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

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.

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The code capacity threshold for error correction using biased-noise qubits is known to be higher than with qubits without such structured noise. However, realistic circuit-level noise severely restricts these improvements. This is because gate operations, such as a controlled-NOT (CX) gate, which do not commute with the dominant error, unbias the noise channel. Here, we overcome the challenge of implementing a bias-preserving CX gate using biased-noise stabilized cat qubits in driven nonlinear oscillators. This continuous-variable gate relies on nontrivial phase space topology of the cat states. Furthermore, by following a scheme for concatenated error correction, we show that the availability of bias-preserving CX gates with moderately sized cats improves a rigorous lower bound on the fault-tolerant threshold by a factor of two and decreases the overhead in logical Clifford operations by a factor of five. Our results open a path toward high-threshold, low-overhead, fault-tolerant codes tailored to biased-noise cat qubits.