2023Activity reportProjectTeamQUANTIC
RNSR: 201321103L Research center Inria Paris Centre at Sorbonne University
 In partnership with:CNRS, Ecole normale supérieure de Paris, Mines ParisTech, Sorbonne Université
 Team name: QUANTum Information Circuits
 In collaboration with:Laboratoire Pierre Aigrain, Centre Automatique et Systèmes
 Domain:Applied Mathematics, Computation and Simulation
 Theme:Optimization and control of dynamic systems
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, Senior Researcher, from Oct 2023, HDR]
 Alain Sarlette [INRIA, Researcher, until Sep 2023, HDR]
Faculty Members
 Zaki Leghtas [ENSMP, Associate Professor]
 TudorAlexandru Petrescu [MINESPARISTECH]
 Pierre Rouchon [ARMINES, Professor, HDR]
 Antoine Tilloy [MINESPARISTECH]
PostDoctoral Fellows
 Yousra Bouasria [INRIA, PostDoctoral Fellow, until Feb 2023]
 Edoardo Lauria [ENSMP, PostDoctoral Fellow, from Mar 2023]
 Vincent Lienhard [ENS PARIS, PostDoctoral Fellow]
 Remi Robin [ARMINES, PostDoctoral Fellow]
PhD Students
 Thiziri Aissaoui [ALICE ET BOB, CIFRE]
 Brieuc Beauseigneur [ARMINES, from Sep 2023]
 Leon Carde [ARMINES, CIFRE]
 Ronan Gautier [INRIA, until Aug 2023]
 Linda Greggio [INRIA]
 Pierre Guilmin [ALICE ET BOB]
 Anissa Jacob [ALICE ET BOB, CIFRE, from Nov 2023]
 FrançoisMarie Le Regent [ALICE ET BOB]
 Louis Paletta [INRIA]
 Matthieu Praquin [ENS PARIS]
 Angela Riva [INRIA]
 Emilio Rui [ALICE ET BOB, from Oct 2023]
 Diego Ruiz [ALICE ET BOB, CIFRE]
 LevArcady Sellem [ARMINES, from Sep 2023]
 LevArcady Sellem [ENS PARISSACLAY, until Aug 2023]
 Christian Siegele [INRIA, until Feb 2023]
 Karanbir Tiwana [ARMINES]
 Aron Vanselow [INRIA]
Interns and Apprentices
 Brieuc Beauseigneur [INRIA, Intern, from Apr 2023 until Jul 2023]
 Leonardo Cardinale [ENSMP, Intern, from May 2023 until Aug 2023]
 Vania Clesca Dedieu [MINESPARISTECH, Intern, until Feb 2023]
 Robin Geiller [ENS DE LYON, Intern, from May 2023 until Jul 2023]
 Philippe Gigon [INRIA, Intern, from Feb 2023 until Jun 2023]
 Danilo Nascimento Guimaraes [INRIA, Intern, from Apr 2023 until Jul 2023]
 Etienne Parcollet [ENS Paris, Intern, from Apr 2023 until Jun 2023]
 Tom Perrin [MINESPARISTECH, Intern, from Dec 2023]
Administrative Assistant
 Derya Gok [INRIA]
Visiting Scientists
 Henrik Clausen [UNIV AALBORG, from Sep 2023]
 Birgitta Whaley [UNIV BERKELEY, from May 2023 until Jul 2023]
External Collaborators
 Ronan Gautier [ALICE ET BOB, from Sep 2023]
 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 109, 113. 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 81. 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 78, 71 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 80. Through a recent experimental work 120, 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 114 has recently led to a first experimental realization of a full quantum error correcting code improving the coherence time of quantum information 7. 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 51. In the context of cavity quantum electrodynamics (cavity QED) with Rydberg atoms 74, a first experiment on continuous QND measurements of the number of microwave photons was performed by the group at Laboratoire KastlerBrossel (ENS) 72. 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 9 (this groundbreaking work was mentioned in the Nobel prize attributed to Serge Haroche). The QUANTIC team contributed to the theoretical work behind this experiment 62, 40, 112, 42. 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) 60, recent advances in quantumlimited amplifiers 102, 117 have opened doors to highfidelity nondemolition measurements and realtime feedback for superconducting qubits 75. This ability to perform highfidelity nondemolition measurements of a quantum signal has very recently led to quantum feedback experiments with quantum superconducting circuits 117, 101, 53. 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 89.
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 96 and the closely related coherent feedback 86 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 70, singlequbit state stabilization 90, and the creation 45 and stabilization 79, 85, 108 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 70, 83, 58, 61. The experimental results based on these protocols have illustrated the efficiency of the approach 70, 108. Through these experiments, we exploit the strong dispersive interaction 106 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 104, 766, 4.
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 103, 40, 111, 105, 112, 428 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 42: 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 121; 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 93. 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 75, 53.
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 47 and is related to quantum trajectories 54, 59. A modern and mathematical exposure of the diffusive models is given in 44. In 73 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 103, 111. This stability result is extended to a large class of continuoustime filters in 41. Further efforts are required to characterize asymptotic and exponential stability. Estimations of convergence rates are available only for quantum nondemolition measurements 48. Parameter estimations based on measurement data of quantum trajectories can be formulated within such quantum filtering framework 65, 91.
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 52 that postselection statistics and “past quantum” state analysis 66 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 96, 86. 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 8108, 70, 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 95). 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., 43[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 8, 6108, 70 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 69, 68. We will start with 107 and 99 where, based on a theorem due to Birkhoff 49, 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 116 and by our recent generalized extension of the latter synchronizing interactions to quantum systems 88.
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 92, 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 119, 84 with experimental limitations. The characterization of Kraus maps to stabilize any types of states has also been established 50, 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 74, 67. 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 82.
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 55 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 63 and invariant manifold techniques 56 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 77 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 70, 108, 83.
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.
3.7 Onchip microwave engineering
The rapid development of circuitQED over the past 20 years was enabled by commercially available microwave components such as filters, switches and circulators, which allow experimentalists to shape and route measurement and control signals in and out of quantum systems. However, these components are intrinsically bulky, lossy and are imperfectly impedancematched, leading to spurious reflections at their ports. In order to implement a fullscale quantum computer based on superconducting circuits, it is crucial that these functionalities be enabled reliably onchip.
Onchip filters commonly used in circuitQED experiments are far from the level of variety and refinement of commercially available components. The near exclusive strategy known as "Purcellfiltering" 100 consists in placing $\lambda /4$ stubs 97 on all feed lines. This cancels the admittance of the environment seen by a superconducting qubit at its resonance frequency, inhibiting spontaneous relaxation. An issue with this strategy is that given the modest width of the stub stopband, performances are degraded as soon as the qubit is not perfectly in resonance. Moreover, this approach is not suited for multiplexed control and measurements, in which a single feed line addresses simultaneously several qubits. Notable alternatives include highpass waveguide filters only available in 3D circuitQED 98, and a recent implementation of a bandpass filter 64.
Onchip nonreciprocal elements, such as isolators, circulators and gyrators are at a very early stage of development. So far, the most promising approach to break reciprocity without resorting to strong magnetic fields—which are incompatible with superconducting circuit technology—relies on the differential phase impinged on a signal during parametric downconversion with respect to the reverse process of upconversion. Combining coherently several conversion paths with wellchosen phases, one obtains a constructive forward interference, and a destructive backward one. In circuitQED, frequency conversion is enabled by a nonlinear Josephson circuit 110, 57, 39, or by electromechanical coupling to nanoresonators 46, 94. A serious drawback of this approach is that it relies on a destructive interference effect to obtain the reverse isolation, which limits the operational bandwidth: the highest value reported so far is a 23 dB isolation over a 8 MHz band 39. For completeness, we mention a recent implementation of a forward amplifier based on resistively shunted Josephson junctions 115 that reaches a 100 MHz bandwidth at the cost of added noise, and the long term prospect of harnessing the anomalous Hall effect to implement a gyrator 118, 87.
In this project, we propose to develop novel onchip filters and isolators based on 1D photonic crystals, which could reach unprecedented bandwidth, tunable range and on/off or forward/backward transmission ratios. The central idea is that a microwave transmission line with periodically modulated electrical properties behaves as a robust stopband filter, with attenuation scaling exponentially with the line length.
By fabricating lines whose properties are modulated by design, we plan to demonstrate the efficiency of this novel type of stopband filters. These lines will be fabricated in a highkinetic inductance material (such as chains of Josephson junctions or granular aluminium), we will overcome the main weakness of this approach, which is the large onchip footprint required when fabricating with conventional superconductors. Extending the numerical simulation methods developed in thsi work, we plan to design other types of filters (bandpass, highpass, lowpass) based on a similar technology.
We will then change perspective and modulate a line properties parametrically instead of by design to implement nonreciprocal elements. The idea is to design a line that possesses two traveling mode 1 and 2, with different propagation phasevelocity ${v}_{1}\ll {v}_{2}$. If a lowfrequency pump wave propagates on 1 modulates the electrical properties of the mode 2 thanks to shared nonlinear inductance—for instance Josephson junction participating in both modes inductance—we obtain a situation in which mode 2 has spatially modulated electrical properties, with the phase of this modulation slowly evolving in time with at the pump period. A stopband appears in the transmission of a probe signal on mode 2, whose central frequency depends on whether the probe propagates in the same direction as the pump or not. It is then straightforward to turn this nonreciprocal filter into an isolator on circulator.
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
 Successful PhD defense of Christian Siegele on February 24th.
 Successful PhD defense of Camille Berdou on March 14th.
 Successful PhD defense of Marius Villiers on May 15th.
 Successful PhD defense of Ronan Gautier on December 4th.
 Installation of a new experimental laboratory space at ENS Paris with a new dilution cryostat.
 Promotion of Alain Sarlette to DR2.
6 New results
6.1 Designing HighFidelity Zeno Gates for Dissipative Cat Qubits
Participants: Ronan Gautier, Mazyar Mirrahimi, Alain Sarlette.
This paper 15, published in PRX Quantum, gathers several different system design possibilities in order to perform fast operations on cat qubits while strongly reducing the decoherence induced by the fast actuation on the system. The decoherence is due to the fact that a dissipative element is present on purpose in order to stabilize the qubit, while simple enough actuation Hamiltonians cannot drive all necessary system operations without activating this stabilizing dissipation; in this way, fast actuation would also induce rather fast information loss. One proposed solution, also submitted for patenting, is to introduce backaction from the output of the main dissipation channel, thus reinjecting the part of the evacuated entropy which correlates to qubit information which we want to maintain. Other proposed solutions include the engineering of small nonlinearities in the actuation Hamiltonian, or designing jumpactivated operations. This has been a major part of the thesis of Ronan Gautier, successfully defended in December 2023.
6.2 Stabilization of approximate GHZ states with quasilocal couplings
Participants: Vincent Martin, Alain Sarlette.
This work 30, currently under review, comprises essentially 3 contributions. First, we solve the objective of stabilizing intrinsically longrange entanglement, for which impossibility results have been known, on a prototypical example. Second, our solution proposes and investigates the use of engineered reservoirs — thus an autonomous system with steady couplings — whose aim is to mimic the behavior of automata with little but nonzero memory. We establish the effectiveness of several options and workings, involving: introducing one auxiliary system per data qubit; adding a third level to the data qubit for operation purposes ; and inducing waves of effects propagating through the overall system at stochastic speeds. A third contribution concerns the analysis method, where we bound the Lindblad master equation evolution by a Markov chain involving hypothetical measurements of the dissipation channels, and then analyze the associated classical Markov chain.
6.3 Stability and decoherence rates of a GKP qubit protected by dissipation
Participants: LevArcady Sellem, Rémi Robin, Philippe CampagneIbarcq, Pierre Rouchon.
In this communication at the IFAC WC 202320, We analyze an experimentally accessible Lindblad master equation for a quantum harmonic oscillator. It approximately stabilizes finiteenergy periodic grid states called GottesmanKitaevPreskill (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 2Dtorus 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.
6.4 Complete Positivity Violation in Higherorder Quantum Adiabatic Elimination
Participants: Masaaki Tokieda, Alain Sarlette, Pierre Rouchon.
This communication at the IFAC WC 2023 21 adresses quantum adiabatic elimination and completepositivity. When a composite Lindblad system consists of weakly coupled subsystems with fast and slow timescales, the description of slow dynamics can be simplified by discarding fast degrees of freedom. This model reduction technique is called adiabatic elimination. While secondorder perturbative expansion with respect to the timescale separation has revealed that the evolution of a reduced state is completely positive, this paper presents an example exhibiting complete positivity violation in the fourthorder expansion. Despite the nonuniqueness of slow dynamics parametrization, we prove that complete positivity cannot be ensured in any parametrization. The violation stems from correlation in the initial state.
6.5 Correlation functions for realistic continuous quantum measurement
Participants: Pierre Guilmin, Pierre Rouchon, Antoine Tilloy.
This communication at the IFAC WC 2023 25 proposes a selfcontained and accessible derivation of an exact formula for the npoint correlation functions of the signal measured when continuously observing a quantum system. The expression depends on the initial quantum state and on the Stochastic Master Equation (SME) governing the dynamics. This derivation applies to both jump and diffusive evolutions and takes into account common imperfections of realistic measurement devices. We show how these correlations can be efficiently computed numerically for commonly filtered and integrated signals available in practice.
6.6 Heisenberg formulation of adiabatic elimination for open quantum systems with two timescales
Participants: FrançoisMarie Le Regent, Pierre Rouchon.
This communication at the IEEE CDC 2023 19,considers an open quantum system governed by a Gorini, Kossakowski, Sudarshan, Lindblad (GKSL) master equation with two timesscales: a fast one, exponentially converging towards a linear subspace of quasiequilibria; a slow one resulting from perturbations (small arbitrary decoherence and Hamiltonian dynamics). Usually adiabatic elimination is performed in the Schrödinger picture. We propose here an Heisenberg formulation where the invariant operators attached to the fast decay dynamics towards the quasiequilibria subspace play a key role. Based on geometric singular perturbations, asympotic expansions of the Heisenberg slow dynamics and of the fast invariant linear subspaces are proposed. They exploit Carr's approximation lemma from centermanifold and bifurcation theory. Secondorder expansions are detailed and shown to ensure preservation, up to secondorder terms, of the trace and complete positivity for the slow dynamics on a slow timescale. Such expansions can be exploited numerically.
6.7 Adiabatic elimination for composite open quantum systems: reduced model formulation and numerical simulations
Participants: FrançoisMarie Le Régent, Pierre Rouchon.
This preprint under review in Physical Review A 32 proposes a numerical method for simulation of composite open quantum systems. It is based on Lindblad master equations and adiabatic elimination. Each subsystem is assumed to converge exponentially towards a stationary subspace, slightly impacted by some decoherence channels and weakly coupled to the other subsystems. This numerical method is based on a perturbation analysis with an asymptotic expansion. It exploits the formulation of the slow dynamics with reduced dimension. It relies on the invariant operators of the local and nominal dissipative dynamics attached to each subsystem. Secondorder expansion can be computed only with local numerical calculations. It avoids computations on the tensorproduct Hilbert space attached to the full system. This numerical method is particularly well suited for autonomous quantum error correction schemes. Simulations of such reduced models agree with complete full model simulations for typical gates acting on one and two catqubits (Z, ZZ and CNOT) when the mean photon number of each catqubit is less than 8. For larger mean photon numbers and gates with three catqubits (ZZZ and CCNOT), full model simulations are almost impossible whereas reduced model simulations remain accessible. In particular, they capture both the dominant phaseflip errorrate and the very small bitflip errorrate with its exponential suppression versus the mean photon number.
6.8 Convergence of bipartite open quantum systems stabilized by reservoir engineering
Participants: Rémi Robin, Pierre Rouchon, LevArcady Sellem.
This preprint 34 considers a generic family of Lindblad master equations modeling bipartite open quantum systems, where one tries to stabilize a quantum system by carefully designing its interaction with another, dissipative, quantum system  a strategy known as quantum reservoir engineering. We provide sufficient conditions for convergence of the considered Lindblad equations; our setting accommodates the case where steady states are not unique but rather supported on a given subspace of the underlying Hilbert space. We apply our result to a Lindblad master equation proposed for the stabilization of socalled cat qubits, a system that received considerable attention in recent years due to its potential applications in quantum information processing.
6.9 Highsensitivity ACcharge detection with a MHzfrequency fluxonium qubit
Participants: Angela Riva, Marius Villiers, Alain Sarlette, Zaki Leghtas.
This work 31, provisionally accepted in PRX, is the first step of results in the framework of the ANR project MecaFlux led by Samuel Deléglise, whose ultimate goal is to stabilize quantum states in a mechanical oscillator thanks to its coupling to a superconducting circuit in socalled "heavy fluxonium" configuration. The first step demonstrates the capabilities of such a fluxonium, in terms of actively cooling it down to the ground state, maintaining qubittype quantum states, reading out such state. It also demonstrates experimentally how this circuit does couple to the degree of freedom (MHz charge fluctuations) where the mechanical oscillator is to be placed; thus we have a fully operational building block for the overall project, and which may be of independent interest.
6.10 Quantum control of a catqubit with bitflip times exceeding ten seconds
Participants: Ulysse Reglade, Adrien Bocquet, Ronan Gautier, LevArcady Sellem, Pierre Rouchon, Alain Sarlette, Mazyar Mirrahimi, Philippe CampagneIbarcq, Zaki Leghtas.
Binary classical information is routinely encoded in the two metastable states of a dynamical system. Since these states may exhibit macroscopic lifetimes, the encoded information inherits a strong protection against bitflips. A recent qubit  the catqubit  is encoded in the manifold of metastable states of a quantum dynamical system, thereby acquiring bitflip protection. An outstanding challenge is to gain quantum control over such a system without breaking its protection. If this challenge is met, significant shortcuts in hardware overhead are forecast for quantum computing. In this experiment, we implement a catqubit with bitflip times exceeding ten seconds. This is a four order of magnitude improvement over previous catqubit implementations, and six orders of magnitude enhancement over the single photon lifetime that compose this dynamical qubit. This was achieved by introducing a quantum tomography protocol that does not break bitflip protection. We prepare and image quantum superposition states, and measure phaseflip times above 490 nanoseconds. Most importantly, we control the phase of these superpositions while maintaining the bitflip time above ten seconds. This work demonstrates quantum operations that preserve macroscopic bitflip times, a necessary step to scale these dynamical qubits into fully protected hardwareefficient architectures.
This work has given rise to the preprint 33.
6.11 Observation of Josephson Harmonics in Tunnel Junctions
Participants: Zaki Leghtas.
Superconducting quantum processors have a long road ahead to reach faulttolerant quantum computing. One of the most daunting challenges is taming the numerous microscopic degrees of freedom ubiquitous in solidstate devices. Stateoftheart technologies, including the world's largest quantum processors, employ aluminum oxide (AlOx) tunnel Josephson junctions (JJs) as sources of nonlinearity, assuming an idealized pure sinφ currentphase relation (CφR). However, this celebrated sinφ CφR is only expected to occur in the limit of vanishingly lowtransparency channels in the AlOx barrier. Here we show that the standard CφR fails to accurately describe the energy spectra of transmon artificial atoms across various samples and laboratories. Instead, a mesoscopic model of tunneling through an inhomogeneous AlOx barrier predicts $\%$level contributions from higher Josephson harmonics. By including these in the transmon Hamiltonian, we obtain orders of magnitude better agreement between the computed and measured energy spectra. The reality of Josephson harmonics transforms qubit design and prompts a reevaluation of models for quantum gates and readout, parametric amplification and mixing, Floquet qubits, protected Josephson qubits, etc. As an example, we show that engineered Josephson harmonics can reduce the charge dispersion and the associated errors in transmon qubits by an order of magnitude, while preserving anharmonicity.
This work has led to the preprint 38.
6.12 Robust suppression of noise propagation in GottesmanKitaevPreskill error correction
Participants: Christian Siegele, Philippe Campagne Ibarcq.
The recently reported generation and errorcorrection 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 errorsyndrome detection via an auxiliary physical qubit, whose noise may propagate and corrupt the encoded GKP qubit. In [Siegele, C., & CampagneIbarcq, P. (2023). Robust suppression of noise propagation in GottesmanKitaevPreskill error correction. Physical Review A, 108(4), 042427.], 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 errorcorrected 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 phasespace may be exponentially suppressed as the noise strength over individual operations is decreased. Our approach circumvents the main roadblock towards faulttolerant quantum computation with GKP qubits.
This work was published in Phys. Rev. A 18.
6.13 A GKP qubit protected by dissipation in a highimpedance superconducting circuit driven by a microwave frequency comb
Participants: LevArcady Sellem, Alain Sarlette, Zaki Leghtas, Mazyar Mirrahimi, Pierre Rouchon, Philippe CampagneIbarcq.
In the preprint 35, 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 finiteenergy GKP code. The encoded GKP qubit is robustly protected against all dominant decoherence channels plaguing superconducting circuits but quasiparticle poisoning. In particular, noise from ancillary modes leveraged for dissipation engineering does not propagate at the logical level. In a stateoftheart experimental setup, we estimate that the encoded qubit lifetime could extend two orders of magnitude beyond the breakeven 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 faulttolerant quantum computing architecture.
6.14 Robust sparse IQP sampling in constant depth
Participants: Louis Paletta, Alain Sarlette, Mazyar Mirrahimi.
Between NISQ (noisy intermediate scale quantum) approaches without any proof of robust quantum advantage and fully faulttolerant quantum computation, we propose a scheme to achieve a provable superpolynomial quantum advantage (under some widely accepted complexity conjectures) that is robust to noise with minimal error correction requirements. We choose a class of sampling problems with commuting gates known as sparse IQP (Instantaneous Quantum Polynomialtime) circuits and we ensure its faulttolerant implementation by introducing the tetrahelix code. This new code is obtained by merging several tetrahedral codes (3D color codes) and has the following properties: each sparse IQP gate admits a transversal implementation, and the depth of the logical circuit can be traded for its width. Combining those, we obtain a depth1 implementation of any sparse IQP circuit up to the preparation of encoded states. This comes at the cost of a space overhead which is only polylogarithmic in the width of the original circuit. We furthermore show that the state preparation can also be performed in constant depth with a single step of feedforward from classical computation. Our construction thus exhibits a robust superpolynomial quantum advantage for a sampling problem implemented on a constant depth circuit with a single round of measurement and feedforward.
This work 23 has been accepted for a plenary presentation at the conference QIP 2024 in Taipei and is under revision for Quantum.
7 Bilateral contracts and grants with industry
7.1 Bilateral contracts with industry
 New PhD contracts with Alice&Bob for 2 new students: Emilio Rui , Anissa Jacob .
8 Partnerships and cooperations
8.1 International research visitors
8.1.1 Visits of international scientists
Other international visits to the team
Birgitta Whaley

Status:
Professor

Institution of origin:
Univ. of Berkeley

Country:
USA

Dates:
May 5th to June 18th.

Context of the visit:
Collaborations with Pierre Rouchon.
Paulo Sérgio Pereira da Silva

Status:
Professor

Institution of origin:
Univ. of Sao Paolo

Country:
Brasil
 Dates:

Context of the visit:
Collaborations with Pierre Rouchon.
Philippe Lewalle

Status:
PhD

Institution of origin:
Univ. of Berkeley

Country:
USA

Dates:
May 4th to June 18th.

Context of the visit:
Collaborations with Pierre Rouchon.
Henrik Clausen

Status:
PhD

Institution of origin:
Aalborg University

Country:
Denmark

Dates:
Sept 11th to Dec 8th.

Context of the visit:
Collaborations with Pierre Rouchon.
8.2 European initiatives
8.2.1 Horizon Europe
DANCINGFOOL
Participants: Philippe Campagne Ibarcq, Matthieu Praquin, Vincent Lienhard, Aron Vanselow.
DANCINGFOOL project on cordis.europa.eu

Title:
Highimpedance Superconducting Circuits Enabling Faulttolerant Quantum Computing by Wideband Microwave Control

Duration:
From December 1, 2022 to November 30, 2027

Partners:
 INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET AUTOMATIQUE (INRIA), France
 ECOLE NORMALE SUPERIEURE (ENS), France

Inria contact:
Philippe CampagneIbarcq

Coordinator:
Philippe Campagne Ibarcq

Summary:
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 filters, tunable couplers and isolators based on periodically modulated, highimpedance, transmission lines. These onchip components would find a wide range of applications in quantum technologies, and favor the advent of largescale quantum computing platforms.
QFT.zip
Participants: Antoine Tilloy, Karanbir Tiwana, Edoardo Lauria.
QFT.zip project on cordis.europa.eu

Title:
Compressing manybody quantum states in continuous spacetime with tensor networks

Duration:
From January 1, 2023 to December 31, 2027

Partners:
 ARMINES, France
 INRIA, France

Coordinator:
Antoine Tilloy

Summary:
Manybody quantum systems with strong correlations are particularly difficult to understand in the continuum, where nonperturbative techniques are in scarce supply. Direct diagonalization methods are not available, since the Hilbert space is simply too large to be manageable. This inhibits progress in high energy physics, nuclear physics, and in the study of exotic topological phases of matter. On the lattice, tensor network states, a variational class of wavefunctions coming from quantum information theory, have allowed to compress exponentially large Hilbert spaces down to a smaller numerically manageable corner. This has allowed substantial theoretical and numerical advance on the manybody problem on the lattice. This project will develop continuous tensor network states, a new framework to extend the recent lattice progress to the continuum and quantum field theory (QFT). The originality of the approach is that it will not rely on any discretization of spacetime. We will work directly in the continuum, without any cutoff. Low energy states of quantum field theories, which a priori live in a continuously infinite dimensional Hilbert space, will be compressed down to a finite and small number of parameters. This will then allow to solve numerically very generic (nonintegrable) strongly coupled theories in a fully nonperturbative manner. Such a compression was long thought to be impossible, in particular in the relativistic case, but we overcame crucial theoretical hurdles in the past year, making the proposal particularly timely. We will construct this framework with 3 main applications in mind: i) nonrelativistic problems in 2 space dimensions and more, including e.g. fractional quantum Hall states, ii) relativistic QFT, starting with 1+1 dimensional toy model and gradually increasing complexity to get closer to nonabelian gauge theories, iii) critical quantum systems (and classical statistical mechanics).
8.2.2 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).
8.3 National initiatives
 PEPR NISQ2LSQ: Quantic is a PI and the coordinator of WP1 of this PEPR project which started in 2022. The goal is to accelerate French research on the topic of bosonic and LDPC codes for preparing the ground for hardwareefficient and faulttolerant quantum computation.
 PEPR RobustSuperQ: Quantic is a PI and the coordinator of WP1 of this PEPR project which started in 2022. The goal is to accelerate French research on the topic of high quality, noise resilient, superconducting qubits.
 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 Mecaflux: Alain Sarlette is a PI of this ANR Grant that started in 2022 and runs for 4 years. 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. The PhD thesis of Angela Riva is funded on this ANR.
 ANR project OCTAVES: Mazyar Mirrahimi is a PI of this ANR Grant that started in 2022 and will run for 4 years. 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). The PhD thesis of Linda Greggio is funded on this ANR.
8.4 Regional initiatives
 Alain Sarlette is a member of the steering committee of DIM Quantip.
9 Dissemination
9.1 Promoting scientific activities
9.1.1 Scientific events: selection
Reviewer
 Pierre Rouchon was a referee for IEEE and IFAC conferences.
 Alain Sarlette was a referee for IEEE and IFAC conferences.
9.1.2 Journal
Member of the editorial boards
 Pierre Rouchon is a member of the editorial board of Annual Reviews in Control.
Reviewer  reviewing activities
 Philippe Campagne Ibarcq was a referee for PRL, PRX, Nature Physics.
 Zaki Leghtas was a referee for PRL and Nature Communications.
 Mazyar Mirrahimi was a referee for PRL and npj Quantum Information.
 TudorAlexandru Petrescu was a referee for Physical Review and Quantum.
 Pierre Rouchon was a referee for Journal of differential equation, PRX.
 Alain Sarlette was a referee for IEEE Transactions, PRL, PRX Quantum, PRA.
 Antoine Tilloy was a referee for PRL, PRX, Physics Letters A, and Foundations of Physics.
9.1.3 Invited talks
 Philippe Campagne Ibarcq : BBQ conference in Boulder, Colorado, on bosonic error correction.
 Philippe Campagne Ibarcq : Erlangen University (remote).
 Philippe Campagne Ibarcq : Chalmers University.
 Zaki Leghtas : Hebrew University of Jerusalem. Invited by Alex Retzker (2023).
 Zaki Leghtas : Weizmann Instiute. Invited by Fabien Lafont and Serge Rosenblum (2023).
 Zaki Leghtas : Institut Néel. Invited by Jérémie Viennot (2023).
 Zaki Leghtas : C2N. Invited by Rebeca RibeiroPalau (2023)
 Zaki Leghtas : Les Houches. France. Invited by Benjamin Sacépé (2023)
 Mazyar Mirrahimi : BBQ conference in Boulder, Colorado, on bosonic error correction.
 Mazyar Mirrahimi : ENSMitsubishi Heavy Industry meeting.
 Mazyar Mirrahimi : University of Glasgow Centre for Quantum Technology (remote) invited by Kaveh Delfanazari.
 Mazyar Mirrahimi : Workshop "Feedback control of Quantum Machines", Okinawa Institute for Science and Technology, Japan.
 Mazyar Mirrahimi : Conference "Quantum PhotoniX", Jinhua, China.
 TudorAlexandru Petrescu : APS March Meeting 2023 in New Orleans
 TudorAlexandru Petrescu : MPQ
 TudorAlexandru Petrescu : Université ParisCité
 TudorAlexandru Petrescu : Institut Néel, Grenoble
 Pierre Rouchon : Feedback and autonomous Quantum Error Correction (QEC). Innovations in Nonlinear Control 1317 February 2023, IIT Bombay.
 Pierre Rouchon : Feedback and quantum error corrections: Séminaire de Mathématiques Appliquées, Collège de France, 9 juin 2023.
 Pierre Rouchon : Feedback et correction d'erreurs quantiques: Stratégie Nationale Quantique Journée du 30 mars 2023, Paris; Académie des Technologies, 21 novembre 2023, Paris.
 Antoine Tilloy : IFT Madrid.
 Antoine Tilloy : Capital Fund Management.
 Antoine Tilloy : "Entangle This 5" conference in Benasque.
 Antoine Tilloy : Maspin event in Besançon.
9.1.4 Leadership within the scientific community
 Alain Sarlette is a board member of the new DIM "QUANTIP" (Quantum Technologies in Paris Region), which has been launched in 2022.
 Mazyar Mirrahimi is leader for WP1 of the PEPR NISQ2LSQ.
 Zaki Leghtas is coleader for WP1 of the PEPR RobustSuperQ.
9.1.5 Scientific expertise
 Pierre Rouchon was a member of ANR Comité d’Evaluation Scientifique on Quantum Technologies (CS 47).
 Philippe CampagneIbarcq, Zaki Leghtas and Mazyar Mirrahimi are members of the scientific board of the Startup Alice and Bob.
9.1.6 Research administration
 Alain Sarlette has been a member of the selection committees at inria Paris (invited professors, PhD and postdoc grants).
 Philippe CampagneIbarcq has been a member of the selection committee at Inria ParisSaclay for Junior Researcher (CRCN) positions.
 Mazyar Mirrahimi has been a member of the selection committee at Inria Paris for Junior Researcher (CRCN) positions.
9.2 Teaching  Supervision  Juries
9.2.1 Teaching
 Mazyar Mirrahimi: Quantum Feedback at Ecole Polytechnique (60 hours), Ecole Polytechnique bachelor program thesis instructor (20 hours).
 Alexandru Petrescu and Pierre Rouchon give the M1 course (Mathematical method for quantum engineering) of the new PSL Master Quantum Engineering.
 Alexandru Petrescu and Pierre Rouchon give the M2 course (dynamics and control of open quantum systems) of the PSL Master Quantum Engineering.
 Philippe CampagneIbarcq : 12h of préceptorat at ESPCI.
 Alex Petrescu Automatics at Mines ParisTech (12 hours).
 Pierre Rouchon is a member of the steering commitee of PSL master of Quantum Engineering with ENSParis.
 Pierre Rouchon is in charge of the "Mathematics and Automatics" specialty within the ISMME621 doctoral school.
 Antoine Tilloy: Mines Paris, TDs of Optimisation, TDs of Introduction to quantum mechanics.
 Alain Sarlette and Antoine Tilly: M1 Course (Introduction to quantum computing) of the new PSL Master Quantum Engineering, 20 hours.
 Alain Sarlette: Robotics at Ghent university (24 hours).
 Alain Sarlette : M2 Course (Quantum Information Theory) of the new PSL Master Quantum Engineering, 10 hours.
9.2.2 Supervision

PhD defended in 2023
Marius Villiers. Enhancing qubitphoton interactions with antisqueezing. Supervision Zaki Leghtas.

PhD defended in 2023
Camille Berdou. Catqubit: macroscopic bitflip times. Supervision of Zaki Leghtas.

PhD defended in 2023
Christian Siegele. Quantum error correction with grid states of light. Supervision of Philippe CampagneIbarcq and Mazyar Mirrahimi.

PhD defended in 2023
Ronan Gautier. Efficient protection and fast gates in bosonic quantum codes. Supervision of Alain Sarlette.

PhD in progress
Alvise Borgognoni. Nonlocal quantum dynamics. Supervision of Zaki Leghtas.

PhD in progress
Ulysse Réglade. 3D catqubit. Supervision of Zaki Leghtas and Raphaël Lescanne.

PhD in progress
Adrien Bocquet. Catqubit: quantum coherence and macroscopic bitflip times. Supervision of Zaki Leghtas and Raphaël Lescanne.

PhD in progress
Erwan Roverch’. Protected qubits. Supervision of Zaki Leghtas.

PhD in progress
Angela Riva. Dynamics and control of a mechanical quantum oscillator quasiresonantly coupled to a Heavy Fluxonium qubit. Supervision of Alain Sarlette.

PhD in progress
Aron Vanselow. Autonomous stabilization of GKP grid states in high impedance circuits. Supervision of Philippe CampagneIbarcq and Mazyar Mirrahimi.

PhD in progress
FrançoisMarie Le Régent. Quantum computing architecture with catqubits. Supervision of Mazyar Mirrahimi and Jérémie Guillaud

PhD in progress
LevArcady Sellem. Mathematical methods for simulation and control of open quantum systems. Supervision of Claude Le Bris and Pierre Rouchon.

PhD in progress
Thiziei Aissaoui. Onchip biasing of superconducting circuits. Supervision of Alain Sarlette and Anil Murani.

PhD in progress
Matthieu Praquin, Reconfigurable microwave photonics crystals based on granular superconductors. Supervision of Philippe CampagneIbarcq and Zaki Leghtas.

PhD in progress
Linda Greggio. Strong drive effects in quantum superconducting circuits. Supervision of Alexandru Petrescu and Mazyar Mirrahimi.

PhD in progress
Diego Ruiz. Scaling up a bosonic quantum processor. Supervision of Mazyar Mirrahimi and Jérémie Guillaud.

PhD in progress
Louis Paletta. Autonomous quantum error correction with cat qubits. Supervision of Mazyar Mirrahimi, Anthony Leverrier, Christophe Vuillot and Alain Sarlette.

PhD in progress
Pierre Guilmin. Quantum estimation and control of catqubit. Supervision of Pierre Rouchon and Antoine Tilloy.

PhD in progress
Leon Carde. Control and fast preparation of cat qubits, supervision of Joachim Cohen, Alexandru Petrescu, Pierre Rouchon.

PhD in progress
Karanbir Singh Tiwana. Tensor networks for quantum field theory. Supervision of Antoine Tilloy.

PhD in progress
Emilio Rui. Cifre with Alice and Bob, Tensor network simulations for superconducting quantum circuit design. Supervision of Alexandru Petrescu and Pierre Rouchon.

PhD in progress
Brieuc Beauseigneur. Supported by ERC QFeedback, Design and characterization of highimpedance superconducting circuits for autonomous errorcorrection. Supervision of Philippe CampagneIbarcq and Pierre Rouchon.

PhD in progress
Anissa Jacob. Parametric pumping of Josephson circuits for quantum errorcorrection. Supervision of Philippe CampagneIbarcq and Anil Murani (Alice and Bob).
9.2.3 Juries
 Zaki Leghtas has been referee for the HDR of JeanDamien Pillet.
 Mazyar Mirrahimi has been the jury president for the PhD defense of Valentin Heyraud at Univ. Paris Cité.
 Mazyar Mirrahimi was a jury member for the PhD defense of Leo Balembois at Univ. Paris Saclay.
 Alexandru Petrescu was external reader for the PhD thesis of Xu Xiao at Yale University.
 Pierre Rouchon was referee of the "mémoire d’habilitation de François Ollivier" on Effective Formal Resolution of Systems of Algebraic Differential Equations at Université ParisSaclay.
 Pierre Rouchon was member of the jury of the "mémoire d’habilitation de Amaury Hayat" on Stabilization of 1D evolution systems at Université ParisDauphine.
 Antoine Tilloy was in the PhD committee of Manuel Campos, UNIVERSIDAD AUTÓNOMA DE MADRID.
9.3 Popularization
9.3.1 Interventions
 Alain Sarlette participated in an afternoon roundtable discussion on "quantum computing" at EFREI, Villejuif.
10 Scientific production
10.1 Major publications
 1 articleQuantum FastForwarding: Markov chains and graph property testing.Quantum Information & ComputationApril 2019HAL
 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 2020HAL
 3 articleRepetition Cat Qubits for FaultTolerant Quantum Computation.Physical Review Xhttps://arxiv.org/abs/1904.09474  22 pages, 11 figuresDecember 2019HALDOI
 4 articleConfining the state of light to a quantum manifold by engineered twophoton loss.Science3476224February 2015, 853857HALDOIback to text
 5 articleExponential suppression of bitflips in a qubit encoded in an oscillator.Nature PhysicsMarch 2020HALDOI
 6 articleDynamically protected catqubits: a new paradigm for universal quantum computation.New Journal of Physics164apr 2014, 045014back to textback to text
 7 articleExtending the lifetime of a quantum bit with error correction in superconducting circuits.Nature5362016, 5back to text
 8 articleStabilization of nonclassical states of the radiation field in a cavity by reservoir engineering.Phys. Rev. Lett.1070104022011back to textback to textback to text
 9 articleRealtime quantum feedback prepares and stabilizes photon number states.Nature4772011, 7377back to text
 10 articleMagnifying quantum phase fluctuations with Cooperpair pairing.Physical Review X122April 2022, 021002HALDOI
10.2 Publications of the year
International journals
 11 articleOne hundred second bitflip time in a twophoton dissipative oscillator.PRX Quantum422023, 020350HALDOI
 12 articleReminiscence of classical chaos in driven transmons.PRX Quantum422023, 020312HALDOI
 13 articleInhomogeneous magnetic fields interacting with spinful states in a double quantum dot: Evidence for a staggered spinorbit interaction.Physical Review B1078February 2023, 085152HALDOI
 14 articleSymmetries and field tensor network states.Physical Review B10715April 2023, 155102HALDOI
 15 articleDesigning HighFidelity Zeno Gates for Dissipative Cat Qubits.PRX Quantum44October 2023, 040316HALDOIback to text
 16 articleHighperformance repetition cat code using fast noisy operations.Quantum7December 2023, 1198HALDOI
 17 articleTwophoton driven Kerr quantum oscillator with multiple spectral degeneracies.Physical Review A1074April 2023, 042407HALDOI
 18 articleRobust suppression of noise propagation in GKP errorcorrection.Phys.Rev.A10842023, 042427HALDOIback to text
International peerreviewed conferences
 19 inproceedingsHeisenberg formulation of adiabatic elimination for open quantum systems with two timescales.62nd IEEE Conference on Decision and ControlSingapore, Singapore2023HALback to text
 20 inproceedingsStability and decoherence rates of a GKP qubit protected by dissipation.22nd World Congress of the International Federation of Automatic ControlIFAC 2023  22nd World Congress of the International Federation of Automatic ControlYokohama, JapanApril 2023HALDOIback to text
 21 inproceedingsComplete Positivity Violation in Higherorder Quantum Adiabatic Elimination.IFACPapersOnLineIFAC 2023  22nd World Congress of the International Federation of Automatic Control562Yokohama, Japan2023, 13331338HALDOIback to text
Conferences without proceedings
 22 inproceedingsRobust sparse IQP sampling in constant depth.QIP2024Taipei, TaiwanJuly 2023HAL
 23 inproceedingsRobust sparse IQP sampling in constant depth.QIP2024 The international conference on Quantum Information ProcessingTaipei, TaiwanJuly 2023HALback to text
Scientific book chapters
Edition (books, proceedings, special issue of a journal)
 25 proceedingsP.Pierre GuilminP.Pierre RouchonA.Antoine TilloyCorrelation functions for realistic continuous quantum measurement.IFACPapersOnLine562November 2023, 51645170HALDOIback to text
Doctoral dissertations and habilitation theses
 26 thesisMacroscopic bitflip times in twophoton dissipative oscillator.Université Paris sciences et lettresMarch 2023HAL
 27 thesisHighFidelity Control and Stabilization of Cat Qubits.Inria ParisDecember 2023HAL
 28 thesisRobust suppression of backpropagating noise in GKP error syndrome measurements.ENS Paris  Ecole Normale Supérieure de ParisFebruary 2023HAL
 29 thesisDynamically Enhancing QubitPhoton Interactions with AntiSqueezing.Sorbonne UniversitéMay 2023HAL
Reports & preprints
 30 miscStabilization of approximate GHZ state with quasilocal couplings.June 2023HALback to text
 31 miscHighsensitivity ACcharge detection with a MHzfrequency fluxonium qubit.July 2023HALback to text
 32 miscAdiabatic elimination for composite open quantum systems: reduced model formulation and numerical simulations.January 2024HALback to text
 33 miscQuantum control of a catqubit with bitflip times exceeding ten seconds.July 2023HALback to text
 34 miscConvergence of bipartite open quantum systems stabilized by reservoir engineering.November 2023HALback to text
 35 miscA GKP qubit protected by dissipation in a highimpedance superconducting circuit driven by a microwave frequency comb.April 2023HALDOIback to text
 36 miscSpectral signature of highorder photon processes mediated by Cooperpair pairing.December 2023HAL
 37 miscDynamically enhancing qubitoscillator interactions with antisqueezing.January 2023HAL
 38 miscObservation of Josephson Harmonics in Tunnel Junctions.August 2023HALback to text
10.3 Cited publications
 39 articleOnchip singlepump interferometric Josephson isolator for quantum measurements.arXiv preprint arXiv:2006.019182020back to textback to text
 40 articleStabilization of a delayed quantum system: the Photon Box casestudy.IEEE Trans. Automatic Control5782012, 19181930back to textback to text
 41 articleStability of continuoustime quantum filters with measurement imperfections.Russian Journal of Mathematical Physics212014, 297315back to text
 42 articleFeedback stabilization of discretetime quantum systems subject to nondemolition measurements with imperfections and delays.Automatica4992013, 26832692back to textback to textback to text
 43 bookS.S. AttalA.A. JoyeC.A.C.A. PilletOpen Quantum Systems III: Recent Developments.Springer, Lecture notes in Mathematics 18802006back to text
 44 bookQuantum Trajectories and Measurements in Continuous Time: the Diffusive Case.Springer Verlag2009back to text
 45 articleAn opensystem quantum simulator with trapped ions.Nature4704862011back to text
 46 articleMechanical onchip microwave circulator.Nature communications812017, 17back to text
 47 articleQuantum stochastic calculus and quantum nonlinear filtering.Journal of Multivariate Analysis4221992, 171201back to text
 48 articleLarge Time Behavior and Convergence Rate for Quantum Filters Under Standard Non Demolition Conditions.Communications in Mathematical Physics2014, 121URL: http://dx.doi.org/10.1007/s0022001420296back to text
 49 articleExtensions of Jentzch's theorem.Trans. Amer. Math. Soc.851957, 219227back to text
 50 articleEngineering stable discretetime quantum dynamics via a canonical QR decomposition.IEEE Trans. Autom. Control552010back to text
 51 bookQuantum Measurements.Cambridge University Press1992back to text
 52 articleObserving Interferences between Past and Future Quantum States in Resonance Fluorescence.Phys. Rev. Lett.11218040218May 2014, URL: http://link.aps.org/doi/10.1103/PhysRevLett.112.180402DOIback to text
 53 articlePersistent Control of a Superconducting Qubit by Stroboscopic Measurement Feedback.Phys. Rev. X30210082013back to textback to text
 54 bookAn Open Systems Approach to Quantum Optics.SpringerVerlag1993back to text
 55 bookStatistical Methods in Quantum Optics 2: NonClassical Fields.Spinger2007back to text
 56 bookApplication of Center Manifold Theory.Springer1981back to text
 57 articleWidely tunable onchip microwave circulator for superconducting quantum circuits.Physical Review X742017, 041043back to text
 58 articleDissipationinduced continuous quantum error correction for superconducting circuits.Phys. Rev. A902014, 062344back to text
 59 articleWavefunction approach to dissipative processes in quantum optics.Phys. Rev. Lett.6851992, 580583back to text
 60 miscSuperconducting Qubits: A Short Review.arXiv:condmat/04111742004back to text
 61 articleRemote entanglement stabilization and concentration by quantum reservoir engineering.Physical Review Ahttps://arxiv.org/abs/1703.03379  5 pages, 4 figuresJuly 2018HALDOIback to text
 62 articleQuantum feedback by discrete quantum nondemolition measurements: towards ondemand generation of photonnumber states.Physical Review A80: 0138050138132009back to text
 63 articleGeometric singular perturbation theory for ordinary differential equations.J. Diff. Equations311979, 5398back to text
 64 articleCollapse and Revival of an Artificial Atom Coupled to a Structured Photonic Reservoir.arXiv preprint arXiv:2001.032402020back to text
 65 articleState and dynamical parameter estimation for open quantum systems.Phys. Rev. A644042105September 2001, URL: http://link.aps.org/doi/10.1103/PhysRevA.64.042105back to text
 66 articlePast Quantum States of a Monitored System.Phys. Rev. Lett.11116160401October 2013, URL: http://link.aps.org/doi/10.1103/PhysRevLett.111.160401back to text
 67 bookQuantum Noise.Springer2010back to text
 68 articleChecking the strict positivity of Kraus maps is NPhard.arXiv:1402.14292014back to text
 69 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/S0022039614000424back to text
 70 articleDemonstrating a Driven Reset Protocol of a Superconducting Qubit.Phys. Rev. Lett.1101205012013back to textback to textback to textback to textback to textback to text
 71 articleEncoding a qubit in an oscillator.Phys. Rev. A640123102001back to text
 72 articleProgressive fieldstate collapse and quantum nondemolition photon counting.Nature4482007, 889893back to text
 73 articleThe stability of quantum Markov filters.Infin. Dimens. Anal. Quantum Probab. Relat. Top.122009, 153172back to text
 74 bookExploring the Quantum: Atoms, Cavities and Photons.Oxford University Press2006back to textback to text
 75 articleQuantum backaction of an individual variablestrength measurement.Science3392013, 178181back to textback to text
 76 articleSinglephotonresolved crossKerr interaction for autonomous stabilization of photonnumber states.Phys. Rev. Lett.1152015, 180501back to text
 77 bookPerturbation Theory for Linear Operators.Springer1966back to text
 78 articleA scheme for efficient quantum computation with linear optics.Nature409462001back to text
 79 articleEntanglement Generated by Dissipation and Steady State Entanglement of Two Macroscopic Objects.Phys. Rev. Lett.1070805032011back to text
 80 articleDeterministic protocol for mapping a qubit to coherent state superpositions in a cavity.Phys. Rev. A870423152013back to text
 81 articleHardwareefficient autonomous quantum memory protection.Phys. Rev. Lett.1111205012013back to text
 82 articleAdiabatic passage and ensemble control of quantum systems.J. Phys. B441540172011back to text
 83 articleStabilizing a Bell state of two superconducting qubits by dissipation engineering.Phys. Rev. A880238492013back to textback to text
 84 articleEnsemble control of Bloch equations.IEEE Trans. Autom. Control542009, 528536back to text
 85 articleDissipative production of a maximally entangled steady state of two quantum bits.Nature5042013, 415418back to text
 86 articleCoherent quantum feedback.Phys. Rev. A620221082000back to textback to text
 87 articleZerofield edge plasmons in a magnetic topological insulator.Nature communications812017, 17back to text
 88 articleConsensus for quantum networks: from symmetry to gossip iterations.IEEE Trans. Automat. Controlin press2014back to text
 89 conferenceStrong measurement and quantum feedback for persistent Rabi oscillations in circuit QED experiments.IEEE Conference on Decision and ControlIEEE Conference on Decision and Control2012back to text
 90 articleCavityassisted quantum bath engineering.Phys. Rev. Lett.1091836022012back to text
 91 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=125002back to text
 92 articleCoherent quantum LQG control.Automatica452009, 18371846back to text
 93 articleAdaptive Quantum Nondemolition Measurement of a Photon Number.Phys. Rev. Lett.1128080401Feb 2014, URL: http://link.aps.org/doi/10.1103/PhysRevLett.112.080401DOIback to text
 94 articleDemonstration of efficient nonreciprocity in a microwave optomechanical circuit.Physical Review X732017, 031001back to text
 95 articleMonotone Metrics on matrix spaces.Linear Algebra and its Applications2441996, 8196back to text
 96 articleQuantum Reservoir Engineering with Laser Cooled Trapped Ions.Phys. Rev. Lett.77231996, 47284731back to textback to text
 97 bookMicrowave engineering.John wiley & sons2011back to text
 98 articleQuantum memory with millisecond coherence in circuit QED.Physical Review B9412016, 014506back to text
 99 articleHilbert's projective metric in quantum information theory.Journal of Mathematical Physics528082201August 2011, URL: http://dx.doi.org/10.1063/1.3615729back to text
 100 articleFast reset and suppressing spontaneous emission of a superconducting qubit.Applied Physics Letters96202010, 203110back to text
 101 articleInitialization by measurement of a superconducting quantum bit circuit.Phys. Rev. Lett.1090505072012back to text
 102 articleWidely tunable, nondegenerate threewave mixing microwave device operating near the quantum limit.Phys. Rev. Lett.1081477012012back to text
 103 articleFidelity is a SubMartingale for DiscreteTime Quantum Filters.IEEE Transactions on Automatic Control56112011, 27432747back to textback to text
 104 articleContinuous generation and stabilization of mesoscopic field superposition states in a quantum circuit.Phys. Rev. A912015, 013810back to text
 105 articleStabilization of nonclassical states of one and twomode radiation fields by reservoir engineering.Phys. Rev. A860121142012back to text
 106 articleResolving photon number states in a superconducting circuit.Nature4452007, 515518back to text
 107 inproceedingsConsensus in noncommutative spaces.Decision and Control (CDC), 2010 49th IEEE Conference on2010, 65966601back to text
 108 articleAutonomously stabilized entanglement between two superconducting quantum bits.Nature5042013, 419422back to textback to textback to textback to textback to text
 109 articleScheme for reducing decoherence in quantum memory.Phys. Rev. A521995, 24932496back to text
 110 articleReconfigurable Josephson circulator/directional amplifier.Physical Review X542015, 041020back to text
 111 inproceedingsDesign and Stability of DiscreteTime Quantum Filters with Measurement Imperfections.American Control Conference2012, 50845089back to textback to text
 112 articleApproximate stabilization of infinite dimensional quantum stochastic system.Reviews in Mathematical Physics2513500012013back to textback to text
 113 articleError Correcting Codes in Quantum Theory.Phys. Rev. Lett7751996back to textback to text
 114 articleTracking photon jumps with repeated quantum nondemolition parity measurements.Nature5112014, 444448back to text
 115 articleReverse isolation and backaction of the SLUG microwave amplifier.Physical Review Applied852017, 054007back to text
 116 articleProblems in decentralized decision making and computation.PhD Thesis, MIT1984back to text
 117 articleStabilizing Rabi oscillations in a superconducting qubit using quantum feedback.Nature4902012, 7780back to textback to text
 118 articleHall effect gyrators and circulators.Physical Review X422014, 021019back to text
 119 articleDynamical decoupling of open quantum system.Phys. Rev. Lett.821999, 24172421back to text
 120 articleDeterministically encoding quantum information using 100photon Schrödinger cat states.Science3422013, 607610back to text
 121 articleField locked to Fock state by quantum feedback with single photon corrections.Physical Review Letter1082436022012back to text