Keywords
Computer Science and Digital Science
 A1.1.11. Quantum architectures
 A4.2. Correcting codes
 A6. Modeling, simulation and control
 A6.1. Methods in mathematical modeling
 A6.1.1. Continuous Modeling (PDE, ODE)
 A6.1.2. Stochastic Modeling
 A6.1.3. Discrete Modeling (multiagent, people centered)
 A6.1.4. Multiscale modeling
 A6.2. Scientific computing, Numerical Analysis & Optimization
 A6.2.1. Numerical analysis of PDE and ODE
 A6.2.3. Probabilistic methods
 A6.2.6. Optimization
 A6.3.1. Inverse problems
 A6.3.2. Data assimilation
 A6.3.3. Data processing
 A6.3.4. Model reduction
 A6.4. Automatic control
 A6.4.1. Deterministic control
 A6.4.2. Stochastic control
 A6.4.3. Observability and Controlability
 A6.4.4. Stability and Stabilization
Other Research Topics and Application Domains
 B5.3. Nanotechnology
 B5.4. Microelectronics
 B6.5. Information systems
 B9.10. Privacy
1 Team members, visitors, external collaborators
Research Scientists
 Mazyar Mirrahimi [Team leader, Inria, Senior Researcher]
 Philippe Campagne Ibarcq [Inria, Researcher]
 Alain Sarlette [Inria, Researcher]
Faculty Members
 Zaki Leghtas [École Nationale Supérieure des Mines de Paris , Associate Professor]
 Pierre Rouchon [École Nationale Supérieure des Mines de Paris , Professor, HDR]
 Antoine Tilloy [Mines ParisTech, from Oct 2021]
PostDoctoral Fellows
 Cyril Elouard [Inria, until Sep 2021]
 Vincent Lienhard [École Normale Supérieure de Paris, from Feb 2021]
 Mathys Rennela [Inria, from Mar 2021]
 William Smith [Ecole normale supérieure ParisSaclay]
PhD Students
 Michiel Burgelman [Inria]
 Ronan Gautier [Inria]
 Francois Marie Le Regent [Alice et Bob]
 Vincent Martin [Inria]
 Matthieu Praquin [École Normale Supérieure de Paris, from Sep 2021]
 Lev Arcady Sellem [Ecole normale supérieure ParisSaclay]
 Christian Siegele [Inria]
 Aron Vanselow [Inria]
Interns and Apprentices
 Pablo Bermejo [Inria, from Apr 2021 until Jun 2021]
 Leon Carde [École Normale Supérieure de Paris, from Apr 2021 until Jun 2021]
 Louis Paletta [Inria, from Sep 2021]
Administrative Assistant
 Derya Gök [Inria, until Nov 2021]
External Collaborator
 Jeremie Guillaud [Alice et Bob]
2 Overall objectives
2.1 Overall objectives
The research activities of QUANTIC team lie at the border between theoretical and experimental efforts in the emerging field of quantum systems engineering. Our research topics are in direct continuation of a historic research theme of INRIA, classical automatic control, while opening completely new perspectives toward quantum control: by developing a new mathematical system theory for quantum circuits, we will realize the components of a future quantum information processing unit.
One of the unique features of our team concerns the large spectrum of our subjects going from the mathematical analysis of the physical systems (development of systematic mathematical methods for control and estimation of quantum systems), and the numerical analysis of the proposed solutions, to the experimental implementation of the quantum circuits based on these solutions. This is made possible by the constant and profound interaction between the applied mathematicians and the physicists in the group. Indeed, this close collaboration has already brought a significant acceleration in our research efforts. In a long run, this synergy should lead to a deeper understanding of the physical phenomena behind these emerging technologies and the development of new research directions within the field of quantum information processing.
Towards this ultimate task of practical quantum digital systems, the approach of the QUANTIC team is complementary to the one taken by teams with expertise in quantum algorithms. Indeed, we start from the specific controls that can be realistically applied on physical systems, to propose designs which combine them into hardware shortcuts implementing robust behaviors useful for quantum information processing. Whenever a significant new element of quantum engineering architecture is developed, the initial motivation is to prove an enabling technology with major impact for the groups working one abstraction layer higher: on quantum algorithms but also on e.g. secure communication and metrology applications.
3 Research program
3.1 Hardwareefficient quantum information processing
In this scientific program, we will explore various theoretical and experimental issues concerning protection and manipulation of quantum information. Indeed, the next, critical stage in the development of Quantum Information Processing (QIP) is most certainly the active quantum error correction (QEC). Through this stage one designs, possibly using many physical qubits, an encoded logical qubit which is protected against major decoherence channels and hence admits a significantly longer effective coherence time than a physical qubit. Reliable (faulttolerant) computation with protected logical qubits usually comes at the expense of a significant overhead in the hardware (up to thousands of physical qubits per logical qubit). Each of the involved physical qubits still needs to satisfy the best achievable properties (coherence times, coupling strengths and tunability). More remarkably, one needs to avoid undesired interactions between various subsystems. This is going to be a major difficulty for qubits on a single chip.
The usual approach for the realization of QEC is to use many qubits to obtain a larger Hilbert space of the qubit register 86, 89. By redundantly encoding quantum information in this Hilbert space of larger dimension one make the QEC tractable: different error channels lead to distinguishable error syndromes. There are two major drawbacks in using multiqubit registers. The first, fundamental, drawback is that with each added physical qubit, several new decoherence channels are added. Because of the exponential increase of the Hilbert's space dimension versus the linear increase in the number of decay channels, using enough qubits, one is able to eventually protect quantum information against decoherence. However, multiplying the number of possible errors, this requires measuring more error syndromes. Note furthermore that, in general, some of these new decoherence channels can lead to correlated action on many qubits and this needs to be taken into account with extra care: in particular, such kind of nonlocal error channels are problematic for surface codes. The second, more practical, drawback is that it is still extremely challenging to build a register of more than on the order of 10 qubits where each of the qubits is required to satisfy near the best achieved properties: these properties include the coherence time, the coupling strengths and the tunability. Indeed, building such a register is not merely only a fabrication task but rather, one requirers to look for architectures such that, each individual qubit can be addressed and controlled independently from the others. One is also required to make sure that all the noise channels are wellcontrolled and uncorrelated for the QEC to be effective.
We have recently introduced a new paradigm for encoding and protecting quantum information in a quantum harmonic oscillator (e.g. a highQ mode of a 3D superconducting cavity) instead of a multiqubit register 63. The infinite dimensional Hilbert space of such a system can be used to redundantly encode quantum information. The power of this idea lies in the fact that the dominant decoherence channel in a cavity is photon damping, and no more decay channels are added if we increase the number of photons we insert in the cavity. Hence, only a single error syndrome needs to be measured to identify if an error has occurred or not. Indeed, we are convinced that most early proposals on continuous variable QIP 60, 52 could be revisited taking into account the design flexibilities of Quantum Superconducting Circuits (QSC) and the new coupling regimes that are provided by these systems. In particular, we have illustrated that coupling a qubit to the cavity mode in the strong dispersive regime provides an important controllability over the Hilbert space of the cavity mode 62. Through a recent experimental work 94, we benefit from this controllability to prepare superpositions of quasiorthogonal coherent states, also known as Schrödinger cat states.
In this Scheme, the logical qubit is encoded in a fourcomponent Schrödinger cat state. Continuous quantum nondemolition (QND) monitoring of a single physical observable, consisting of photon number parity, enables then the tractability of single photon jumps. We obtain therefore a firstorder quantum error correcting code using only a single highQ cavity mode (for the storage of quantum information), a single qubit (providing the nonlinearity needed for controllability) and a single lowQ cavity mode (for reading out the error syndrome). An earlier experiment on such QND photonnumber parity measurements 90 has recently led to a first experimental realization of a full quantum error correcting code improving the coherence time of quantum information 8. As shown in Figure 1, this leads to a significant hardware economy for realization of a protected logical qubit. Our goal here is to push these ideas towards a reliable and hardwareefficient paradigm for universal quantum computation.
3.2 Reservoir (dissipation) engineering and autonomous stabilization of quantum systems
Being at the heart of any QEC protocol, the concept of feedback is central for the protection of quantum information, enabling manyqubit quantum computation or longdistance quantum communication. However, such a closedloop control which requires a realtime and continuous measurement of the quantum system has been for long considered as counterintuitive or even impossible. This thought was mainly caused by properties of quantum measurements: any measurement implies an instantaneous strong perturbation to the system's state. The concept of quantum nondemolition (QND) measurement has played a crucial role in understanding and resolving this difficulty 34. In the context of cavity quantum electrodynamics (cavity QED) with Rydberg atoms 56, a first experiment on continuous QND measurements of the number of microwave photons was performed by the group at Laboratoire KastlerBrossel (ENS) 53. Later on, this ability of performing continuous measurements allowed the same group to realize the first continuous quantum feedback protocol stabilizing highly nonclassical states of the microwave field in the cavity, the socalled photon number states 10 (this groundbreaking work was mentioned in the Nobel prize attributed to Serge Haroche). The QUANTIC team contributed to the theoretical work behind this experiment 44, 22, 88, 24. These contributions include the development and optimization of the quantum filters taking into account the quantum measurement backaction and various measurement noises and uncertainties, the development of a feedback law based on control Lyapunov techniques, and the compensation of the feedback delay.
In the context of circuit quantum electrodynamics (circuit QED) 42, recent advances in quantumlimited amplifiers 79, 92 have opened doors to highfidelity nondemolition measurements and realtime feedback for superconducting qubits 57. This ability to perform highfidelity nondemolition measurements of a quantum signal has very recently led to quantum feedback experiments with quantum superconducting circuits 92, 78, 36. Here again, the QUANTIC team has participated to one of the first experiments in the field where the control objective is to track a dynamical trajectory of a single qubit rather than stabilizing a stationary state. Such quantum trajectory tracking could be further explored to achieve metrological goals such as the stabilization of the amplitude of a microwave drive 70.
While all this progress has led to a strong optimism about the possibility to perform active protection of quantum information against decoherence, the rather short dynamical time scales of these systems limit, to a great amount, the complexity of the feedback strategies that could be employed. Indeed, in such measurementbased feedback protocols, the timeconsuming data acquisition and posttreatment of the output signal leads to an important latency in the feedback procedure.
The reservoir (dissipation) engineering 76 and the closely related coherent feedback 68 are considered as alternative approaches circumventing the necessity of a realtime data acquisition, signal processing and feedback calculations. In the context of quantum information, the decoherence, caused by the coupling of a system to uncontrolled external degrees of freedom, is generally considered as the main obstacle to synthesize quantum states and to observe quantum effects. Paradoxically, it is possible to intentionally engineer a particular coupling to a reservoir in the aim of maintaining the coherence of some particular quantum states. In a general viewpoint, these approaches could be understood in the following manner: by coupling the quantum system to be stabilized to a strongly dissipative ancillary quantum system, one evacuates the entropy of the main system through the dissipation of the ancillary one. By building the feedback loop into the Hamiltonian, this type of autonomous feedback obviates the need for a complicated external control loop to correct errors. On the experimental side, such autonomous feedback techniques have been used for qubit reset 51, singlequbit state stabilization 71, and the creation 29 and stabilization 61, 67, 85 of states of multipartite quantum systems.
Such reservoir engineering techniques could be widely revisited exploring the flexibility in the Hamiltonian design for QSC. We have recently developed theoretical proposals leading to extremely efficient, and simple to implement, stabilization schemes for systems consisting of a single, two or three qubits 51, 65, 40, 43. The experimental results based on these protocols have illustrated the efficiency of the approach 51, 85. Through these experiments, we exploit the strong dispersive interaction 83 between superconducting qubits and a single lowQ cavity mode playing the role of a dissipative reservoir. Applying continuouswave (cw) microwave drives with wellchosen fixed frequencies, amplitudes, and phases, we engineer an effective interaction Hamiltonian which evacuates the entropy of the system interacting with a noisy environment: by driving the qubits and cavity with continuouswave drives, we induce an autonomous feedback loop which corrects the state of the qubits every time it decays out of the desired target state. The schemes are robust against small variations of the control parameters (drives amplitudes and phase) and require only some basic calibration. Finally, by avoiding resonant interactions between the qubits and the lowQ cavity mode, the qubits remain protected against the Purcell effect, which would reduce the coherence times. We have also investigated both theoretically and experimentally the autonomous stabilization of nonclassical states (such as Schrodinger cat states and Fock states) of microwave field confined in a highQ cavity mode 81, 587, 5.
3.3 System theory for quantum information processing
In parallel and in strong interactions with the above experimental goals, we develop systematic mathematical methods for dynamical analysis, control and estimation of composite and open quantum systems. These systems are built with several quantum subsystems whose irreversible dynamics results from measurements and/or decoherence. A special attention is given to spin/spring systems made with qubits and harmonic oscillators. These developments are done in the spirit of our recent contributions 80, 22, 87, 82, 88, 249 resulting from collaborations with the cavity quantum electrodynamics group of Laboratoire Kastler Brossel.
3.4 Stabilization by measurementbased feedback
The protection of quantum information via efficient QEC is a combination of (i) tailored dynamics of a quantum system in order to protect an informational qubit from certain decoherence channels, and (ii) controlled reaction to measurements that efficiently detect and correct the dominating disturbances that are not rejected by the tailored quantum dynamics.
In such feedback scheme, the system and its measurement are quantum objects whereas the controller and the control input are classical. The stabilizing control law is based on the past values of the measurement outcomes. During our work on the LKB photon box, we have developed, for single input systems subject to quantum nondemolition measurement, a systematic stabilization method 24: it is based on a discretetime formulation of the dynamics, on the construction of a strict control Lyapunov function and on an explicit compensation of the feedbackloop delay. Keeping the QND measurement assumptions, extensions of such stabilization schemes will be investigated in the following directions: finite set of values for the control input with application to the convergence analysis of the atomic feedback scheme experimentally tested in 95; multiinput case where the construction by inversion of a Metzler matrix of the strict Lyapunov function is not straightforward; continuoustime systems governed by diffusive master equations; stabilization towards a set of density operators included in a target subspace; adaptive measurement by feedback to accelerate the convergence towards a stationary state as experimentally tested in 74. Without the QND measurement assumptions, we will also address the stabilization of nonstationary states and trajectory tracking, with applications to systems similar to those considered in 57, 36.
3.5 Filtering, quantum state and parameter estimations
The performance of every feedback controller crucially depends on its online estimation of the current situation. This becomes even more important for quantum systems, where full state measurements are physically impossible. Therefore the ultimate performance of feedback correction depends on fast, efficient and optimally accurate state and parameter estimations.
A quantum filter takes into account imperfection and decoherence and provides the quantum state at time $t\ge 0$ from an initial value at $t=0$ and the measurement outcomes between 0 and $t$. Quantum filtering goes back to the work of Belavkin 30 and is related to quantum trajectories 37, 41. A modern and mathematical exposure of the diffusive models is given in 28. In 55 a first convergence analysis of diffusive filters is proposed. Nevertheless the convergence characterization and estimation of convergence rate remain open and difficult problems. For discrete time filters, a general stability result based on fidelity is proven in 80, 87. This stability result is extended to a large class of continuoustime filters in 23. Further efforts are required to characterize asymptotic and exponential stability. Estimations of convergence rates are available only for quantum nondemolition measurements 31. Parameter estimations based on measurement data of quantum trajectories can be formulated within such quantum filtering framework 46, 72.
We will continue to investigate stability and convergence of quantum filtering. We will also exploit our fidelitybased stability result to justify maximum likelihood estimation and to propose, for open quantum system, parameter estimation algorithms inspired of existing estimation algorithms for classical systems. We will also investigate a more specific quantum approach: it is noticed in 35 that postselection statistics and “past quantum” state analysis 47 enhance sensitivity to parameters and could be interesting towards increasing the precision of an estimation.
3.6 Stabilization by interconnections
In such stabilization schemes, the controller is also a quantum object: it is coupled to the system of interest and is subject to decoherence and thus admits an irreversible evolution. These stabilization schemes are closely related to reservoir engineering and coherent feedback 76, 68. The closedloop system is then a composite system built with the original system and its controller. In fact, and given our particular recent expertise in this domain 985, 51, this subsection is dedicated to further developing such stabilization techniques, both experimentally and theoretically.
The main analysis issues are to prove the closedloop convergence and to estimate the convergence rates. Since these systems are governed by Lindblad differential equations (continuoustime case) or Kraus maps (discretetime case), their stability is automatically guaranteed: such dynamics are contractions for a large set of metrics (see 75). Convergence and asymptotic stability is less well understood. In particular most of the convergence results consider the case where the target steadystate is a density operator of maximum rank (see, e.g., 27[chapter 4, section 6]). When the goal steadystate is not full rank very few convergence results are available.
We will focus on this geometric situation where the goal steadystate is on the boundary of the cone of positive Hermitian operators of finite trace. A specific attention will be given to adapt standard tools (Lyapunov function, passivity, contraction and Lasalle's invariance principle) for infinite dimensional systems to spin/spring structures inspired of 9, 785, 51 and their associated FokkerPlanck equations for the Wigner functions.
We will also explore the Heisenberg point of view in connection with recent results of the INRIA projectteam MAXPLUS (algorithms and applications of algebras of maxplus type) relative to PerronFrobenius theory 50, 49. We will start with 84 and 77 where, based on a theorem due to Birkhoff 32, dual Lindblad equations and dual Kraus maps governing the Heisenberg evolution of any operator are shown to be contractions on the cone of Hermitian operators equipped with Hilbert's projective metric. As the Heisenberg picture is characterized by convergence of all operators to a multiple of the identity, it might provide a mean to circumvent the rank issues. We hope that such contraction tools will be especially well adapted to analyzing quantum systems composed of multiple components, motivated by the facts that the same geometry describes the contraction of classical systems undergoing synchronizing interactions 91 and by our recent generalized extension of the latter synchronizing interactions to quantum systems 69.
Besides these analysis tasks, the major challenge in stabilization by interconnections is to provide systematic methods for the design, from typical building blocks, of control systems that stabilize a specific quantum goal (state, set of states, operation) when coupled to the target system. While constructions exist for socalled linear quantum systems 73, this does not cover the states that are more interesting for quantum applications. Various strategies have been proposed that concatenate iterative control steps for openloop steering 93, 66 with experimental limitations. The characterization of Kraus maps to stabilize any types of states has also been established 33, but without considering experimental implementations. A viable stabilization by interaction has to combine the capabilities of these various approaches, and this is a missing piece that we want to address.
3.6.1 Perturbation methods
With this subsection we turn towards more fundamental developments that are necessary in order to address the complexity of quantum networks with efficient reduction techniques. This should yield both efficient mathematical methods, as well as insights towards unravelling dominant physical phenomena/mechanisms in multipartite quantum dynamical systems.
In the Schrödinger point of view, the dynamics of open quantum systems are governed by master equations, either deterministic or stochastic 56, 48. Dynamical models of composite systems are based on tensor products of Hilbert spaces and operators attached to the constitutive subsystems. Generally, a hierarchy of different timescales is present. Perturbation techniques can be very useful to construct reliable models adapted to the timescale of interest.
To eliminate high frequency oscillations possibly induced by quasiresonant classical drives, averaging techniques are used (rotating wave approximation). These techniques are well established for closed systems without any dissipation nor irreversible effect due to measurement or decoherence. We will consider in a first step the adaptation of these averaging techniques to deterministic Lindblad master equations governing the quantum state, i.e. the system density operator. Emphasis will be put on first order and higher order corrections based on noncommutative computations with the different operators appearing in the Lindblad equations. Higher order terms could be of some interest for the protected logical qubit of figure 1b. In future steps, we intend to explore the possibility to explicitly exploit averaging or singular perturbation properties in the design of coherent quantum feedback systems; this should be an opensystems counterpart of works like 64.
To eliminate subsystems subject to fast convergence induced by decoherence, singular perturbation techniques can be used. They provide reduced models of smaller dimension via the adiabatic elimination of the rapidly converging subsystems. The derivation of the slow dynamics is far from being obvious (see, e.g., the computations of page 142 in 38 for the adiabatic elimination of lowQ cavity). Conversely to the classical composite systems where we have to eliminate one component in a Cartesian product, we here have to eliminate one component in a tensor product. We will adapt geometric singular perturbations 45 and invariant manifold techniques 39 to such tensor product computations to derive reduced slow approximations of any order. Such adaptations will be very useful in the context of quantum Zeno dynamics to obtain approximations of the slow dynamics on the decoherencefree subspace corresponding to the slow attractive manifold.
Perturbation methods are also precious to analyze convergence rates. Deriving the spectrum attached to the Lindblad differential equation is not obvious. We will focus on the situation where the decoherence terms of the form $L\rho {L}^{\u2020}({L}^{\u2020}L\rho +\rho {L}^{\u2020}L)/2$ are small compared to the conservative terms $i[H/\hslash ,\rho ]$. The difficulty to overcome here is the degeneracy of the unperturbed spectrum attached to the conservative evolution $\frac{d}{dt}\rho =i[H/\hslash ,\rho ]$. The degree of degeneracy of the zero eigenvalue always exceeds the dimension of the Hilbert space. Adaptations of usual perturbation techniques 59 will be investigated. They will provide estimates of convergence rates for slightly open quantum systems. We expect that such estimates will help to understand the dependence on the experimental parameters of the convergence rates observed in 51, 85, 65.
As particular outcomes for the other subsections, we expect that these developments towards simpler dominant dynamics will guide the search for optimal control strategies, both in openloop microwave networks and in autonomous stabilization schemes such as reservoir engineering. It will further help to efficiently compute explicit convergence rates and quantitative performances for all the intended experiments.
4 Application domains
4.1 Quantum engineering
A new field of quantum systems engineering has emerged during the last few decades. This field englobes a wide range of applications including nanoelectromechanical devices, nuclear magnetic resonance applications, quantum chemical synthesis, high resolution measurement devices and finally quantum information processing devices for implementing quantum computation and quantum communication. Recent theoretical and experimental achievements have shown that the quantum dynamics can be studied within the framework of estimation and control theory, but give rise to new models that have not been fully explored yet.
The QUANTIC team's activities are defined at the border between theoretical and experimental efforts of this emerging field with an emphasis on the applications in quantum information, computation and communication. The main objective of this interdisciplinary team is to develop quantum devices ensuring a robust processing of quantum information.
On the theory side, this is done by following a system theory approach: we develop estimation and control tools adapted to particular features of quantum systems. The most important features, requiring the development of new engineering methods, are related to the concept of measurement and feedback for composite quantum systems. The destructive and partial 1 nature of measurements for quantum systems lead to major difficulties in extending classical control theory tools. Indeed, design of appropriate measurement protocols and, in the sequel, the corresponding quantum filters estimating the state of the system from the partial measurement record, are themselves building blocks of the quantum system theory to be developed.
On the experimental side, we develop new quantum information processing devices based on quantum superconducting circuits. Indeed, by realizing superconducting circuits at low temperatures and using microwave measurement techniques, the macroscopic and collective degrees of freedom such as the voltage and the current are forced to behave according to the laws of quantum mechanics. Our quantum devices are aimed to protect and process quantum information through these integrated circuits.
5 Highlights of the year
 Hiring of two new permanent members TudorAlexandru Petrescu and Antoine Tilloy as associate professors at Mines Paristech.
 Philippe CampagneIbarcq has obtained an ERC starting grant entitled DANCINGFOOL "Highimpedance Superconducting Circuits Enabling Faulttolerant Quantum Computing by Wideband Microwave Control".
 Alain Sarlette is a PI for a new ANR Grant entitled Mecaflux "Control quantique d'un résonateur mécanique ultracohérent à l'aide d'un qubit fluxonium".
 Mazyar Mirrahimi is a PI for a new ANR Grant entitled OCTAVES "Dépasser les limites dans le forçage cohérent des circuits supraconducteurs quantiques".
 Antoine Tilloy has receieved a PSL jeune equipe Grant to start his activity in his new permanent position.
 Quantic team is participating in two "Plan Quantique" PEPRs entitled RobustSuperQ and NISQ2LSQ.
 Successful PhD defense of Jérémie Guillaud on April 13th.
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6 New results
6.1 Error Rates and Resource Overheads of Repetition Cat Qubits
Participants: J. Guillaud, M. Mirrahimi
In 13, we estimate and analyze the error rates and the resource overheads of the repetition cat qubit approach introduced in 54. While in 54, we show that using only biaspreserving gates on the catqubits, it is possible to build a universal set of faulttolerant logical gates at the level of the repetition cat qubit, in this work, we perform MonteCarlo simulations of all the circuits implementing the protected logical gates, using a circuitlevel error model. Furthermore, we analyze two different approaches to implement a faulttolerant Toffoli gate on repetition cat qubits. These numerical simulations indicate that very low logical error rates could be achieved with a reasonable resource overhead, and with parameters that are within the reach of nearterm circuit QED experiments.
6.2 Combined Dissipative and Hamiltonian Confinement of Cat Qubits
Participants: R. Gautier, M. Mirrahimi, A. Sarlette
To confine the state of an oscillator to the cat qubit manifold, two main approaches have been considered so far: a Kerrbased Hamiltonian confinement with high gate performances, and a dissipative confinement with robust protection against a broad range of noise mechanisms. In our recent work 18, we introduce a new combined dissipative and Hamiltonian confinement scheme based on twophoton dissipation together with a TwoPhoton Exchange (TPE) Hamiltonian. The TPE Hamiltonian is similar to Kerr nonlinearity, but unlike the Kerr it only induces a bounded distinction between even and oddphoton eigenstates, a highly beneficial feature for protecting the cat qubits with dissipative mechanisms. Using this combined confinement scheme, we demonstrate fast and biaspreserving gates with drastically improved performance compared to dissipative or Hamiltonian schemes. In addition, this combined scheme can be implemented experimentally with only minor modifications of existing dissipative cat qubit experiments.
6.3 Multiplexed Photon Number Measurement
Participants: A. Sarlette, P. Rouchon, Z. Leghtas
In 12, a collaboration with our longstanding experimental partner and former QUANTIC member Benjamin Huard, a new scheme is proposed to measure the photon number operator on a superconducting cavity, thanks to a qubit dispersively coupled to the cavity and probed with a frequency comb. Thanks to this frequency multiplexing, the measurement time can be decreased compared to existing schemes. Our contribution has been to analyze the master equations modeling this experiment, in order to evaluate its benefits and clarify key elements. From a practical perspective, this has enabled us to refine the claim of the paper, identifying the use of socalled heterodyne field detectors as the main current bottleneck. From a theoretical perspective, this model has served as inspiration for developing adiabatic elimination expansions at infinite order and whose generalization we are still pursuing.
6.4 Characterizing limits and opportunities in speeding up Markov chain mixing
Participants: A. Sarlette
In our recent work 11 as a followup to our previously published work in 26, 25, we investigate an astounding point of doubt: can discretetime quantum walks allow information to spread faster (i.e. in less steps) on a graph, than classical processes? Indeed, "quantum accelerated transport" for instance has sometimes been claimed to lay behind some fast processes, but there also exist acceleration methods for classical dynamics. We have clarified this situation by following the framework of "Lifted Markov Chains", which adds classical memory to the walker much like the quantum walk also does. We have shown that, in principle, for any evolution described by a Quantum Random Walk on a graph, there exists a classical Lifted Markov chain reproducing the same distribution over nodes in time. We have also shown that instances where particularly fast mixing Quantum Walks were known, are also those where rather intuitive Lifted Markov chains can be designed for accelerated mixing. Finally, we have clarified how much the context / constraints imposed on the Walker setting, can influence the ultimately achievable performance, hence explaining why there have been various seemingly contradictory claims in the literature. For instance, reaching a distribution that should be invariant at all time steps can be an important constraint, as well as assuming that the lifted system would start in a particular subspace of all possible distributions. This somewhat demystifies the "accelerated" evolution of Quantum Random Walks.
7 Bilateral contracts and grants with industry
7.1 Bilateral contracts with industry
 New agreement with the startup Alice&Bob defining the perimeters of our collaboration on cat qubits.
 New PhD contracts with Alice&Bob for two new students: T. Aissaoui and P. Guilmin.
 InriaMicrosoft lab funding to further study "quantum random walks". This funding has led to the hiring of Mathys Rennela.
8 Partnerships and cooperations
8.1 European initiatives
8.1.1 Horizon Europe
ERC Starting Grant ECLIPSE
 Program: Horizon Europe
 Type: ERC
 Project acronym: DANCINGFOOL
 Project title: Highimpedance Superconducting Circuits Enabling Faulttolerant Quantum Computing by Wideband Microwave Control.
 Duration: 20222027
 Coordinator: Philippe CampagneIbarcq, Inria

Abstract: A physical system implementing a quantum bit (qubit) is never perfectly isolated from an uncontrolled environment. The system dynamics is thus noisy, modifying randomly the qubit state. This phenomenon of decoherence is the main roadblock to build a stable quantum computing platform. In order to mitigate decoherence, quantum error correction employs only a few code states within a much larger informational space, so that noiseinduced dynamics can be detected and corrected before the encoded information gets corrupted. Unfortunately, most known protocols require to control dauntingly complex systems, with a degree of coherence currently out of reach.
Our project is to build autonomously errorcorrected qubits encoded in highimpedance superconducting circuits. In our protocol, a qubit is encoded in the vast phasespace of the quantum oscillator implemented by each circuit, in the form of GottesmanKitaevPreskill (GKP) states. The novelty is that the GKP states are fully stabilized by a modular dissipation, induced by the coherent tunneling of charges through a stroboscopically biased Josephson junction. The coherence of the encoded qubit is expected to exceed that of existing superconducting qubits by orders of magnitude. Furthermore, we propose to perform protected logical gates between encoded qubits by varying adiabatically the parameters of the modular dissipation, paving the way toward faulttolerant quantum computing.
The major experimental challenge of our protocol resides in the exquisite level of control needed over a wide band in the microwave range. We propose to address this challenge by developing novel onchip filter, tunable couplers and isolators based on periodically modulated, highimpedence, transmission lines. These onchip components would find a wide range of applications in quantum technologies, and favor the advent of largescale quantum computing platforms.
8.1.2 FP7 & H2020 projects
ERC Starting Grant ECLIPSE
 Program: H2020
 Type: ERC
 Project acronym: ECLIPSE
 Project title: Exotic superconducting circuits to probe and protect quantum states of light and matter
 Duration: 20192024
 Coordinator: Zaki Leghtas, Mines Paristech
 Abstract: Quantum systems can occupy peculiar states, such as superposition or entangled states. These states are intrinsically fragile and eventually get wiped out by inevitable interactions with the environment. Protecting quantum states against decoherence is a formidable and fundamental problem in physics, which is pivotal for the future of quantum computing. The theory of quantum error correction provides a solution, but its current envisioned implementations require daunting resources: a single bit of information is protected by encoding it across tens of thousands of physical qubits. This project intend to encode quantum information in an entirely new type of qubit with two key specificities. First, it will be encoded in a single superconducting circuit resonator whose infinite dimensional Hilbert space can replace large registers of physical qubits. Second, this qubit will be rfpowered, continuously exchanging photons with a reservoir. This approach challenges the intuition that a qubit must be isolated from its environment. Instead, the reservoir acts as a feedback loop which continuously and autonomously corrects against errors. This correction takes place at the level of the quantum hardware, and reduces the need for error syndrome measurements which are resource intensive. The circuits I will develop manipulate quantum states of light, whose utility transcends the long term goal of quantum computing, and can readily be used to probe fundamental properties of matter. In mesoscopic physics where a large number of particles exhibit collective quantum phenomena, the measurement tools to characterize subtle quantum effects are often lacking. Here, the project proposes to measure the spin entanglement of a single Cooper pair, by coupling a superconductor to a circuit composed of microwave resonators and a carbon nanotube. The spin entanglement can be swapped into microwave photons, which can be detected by deploying the arsenal of quantum limited microwave measurement devices.
ERC Advanced Grant QFeedback
 Program: H2020
 Type: ERC
 Project acronym: QFeedback
 Project title: Quantum feedback Engineering
 Duration: 20202025
 Coordinator: Pierre Rouchon, Mines Paristech

Abstract : Quantum technologies, such as quantum computers and simulators, have the potential of revolutionizing our computational speed, communication security and measurement precision.The power of the quantum relies on two key but fragile resources: quantum coherence and entanglement. This promising field is facing a major open question: how to design machines which exploit quantum properties on a large scale, and efficiently protect them fromexternal perturbations (decoherence), which tend to suppress the quantum advantage?
Making a system robust and stable to the influence of external perturbations is one of the core problems in control engineering. The goal of this project is to address the above question from the angle of control systems. The fundamental and scientific ambition is to elaborate theoretical control methods to analyse and design feedback schemes for protecting and stabilizing quantum information. QFeedback develops mathematical methods to harness the inherently stochastic aspects of quantum measurements. Relying on the development of original mathematical perturbation techniques specific to open quantum systems, QFeedback proposes a new hierarchical strategy for quantum feedback modeling, design and analysis.
The building block of a quantum machine is the quantum bit (qubit), a system which can adopt two quantum states. Despite major progress, qubits remain fragile and lose their quantum properties before a meaningful task can be accomplished. For this reason, a qubit must be both protected against external perturbations, and manipulated to perform a task. Today, no such qubit has been built. In collaboration with experimentalists, the practical ambition is to design, relying on the control tools developed here, qubits readily integrable in a quantum processing unit. The physical platform will be Josephson superconducting circuits. QFeedback is expected to demonstrate the crucial role of control engineering in emerging quantum technologies.
Quantera Grant QuCos
 Program: H2020
 Type: Quantera
 Project acronym: QuCos
 Project title: Quantum Computation with Schrödinger cat states
 Duration: 20192023
 Coordinator: Gerhard Kirchmair, University of Innsbruck, Austria.
 Inria contacts: Zaki Leghtas and Mazyar Mirrahimi
 Other partners: ENS Lyon (France), Karlsruhe Institut of Technology (Germany), Quantum Machines (Israel), National Institute for Research and Development of Isotopic and Molecular Technologies, Romania.
 Abstract: This project seeks to establish a radically new, alternative approach to realizing the fundamental building blocks of quantum computers with superconducting qubits. In the next 3 years, we plan to employ only a handful of realistic components to realize robust errorcorrected logical quantum bits. We aim to demonstrate the same level of protection provided by a few hundreds of qubits (with properties beyond the state of the art) in today’s mainstream approach of the socalled surface code architecture. Our alternative approach is known as cat codes, because it employs multiple interconnected high coherence cavity modes with nonlinear dissipation, to encode a qubit in superpositions of Schrödinger cat states. Our project combines realizing the quantum processor architecture as well as the control system and the protocols that drive it, building towards a fullstack errorcorrected quantum computer. The partners in our collaboration form a strong synergetic group that has the full range of expertise needed to design and realize these systems, and to obtain these challenging goals. Furthermore, all partners of our project, including both industry and academia, have worked together and published works in the fields of quantum computing and quantum information processing. We aim to implement error protected qubits, fault tolerant operations, and demonstrate the scalability of this approach by realizing a repetition code. Our project will enable quantum experiments towards the ambitious and welldefined goal of constructing a logical qubit, on which we can perform gates, and most importantly, quantum errorcorrection (QEC).
8.2 National initiatives
 ANR project HAMROQS: In the framework of the ANR program JCJC, Alain Sarlette has received a funding for his research program "Highaccuracy model reduction for open quantum systems". This grant of 212k euros started on april 2019 and will run for 4 years.
 ANR project SYNCAMIL: In the framework of the ANR program JCJC, Philippe CampagneIbarcq has received a funding for his research program "Synthetic NonLocal Hamiltonians for the protection of quantum information". This grant of 380k euros has allowed us to purchase the experimental equipment to perform an experiment on stabilization of GKP grid states at ENS Paris, and to hire a postdoctoral associate, Vincent Lienhard, for 2 years. The project started in December 2020 for 48 months.
 ANR project Mecaflux: Alain Sarlette is a PI of this newly obtained ANR Grant. This project aims to couple mechanical oscillators with superconduncting circuits at the quantum level, using a new circuit architecture allowing nearresonant coupling. The project is coordinated by mechanical oscillators expert Samuel Deléglise (LKB, U.Sorbonne), other project PIs are Alain Sarlette and Zaki Leghtas (QUANTIC projectteam), Emmanuel Flurin and Hélène LeSueur (CEA Saclay). Our new recruit Antoine Tilloy may join with quantum gravity expertise if the level of control attains the objective where those effects become significant.
 ANR project OCTAVES: Mazyar Mirrahimi is a PI of this newly obtained ANR Grant. This project aims in studying the measurement problem in circuit QED (non QND effects in presence of probe drives) as well as limitations to the parametric driving for cat qubit stabilization. The project is coordinated by Olivier Buisson (Institut Néel, Grenoble) and other project PIs are Benjamin Huard (ENS Lyon), Mazyar Mirrahimi (Quantic projectteam), and Dima Shepelyansky (LPT, Toulouse).
8.3 Regional initiatives
 In the framework of the project “DIM SIRTEQ Domaine d’intérêt Majeur: Science et Ingénierie Quantique” of Ile de France Region, we have received two half PhD fellowships covering the PhDs of Jérémie Guillaud and Marius Villiers.
 In the framework of the project “DIM SIRTEQ Domaine d’intérêt Majeur: Science et Ingénierie Quantique” of Ile de France Region, we have received a 109k€ grant to cofund the purchase of a fast laser lithography system in the frame of the collaborative project "ChipIn" which aims at designing and testing onchip microwave components for highcoherence and scalable superconducting circuits.
 PSL Jeune Equipe Grant: Antoine Tilloy has obtained a PSL grant to start his activity as an associate professor in our team working on the topic of tensor networks.
9 Dissemination
9.1 Promoting scientific activities
9.1.1 Journal
Member of the editorial boards
Pierre Rouchon is a member of the editorial board of Annual Reviews in Control.
Reviewer  reviewing activities
 Philippe CampagneIbarcq was a reviewer for Phys. Rev. X.
 Mazyar Mirrahimi was a reviewer for Physical Review and Nature Journals.
 Alain Sarlette has been a reviewer for Physical Review, Journal of the Franklin Insitute, npj Quantum Information, and several control theory conferences.
9.1.2 Invited talks
 Philippe CampagneIbarcq: American Physical Society, March meeting.
 Philippe CampagneIbarcq: ISNTT, Tokyo, Japan.
 Zaki Leghtas: IBM Qiskit
 Cargèse Physique Mésocopique School. Corsica.
 Mazyar Mirrahimi: IBM Qiskit.
 Mazyar Mirrahimi: Inria Alumni.
 Mazyar Mirrahimi: Inria Journées scientifiques.
 Pierre Rouchon: plenary talk at 9th International Conference on Systems and Control (ICSC'2021)
 Alain Sarlette: Launch event InriaMicrosoft lab collaboration on quantum algorithms
9.1.3 Scientific expertise
 Pierre Rouchon is a member of the scientific committee of LAGEP (Laboratoire d'Automatique et de Génie des Procédés) since 2017.
 Pierre Rouchon was a member of ANR Comité d’Evaluation Scientifique on Quantum Technologies.
 Mazyar Mirrahimi was the copresident of Inria's comité des emplois scientifiques.
9.2 Teaching  Supervision  Juries
9.2.1 Teaching
 Zaki Leghtas: Quantum Mechanics and Statistical Physics at Mines ParisTech (12 hours), Quantum Computing at Mines ParisTech (20 hours).
 Philippe CampagneIbarcq: Quantum Mechanics and Statistical Physics at Mines ParisTech (15 hours), Electromagnetism at ESPCI (12 hours).
 Mazyar Mirrahimi: Automatic Control at Ecole Polytechnique (16 hours), Optimization and Control at Ecole Polytechnique (40 hours), Bachelor program thesis instructor (20 hours).
 Pierre Rouchon: Mathematical methods for modeling and control of open quantum systems at International Graduate School on Applied Mathematics LIASFMA (Laboratoire International Associé SinoFrançais de Mathématiques Appliquées) (8 hours)
 Alain Sarlette: Robotics at Ghent university (24 hours).
9.2.2 Supervision
 PhD in progress: Christian Siegele, Quantum error correction with grid states of light, advisors: Philippe CampagneIbarcq and Mazyar Mirrahimi, starting date: Sept 2019.
 PhD in progress: Aron Vanselow, Autonomous stabilization of GKP grid states in high impedance circuits, advisors: Philippe CampagneIbarcq and Mazyar Mirrahimi, starting date: Sept 2020.
 PhD in progress : Michiel Burgelman, A systematic study of strongly driven and dissipative quantum systems towards highaccuracy quantum control designs, advisors: Pierre Rouchon and Alain Sarlette, starting date: Nov 2018.
 PhD defended : Jérémie Guillaud, Repetition catqubits, advisor: Mazyar Mirrahimi, starting date: Nov 2017, Defended: April 2021.
 PhD in progress : Vincent Martin, Faulttolerance of quantum systems under continuoustime feedback stabilization, advisor: Alain Sarlette, starting date: Oct 2018.
 PhD in progress: Marius Villiers, Probing the spin entanglement of single Cooper pair, advisors: Zaki Leghtas and Takis Kontos, starting date: September 2019.
 PhD in progress: Camille Berdou, A catqubit repetition code, advisors: Zaki Leghtas and Pierre Rouchon, starting date Sept 2019.
 PhD in progress: Alvise Borgognoni. Topologically protected qubits. Supervision of Zaki Leghtas.
 PhD in progress: Ulysse Réglade. A 3D catqubit. Supervision of Zaki Leghtas and Raphaël Lescanne.
 PhD in progress: FrançoisMarie Le Régent, Quantum computing architecture with catqubits, advisor: Mazyar Mirrahimi and Jérémie Guillaud, Starting date: Sept 2020.
 PhD in progress: Ronan Gautier, Efficient protection and fast gates in bosonic quantum codes. advisor: Alain Sarlette, Starting date: September 2020.
 PhD in progress: LevArcady Sellem. Mathematical methods for simulation and control of open quantum systems, advisors: Claude Le Bris and Pierre Rouchon, Starting date: October 2020.
 PhD in progress: Thiziei Aissaoui, Onchip biasing of superconducting circuits, Starting date: september 2021, Advisors: Alain Sarlette and Anil Murani (Alice&Bob).
 PhD in progress: Matthieu Praquin, Reconfigurable microwave photonics crystals based on granular superconductors, advisors: Philippe CampagneIbarcq and Zaki Leghtas, starting date: Sept 2021.
9.2.3 Juries
 Zaki Leghtas was a member of the HDR defence jury of Daniel GarciaSanchez (INSP Paris) and the PhD defence jury of Sébastien Leger (Institut Néel Grenoble)
 Mazyar Mirrahimi was a reviewer for two PhD dissertations: Eugenio Pozzoli and Marco FellousAsiani.
 Alain Sarlette was a member of PhD defence jury of Antoine Essig at ENS Lyon.
9.3 Popularization
9.3.1 Articles and contents
Alain Sarlette's popularization article in Interstices 21.
9.3.2 Interventions
Mazyar Mirrahimi interview on podcast "Decode Quantum", 2021.
10 Scientific production
10.1 Major publications
 1 articleQuantum FastForwarding: Markov chains and graph property testing.Quantum Information & ComputationApril 2019
 2 articleQuantum error correction of a qubit encoded in grid states of an oscillator.Nature584Text and figures edited for clarity. The claims of the paper remain the same. Author list fixedAugust 2020
 3 articleExponential stabilization of quantum systems under continuous nondemolition measurements.AutomaticaDecember 2019
 4 articleRepetition Cat Qubits for FaultTolerant Quantum Computation.Physical Review Xhttps://arxiv.org/abs/1904.09474  22 pages, 11 figuresDecember 2019
 5 articleConfining the state of light to a quantum manifold by engineered twophoton loss.Science3476224February 2015, 853857
 6 articleExponential suppression of bitflips in a qubit encoded in an oscillator.Nature PhysicsMarch 2020
 7 articleDynamically protected catqubits: a new paradigm for universal quantum computation.New Journal of Physics164apr 2014, 045014
 8 articleExtending the lifetime of a quantum bit with error correction in superconducting circuits.Nature5362016, 5
 9 articleStabilization of nonclassical states of the radiation field in a cavity by reservoir engineering.Phys. Rev. Lett.1070104022011
 10 articleRealtime quantum feedback prepares and stabilizes photon number states.Nature4772011, 7377
10.2 Publications of the year
International journals
 11 articleCharacterizing limits and opportunities in speeding up Markov chain mixing.Stochastic Processes and their Applications136June 2021, Pages 145191
 12 articleMultiplexed Photon Number Measurement.Physical Review X113August 2021
 13 articleError Rates and Resource Overheads of Repetition Cat Qubits.Physical Review A1032021
 14 articleEnergyparticipation quantization of Josephson circuits.npj Quantum Information71December 2021
 15 articleA fixed point algorithm for improving fidelity of quantum gates.ESAIM: Control, Optimisation and Calculus of Variations272021, S9
 16 articleDetermining the position of a single spin relative to a metallic nanowire.Journal of Applied Physics129142021
Doctoral dissertations and habilitation theses
 17 thesisRepetition Cat Qubits.École Normale SupérieureApril 2021
Reports & preprints
 18 miscCombined Dissipative and Hamiltonian Confinement of Cat Qubits.January 2022
 19 miscMotion planing for an ensemble of Bloch equations towards the south pole with smooth bounded control.August 2021
 20 miscEnergetics of a Single Qubit Gate.October 2021
10.3 Other
Scientific popularization
 21 articleConstruire une machine quantique.IntersticesMarch 2021
10.4 Cited publications
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 44 articleQuantum feedback by discrete quantum nondemolition measurements: towards ondemand generation of photonnumber states.Physical Review A80: 0138050138132009
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 47 articlePast Quantum States of a Monitored System.Phys. Rev. Lett.11116160401October 2013, URL: http://link.aps.org/doi/10.1103/PhysRevLett.111.160401
 48 bookQuantum Noise.Springer2010
 49 articleChecking the strict positivity of Kraus maps is NPhard.arXiv:1402.14292014
 50 articleThe contraction rate in Thompson's part metric of orderpreserving flows on a cone  Application to generalized Riccati equations.Journal of Differential Equations2568April 2014, 29022948URL: http://www.sciencedirect.com/science/article/pii/S0022039614000424
 51 articleDemonstrating a Driven Reset Protocol of a Superconducting Qubit.Phys. Rev. Lett.1101205012013
 52 articleEncoding a qubit in an oscillator.Phys. Rev. A640123102001
 53 articleProgressive fieldstate collapse and quantum nondemolition photon counting.Nature4482007, 889893
 54 articleRepetition Cat Qubits for FaultTolerant Quantum Computation.Physical Review X22 pages, 11 figuresDecember 2019
 55 articleThe stability of quantum Markov filters.Infin. Dimens. Anal. Quantum Probab. Relat. Top.122009, 153172
 56 bookExploring the Quantum: Atoms, Cavities and Photons.Oxford University Press2006
 57 articleQuantum backaction of an individual variablestrength measurement.Science3392013, 178181
 58 articleSinglephotonresolved crossKerr interaction for autonomous stabilization of photonnumber states.Phys. Rev. Lett.1152015, 180501
 59 bookPerturbation Theory for Linear Operators.Springer1966
 60 articleA scheme for efficient quantum computation with linear optics.Nature409462001
 61 articleEntanglement Generated by Dissipation and Steady State Entanglement of Two Macroscopic Objects.Phys. Rev. Lett.1070805032011
 62 articleDeterministic protocol for mapping a qubit to coherent state superpositions in a cavity.Phys. Rev. A870423152013
 63 articleHardwareefficient autonomous quantum memory protection.Phys. Rev. Lett.1111205012013
 64 articleAdiabatic passage and ensemble control of quantum systems.J. Phys. B441540172011
 65 articleStabilizing a Bell state of two superconducting qubits by dissipation engineering.Phys. Rev. A880238492013
 66 articleEnsemble control of Bloch equations.IEEE Trans. Autom. Control542009, 528536
 67 articleDissipative production of a maximally entangled steady state of two quantum bits.Nature5042013, 415418
 68 articleCoherent quantum feedback.Phys. Rev. A620221082000
 69 articleConsensus for quantum networks: from symmetry to gossip iterations.IEEE Trans. Automat. Controlin press2014
 70 conferenceStrong measurement and quantum feedback for persistent Rabi oscillations in circuit QED experiments.IEEE Conference on Decision and ControlIEEE Conference on Decision and Control2012
 71 articleCavityassisted quantum bath engineering.Phys. Rev. Lett.1091836022012
 72 articleEstimation of classical parameters via continuous probing of complementary quantum observables.New Journal of Physics15121250022013, URL: http://stacks.iop.org/13672630/15/i=12/a=125002
 73 articleCoherent quantum LQG control.Automatica452009, 18371846
 74 articleAdaptive Quantum Nondemolition Measurement of a Photon Number.Phys. Rev. Lett.1128080401Feb 2014, URL: http://link.aps.org/doi/10.1103/PhysRevLett.112.080401
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 77 articleHilbert's projective metric in quantum information theory.Journal of Mathematical Physics528082201August 2011, URL: http://dx.doi.org/10.1063/1.3615729
 78 articleInitialization by measurement of a superconducting quantum bit circuit.Phys. Rev. Lett.1090505072012
 79 articleWidely tunable, nondegenerate threewave mixing microwave device operating near the quantum limit.Phys. Rev. Lett.1081477012012
 80 articleFidelity is a SubMartingale for DiscreteTime Quantum Filters.IEEE Transactions on Automatic Control56112011, 27432747
 81 articleContinuous generation and stabilization of mesoscopic field superposition states in a quantum circuit.Phys. Rev. A912015, 013810
 82 articleStabilization of nonclassical states of one and twomode radiation fields by reservoir engineering.Phys. Rev. A860121142012
 83 articleResolving photon number states in a superconducting circuit.Nature4452007, 515518
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