Section: New Results

Quantum engineering: controlled quantum systems

Participants : Joachim Cohen, Loïc Herviou, Mazyar Mirrahimi, Pierre Rouchon, Pierre Six.

Schrödinger cat states and hardware efficient quantum error correction

We introduce a new gate that transfers an arbitrary state of a qubit into a superposition of two quasi-orthogonal coherent states of a cavity mode, with opposite phases [52] . Such a highly non-classical state is often called a Schrödinger cat state. This qcMAP gate is based on conditional qubit and cavity operations exploiting the energy level dispersive shifts, in the regime where they are much stronger than the cavity and qubit linewidths. The generation of multi-component superpositions of quasi-orthogonal coherent states, non-local entangled states of two resonators and multi-qubit GHz states can be efficiently achieved by this gate.

In a second contribution [53] , we propose to use an encoding of a quantum bit of information in a four-component Schrödinger cat state to ensure its protection against the photon loss, being the major source of decoherence for such a quantum harmonic oscillator. This protection is ensured by an efficient quantum error correction scheme employing the nonlinearity provided by a single physical qubit coupled to the cavity. We describe in detail how to implement these operations in a circuit quantum electrodynamics system. This directly addresses the task of building a hardware-efficient quantum memory and can lead to important shortcuts in quantum computing architectures.

As an important step towards the realization of such a protected quantum memory, in a collaboration with the team of Robert J. Schoelkopf at Yale university, we have successfully realized the encoding protocol of [52] using a 3D transom qubit coupled to a waveguide cavity resonator with a highly ideal off-resonant coupling [60] . This dispersive interaction is much greater than decoherence rates and higher-order nonlinearities to allow simultaneous manipulation of hundreds of photons. We created cat states as large as 111 photons and created 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. This important achievement was published in Science and was also highlighted in Science Perspectives [103] .

Quantum reservoir (dissipation) engineering

We have studied the application of dissipation engineering techniques to perform a high-performance and fast qubit reset [64] . Qubit reset is crucial at the start of and during quantum information algorithms. In a collaboration with the team of Michel H. Devoret at Yale university, our protocol, nicknamed DDROP (Double Drive Reset of Population) was experimentally tested on a superconducting transmon qubit and achieves a ground state preparation of at least $99.5\%$ in times less than $3\mu s$; faster and higher fidelity are predicted upon parameter optimization [46] .

Next, we proposed a dissipation engineering scheme that prepares and protects a maximally entangled state of a pair of superconducting qubits [54] . This is done by off-resonantly coupling the two qubits to a low-Q cavity mode playing the role of a dissipative reservoir. We engineer this coupling by applying six continuous-wave microwave drives with appropriate frequencies. The two qubits need not be identical. We show that our approach does not require any fine-tuning of the parameters and requires only that certain ratios between them be large. This protocol was experimentally realized in a collaboration with the team of M. H. Devoret at Yale university [57] . Unlike conventional, measurement-based schemes, this autonomous approach uses engineered dissipation to counteract decoherence, obviating the need for a complicated external feedback loop to correct errors. Instead, the feedback loop is built into the Hamiltonian such that the steady state of the system in the presence of drives and dissipation is a Bell state, an essential building block for quantum information processing. Such autonomous schemes, which are broadly applicable to a variety of physical systems, will be an essential tool for the implementation of quantum error correction. This important result appeared in Nature back-to-back to another paper by the group of D.J. Wineland (2012 Nobel prize winner) at NIST implementing similar ideas on another physical system consisting of trapped ion qubits  [105] .

Quantum measurement and measurement-based feedback

Measuring a quantum system can randomly perturb its state. The strength and nature of this back-action depend on the quantity that is measured. In a partial measurement performed by an ideal apparatus, quantum physics predicts that the system remains in a pure state whose evolution can be tracked perfectly from the measurement record. This property was proved in a collaboration with the group of Michel H. Devoret (Yale university) using a superconducting qubit dispersively coupled to a cavity traversed by a microwave signal [47] . The back-action on the qubit state of a single measurement of both signal quadratures was observed and shown to produce a stochastic operation whose action is determined by the measurement result. This accurate monitoring of a qubit state is an essential prerequisite for measurement-based feedback control of quantum systems. Indeed, in another experiment performed by our collaborators at ENS (team of Benjamin Huard and François Mallet), we demonstrated stabilization of an arbitrary trajectory of a superconducting qubit by such a measurement-based feedback [37] . The protocol benefits from the long coherence time ($T_2>10\mu$s) of the 3D transmon qubit, the high efficiency ($82\%$) of the phase preserving Josephson amplifier, and fast electronics ensuring less than 500 ns delay. At discrete time intervals, the state of the qubit is measured and corrected in case an error is detected. For Rabi oscillations, where the discrete measurements occur when the qubit is supposed to be in the measurement pointer states, we demonstrate an average fidelity of $85\%$ to the targeted trajectory. Incidentally, we demonstrate a fast reset protocol allowing to cool a 3D transmon qubit down to $0.6\%$ in the excited state.