Search

Over the past decade, autonomous stabilization of bosonic qubits has emerged as a promising approach for hardware-efficient protection of quantum information. However, applying these techniques to more complex encodings than the Schr\"odinger cat code requires exquisite control of high-order wave mixing processes. The challenge is to enable specific multiphotonic dissipation channels while avoiding unintended non-linear interactions. In this work, we leverage a genuine six-wave mixing process enabled by a near Kerr-free Josephson element to enforce dissipation of quartets of excitations in a high-impedance superconducting resonator. Owing to residual non-linearities stemming from stray inductances in our circuit, this dissipation channel is only effective when the resonator holds a specific number of photons. Applying it to the fourth excited state of the resonator, we show an order of magnitude enhancement of the state decay rate while only marginally impacting the relaxation and coherence of lower energy states. Given that stray inductances could be strongly reduced through simple modifications in circuit design and that our methods can be adapted to activate even higher-order dissipation channels, these results pave the way toward the dynamical stabilization of four-component Schr\"odinger cat qubits and even more complex bosonic qubits.

Light waves do not interact in vacuum, but may mix through various parametric processes when traveling in a nonlinear medium. In particular, a high-amplitude wave can be leveraged to frequency convert a low-amplitude signal, as long as the overall energy and momentum of interacting photons are conserved. These conditions are typically met when all waves propagate in the medium with identical phase velocity along a particular axis. In this work, we investigate an alternative scheme by which an input microwave signal propagating along a 1-dimensional Josephson metamaterial is converted to an output wave propagating in the opposite direction. The interaction is mediated by a pump wave propagating at low phase velocity. In this novel regime, the input signal is exponentially attenuated as it travels down the device. We exploit this process to implement a robust on-chip microwave isolator that can be reconfigured into a reciprocal and tunable coupler. The device mode of operation is selected in situ, along with its working frequency over a wide microwave range. In the 5.5-8.5 GHz range, we measure an isolation over 15 dB on a typical bandwidth of 100 MHz, on par with the best existing on-chip isolators. Substantial margin for improvement exists through design optimization and by reducing fabrication disorder, opening new avenues for microwave routing and processing in superconducting circuits.

Quantum bits (qubits) are prone to several types of errors due to uncontrolled interactions with their environment. Common strategies to correct these errors are based on architectures of qubits involving daunting hardware overheads. A hopeful path forward is to build qubits that are inherently protected against certain types of errors, so that the overhead required to correct remaining ones is significantly reduced. However, the foreseen benefit rests on a severe condition: quantum manipulations of the qubit must not break the protection that has been so carefully engineered. A recent qubit - the cat-qubit - is encoded in the manifold of metastable states of a quantum dynamical system, thereby acquiring continuous and autonomous protection against bit-flips. Here, in a superconducting circuit experiment, we implement a cat-qubit with bit-flip times exceeding 10 seconds. This is a four order of magnitude improvement over previous cat-qubit implementations. We prepare and image quantum superposition states, and measure phase-flip times above 490 nanoseconds. Most importantly, we control the phase of these quantum superpositions without breaking bit-flip protection. This experiment demonstrates the compatibility of quantum control and inherent bit-flip protection at an unprecedented level, showing the viability of these dynamical qubits for future quantum technologies.

We analyze an experimentally accessible Lindblad master equation for a quantum harmonic oscillator. It approximately stabilizes finite-energy periodic grid states called Gottesman-Kitaev-Preskill (GKP) states, that can be used to encode and protect a logical qubit. We give explicit upper bounds for the energy of the solutions of the Lindblad master equation. Using three periodic observables to define the Bloch sphere coordinates of a logical qubit, we show that their dynamics is governed by a diffusion partial differential equation on a 2D-torus with a Witten Laplacian. We show that the evolution of these logical coordinates is exponentially slow even in presence of small diffusive noise processes along the two quadratures of the phase space. Numerical simulations indicate similar results for other physically relevant noise processes., Comment: 16 pages, 1 figure. This work has been accepted to IFAC for publication under a Creative Commons Licence CC-BY-NC-ND

We propose a novel approach to generate, protect and control GKP qubits. It employs a microwave frequency comb parametrically modulating a Josephson circuit to enforce a dissipative dynamics of a high impedance circuit mode, autonomously stabilizing the finite-energy GKP code. The encoded GKP qubit is robustly protected against all dominant decoherence channels plaguing superconducting circuits but quasi-particle poisoning. In particular, noise from ancillary modes leveraged for dissipation engineering does not propagate at the logical level. In a state-of-the-art experimental setup, we estimate that the encoded qubit lifetime could extend two orders of magnitude beyond the break-even point, with substantial margin for improvement through progress in fabrication and control electronics. Qubit initialization, readout and control via Clifford gates can be performed while maintaining the code stabilization, paving the way toward the assembly of GKP qubits in a fault-tolerant quantum computing architecture., Comment: 66 pages, 20 figures, 1 table. Updated version: reorganization of the paper, several clarifications, new sections about ancilla noise propagation and protected measurement of Pauli operators

Straightforward logical operations contrasting with complex state preparation are the hallmarks of the bosonic encoding proposed by Gottesman, Kitaev and Preskill (GKP). The recently reported generation and error-correction of GKP qubits in trapped ions and superconducting circuits thus holds great promise for the future of quantum computing architectures based on such encoded qubits. However, these experiments rely on error-syndrome detection via an auxiliary physical qubit, whose noise may propagate and corrupt the encoded GKP qubit. We propose a simple module composed of two oscillators and a physical qubit, operated with two experimentally accessible quantum gates and elementary feedback controls to implement an error-corrected GKP qubit protected from such propagating errors. In the idealized setting of periodic GKP states, we develop efficient numerical methods to optimize our protocol parameters and show that errors of the encoded qubit stemming from flips of the physical qubit and diffusion of the oscillators state in phase-space may be exponentially suppressed as the noise strength over individual operations is decreased. Our approach circumvents the main roadblock towards fault-tolerant quantum computation with GKP qubits.

Based on the stabilizer formalism underlying Quantum Error Correction (QEC), the design of an original Lindblad master equation for the density operator of a quantum harmonic oscillator is proposed. This Lindblad dynamics stabilizes exactly the finite-energy grid states introduced in 2001 by Gottesman, Kitaev and Preskill for quantum computation. Stabilization results from an exponential Lyapunov function with an explicit lower-bound on the convergence rate. Numerical simulations indicate the potential interest of such autonomous QEC in presence of non-negligible photon-losses., Comment: Submitted, 15 pages, 1 figures

In quantum error correction, information is encoded in a high-dimensional system to protect it from the environment. A crucial step is to use natural, low-weight operations with an ancilla to extract information about errors without causing backaction on the encoded system. Essentially, ancilla errors must not propagate to the encoded system and induce errors beyond those which can be corrected. The current schemes for achieving this fault-tolerance to ancilla errors come at the cost of increased overhead requirements. An efficient way to extract error syndromes in a fault-tolerant manner is by using a single ancilla with strongly biased noise channel. Typically, however, required elementary operations can become challenging when the noise is extremely biased. We propose to overcome this shortcoming by using a bosonic-cat ancilla in a parametrically driven nonlinear cavity. Such a cat-qubit experiences only bit-flip noise and is stabilized against phase-flips. To highlight the flexibility of this approach, we illustrate the syndrome extraction process in a variety of codes such as qubit-based toric codes, bosonic cat- and Gottesman-Kitaev-Preskill (GKP) codes. Our results open a path for realizing hardware-efficient, fault-tolerant error syndrome extraction.

Modular quantum computing architectures require fast and efficient distribution of quantum information through propagating signals. Here we report rapid, on-demand quantum state transfer between two remote superconducting cavity quantum memories through traveling microwave photons. We demonstrate a quantum communication channel by deterministic transfer of quantum bits with 76% fidelity. Heralding on errors induced by experimental imperfection can improve this to 87% with a success probability of 0.87. By partial transfer of a microwave photon, we generate remote entanglement at a rate that exceeds photon loss in either memory by more than a factor of three. We further show the transfer of quantum error correction code words that will allow deterministic mitigation of photon loss. These results pave the way for scaling superconducting quantum devices through modular quantum networks., Comment: main text 7 pages, 4 figures; supplement 16 pages, 16 figures

Tomography of a quantum state is usually based on positive operator-valued measure (POVM) and on their experimental statistics. Among the available reconstructions, the maximum-likelihood (MaxLike) technique is an efficient one. We propose an extension of this technique when the measurement process cannot be simply described by an instantaneous POVM. Instead, the tomography relies on a set of quantum trajectories and their measurement records. This model includes the fact that, in practice, each measurement could be corrupted by imperfections and decoherence, and could also be associated with the record of continuous-time signals over a finite amount of time. The goal is then to retrieve the quantum state that was present at the start of this measurement process. The proposed extension relies on an explicit expression of the likelihood function via the effective matrices appearing in quantum smoothing and solutions of the adjoint quantum filter. It allows to retrieve the initial quantum state as in standard MaxLike tomography, but where the traditional POVM operators are replaced by more general ones that depend on the measurement record of each trajectory. It also provides, aside the MaxLike estimate of the quantum state, confidence intervals for any observable. Such confidence intervals are derived, as the MaxLike estimate, from an asymptotic expansion of multi-dimensional Laplace integrals appearing in Bayesian Mean estimation. A validation is performed on two sets of experimental data: photon(s) trapped in a microwave cavity subject to quantum non-demolition measurements relying on Rydberg atoms; heterodyne fluorescence measurements of a superconducting qubit., Comment: 11 pages, 4 figures, submitted

Electromagnetic modes are instrumental in building quantum machines. In this experiment, we introduce a method to manipulate these modes by effectively controlling their phase space. Preventing access to a single energy level, corresponding to a number of photons N, confined the dynamics of the field to levels 0 to N-1. Under a resonant drive, the level occupation was found to oscillate in time, similarly to an N-level system. Performing a direct Wigner tomography of the field revealed its nonclassical features, including a Schr\"{o}dinger cat-like state at half period in the evolution. This fine control of the field in its phase space may enable applications in quantum information and metrology., Comment: 5 pages, 4 figures. Supplementary Materials available as an ancillary file here

The dynamics of many open quantum systems are described by stochastic master equations. In the discrete-time case, we recall the structure of the derived quantum filter governing the evolution of the density operator conditioned to the measurement outcomes. We then describe the structure of the corresponding particle quantum filters for estimating constant parameter and we prove their stability. In the continuous-time (diffusive) case, we propose a new formulation of these particle quantum filters. The interest of this new formulation is first to prove stability, and also to provide an efficient algorithm preserving, for any discretization step-size, positivity of the quantum states and parameter classical probabilities. This algorithm is tested on experimental data to estimate the detection efficiency for a superconducting qubit whose fluorescence field is measured using a heterodyne detector., Comment: 8 pages, 3 figures, submitted

In quantum physics, measurement results are random but their statistics can be predicted assuming some knowledge about the system in the past. Additional knowledge from a future measurement deeply changes the statistics in the present and leads to purely quantum features. In particular conditioned average outcomes of a weak measurement, so-called weak values, were shown to go beyond the conventional range, give a way to directly measure complex quantities, and can be used to enhance the sensitivity of quantum meters. Recently, these concepts have been considered in the general case of open quantum systems where decoherence occurs. Then, what are the properties of weak values for the unavoidable measurement associated to decoherence, the one performed by the environment? Here, we answer this question in the simplest open quantum system: a quantum bit in presence of a relaxation channel. We continuously monitor the fluorescence emitted by a superconducting qubit driven at resonance. Conditioned on initial preparation and final single shot measurement outcome of the qubit state, we probe weak values displaying all the above properties. The fluorescence signal exhibits interferences between oscillations associated to past and future quantum states. The measured data are in complete agreement with theory., Comment: 5 pages, 3 figures. Supplemnatary information available as an ancillary file here

Making a system state follow a prescribed trajectory despite fluctuations and errors commonly consists in monitoring an observable (temperature, blood-glucose level...) and reacting on its controllers (heater power, insulin amount ...). In the quantum domain, there is a change of paradigm in feedback since measurements modify the state of the system, most dramatically when the trajectory goes through superpositions of measurement eigenstates. Here, we demonstrate the stabilization of an arbitrary trajectory of a superconducting qubit by measurement based feedback. 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. For Ramsey oscillations, which does not go through pointer states, the average fidelity reaches 75%. Incidentally, we demonstrate a fast reset protocol allowing to cool a 3D transmon qubit down to 0.6% in the excited state., Comment: 7 pages, 3 figures and 1 table. Supplementary information available as an ancilla file

We present the first experimental realization of a widely frequency tunable, non-degenerate three-wave mixing device for quantum signals at GHz frequency. It is based on a new superconducting building-block consisting of a ring of four Josephson junctions shunted by a cross of four linear inductances. The phase configuration of the ring remains unique over a wide range of magnetic fluxes threading the loop. It is thus possible to vary the inductance of the ring with flux while retaining a strong, dissipation-free, and noiseless non-linearity. The device has been operated in amplifier mode and its noise performance has been evaluated by using the noise spectrum emitted by a voltage biased tunnel junction at finite frequency as a test signal. The unprecedented accuracy with which the crossover between zero-point-fluctuations and shot noise has been measured provides an upper-bound for the noise and dissipation intrinsic to the device., Comment: Accepted for Physical Review Letters. Supplementary material can be found in the source package