qubit 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 Schrödinger-cat states of a superconducting resonator. We implement a full QEC protocol by using real-time feedback to encode, monitor naturally occurring errors, decode and correct. As measured by full process tomography, without any post-selection, the corrected qubit lifetime is 320 microseconds, which is longer than the lifetime of any of the parts of the system: 20 times longer than the lifetime of the transmon, about 2.2 times longer than the lifetime of an uncorrected logical encoding and about 1.1 longer than the lifetime of the best physical qubit (the |0〉f and |1〉f Fock states of the resonator). Our results illustrate the benefit of using hardware-efficient qubit encodings rather than traditional QEC schemes. Furthermore, they advance the field of experimental error correction from confirming basic concepts to exploring the metrics that drive system performance and the challenges in realizing a fault-tolerant system.

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 multiphoton driven dissipative processes to encode quantum information in logical bases composed of Schrödinger 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 Schrödinger 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 bitflip 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 Schrödinger 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 Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. amplification, which should allow one to achieve these driven dissipative processes along with the Hamiltonians ensuring the universal operations in an efficient manner.Keywords: quantum superconducting circuits, circuit quantum electrodynamics, quantum error correction, quantum reservoir engineering, universal quantum computation n n n and therefore the measurement of the photon number parity can indicate the occurrence of a photon loss event. While the parity measurements keep track of the photon loss events, the deterministic relaxation of the energy, replacing α by α κ − e / t 2 , remains inevitable. To overcome this relaxation of energy, we need to intervene before the coherent states start to overlap in a significant manner to re-pump energy into the codeword. New J. Phys. 16 (2014) 045014 M Mirrahimi et al New J. Phys. 16 (2014) 045014 M Mirrahimi et al New J. Phys. 16 (2014) 045014 M Mirrahimi et al 7 / aa . Extension to four-photon driven dissipative processSimilarly to the case of the two-photon process, we need to achieve an effective Hamiltonian of the form New J. Phys. 16 (2014) 045014 M Mirrahimi et al 21

In contrast to a single quantum bit, an oscillator can store multiple excitations and coherences provided one has the ability to generate and manipulate complex multiphoton states. We demonstrate multiphoton control by using a superconducting transmon qubit coupled to a waveguide cavity resonator with a highly ideal off-resonant coupling. This dispersive interaction is much greater than decoherence rates and higher-order nonlinearities to allow simultaneous manipulation of hundreds of photons. With a tool set of conditional qubit-photon logic, we mapped an arbitrary qubit state to a superposition of coherent states, known as a "cat state." We created cat states as large as 111 photons and extended this protocol to create superpositions of up to four coherent states. This control creates a powerful interface between discrete and continuous variable quantum computation and could enable applications in metrology and quantum information processing.

Physical systems usually exhibit quantum behavior, such as superpositions and entanglement, only when they are sufficiently decoupled from a lossy environment. Paradoxically, a specially engineered 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 confined the state of a superconducting resonator to the quantum manifold spanned by two coherent states of opposite phases and have observed a Schrödinger cat state spontaneously squeeze out of vacuum before decaying into a classical mixture. This experiment points toward robustly encoding quantum information in multidimensional steady-state manifolds.

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 weak interactions 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 [1, 2]. In these systems, however, one only indirectly controls the quantum state of the light by manipulating the atoms [3]. 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 [4], 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 [5], 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 Schrödinger 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 ability to create and manipulate superpositions of coherent states in such a high quality factor photon mode opens perspectives for combining the physics of continuous variables [6] with superconducting circuits. The single-photon Kerr effect could be employed in QND measurement of photons [7], single photon generation [8], autonomous quantum feedback schemes [9] and quantum logic operations [10].A material whose refractive index depends on the intensity of the light field is called a Kerr medium. A light beam traveling through such a material acquires a phase shift φ Kerr = Kτ I [11] where I is the intensity of the beam, τ is the interaction time of the light field with the material, and K is the Kerr constant. The Kerr effect is a widely used phenomenon in nonlinear quantum optics and has been successfully employed to generate quadrature and amplitude squeezed states [12], parametrically convert frequencies [13], and create ultra-fast pulses [14]. In the field of quantum optics with microwave circuits, the direct analog of the Kerr effect is naturally created by the nonlinear inductance of a Josephson junction (specifically the4 term in the Taylor expansion of the cos φ of the Josephson energy relation) [15,16]. This effect has been used to create Josephson parametric amplifiers [17][18][19] and to generate squeezing of microwave fields [20]. However, in both the microwave and optical dom...

Feedback loops are central to most classical control procedures. A controller compares the signal measured by a sensor (system output) with the target value or set-point. It then adjusts an actuator (system input) to stabilize the signal around the target value. Generalizing this scheme to stabilize a micro-system's quantum state relies on quantum feedback, which must overcome a fundamental difficulty: the sensor measurements cause a random back-action on the system. An optimal compromise uses weak measurements, providing partial information with minimal perturbation. The controller should include the effect of this perturbation in the computation of the actuator's operation, which brings the incrementally perturbed state closer to the target. Although some aspects of this scenario have been experimentally demonstrated for the control of quantum or classical micro-system variables, continuous feedback loop operations that permanently stabilize quantum systems around a target state have not yet been realized. Here we have implemented such a real-time stabilizing quantum feedback scheme following a method inspired by ref. 13. It prepares on demand photon number states (Fock states) of a microwave field in a superconducting cavity, and subsequently reverses the effects of decoherence-induced field quantum jumps. The sensor is a beam of atoms crossing the cavity, which repeatedly performs weak quantum non-demolition measurements of the photon number. The controller is implemented in a real-time computer commanding the actuator, which injects adjusted small classical fields into the cavity between measurements. The microwave field is a quantum oscillator usable as a quantum memory or as a quantum bus swapping information between atoms. Our experiment demonstrates that active control can generate non-classical states of this oscillator and combat their decoherence, and is a significant step towards the implementation of complex quantum information operations.

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Quantum superpositions of distinct coherent states in a single-mode harmonic oscillator, known as "cat states," have been an elegant demonstration of Schrödinger's famous cat paradox. Here, we realize a two-mode cat state of electromagnetic fields in two microwave cavities bridged by a superconducting artificial atom, which can also be viewed as an entangled pair of single-cavity cat states. We present full quantum state tomography of this complex cat state over a Hilbert space exceeding 100 dimensions via quantum nondemolition measurements of the joint photon number parity. The ability to manipulate such multicavity quantum states paves the way for logical operations between redundantly encoded qubits for fault-tolerant quantum computation and communication.