2025Activity reportProject-Team‌QUANTIC

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.​‌

3.1​​ Hardware-efficient 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​​ (fault-tolerant) 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  130​​, 136. 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 multi-qubit​​​‌ 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 non-local​​​‌ 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 well-controlled 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 high-Q mode‌​‌ of a 3D superconducting​​ cavity) instead of a​​​‌ multi-qubit register  97.‌ 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  93​​, 86 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  96. Through‌​‌ a recent experimental work​​  144, we benefit​​​‌ from this controllability to‌ prepare superpositions of quasi-orthogonal‌​‌ coherent states, also known​​ as Schrödinger cat states.​​​‌

In this Scheme, the‌ logical qubit is encoded‌​‌ in a four-component Schrödinger​​ cat state. Continuous quantum​​​‌ non-demolition (QND) monitoring of‌ a single physical observable,‌​‌ consisting of photon number​​ parity, enables then the​​​‌ tractability of single photon‌ jumps. We obtain therefore‌​‌ a first-order quantum error​​ correcting code using only​​​‌ a single high-Q cavity‌ mode (for the storage‌​‌ of quantum information), a​​ single qubit (providing the​​​‌ non-linearity needed for controllability)‌ and a single low-Q‌​‌ cavity mode (for reading​​ out the error syndrome).​​​‌ An earlier experiment on‌ such QND photon-number parity‌​‌ measurements  137 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 hardware-efficient paradigm​‌ for universal quantum computation.​​

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The image depicts a​​​‌ schematic of a series​ of interconnected electronic components.​‌ It features three horizontal​​ rows of symbols. The​​​‌ top row contains wavy​ lines within blue rectangles,​‌ suggesting transmission lines. The​​ middle row has components​​​‌ in green rectangles, representing​ Josephson junctions. The bottom​‌ row contains symbols representing​​ Josephson junctions in a​​​‌ light grey rectangle. Each​ component is connected with​‌ vertical lines, indicating electrical​​ connections between them. The​​​‌ entire assembly is structured​ in a modular or​‌ repeated fashion. (Description generated​​ at January 23rd, 2026​​​‌ by Albert AI with​ the model Mistral-Small-3.2-24B)

The​‌ image depicts a schematic​​ of a series of​​​‌ interconnected electronic components. It​ features three horizontal rows​‌ of symbols. The top​​ row contains wavy lines​​​‌ within blue rectangles, suggesting​ transmission lines. The middle​‌ row has components in​​ green rectangles, representing Josephson​​​‌ junctions. The bottom row​ contains symbols representing Josephson​‌ junctions in a light​​ grey rectangle. Each component​​​‌ is connected with vertical​ lines, indicating electrical connections​‌ between them. The entire​​ assembly is structured in​​​‌ a modular or repeated​ fashion. (Description generated at​‌ January 23rd, 2026 by​​ Albert AI with the​​​‌ model Mistral-Small-3.2-24B)

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 many-qubit quantum computation​​​‌ or long-distance quantum communication.​ However, such a closed-loop​‌ control which requires a​​ real-time and continuous measurement​​​‌ of the quantum system​ has been for long​‌ considered as counter-intuitive 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 non-demolition​‌ (QND) measurement has played​​ a crucial role in​​​‌ understanding and resolving this​ difficulty   65. In​‌ the context of cavity​​ quantum electro-dynamics (cavity QED)​​​‌ with Rydberg atoms  89​, a first experiment​‌ on continuous QND measurements​​ of the number of​​​‌ microwave photons was performed​ by the group at​‌ Laboratoire Kastler-Brossel (ENS)  87​​. Later on, this​​ ability of performing continuous​​​‌ measurements allowed the same‌ group to realize the‌​‌ first continuous quantum feedback​​ protocol stabilizing highly non-classical​​​‌ states of the microwave‌ field in the cavity,‌​‌ the so-called photon number​​ states 11 (this ground-breaking​​​‌ work was mentioned in‌ the Nobel prize attributed‌​‌ to Serge Haroche). The​​ QUANTIC team contributed to​​​‌ the theoretical work behind‌ this experiment  76,‌​‌ 54, 135,​​ 56. These contributions​​​‌ include the development and‌ optimization of the quantum‌​‌ filters taking into account​​ the quantum measurement back-action​​​‌ 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)  74, recent‌ advances in quantum-limited amplifiers‌​‌  123, 140 have​​ opened doors to high-fidelity​​​‌ non-demolition measurements and real-time‌ feedback for superconducting qubits‌​‌  90. This ability​​ to perform high-fidelity non-demolition​​​‌ measurements of a quantum‌ signal has very recently‌​‌ led to quantum feedback​​ experiments with quantum superconducting​​​‌ circuits  140, 122‌, 67. 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  107.‌

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 measurement-based feedback protocols,‌​‌ the time-consuming data acquisition​​ and post-treatment of the​​​‌ output signal leads to‌ an important latency in‌​‌ the feedback procedure.

The​​ reservoir (dissipation) engineering  115​​​‌ and the closely related‌ coherent feedback  103 are‌​‌ considered as alternative approaches​​ circumventing the necessity of​​​‌ a real-time 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  85, single-qubit​​​‌ state stabilization  108,​ and the creation  59​‌ and stabilization  95,​​ 102, 129 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​​​‌  85, 99,​ 72, 75.​‌ The experimental results based​​ on these protocols have​​​‌ illustrated the efficiency of​ the approach  85,​‌ 129. Through these​​ experiments, we exploit the​​​‌ strong dispersive interaction  127​ between superconducting qubits and​‌ a single low-Q cavity​​ mode playing the role​​​‌ of a dissipative reservoir.​ Applying continuous-wave (cw) microwave​‌ drives with well-chosen 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 continuous-wave 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 low-Q 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 non-classical states​​​‌ (such as Schrodinger cat​ states and Fock states)​‌ of microwave field confined​​ in a high-Q cavity​​​‌ mode  125, 91​6, 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​  124, 54,​‌ 134, 126,​​ 135, 5610​​​‌ resulting from collaborations with​ the cavity quantum electrodynamics​‌ group of Laboratoire Kastler​​ Brossel.

3.4 Stabilization by​​​‌ measurement-based 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​​ non-demolition measurement, a systematic​​ stabilization method  56:​​​‌ it is based on‌ a discrete-time formulation of‌​‌ the dynamics, on the​​ construction of a strict​​​‌ control Lyapunov function and‌ on an explicit compensation‌​‌ of the feedback-loop 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  145‌; multi-input case where‌​‌ the construction by inversion​​ of a Metzler matrix​​​‌ of the strict Lyapunov‌ function is not straightforward;‌​‌ continuous-time 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  112​​. Without the QND​​​‌ measurement assumptions, we will‌ also address the stabilization‌​‌ of non-stationary states and​​ trajectory tracking, with applications​​​‌ to systems similar to‌ those considered in  90‌​‌, 67.

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 $\mathrm{t‌}\ge 0$ from an‌​‌ initial value at $\mathrm{t}=0$ and the​​​‌ measurement outcomes between 0‌ and $t$. Quantum‌​‌ filtering goes back to​​ the work of Belavkin​​​‌   61 and is related‌ to quantum trajectories  68‌​‌, 73. A​​ modern and mathematical exposure​​​‌ of the diffusive models‌ is given in  58‌​‌. In  88 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​​​‌  124, 134.‌ This stability result is‌​‌ extended to a large​​ class of continuous-time filters​​​‌ in  55. Further‌ efforts are required to‌​‌ characterize asymptotic and exponential​​ stability. Estimations of convergence​​​‌ rates are available only‌ for quantum non-demolition measurements‌​‌  62. Parameter estimations​​ based on measurement data​​​‌ of quantum trajectories can‌ be formulated within such‌​‌ quantum filtering framework  80​​, 110.

We​​​‌ will continue to investigate‌ stability and convergence of‌​‌ quantum filtering. We will​​ also exploit our fidelity-based​​​‌ 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  66 that‌​‌ post-selection statistics and “past​​ quantum” state analysis  81​​​‌ 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   115, 103​. The closed-loop 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​​​‌ 10129, 85​, this subsection is​‌ dedicated to further developing​​ such stabilization techniques, both​​​‌ experimentally and theoretically.

The​ main analysis issues are​‌ to prove the closed-loop​​ convergence and to estimate​​​‌ the convergence rates. Since​ these systems are governed​‌ by Lindblad differential equations​​ (continuous-time case) or Kraus​​​‌ maps (discrete-time case), their​ stability is automatically guaranteed:​‌ such dynamics are contractions​​ for a large set​​​‌ of metrics (see  114​). Convergence and asymptotic​‌ stability is less well​​ understood. In particular most​​​‌ of the convergence results​ consider the case where​‌ the target steady-state is​​ a density operator of​​​‌ maximum rank (see, e.g.,​ 57[chapter 4, section​‌ 6]). When the goal​​ steady-state is not full​​​‌ rank very few convergence​ results are available.

We​‌ will focus on this​​ geometric situation where the​​​‌ goal steady-state 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 10​​, 6129,​​​‌ 85 and their associated​ Fokker-Planck equations for the​‌ Wigner functions.

We will​​ also explore the Heisenberg​​​‌ point of view in​ connection with recent results​‌ of the INRIA project-team​​ MAXPLUS (algorithms and applications​​​‌ of algebras of max-plus​ type) relative to Perron-Frobenius​‌ theory  84, 83​​. We will start​​​‌ with  128 and  120​ where, based on a​‌ theorem due to Birkhoff​​  63, 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 139​​ and by our recent​​​‌ generalized extension of the​ latter synchronizing interactions to​‌ quantum systems 106.​​

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 so-called​​ linear quantum systems 111​​​‌, this does not​ cover the states that​‌ are more interesting for​​ quantum applications. Various strategies​​ have been proposed that​​​‌ concatenate iterative control steps‌ for open-loop steering 143‌​‌, 101 with experimental​​ limitations. The characterization of​​​‌ Kraus maps to stabilize‌ any types of states‌​‌ has also been established​​ 64, 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.​​

3.7 On-chip​​​‌ 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​ impedance-matched, leading to spurious​‌ reflections at their ports.​​ In order to implement​​​‌ a full-scale quantum computer​ based on superconducting circuits,​‌ it is crucial that​​ these functionalities be enabled​​​‌ reliably on-chip.

On-chip 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​​ "Purcell-filtering" 121 consists in​​​‌ placing $\lambda /\mathrm{4}$ stubs 116 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 119, and​​ a recent implementation of​​​‌ a bandpass filter 79​.

On-chip non-reciprocal 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 down-conversion with respect​‌ to the reverse process​​ of up-conversion. Combining coherently​​ several conversion paths with​​​‌ well-chosen phases, one obtains‌ a constructive forward interference,‌​‌ and a destructive backward​​ one. In circuitQED, frequency​​​‌ conversion is enabled by‌ a non-linear Josephson circuit‌​‌ 131, 71,​​ 53, or by​​​‌ electromechanical coupling to nanoresonators‌ 60, 113.‌​‌ 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 53. For‌ completeness, we mention a‌​‌ recent implementation of a​​ forward amplifier based on​​​‌ resistively shunted Josephson junctions‌ 138 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​​​‌ 142, 104.‌

In this project, we‌​‌ propose to develop novel​​ on-chip filters and isolators​​​‌ based on 1D Josephson‌ metamaterials, which could reach‌​‌ unprecedented bandwidth, tunable range​​ and on/off or forward/backward​​​‌ transmission ratios. Such metamaterials‌ have been routinely used‌​‌ over the past decade​​ to amplify weak quantum​​​‌ signals 77. It‌ was attempted to adapt‌​‌ these devices to route​​ microwave non-reciprocally (with or​​​‌ without amplification). However, up‌ to now, these devices‌​‌ suffer from limited bandwidth,​​ isolation and pump leakage​​​‌ preventing their integration in‌ quantum circuits 118,‌​‌ 105, 94.​​ In our approach, we​​​‌ propose to route microwaves‌ non-reciprocally by mixing them‌​‌ in a Josephson metamaterial​​ with a pump wave​​​‌ propagating at a much‌ smaller phase velocity. This‌​‌ allows us to activate​​ photon conversion processes that​​​‌ were not previously observed,‌ by which a propagating‌​‌ signal is converted into​​ a counterpropagating one. In​​​‌ this configuration, the signal‌ is exponentially attenuated as‌​‌ it travels down the​​ metamaterial, providing an effective​​​‌ circulator with robust isolation.‌ The device can be‌​‌ reconfigured into a reciprocal​​ and adjustable filter. While​​​‌ the first device’s generation‌ we tested suffers from‌​‌ limited working bandwidth of​​ about 200 MHz and​​​‌ pump leakage 117,‌ we are currently developing‌​‌ a second generation to​​ remedy these shortcomings.

3.8​​​‌ Exotic circuits for qubit‌ protection

Developing error-resilient qubits‌​‌ remains one of the​​ central challenges in superconducting​​​‌ quantum circuits. In addition‌ to error correction protocols‌​‌ and efforts to mitigate​​ environment-induced decoherence, another avenue​​​‌ is to design circuits‌ whose lowest-energy eigenstates are‌​‌ intrinsically protected, usually thanks​​ to symmetry properties. Prominent​​​‌ examples are the so-called‌ $``0-‌‌\pi "$ qubit and​​ the “bi-fluxon" qubit. In​​​‌ this project, we implement‌ another protected qubit candidate:‌​‌ the “$cos(2\phi )$"​​​‌ Transmon 132. It‌ consists of a superconducting‌​‌ island connected to ground​​ through a tunnelling element​​​‌ through which Cooper pairs‌ are only allowed to‌​‌ travel by pairs. This​​ Cooper-pair pairing enforces the​​​‌ conservation of Cooper-pair parity,‌ making the qubit protected‌​‌ against certain error channels.​​ We explore an implementation​​​‌ of this qubit using‌ traditional circuit elements where‌​‌ the parity-preservation property is​​​‌ achieved through Aharonov-Bohm interference​ in a SQUID-like loop,​‌ known as the Kite.​​ On the path towards​​​‌ the construction of our​ protected qubit, we have​‌ realized two experiments showcasing​​ the physics of Cooper-pairing​​​‌ through a Kite element.​

The first one, led​‌ by post-doc Clarke Smith,​​ demonstrated that pairing Cooper​​​‌ pairs magnifies quantum phase​ fluctuations 133. We​‌ accomplish this by tuning​​ our Kite between two​​​‌ operating points where the​ potential is $2\mathrm{\pi ‌}$- or $\pi$-periodic.​​ This doubles the frequency​​​‌ of the corrugation and​ hence also the number​‌ of sites accessible to​​ the ground state. We​​​‌ refer to this change​ of scale, resulting from​‌ a denser packing of​​ the Josephson wells, as​​​‌ a magnification. The second​ one, led by PhD​‌ student Alvise Borgognoni and​​ post-doc Clarke Smith, demonstrated​​​‌ that Cooper-pair pairing enhances​ photon-photon interactions 27.​‌ Indeed, the doubling of​​ the frequency of the​​​‌ Josephson corrugation results in​ a $2^{2\mathrm{n‌}}$ enhancement for $n-$ photon interaction energies. From​​​‌ spectroscopy, we extracted two-,​ three-, and four-photon (eight​‌ wave-mixing) interaction energies, all​​ of similar strength and​​​‌ exceeding the photon loss​ rate. Our results open​‌ a new regime of​​ high-order photon interactions in​​​‌ microwave quantum optics. Finally,​ the implementation of the​‌ “$cos\left(\mathrm{2}\phi \right)$" Transmon​​​‌ is in progress. We​ are in the process​‌ of measuring a device​​ where we observe a​​​‌ doubling of the eigenspectrum​ into two parity sectors,​‌ a hallmark of quartet​​ tunneling. We are analyzing​​​‌ the decay mechanisms of​ the ground state doublet​‌ as a function of​​ flux and charge and​​​‌ reflecting on the prospects​ of the $cos(‌2\phi )$ Transmon​​ as a protected qubit​​​‌ or charge detector.

3.9​ Quantum sensing

Heavy and​‌ low frequency vibrating membranes​​ are promising probes of​​​‌ gravitational effects on quantum​ mechanics, such as gravitational-induced​‌ decoherence. A leading expert​​ on such membranes is​​​‌ Samuel Deléglise from LKB​ CNRS, and in recent​‌ years our team has​​ collaborated with Deléglise to​​​‌ couple a membrane to​ a superconducting circuit. This​‌ coupling would allow not​​ only the quantum sensing​​​‌ of the membrane, but​ also its full quantum​‌ control. The goal further​​ ahead will then be​​​‌ to prepare large cat-states​ of phonons and seek​‌ the presence or absence​​ of gravitational-induced decoherence. During​​​‌ the evaluation period, we​ demonstrated repeated, and high-fidelity​‌ interactions between a 4​​ MHz suspended silicon nitride​​​‌ membrane and a resonant​ superconducting heavy-fluxonium qubit. The​‌ qubit is initialized at​​ an effective temperature of​​​‌ 27 $\mu$K and​ read out in a​‌ single-shot with 77% fidelity.​​ During the membrane's 6​​​‌ ms lifetime, the two​ systems swap excitations more​‌ than 300 times 109​​, 35.

Another​​​‌ attempt at quantum sensing​ was led by Zaki​‌ Leghtas and PhD student​​ Marius Villiers in collaboration​​​‌ with Takis Kontos. The​ long-term goal was to​‌ couple a spin qubit​​ suspended in a carbon-nanotube​​​‌ to a superconducting circuit.​ Anticipating that such a​‌ coupling would be weak,​​ we implemented an idea​​ of Leroux and Clerk​​​‌ 100, stipulating that‌ such couplings could be‌​‌ amplified by antisqueezing the​​ superconducting resonator. In our​​​‌ proof-of-principle implementation 141,‌ we emulated the spin-qubit‌​‌ with a Transmon qubit,​​ and demonstrated a two-fold​​​‌ amplification of the dispersive‌ coupling to a resonator‌​‌ at 5.5 dB of​​ squeezing. Moreover, in addition​​​‌ to interaction amplification, we‌ uncovered the detrimental effects‌​‌ of squeezing-induced qubit decoherence.​​ This effect dissuaded us​​​‌ from deploying this technique‌ to the more complex‌​‌ samples containing actual carbon​​ nanotubes.

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 nano-electromechanical 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.

6.1 Latest​​​‌ software developments

7.1 Dissipative protection​‌ of a GKP qubit​​ in a high-impedance superconducting​​ circuit driven by a​​​‌ microwave frequency comb

Participants:‌ Lev-Arcady Sellem, Alain‌​‌ Sarlette, Zaki Leghtas​​, Mazyar Mirrahimi,​​​‌ Pierre Rouchon, Philippe‌ Campagne-Ibarcq.

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

This work‌​‌ was published in PRX​​ 26.

7.2 Tensor-network​​​‌ representation of excitations in‌ Josephson junction arrays

Participants:‌​‌ Emilio Rui, Joachim​​ Cohen, Alexandru Petrescu​​​‌.

We present a‌ nonperturbative tensor-network approach to‌​‌ the excitation spectra of​​ superconducting circuits based on​​​‌ Josephson junction arrays 47‌. These arrays provide‌​‌ the large lumped inductances​​ required for qubit designs,​​​‌ yet their intrinsically many-body‌ nature is typically reduced‌​‌ to effective single-mode descriptions.​​ Perturbative treatments attempt to​​​‌ include the collective array‌ modes neglected in these‌​‌ approximations, but a fully​​ nonperturbative analysis is challenging​​​‌ due to the many-body‌ structure and the collective‌​‌ character of these modes.​​ We overcome this difficulty​​​‌ using the DMRG-X algorithm,‌ which extends tensor-network methods‌​‌ to excited states. Our​​ key advance is a​​​‌ construction of trial states‌ from the linearized mode‌​‌ structure, enabling direct computation​​ of excitations, even in​​​‌ degenerate manifolds, which was‌ previously inaccessible. Our results‌​‌ reveal significant deviations from,​​ and allow us to​​​‌ improve upon, previous perturbative‌ treatments in the regime‌​‌ of low array junction​​ impedance.

7.3 Suppression of​​​‌ measurement-induced state transitions in‌ $cos\phi$-coupling transmon‌​‌ readout

Participants: Alexandru Petrescu​​.

Drive-induced unwanted state​​​‌ transitions (DUST) are limiting‌ both for microwave readout‌​‌ and parametric operations of​​ superconducting qubits. Among them,​​​‌ measurement-induced state transitions (MIST)‌ are due to intrinsic‌​‌ resonances described by the​​ readout Hamiltonian. They were​​​‌ previously studied with a‌ qubit linearly coupled to‌​‌ its readout mode, which​​ constitutes the usual readout​​​‌ Hamiltonian. Since MIST can‌ appear even at moderate‌​‌ powers, they limit the​​ readout SNR and the​​​‌ QND readout fidelity. In‌ this work 40,‌​‌ we study the high-power​​​‌ readout regime in a​ different transmon readout scheme,​‌ implementing a nonlinear coupling​​ called the $cos\mathrm{\phi ‌}$-coupling. This coupling stems​ from a transmon molecule​‌ circuit and has symmetry​​ properties that suppress nonparity-conserving​​​‌ MIST. We succeed in​ performing multi-state single-shot readout​‌ up to the fifth​​ excited state of the​​​‌ transmon, which enables us​ to identify leakage pathways​‌ from the computational subspace.​​ The measurements indicate that​​​‌ the system is free​ of MIST up to​‌ high powers, with more​​ than 300 photons in​​​‌ the readout mode. The​ MIST can be controllably​‌ turned on by breaking​​ the parity symmetry of​​​‌ the coupling using flux-tuning.​ These experimental results are​‌ corroborated by branch analysis​​ and simulations of the​​​‌ classical chaotic dynamics, showing​ that the $cos\mathrm{\phi ‌}$-coupling is very robust​​ to readout photons compared​​​‌ to the usual transverse​ coupling.

7.4 Non-perturbative switching​‌ rates in bistable open​​ quantum systems: from driven​​​‌ Kerr oscillators to dissipative​ cat qubits

Participants: Léon​‌ Carde, Joachim Cohen​​, Alexandru Petrescu.​​​‌

Accepted in Physical Review​ Letters.

In this work​‌ 19, we use​​ path integral techniques to​​​‌ predict the switching rate​ in a single-mode bistable​‌ open quantum system. While​​ analytical expressions are well-known​​​‌ to be accessible for​ systems subject to Gaussian​‌ noise obeying classical detailed​​ balance, we generalize this​​​‌ approach to a class​ of quantum systems, those​‌ which satisfy the recently-introduced​​ hidden time-reversal symmetry. In​​​‌ particular, in the context​ of quantum computing, we​‌ deliver precise estimates of​​ bit-flip error rates in​​​‌ cat-qubit architectures, circumventing the​ need for costly numerical​‌ simulations. Our results open​​ new avenues for exploring​​​‌ switching phenomena in multistable​ single- and many-body open​‌ quantum systems.

7.5 Optimal​​ absorption and emission of​​​‌ itinerant fields into a​ spin ensemble memory

Participants:​‌ Linda Greggio, Mazyar​​ Mirrahimi, Alexandru Petrescu​​​‌.

Quantum memories integrated​ in a modular quantum​‌ processing architecture can rationalize​​ the resources required for​​​‌ quantum computation. This work​ 36 focuses on spin-based​‌ quantum memories, where itinerant​​ electromagnetic fields are stored​​​‌ in large ensembles of​ effective two-level systems, such​‌ as atomic or solid-state​​ spin ensembles, embedded in​​​‌ a cavity. Using a​ mean-field framework, we model​‌ the ensemble as an​​ effective spin communication channel​​​‌ and develop a cascaded​ quantum model to describe​‌ both absorption and emission​​ processes. We derive optimal​​​‌ time-dependent modulations of the​ cavity linewidth that maximize​‌ storage and retrieval efficiency​​ for finite-duration wavepackets. Our​​​‌ analysis yields an upper​ bound on efficiency, which​‌ can be met in​​ the narrow bandwidth regime.​​​‌ It also shows the​ existence of a critical​‌ bandwidth above which the​​ efficiency severely decreases. Numerical​​​‌ simulations are presented in​ the context of microwave-frequency​‌ quantum memories interfaced with​​ superconducting quantum processors, highlighting​​​‌ the protocol's relevance for​ modular quantum architectures.

7.6​‌ Strongly driven transmon as​​ an incoherent noise source​​​‌

Participants: Linda Greggio,​ Rémi Robin, Mazyar​‌ Mirrahimi, Alexandru Petrescu​​.

Under strong drives,​​​‌ which are becoming necessary​ for fast high-fidelity operations,​‌ transmons can be structurally​​ unstable. Due to chaotic​​ effects, the computational manifold​​​‌ is no longer well‌ separated from the remainder‌​‌ of the spectrum, which​​ correlates with enhanced offset-charge​​​‌ sensitivity and destructive effects‌ in readout. We show‌​‌ here 37 that these​​ detrimental effects can further​​​‌ propagate to other degrees‌ of freedom, for example‌​‌ to neighboring qubits in​​ a multi-qubit system. Specifically,​​​‌ a coherently driven transmon‌ can act as a‌​‌ source of incoherent noise​​ to another circuit element​​​‌ coupled to it. By‌ using a full quantum‌​‌ model and a semiclassical​​ analysis, we perform the​​​‌ noise spectroscopy of the‌ driven transmon coupled to‌​‌ a spectator two-level system​​ (TLS), and we show​​​‌ that, in a certain‌ limit, the interaction with‌​‌ the driven transmon can​​ be modeled as a​​​‌ stochastic diffusive process driving‌ the TLS.

7.7 Convergence‌​‌ Analysis of Galerkin Approximations​​ for the Lindblad Master​​​‌ Equation

Participants: Rémi Robin‌, Pierre Rouchon.‌​‌

This paper 43 analyzes​​ the numerical approximation of​​​‌ the Lindblad master equation‌ on infinite-dimensional Hilbert spaces.‌​‌ We employ a classical​​ Galerkin approach for spatial​​​‌ discretization and investigate the‌ convergence of the discretized‌​‌ solution to the exact​​ solution. Using a priori​​​‌ estimates, we derive explicit‌ convergence rates and demonstrate‌​‌ the effectiveness of our​​ method through examples motivated​​​‌ by autonomous quantum error‌ correction.

7.8 Unconditionally stable‌​‌ time discretization of Lindblad​​ master equations in infinite​​​‌ dimension using quantum channels‌

Participants: Rémi Robin,‌​‌ Pierre Rouchon, Lev-Arcady​​ Sellem.

We examine​​​‌ the time discretization of‌ Lindblad master equations in‌​‌ infinite-dimensional Hilbert spaces. Our​​ study is motivated by​​​‌ the fact that, with‌ unbounded Lindbladian, projecting the‌​‌ evolution onto a finite-dimensional​​ subspace using a Galerkin​​​‌ approximation inherently introduces stiffness,‌ leading to a Courant–Friedrichs–Lewy‌​‌ type condition for explicit​​ integration schemes.

We propose​​​‌ and establish the convergence‌ of a family of‌​‌ explicit numerical schemes for​​ time discretization adapted to​​​‌ infinite dimension 44.‌ These schemes correspond to‌​‌ quantum channels and thus​​ preserve the physical properties​​​‌ of quantum evolutions on‌ the set of density‌​‌ operators: linearity, complete positivity​​ and trace. Numerical experiments​​​‌ inspired by bosonic quantum‌ codes illustrate the practical‌​‌ interest of this approach​​ when approximating the solution​​​‌ of infinite dimensional problems‌ by that of finite‌​‌ dimensional problems of increasing​​ dimension.

7.9 A posteriori​​​‌ error estimates for the‌ Lindblad master equation

Participants:‌​‌ Paul-Louis Etienney, Rémi​​ Robin, Pierre Rouchon​​​‌.

We are interested‌ in the simulation of‌​‌ open quantum systems governed​​ by the Lindblad master​​​‌ equation in an infinite-dimensional‌ Hilbert space. To simulate‌​‌ the solution of this​​ equation, the standard approach​​​‌ involves two sequential approximations:‌ first, we truncate the‌​‌ Hilbert space to derive​​ a differential equation in​​​‌ a finite-dimensional subspace. Then,‌ we use discrete time-step‌​‌ to obtain a numerical​​ solution to the finite-dimensional​​​‌ evolution.

In this paper‌ 34, we establish‌​‌ bounds for these two​​ approximations that can be​​​‌ explicitly computed to guarantee‌ the accuracy of the‌​‌ numerical results. Through numerical​​ examples, we demonstrate the​​​‌ efficiency of our method,‌ empirically highlighting the tightness‌​‌ of the upper bound.​​​‌ While adaptive time-stepping is​ already a common practice​‌ in the time discretization​​ of the Lindblad equation,​​​‌ we extend this approach​ by showing how to​‌ dynamically adjust the truncation​​ of the Hilbert space.​​​‌ This enables fully adaptive​ simulations of the density​‌ matrix. For large-scale simulations,​​ this approach can significantly​​​‌ reduce computational time and​ relieves users of the​‌ challenge of selecting an​​ appropriate truncation.

7.10 Diffusive​​​‌ Stochastic Master Equation (SME)​ with dispersive qubit/cavity coupling​‌

Participants: Pierre Rouchon.​​

A detailed analysis of​​​‌ the diffusive Stochastic Master​ Equation (SME) for qubit/cavity​‌ systems with dispersive coupling​​ is provided 45.​​​‌ This analysis incorporates clas-​ sical input signals and​‌ output signals (measurement outcomes​​ through homodyne detec- tion).​​​‌ The dynamics of the​ qubit/cavity density operator is​‌ shown to converge exponen-​​ tially towards a slow​​​‌ invariant manifold, parameterized via​ a time-varying deterministic Kraus​‌ map by the density​​ operator of a fictitious​​​‌ qubit. This fictitious qubit​ is governed by a​‌ SME incorporating the classical​​ input/output signals. Extension is​​​‌ provided where the qubit​ is replaced by any​‌ qudit dispersively coupled to​​ an arbitrary set of​​​‌ modes with collective input/output​ classical signals.

7.11 Quantum​‌ Zeno dragging with application​​ to solving the k-SAT​​​‌ problem

Participants: Alain Sarlette​, Artem Mamichev.​‌

We have initialized this​​ line of work with​​​‌ the group of B.Whaley​ at Berkeley and obtained​‌ first results in 51​​. Quantum Zeno dragging​​​‌ is an alternative to​ quantum adiabatic computing, where​‌ a solution to a​​ combinatorial problem is progressively​​​‌ distilled among $2^{\mathrm{n}}$ possibilities represented by qubits.​‌ The scheme works by​​ initially representing 0 and​​​‌ 1 logical states by​ the same physical state​‌ vectors, and progressively separating​​ those vectors until they​​​‌ become orthogonal and the​ result can be read​‌ out. The system is​​ measured all along this​​​‌ process, ensuring that it​ remains on a solution​‌ state if we move​​ the vectors slowly enough.​​​‌ In 51, we​ study this scheme under​‌ continuous weak measurement and​​ show that the critical​​​‌ value is a spectral​ gap of a related​‌ Hamiltonian, drawing a clear​​ link with (non-standard) adiabatic​​​‌ computing. We also report​ a preliminary attempt at​‌ optimal scheduling of the​​ time-dependent vectors. Our longer-term​​​‌ goals are to improve​ this k-SAT study in​‌ two ways: (i) find​​ highly informed schedules —​​​‌ possibly qubit-dependent, constraint-dependent,... ,​ taking all insights on​‌ the k-SAT problem into​​ account ; (ii) integrate​​​‌ feedback mechanisms, thanks to​ the weak measurement outputs,​‌ which would allow us​​ to go beyond the​​​‌ performance of adiabatic computing.​

7.12 Confinement to deterministic​‌ manifolds and low-dimensional solution​​ formulas for continuously measured​​​‌ quantum systems

Participants: Alain​ Sarlette, Pierre Rouchon​‌.

Several years ago,​​ we had observed that​​​‌ the state of a​ qubit, conditioned on weak​‌ continuous measurement results, appears​​ to jump randomly not​​​‌ inside the Bloch sphere​ but rather inside a​‌ time-dependent ellipsoid surface. Over​​ the years, we have​​​‌ accumulated knowledge about other​ types of systems that​‌ appear in typical quantum​​ systems and behave in​​ this way. This includes​​​‌ infinite-dimensional systems (q.harmonic oscillators)‌ and bi-partite quantum systems‌​‌ where only one subsystem​​ is observed. We summarize​​​‌ all these findings in‌ the paper 25 published‌​‌ in Phys.Rev.A. The first​​ main point is an​​​‌ algebraic criterion, translated from‌ nonlinear control theory, allowing‌​‌ to quickly check if​​ a continuously monitored quantum​​​‌ system is confined to‌ a low-dimensional, measurement outcome-independent‌​‌ manifold. As a second​​ contribution, for a set​​​‌ of cases where such‌ confinement does hold true,‌​‌ we take advantage of​​ the low dimension in​​​‌ order to provide explicit‌ and exact formulas for‌​‌ the state at time​​ t as a function​​​‌ of the measurement outcomes.‌

7.13 Time-averaged continuous quantum‌​‌ measurement

Participants: Pierre Guilmin​​, Pierre Rouchon,​​​‌ Antoine Tilloy.

The‌ theory of continuous quantum‌​‌ measurement allows to reconstruct​​ the stateof a system​​​‌ from a continuous stochastic‌ measurement record. However, this‌​‌ truly continuous-time signal is​​ never available in practice.​​​‌ In experiments, one generally‌ has access to its‌​‌ digitization, i.e., to a​​ series of time averages​​​‌ over finite intervals of‌ duration $\Delta t$.‌​‌ This contribution takes this​​ digitization seriously and defines​​​‌ the best Bayesian estimate‌ of the quantum state‌​‌ given (only) a digitized​​ record. This allows reconstructing​​​‌ quantum trajectories in regimes‌ of coarse $\Delta \mathrm{t‌‌}$ where existing methods fail,​​ estimating parameters at fixed​​​‌ $\Delta t$ without bias,‌ and directly sampling digitized‌​‌ quantum trajectories with schemes​​ of arbitrarily high order​​​‌ 38.

7.14 A‌ relativistic continuous matrix product‌​‌ state study of field​​ theories with defects

Participants:​​​‌ Karanbir Tiwana, Edoardo‌ Lauria, Antoine Tilloy‌​‌.

In this work​​ 28, we present​​​‌ a method to compute‌ expectation values in $\mathrm{1‌‌}+1$-dimensional massive​​ Quantum Field Theories (QFTs)​​​‌ with line defects using‌ Relativistic Continuous Matrix Product‌​‌ State (RCMPS). Exploiting Euclidean​​ invariance, we use a​​​‌ quantization scheme where (imaginary)‌ time runs perpendicularly to‌​‌ the defect. With this​​ choice, correlation functions of​​​‌ local operators in the‌ presence of the defect‌​‌ can be computed as​​ expectation values of extended​​​‌ operators in the no-defect‌ vacuum, which can be‌​‌ approximated by a homogeneous​​ RCMPS. We demonstrate the​​​‌ effectiveness of this machinery‌ by computing correlation functions‌​‌ of local bulk and​​ defect operators in the​​​‌ self interacting scalar theory‌ with a magnetic line‌​‌ defect, in perturbative, strong​​ coupling, critical, and symmetry-broken​​​‌ regimes.

7.15 Multi-Field Relativistic‌ Continuous Matrix Product States‌​‌

Participants: Karanbir Tiwana,​​ Antoine Tilloy.

Relativistic​​​‌ continuous matrix product states‌ (RCMPS) are a powerful‌​‌ variational ansatz for quantum​​ field theories of a​​​‌ single field. However, they‌ inherit a property of‌​‌ their non-relativistic counterpart that​​ makes them divergent for​​​‌ models with multiple fields,‌ unless a regularity condition‌​‌ is satisfied. This has​​ so far restricted the​​​‌ use of RCMPS to‌ toy models with a‌​‌ single self-interacting field. We​​ address this long standing​​​‌ problem by introducing a‌ Riemannian optimization framework, that‌​‌ allows to minimize the​​ energy density over the​​​‌ regular submanifold of multi-field‌ RCMPS, and thus to‌​‌ retain purely variational results.​​​‌ We demonstrate its power​ on a model of​‌ two interacting scalar fields​​ in dimensions 49.​​​‌ The method captures distinct​ symmetry-breaking phases, and the​‌ signature of a Berezinskii-Kosterlitz-Thouless​​ (BKT) transition along an​​​‌ -symmetric parameter line. This​ makes RCMPS usable for​‌ a far larger class​​ of problems than before.​​​‌

7.16 Extracting quantum field​ theory dynamics from an​‌ approximate ground state

Participants:​​ Sophie Mutzel, Antoine​​​‌ Tilloy.

This contribution​ 41 puts forward a​‌ linear-programming method to extract​​ dynamical information from static​​​‌ ground-state correlators in quantum​ field theory. We recast​‌ the Källén-Lehmann inversion as​​ a convex optimization problem,​​​‌ in a spirit similar​ to the recent approach​‌ of Lawrence [arXiv:2408.11766]. This​​ produces robust estimates of​​​‌ the smeared spectral density,​ the real-time propagator, and​‌ the mass gap directly​​ from an approximate equal-time​​​‌ two-point function, and simultaneously​ yields an a posteriori​‌ lower bound on the​​ correlation-function error. We test​​​‌ the method on the​ -dimensional model, using a​‌ variational approximation to the​​ vacuum – relativistic continuous​​​‌ matrix product states –​ that provides accurate correlators​‌ in the continuum and​​ thermodynamic limits. The resulting​​​‌ mass gaps agree with​ renormalized Hamiltonian truncation and​‌ Borel-resummed perturbation theory across​​ a wide range of​​​‌ couplings, demonstrating that accurate​ dynamical data can be​‌ recovered from a single​​ equal-time slice.

7.17 High-performance​​​‌ local decoders for defect​ matching in 1D

Participants:​‌ Louis Paletta, Mazyar​​ Mirrahimi.

Local decoders,​​​‌ also known as cellular-automaton​ decoders, offer a promising​‌ path toward real-time quantum​​ error correction by replacing​​​‌ centralized classical decoding, with​ inherent hardware constraints, by​‌ a natively parallel and​​ streamlined architecture from a​​​‌ simple local transition rule.​ In a collaboration with​‌ Anthony Leverrier from Inria​​ Cosmiq team and Christophe​​​‌ Vuillot from Inria Mocqua​ team, we propose two​‌ new types of local​​ decoders for the quantum​​​‌ repetition code in one​ dimension 42. The​‌ signal-rule decoders interpret odd​​ parities between neighboring qubits​​​‌ as defects, attracted to​ each other via the​‌ exchange of classical point-like​​ excitations, represented by a​​​‌ few bits of local​ memory. We prove the​‌ existence of a threshold​​ in the code-capacity model​​​‌ and present numerical evidence​ of exponential logical error​‌ suppression under a phenomenological​​ noise model, with data​​​‌ and measurement errors at​ each error correction cycle.​‌ Compared to previously known​​ local decoders that suffer​​​‌ from sub-optimal threshold and​ scaling, our construction significantly​‌ narrows the gap with​​ global decoders for practical​​​‌ system sizes and error​ rates. Implementation requirements can​‌ be further reduced by​​ eliminating the need for​​​‌ local classical memories, with​ a new rule defined​‌ on two rows of​​ qubits. This shearing-rule works​​​‌ well at relevant system​ sizes making it an​‌ appealing short-term solution. When​​ combined with biased-noise qubits,​​​‌ such as cat qubits,​ these decoders enable a​‌ fully local quantum memory​​ in one dimension.

7.18​​​‌ Unfolded distillation: very low-cost​ magic state preparation for​‌ biased-noise qubits

Participants: Diego​​ Ruiz, Jeremie Guillaud​​​‌, Mazyar Mirrahimi.​

Magic state distillation enables​‌ universal fault-tolerant quantum computation​​ by implementing non-Clifford gates​​ via the preparation of​​​‌ high-fidelity magic states. However,‌ it comes at the‌​‌ cost of substantial logical-level​​ overhead in both space​​​‌ and time. In this‌ work 48, in‌​‌ collaboration with Christophe Vuillot​​ from Alice&Bob, we propose​​​‌ a very low-cost magic‌ state distillation scheme for‌​‌ biased-noise qubits. By leveraging​​ the noise bias, our​​​‌ scheme enables the preparation‌ of a magic state‌​‌ with a logical error​​ rate of $3\times ‌{10}^{-7}$,‌ using only 53 qubits‌​‌ and 5.5 error correction​​ rounds, under a noise​​​‌ bias of $\eta \gtrsim ‌5\times {10}^{\mathrm{6‌‌}}$ and a phase-flip noise​​ rate of $0.‌1\%$. This‌ reduces the circuit volume‌​‌ by more than one​​ order of magnitude relative​​​‌ to magic state cultivation‌ for unbiased-noise qubits and‌​‌ by more than two​​ orders of magnitude relative​​​‌ to standard magic state‌ distillation. Moreover, our scheme‌​‌ provides three key advantages​​ over previous proposals for​​​‌ biased-noise qubits. First, it‌ only requires nearest-neighbor two-qubit‌​‌ gates on a 2D​​ lattice. Second, the logical​​​‌ fidelity remains nearly identical‌ even at a more‌​‌ modest noise bias of​​ $\eta \gtrsim 80$,​​​‌ at the cost of‌ a slightly increased circuit‌​‌ volume. Third, the scheme​​ remains effective even at​​​‌ high physical phase-flip rates,‌ in contrast to previously‌​‌ proposed approaches whose circuit​​ volume grows exponentially with​​​‌ the error rate. Our‌ construction is based on‌​‌ unfolding the $X$ stabilizer​​ group of the Hadamard​​​‌ 3D quantum Reed-Muller code‌ in 2D, enabling distillation‌​‌ at the physical level​​ rather than the logical​​​‌ level, and is therefore‌ referred to as unfolded‌​‌ distillation.

7.19 LDPC-cat codes​​ for low-overhead quantum computing​​​‌ in 2D

Participants: Diego‌ Ruiz, Jeremie Guillaud‌​‌, Mazyar Mirrahimi.​​

The main obstacle to​​​‌ large scale quantum computing‌ are the errors present‌​‌ in every physical qubit​​ realization. Correcting these errors​​​‌ requires a large number‌ of additional qubits. Two‌​‌ main avenues to reduce​​ this overhead are (i)​​​‌ low-density parity check (LDPC)‌ codes requiring very few‌​‌ additional qubits to correct​​ errors (ii) cat qubits​​​‌ where bit-flip errors are‌ exponentially suppressed by design.‌​‌ In this work 24​​ published in Nature Communications,​​​‌ and in collaboration with‌ Anthony Leverrier (Inria Cosmiq‌​‌ team) and Christophe Vuillot​​ (Inria Mocqua team), we​​​‌ combine both approaches to‌ obtain an extremely low‌​‌ overhead architecture. Assuming a​​ physical phase-flip error probability​​​‌ $ϵ\approx 0.‌1\%$ per qubit‌​‌ and operation, one hundred​​ logical qubits can be​​​‌ implemented on a 758‌ cat qubit chip, with‌​‌ a total logical error​​ probability per cycle and​​​‌ per logical qubit ${\mathrm{ϵ‌}}_L\le {10}^{-‌‌8}$. Our architecture​​ also features two major​​​‌ advantages. First, the hardware‌ implementation of the code‌​‌ can be realised with​​ short-range qubit interactions in​​​‌ 2D and low-weight stabilizers,‌ under constraints similar to‌​‌ those of the popular​​ surface code architecture. Second,​​​‌ we demonstrate how to‌ implement a fault-tolerant universal‌​‌ set of logical gates​​ with an additional layer​​​‌ of routing cat qubits‌ stacked on top of‌​‌ the LDPC layer, while​​​‌ maintaining the local connectivity.​ Furthermore, our architecture benefits​‌ from a high capacity​​ of parallelization for these​​​‌ logical gates.

8.1 Bilateral contracts​​ with industry

• Four new​​​‌ PhD contracts with Alice&Bob:​ Armelle Celarier , Louis​‌ Lattier , Theo Malas​​ Danze , Roberto Negrin​​​‌ .
• One new PhD​ contract with C12: Gregoire​‌ Charleux .

8.2 Grants​​ with industry

• BPIFrance i-Démo​​​‌
  This project is of​ 22 million EUR is​‌ a partnership between Alice&Bob​​ (20 MEUR), ENS Lyon​​​‌ (1.1 MEUR) and Quantic​ (1.1 MEUR). Zaki Leghtas​‌ is leading the effort​​ of Quantic. The project​​​‌ accelerates the development of​ a cat-qubit quantum computer​‌ by setting technological targets​​ that enable the emergence​​​‌ of an industrializable system​ and by reducing system​‌ costs by a factor​​ of ten. The project’s​​​‌ key work packages include​ nanofabrication, chip design, simulation​‌ tools, and control electronics.​​

9.1 European initiatives

9.2 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​ hardware-efficient and fault-tolerant 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.
• Junior Research​​​‌ Leader chair, NISQ2LSQ:​ In the framework of​‌ the PEPR NISQ2LSQ, Rémi​​ Robin has obtained a​​​‌ Junior Research Leader chair​ consisting of 312k euros​‌ for 4 years starting​​ in 2024.
• 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​‌ near-resonant coupling. The project​​ is coordinated by mechanical​​​‌ oscillators expert Samuel Deléglise​ (LKB, U.Sorbonne), other project​‌ PIs are Alain Sarlette​​ and Zaki Leghtas (QUANTIC​​​‌ project-team), 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 runs 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 project-team), and​‌ Dima Shepelyansky (LPT, Toulouse).​​ The PhD thesis of​​​‌ Linda Greggio is funded​ on this ANR.

9.3​‌ Regional initiatives

Alain Sarlette​​ is amember of the​​ steering committee of DIM​​​‌ Quantip.

10.1‌ Promoting scientific activities

10.2 Teaching -​​ Supervision - Juries -​​​‌ Educational and pedagogical outreach​

10.3 Popularization

11.1​‌ Major publications

• 1 article​​S.Simon Apers and​​​‌ A.Alain Sarlette.​ Quantum Fast-Forwarding: Markov chains​‌ and graph property testing​​.Quantum Information &​​​‌ ComputationApril 2019HAL​
• 2 articleP.Philippe​‌ Campagne-Ibarcq, A.Alec​​ Eickbusch, S.Steven​​​‌ Touzard, E.Evan​ Zalys-Geller, N. E.​‌Nicholas E. Frattini,​​ V. V.Volodymyr V.​​​‌ Sivak, P.Philip​ Reinhold, S.Shruti​‌ Puri, S.Shyam​​ Shankar, R. J.​​​‌Robert J. Schoelkopf,​ L.Luigi Frunzio,​‌ M.Mazyar Mirrahimi and​​ M. H.Michel H.​​​‌ Devoret. Quantum 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 article​‌J.Jérémie Guillaud and​​ M.Mazyar Mirrahimi.​​​‌ Repetition Cat Qubits for​ Fault-Tolerant Quantum Computation.​‌Physical Review Xhttps://arxiv.org/abs/1904.09474​​ - 22 pages, 11​​​‌ figuresDecember 2019HAL​DOI
• 4 articleZ.​‌Zaki Leghtas, S.​​Steven Touzard, I.​​​‌ M.Ioan M. Pop​, A.Angela Kou​‌, B.Brian Vlastakis​​, A.Andrei Petrenko​​​‌, K. M.Katrina​ M. Sliwa, A.​‌Anirudh Narla, S.​​Shyam Shankar, M.​​​‌ J.Michael J. Hatridge​, M.Matthew Reagor​‌, L.Luigi Frunzio​​, R. J.Robert​​​‌ J. Schoelkopf, M.​Mazyar Mirrahimi and M.​‌ H.Michel H. Devoret​​. Confining the state​​​‌ of light to a​ quantum manifold by engineered​‌ two-photon loss.Science​​3476224February 2015​​​‌, 853-857HALDOI​back to text
• 5​‌ articleR.Raphaël Lescanne​​, M.Marius Villiers​​​‌, T.Théau Peronnin​, A.Alain Sarlette​‌, M.Matthieu Delbecq​​, B.Benjamin Huard​​​‌, T.Takis Kontos​, M.Mazyar Mirrahimi​‌ and Z.Zaki Leghtas​​. Exponential suppression of​​​‌ bit-flips in a qubit​ encoded in an oscillator​‌.Nature PhysicsMarch​​ 2020HALDOI
• 6​​​‌ articleM.Mazyar Mirrahimi​, Z.Zaki Leghtas​‌, V. V.Victor​​ V Albert, S.​​​‌Steven Touzard, R.​ J.Robert J Schoelkopf​‌, L.Liang Jiang​​ and M. H.Michel​​​‌ H Devoret. Dynamically​ protected cat-qubits: a new​‌ paradigm for universal quantum​​ computation.New Journal​​​‌ of Physics164​apr 2014, 045014​‌back to textback​​ to text
• 7 article​​N.N. Ofek,​​​‌ A.A. Petrenko,‌ R.R. Heeres,‌​‌ P.P. Reinhold,​​ Z.Z. Leghtas,​​​‌ B.B. Vlastakis,‌ Y.Y. Liu,‌​‌ L.L. Frunzio,​​ S.S.M. Girvin,​​​‌ L.L. Jiang,‌ M.M. Mirrahimi,‌​‌ M. H.M. H.​​ Devoret and R. J.​​​‌R. J. Schoelkopf.‌ Extending the lifetime of‌​‌ a quantum bit with​​ error correction in superconducting​​​‌ circuits.Nature536‌2016, 5back‌​‌ to text
• 8 article​​U.Ulysse Reglade,​​​‌ A.Adrien Bocquet,‌ R.Ronan Gautier,‌​‌ J.Joachim Cohen,​​ A.Antoine Marquet,​​​‌ E.Emanuele Albertinale,‌ N.Natalia Pankratova,‌​‌ M.Mattis Hallén,​​ F.Felix Rautschke,​​​‌ L.-A.Lev-Arcady Sellem,‌ P.Pierre Rouchon,‌​‌ A.Alain Sarlette,​​ M.Mazyar Mirrahimi,​​​‌ P.Philippe Campagne-Ibarcq,‌ R.Raphaël Lescanne,‌​‌ S.Sébastien Jézouin and​​ Z.Zaki Leghtas.​​​‌ Quantum control of a‌ cat qubit with bit-flip‌​‌ times exceeding ten seconds​​.Nature6298013​​​‌2024, 778-783HAL‌DOI
• 9 articleD.‌​‌Diego Ruiz, J.​​Jérémie Guillaud, A.​​​‌Anthony Leverrier, M.‌Mazyar Mirrahimi and C.‌​‌Christophe Vuillot. LDPC-cat​​ codes for low-overhead quantum​​​‌ computing in 2D.‌Nature Communications161‌​‌January 2025, 1040​​HALDOI
• 10 article​​​‌A.A. Sarlette,‌ J.-M.J.-M. Raimond,‌​‌ M.M. Brune and​​ P.P. Rouchon.​​​‌ Stabilization of nonclassical states‌ of the radiation field‌​‌ in a cavity by​​ reservoir engineering.Phys.​​​‌ Rev. Lett.107010402‌2011back to text‌​‌back to textback​​ to text
• 11 article​​​‌C.C. Sayrin,‌ I.I. Dotsenko,‌​‌ X.X. Zhou,​​ B.B. Peaudecerf,​​​‌ T.T. Rybarczyk,‌ S.S. Gleyzes,‌​‌ P.P. Rouchon,​​ M.M. Mirrahimi,​​​‌ H.H. Amini,‌ M.M. Brune,‌​‌ J.-M.J.-M. Raimond and​​ S.S. Haroche.​​​‌ Real-time quantum feedback prepares‌ and stabilizes photon number‌​‌ states.Nature477​​2011, 73--77back​​​‌ to text
• 12 article‌L.-A.Lev-Arcady Sellem,‌​‌ A.Alain Sarlette,​​ Z.Zaki Leghtas,​​​‌ M.Mazyar Mirrahimi,‌ P.Pierre Rouchon and‌​‌ P.Philippe Campagne-Ibarcq.​​ Dissipative Protection of a​​​‌ GKP Qubit in a‌ High-Impedance Superconducting Circuit Driven‌​‌ by a Microwave Frequency​​ Comb.Physical Review​​​‌ X1512025‌, 011011HALDOI‌​‌
• 13 articleW. C.​​William C Smith,​​​‌ A.Alvise Borgognoni,‌ M.Marius Villiers,‌​‌ E.E. Roverc’h,​​ J.José Palomo,​​​‌ M.Matthieu Delbecq,‌ T.Takis Kontos,‌​‌ P.Philippe Campagne-Ibarcq,​​ B.Benoît Douçot and​​​‌ Z.Zaki Leghtas.‌ Spectral signature of high-order‌​‌ photon processes mediated by​​ Cooper-pair pairing.Nature​​​‌ Communications161September‌ 2025, 8359HAL‌​‌DOI
• 14 articleW.​​ C.William C Smith​​​‌, M.Marius Villiers‌, A.Antoine Marquet‌​‌, J.Jose Palomo​​, M.Matthieu Delbecq​​​‌, T.Takis Kontos‌, P.Philippe Campagne-Ibarcq‌​‌, B.Benoît Douçot​​​‌ and Z.Zaki Leghtas​. Magnifying quantum phase​‌ fluctuations with Cooper-pair pairing​​.Physical Review X​​​‌122April 2022​, 021002HALDOI​‌

11.2 Publications of the​​ year

11.3 Cited‌​‌ publications

• 53 articleB.​​Baleegh Abdo, O.​​​‌Oblesh Jinka, N.‌ T.Nicholas T Bronn‌​‌, S.Salvatore Olivadese​​ and M.Markus Brink​​​‌. On-chip single-pump interferometric‌ Josephson isolator for quantum‌​‌ measurements.arXiv preprint​​ arXiv:2006.019182020back to​​​‌ textback to text‌
• 54 articleH.H.‌​‌ Amini, M.M.​​ Mirrahimi and P.P.​​​‌ Rouchon. Stabilization of‌ a delayed quantum system:‌​‌ the Photon Box case-study​​.IEEE Trans. Automatic​​​‌ Control5782012‌, 1918--1930back to‌​‌ textback to text​​
• 55 articleH.H​​​‌ Amini, C.C.‌ Pellegrini and P.P.‌​‌ Rouchon. Stability of​​ continuous-time quantum filters with​​​‌ measurement imperfections.Russian‌ Journal of Mathematical Physics‌​‌212014, 297--315​​back to text
• 56​​​‌ articleH.H. Amini‌, A.A. Somaraju‌​‌, I.I. Dotsenko​​, C.C. Sayrin​​​‌, M.M. Mirrahimi‌ and P.P. Rouchon‌​‌. Feedback stabilization of​​ discrete-time quantum systems subject​​​‌ to non-demolition measurements with‌ imperfections and delays.‌​‌Automatica4992013​​, 2683--2692back to​​​‌ textback to text‌back to text
• 57‌​‌ bookS.S. Attal​​, A.A. Joye​​​‌ and C.-A.C.-A. Pillet‌, eds. Open Quantum‌​‌ Systems III: Recent Developments​​.Springer, Lecture notes​​​‌ in Mathematics 18802006‌back to text
• 58‌​‌ bookA.A. Barchielli​​ and M.M. Gregoratti​​​‌. Quantum Trajectories and‌ Measurements in Continuous Time:‌​‌ the Diffusive Case.​​Springer Verlag2009back​​​‌ to text
• 59 article‌J.J.T. Barreiro,‌​‌ M.M. Muller,​​​‌ P.P. Schindler,​ D.D. Nigg,​‌ T.T. Monz,​​ M.M. Chwalla,​​​‌ M.M. Hennrich,​ C.C.F. Roos,​‌ P.P. Zoller and​​ R.R. Blatt.​​​‌ An open-system quantum simulator​ with trapped ions.​‌Nature4704862011​​back to text
• 60​​​‌ articleS.Shabir Barzanjeh​, M.Matthias Wulf​‌, M.Matilda Peruzzo​​, M.Mahmoud Kalaee​​​‌, P.PB Dieterle​, O.Oskar Painter​‌ and J. M.Johannes​​ M Fink. Mechanical​​​‌ on-chip microwave circulator.​Nature communications81​‌2017, 1--7back​​ to text
• 61 article​​​‌V.V.P. Belavkin.​ Quantum stochastic calculus and​‌ quantum nonlinear filtering.​​Journal of Multivariate Analysis​​​‌4221992,​ 171--201back to text​‌
• 62 articleT.T.​​ Benoist and C.C.​​​‌ Pellegrini. Large Time​ Behavior and Convergence Rate​‌ for Quantum Filters Under​​ Standard Non Demolition Conditions​​​‌.Communications in Mathematical​ Physics2014, 1-21​‌URL: http://dx.doi.org/10.1007/s00220-014-2029-6back to​​ text
• 63 articleG.​​​‌G. Birkhoff. Extensions​ of Jentzch's theorem.​‌Trans. Amer. Math. Soc.​​851957, 219--227​​​‌back to text
• 64​ articleS.S. Bolognani​‌ and F.F. Ticozzi​​. Engineering stable discrete-time​​​‌ quantum dynamics via a​ canonical QR decomposition.​‌IEEE Trans. Autom. Control​​552010back to​​​‌ text
• 65 bookV.​V. Braginski and F.​‌F. Khalili. Quantum​​ Measurements.Cambridge University​​​‌ Press1992back to​ text
• 66 articleP.​‌P. Campagne-Ibarcq, L.​​L. Bretheau, E.​​​‌E. Flurin, A.​A. Auffèves, F.​‌F. Mallet and B.​​B. Huard. Observing​​​‌ Interferences between Past and​ Future Quantum States in​‌ Resonance Fluorescence.Phys.​​ Rev. Lett.112180402​​​‌18May 2014,​ URL: http://link.aps.org/doi/10.1103/PhysRevLett.112.180402DOIback​‌ to text
• 67 article​​P.P. Campagne-Ibarcq,​​​‌ E.E. Flurin,​ N.N. Roch,​‌ D.D. Darson,​​ P.P. Morfin,​​​‌ M.M. Mirrahimi,​ M. H.M. H.​‌ Devoret, F.F.​​ Mallet and B.B.​​​‌ Huard. Persistent Control​ of a Superconducting Qubit​‌ by Stroboscopic Measurement Feedback​​.Phys. Rev. X​​​‌30210082013back​ to textback to​‌ text
• 68 bookH.​​H. Carmichael. An​​​‌ Open Systems Approach to​ Quantum Optics.Springer-Verlag​‌1993back to text​​
• 69 bookH.H.​​​‌ Carmichael. Statistical Methods​ in Quantum Optics 2:​‌ Non-Classical Fields.Spinger​​2007back to text​​​‌
• 70 bookJ.J.​ Carr. Application of​‌ Center Manifold Theory.​​Springer1981back to​​​‌ text
• 71 articleB.​ J.Benjamin J Chapman​‌, E. I.Eric​​ I Rosenthal, J.​​​‌Joseph Kerckhoff, B.​ A.Bradley A Moores​‌, L. R.Leila​​ R Vale, J.​​​‌JAB Mates, G.​ C.Gene C Hilton​‌, K.Kevin Lalumiere​​, A.Alexandre Blais​​​‌ and K.KW Lehnert​. Widely tunable on-chip​‌ microwave circulator for superconducting​​ quantum circuits.Physical​​​‌ Review X74​2017, 041043back​‌ to text
• 72 article​​J.J. Cohen and​​ M.M. Mirrahimi.​​​‌ Dissipation-induced continuous quantum error‌ correction for superconducting circuits‌​‌.Phys. Rev. A​​902014, 062344​​​‌back to text
• 73‌ articleJ.J. Dalibard‌​‌, Y.Y. Castin​​ and K.K. Mölmer​​​‌. Wave-function approach to‌ dissipative processes in quantum‌​‌ optics.Phys. Rev.​​ Lett.6851992​​​‌, 580--583back to‌ text
• 74 miscM.‌​‌ H.M. H. Devoret​​, A.A. Wallraff​​​‌ and J.J.M. Martinis‌. Superconducting Qubits: A‌​‌ Short Review.arXiv:cond-mat/0411174​​2004back to text​​​‌
• 75 articleN.Nicolas‌ Didier, J.Jérémie‌​‌ Guillaud, S.Shyam​​ Shankar and M.Mazyar​​​‌ Mirrahimi. Remote entanglement‌ stabilization and concentration by‌​‌ quantum reservoir engineering.​​Physical Review Ahttps://arxiv.org/abs/1703.03379​​​‌ - 5 pages, 4‌ figuresJuly 2018HAL‌​‌DOIback to text​​
• 76 articleI.I.​​​‌ Dotsenko, M.M.‌ Mirrahimi, M.M.‌​‌ Brune, S.S.​​ Haroche, J.-M.J.-M.​​​‌ Raimond and P.P.‌ Rouchon. Quantum feedback‌​‌ by discrete quantum non-demolition​​ measurements: towards on-demand generation​​​‌ of photon-number states.‌Physical Review A80:‌​‌ 013805-0138132009back to​​ text
• 77 articleM.​​​‌Martina Esposito, A.‌Arpit Ranadive, L.‌​‌Luca Planat and N.​​Nicolas Roch. Perspective​​​‌ on traveling wave microwave‌ parametric amplifiers.Applied‌​‌ Physics Letters11912​​2021back to text​​​‌
• 78 articleN.N.‌ Fenichel. Geometric singular‌​‌ perturbation theory for ordinary​​ differential equations.J.​​​‌ Diff. Equations311979‌, 53--98back to‌​‌ text
• 79 articleV.​​ S.Vinicius S Ferreira​​​‌, J.Jash Banker‌, A.Alp Sipahigil‌​‌, M. H.Matthew​​ H Matheny, A.​​​‌ J.Andrew J Keller‌, E.Eunjong Kim‌​‌, M.Mohammad Mirhosseini​​ and O.Oskar Painter​​​‌. Collapse and Revival‌ of an Artificial Atom‌​‌ Coupled to a Structured​​ Photonic Reservoir.arXiv​​​‌ preprint arXiv:2001.032402020back‌ to text
• 80 article‌​‌J.J. Gambetta and​​ H. M.H. M.​​​‌ Wiseman. State and‌ dynamical parameter estimation for‌​‌ open quantum systems.​​Phys. Rev. A64​​​‌4042105September 2001‌, URL: http://link.aps.org/doi/10.1103/PhysRevA.64.042105back‌​‌ to text
• 81 article​​S.S. Gammelmark,​​​‌ B.B. Julsgaard and‌ K.K. Mölmer.‌​‌ Past Quantum States of​​ a Monitored System.​​​‌Phys. Rev. Lett.111‌16160401October 2013‌​‌, URL: http://link.aps.org/doi/10.1103/PhysRevLett.111.160401back​​ to text
• 82 book​​​‌C.C.W. Gardiner and‌ P.P. Zoller.‌​‌ Quantum Noise.Springer​​2010back to text​​​‌
• 83 articleS.S.‌ Gaubert and Z.Z.‌​‌ Qu. Checking the​​ strict positivity of Kraus​​​‌ maps is NP-hard.‌arXiv:1402.14292014back to‌​‌ text
• 84 articleS.​​S. Gaubert and Z.​​​‌Z. Qu. The‌ contraction rate in Thompson's‌​‌ part metric of order-preserving​​ flows on a cone​​​‌ - Application to generalized‌ Riccati equations.Journal‌​‌ of Differential Equations256​​8April 2014,​​​‌ 2902--2948URL: http://www.sciencedirect.com/science/article/pii/S0022039614000424back‌ to text
• 85 article‌​‌K.K. Geerlings,​​ Z.Z. Leghtas,​​​‌ I.I.M. Pop,‌ S.S. Shankar,‌​‌ L.L. Frunzio,​​​‌ R.R.J. Schoelkopf,​ M.M. Mirrahimi and​‌ M. H.M. H.​​ Devoret. Demonstrating a​​​‌ Driven Reset Protocol of​ a Superconducting Qubit.​‌Phys. Rev. Lett.110​​1205012013back to​​​‌ textback to text​back to textback​‌ to textback to​​ textback to text​​​‌
• 86 articleD.D.​ Gottesman, A.A.​‌ Kitaev and J.J.​​ Preskill. Encoding a​​​‌ qubit in an oscillator​.Phys. Rev. A​‌640123102001back​​ to text
• 87 article​​​‌C.C. Guerlin,​ J.J. Bernu,​‌ S.S. Deléglise,​​ C.C. Sayrin,​​​‌ S.S. Gleyzes,​ S.S. Kuhr,​‌ M.M. Brune,​​ J.-M.J.-M. Raimond and​​​‌ S.S. Haroche.​ Progressive field-state collapse and​‌ quantum non-demolition photon counting​​.Nature4482007​​​‌, 889-893back to​ text
• 88 articleR.​‌R. van Handel.​​ The stability of quantum​​​‌ Markov filters.Infin.​ Dimens. Anal. Quantum Probab.​‌ Relat. Top.122009​​, 153--172back to​​​‌ text
• 89 bookS.​S. Haroche and J.-M.​‌J.-M. Raimond. Exploring​​ the Quantum: Atoms, Cavities​​​‌ and Photons.Oxford​ University Press2006back​‌ to textback to​​ text
• 90 articleM.​​​‌M. Hatridge, S.​S. Shankar, M.​‌M. Mirrahimi, F.​​F. Schackert, K.​​​‌K. Geerlings, T.​T. Brecht, K.​‌K.M. Sliwa, B.​​B. Abdo, L.​​​‌L. Frunzio, S.​S.M. Girvin, R.​‌R.J. Schoelkopf and M.​​ H.M. H. Devoret​​​‌. Quantum back-action of​ an individual variable-strength measurement​‌.Science3392013​​, 178--181back to​​​‌ textback to text​
• 91 articleE.E.T.​‌ Holland, B.B.​​ Vlastakis, R.R.W.​​​‌ Heeres, M. J.​M. J. Reagor,​‌ U.U. Vool,​​ Z.Z. Leghtas,​​​‌ L.L. Frunzio,​ G.G. Kirchmair,​‌ M. H.M. H.​​ Devoret, M.M.​​​‌ Mirrahimi and R.R.J.​ Schoelkopf. Single-photon-resolved cross-Kerr​‌ interaction for autonomous stabilization​​ of photon-number states.​​​‌Phys. Rev. Lett.115​2015, 180501back​‌ to text
• 92 book​​T.T. Kato.​​​‌ Perturbation Theory for Linear​ Operators.Springer1966​‌back to text
• 93​​ articleE.E. Knill​​​‌, R.R. Laflamme​ and G.G.J. Milburn​‌. A scheme for​​ efficient quantum computation with​​​‌ linear optics.Nature​409462001back​‌ to text
• 94 article​​C.CS Kow and​​​‌ M.MT Bell.​ Traveling-Wave Parametric Amplifier with​‌ Passive Reverse Isolation.​​arXiv preprint arXiv:2505.040592025​​​‌back to text
• 95​ articleH.H. Krauter​‌, C.C.A. Muschik​​, K.K. Jensen​​​‌, W.W. Wasilewski​, J.J.M. Petersen​‌, J.J.I. Cirac​​ and E.E.S. Polzik​​​‌. Entanglement Generated by​ Dissipation and Steady State​‌ Entanglement of Two Macroscopic​​ Objects.Phys. Rev.​​​‌ Lett.1070805032011​back to text
• 96​‌ articleZ.Z. Leghtas​​, G.G. Kirchmair​​​‌, B.B. Vlastakis​, M. H.M.​‌ H. Devoret, R.​​ J.R. J. Schoelkopf​​ and M.M. Mirrahimi​​​‌. Deterministic protocol for‌ mapping a qubit to‌​‌ coherent state superpositions in​​ a cavity.Phys.​​​‌ Rev. A87042315‌2013back to text‌​‌
• 97 articleZ.Z.​​ Leghtas, G.G.​​​‌ Kirchmair, B.B.‌ Vlastakis, R. J.‌​‌R. J Schoelkopf,​​ M. H.M. H.​​​‌ Devoret and M.M.‌ Mirrahimi. Hardware-efficient autonomous‌​‌ quantum memory protection.​​Phys. Rev. Lett.111​​​‌1205012013back to‌ text
• 98 articleZ.‌​‌Z. Leghtas, A.​​A. Sarlette and P.​​​‌P. Rouchon. Adiabatic‌ passage and ensemble control‌​‌ of quantum systems.​​J. Phys. B44​​​‌1540172011back to‌ text
• 99 articleZ.‌​‌Z. Leghtas, U.​​U. Vool, S.​​​‌S. Shankar, M.‌M. Hatridge, S.‌​‌S.M. Girvin, M.​​ H.M. H. Devoret​​​‌ and M.M. Mirrahimi‌. Stabilizing a Bell‌​‌ state of two superconducting​​ qubits by dissipation engineering​​​‌.Phys. Rev. A‌880238492013back‌​‌ to textback to​​ text
• 100 articleC.​​​‌C. Leroux, L.‌ C.L. C. G.‌​‌ Govia and A. A.​​A. A. Clerk.​​​‌ Enhancing Cavity Quantum Electrodynamics‌ via Antisqueezing: Synthetic Ultrastrong‌​‌ Coupling.Phys. Rev.​​ Lett.1209Mar​​​‌ 2018, 093602URL:‌ https://link.aps.org/doi/10.1103/PhysRevLett.120.093602DOIback to‌​‌ text
• 101 articleJ.-S.​​Jr-Shin Li and N.​​​‌N. Khaneja. Ensemble‌ control of Bloch equations‌​‌.IEEE Trans. Autom.​​ Control542009,​​​‌ 528--536back to text‌
• 102 articleY.Y.‌​‌ Lin, J.J.P.​​ Gaebler, F.F.​​​‌ Reiter, T.T.R.‌ Tan, R.R.‌​‌ Bowler, A.A.S.​​ Sorensen, D.D.​​​‌ Leibfried and D.D.J.‌ Wineland. Dissipative production‌​‌ of a maximally entangled​​ steady state of two​​​‌ quantum bits.Nature‌5042013, 415--418‌​‌back to text
• 103​​ articleS.S. Lloyd​​​‌. Coherent quantum feedback‌.Phys. Rev. A‌​‌620221082000back​​ to textback to​​​‌ text
• 104 articleA.‌ C.Alice C Mahoney‌​‌, J. I.James​​ I Colless, L.​​​‌Lucas Peeters, S.‌ J.Sebastian J Pauka‌​‌, E. J.Eli​​ J Fox, X.​​​‌Xufeng Kou, L.‌Lei Pan, K.‌​‌ L.Kang L Wang​​, D.David Goldhaber-Gordon​​​‌ and D. J.David‌ J Reilly. Zero-field‌​‌ edge plasmons in a​​ magnetic topological insulator.​​​‌Nature communications81‌2017, 1--7back‌​‌ to text
• 105 article​​M.M Malnou,​​​‌ B.BT Miller,‌ J.JA Estrada,‌​‌ K.K Genter,​​ K.K Cicak,​​​‌ J.JD Teufel,‌ J.J Aumentado and‌​‌ F.F Lecocq.​​ A travelling-wave parametric amplifier​​​‌ and converter.Nature‌ Electronics8112025‌​‌, 1082--1088back to​​ text
• 106 articleL.​​​‌L. Mazzarella, A.‌A. Sarlette and F.‌​‌F. Ticozzi. Consensus​​ for quantum networks: from​​​‌ symmetry to gossip iterations‌.IEEE Trans. Automat.‌​‌ Controlin press2014​​back to text
• 107​​​‌ conferenceM.M. Mirrahimi‌, B.B. Huard‌​‌ and M. H.M.​​​‌ H. Devoret. Strong​ 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
• 108 articleK.​K.W. Murch, U.​‌U. Vool, D.​​D. Zhou, S.​​​‌S.J. Weber, S.​S.M. Girvin and I.​‌I. Siddiqi. Cavity-assisted​​ quantum bath engineering.​​​‌Phys. Rev. Lett.109​1836022012back to​‌ text
• 109 articleB.-L.​​Baldo-Luis Najera-Santos, R.​​​‌Rémi Rousseau, K.​Kyrylo Gerashchenko, H.​‌Himanshu Patange, A.​​Angela Riva, M.​​​‌Marius Villiers, T.​Tristan Briant, P.-F.​‌Pierre-François Cohadon, A.​​Antoine Heidmann, J.​​​‌José Palomo, M.​Michael Rosticher, H.​‌Hélène Le Sueur,​​ A.Alain Sarlette,​​​‌ W. C.William C.​ Smith, Z.Zaki​‌ Leghtas, E.Emmanuel​​ Flurin, T.Thibaut​​​‌ Jacqmin and S.Samuel​ Deléglise. High-Sensitivity ac-Charge​‌ Detection with a MHz-Frequency​​ Fluxonium Qubit.Physical​​​‌ Review X141​2024, 011007HAL​‌DOIback to text​​
• 110 articleA.A​​​‌ Negretti and K.K​ Mölmer. Estimation of​‌ classical parameters via continuous​​ probing of complementary quantum​​​‌ observables.New Journal​ of Physics1512​‌1250022013, URL:​​ http://stacks.iop.org/1367-2630/15/i=12/a=125002back to text​​​‌
• 111 articleH.H.I.​ Nurdin, M.M.R.​‌ James and I.I.R.​​ Petersen. Coherent quantum​​​‌ LQG control.Automatica​452009, 1837--1846​‌back to text
• 112​​ articleB.B. Peaudecerf​​​‌, T.T. Rybarczyk​, S.S. Gerlich​‌, S.S. Gleyzes​​, J.-M.J.-M. Raimond​​​‌, S.S. Haroche​, I.I. Dotsenko​‌ and M.M. Brune​​. Adaptive Quantum Nondemolition​​​‌ Measurement of a Photon​ Number.Phys. Rev.​‌ Lett.1128080401​​Feb 2014, URL:​​​‌ http://link.aps.org/doi/10.1103/PhysRevLett.112.080401DOIback to​ text
• 113 articleG.​‌ A.Gabriel A Peterson​​, F.F Lecocq​​​‌, K.Katarina Cicak​, R. W.Raymond​‌ W Simmonds, J.​​J Aumentado and J.​​​‌ D.John D Teufel​. Demonstration of efficient​‌ nonreciprocity in a microwave​​ optomechanical circuit.Physical​​​‌ Review X73​2017, 031001back​‌ to text
• 114 article​​D.D. Petz.​​​‌ Monotone Metrics on matrix​ spaces.Linear Algebra​‌ and its Applications244​​1996, 81--96back​​​‌ to text
• 115 article​J.J.F. Poyatos,​‌ J.J.I. Cirac and​​ P.P. Zoller.​​​‌ Quantum Reservoir Engineering with​ Laser Cooled Trapped Ions​‌.Phys. Rev. Lett.​​77231996,​​​‌ 4728--4731back to text​back to text
• 116​‌ bookD. M.David​​ M Pozar. Microwave​​​‌ engineering.John wiley​ & sons2011back​‌ to text
• 117 article​​M.Matthieu Praquin,​​​‌ V.Vincent Lienhard,​ A.Anthony Giraudo,​‌ A.Aron Vanselow,​​ Z.Zaki Leghtas and​​​‌ P.Philippe Campagne-Ibarcq.​ Mixing of counterpropagating signals​‌ in a traveling-wave Josephson​​ device.arXiv preprint​​​‌ arXiv:2406.197512024back to​ text
• 118 articleA.​‌Arpit Ranadive, B.​​Bekim Fazliji, G.​​Gwenael Le Gal,​​​‌ G.Giulio Cappelli,‌ G.Guilliam Butseraen,‌​‌ E.Edgar Bonet,​​ E.Eric Eyraud,​​​‌ S.Sina Böhling,‌ L.Luca Planat,‌​‌ A.A Metelmann and​​ others. A travelling-wave​​​‌ parametric amplifier isolator.‌Nature Electronics2025,‌​‌ 1--10back to text​​
• 119 articleM.Matthew​​​‌ Reagor, W.Wolfgang‌ Pfaff, C.Christopher‌​‌ Axline, R. W.​​Reinier W Heeres,​​​‌ N.Nissim Ofek,‌ K.Katrina Sliwa,‌​‌ E.Eric Holland,​​ C.Chen Wang,​​​‌ J.Jacob Blumoff,‌ K.Kevin Chou and‌​‌ others. Quantum memory​​ with millisecond coherence in​​​‌ circuit QED.Physical‌ Review B941‌​‌2016, 014506back​​ to text
• 120 article​​​‌D.D. Reeb,‌ M. ..M .J.‌​‌ Kastoryano and M. ..​​M .M. Wolf.​​​‌ Hilbert's projective metric in‌ quantum information theory.‌​‌Journal of Mathematical Physics​​528082201August​​​‌ 2011, URL: http://dx.doi.org/10.1063/1.3615729‌back to text
• 121‌​‌ articleM. D.Matthew​​ D Reed, B.​​​‌ R.Blake R Johnson‌, A. A.Andrew‌​‌ A Houck, L.​​Leonardo DiCarlo, J.​​​‌ M.Jerry M Chow‌, D. I.David‌​‌ I Schuster, L.​​Luigi Frunzio and R.​​​‌ J.Robert J Schoelkopf‌. Fast reset and‌​‌ suppressing spontaneous emission of​​ a superconducting qubit.​​​‌Applied Physics Letters96‌202010, 203110‌​‌back to text
• 122​​ articleD.D. Ristè​​​‌, J.J.V. Leeuwen‌, H.-S.H.-S. Ku‌​‌, K.K. Lehnert​​ and L.L. Dicarlo​​​‌. Initialization by measurement‌ of a superconducting quantum‌​‌ bit circuit.Phys.​​ Rev. Lett.109050507​​​‌2012back to text‌
• 123 articleN.N.‌​‌ Roch, E.E.​​ Flurin, F.F.​​​‌ Nguyen, P.P.‌ Morfin, P.P.‌​‌ Campagne-Ibarcq, M. H.​​M. H. Devoret and​​​‌ B.B. Huard.‌ Widely tunable, non-degenerate three-wave‌​‌ mixing microwave device operating​​ near the quantum limit​​​‌.Phys. Rev. Lett.‌1081477012012back‌​‌ to text
• 124 article​​P.P. Rouchon.​​​‌ Fidelity is a Sub-Martingale‌ for Discrete-Time Quantum Filters‌​‌.IEEE Transactions on​​ Automatic Control5611​​​‌2011, 2743--2747back‌ to textback to‌​‌ text
• 125 articleA.​​A. Roy, Z.​​​‌Z. Leghtas, A.‌A.D. Stone, M.‌​‌M.H. Devoret and M.​​M. Mirrahimi. Continuous​​​‌ generation and stabilization of‌ mesoscopic field superposition states‌​‌ in a quantum circuit​​.Phys. Rev. A​​​‌912015, 013810‌back to text
• 126‌​‌ articleA.A. Sarlette​​, Z.Z. Leghtas​​​‌, M.M. Brune‌, J.-M.J.-M. Raimond‌​‌ and P.P. Rouchon​​. Stabilization of nonclassical​​​‌ states of one- and‌ two-mode radiation fields by‌​‌ reservoir engineering.Phys.​​ Rev. A86012114​​​‌2012back to text‌
• 127 articleD.D.I.‌​‌ Schuster, A.A.A.​​ Houck, J.J.A​​​‌ Schreier, A.A.‌ Wallraff, J.J.M.‌​‌ Gambetta, A.A.​​ Blais, L.L.​​​‌ Frunzio, J.J.‌ Majer, B.B.‌​‌ Johnson, M. H.​​​‌M. H. Devoret,​ S.S.M. Girvin and​‌ R. J.R. J.​​ Schoelkopf. Resolving photon​​​‌ number states in a​ superconducting circuit.Nature​‌4452007, 515--518​​back to text
• 128​​​‌ inproceedingsR.R. Sepulchre​, A.A. Sarlette​‌ and P.P. Rouchon​​. Consensus in non-commutative​​​‌ spaces.Decision and​ Control (CDC), 2010 49th​‌ IEEE Conference on2010​​, 6596--6601back to​​​‌ text
• 129 articleS.​S. Shankar, M.​‌M. Hatridge, Z.​​Z. Leghtas, K.​​​‌K.M. Sliwa, A.​A. Narla, U.​‌U. Vool, S.​​S.M. Girvin, L.​​​‌L. Frunzio, M.​M. Mirrahimi and M.​‌ H.M. H. Devoret​​. Autonomously stabilized entanglement​​​‌ between two superconducting quantum​ bits.Nature504​‌2013, 419--422back​​ to textback to​​​‌ textback to text​back to textback​‌ to text
• 130 article​​P.P. Shor.​​​‌ Scheme for reducing decoherence​ in quantum memory.​‌Phys. Rev. A52​​1995, 2493--2496back​​​‌ to text
• 131 article​K.KM Sliwa,​‌ M.M Hatridge,​​ A.A Narla,​​​‌ S.S Shankar,​ L.L Frunzio,​‌ R.RJ Schoelkopf and​​ M.MH Devoret.​​​‌ Reconfigurable Josephson circulator/directional amplifier​.Physical Review X​‌542015,​​ 041020back to text​​​‌
• 132 articleW. C.​W. C. Smith,​‌ A.A. Kou,​​ X.X. Xiao,​​​‌ U.U. Vool and​ M. H.M. H.​‌ Devoret. Superconducting circuit​​ protected by two-Cooper-pair tunneling​​​‌.npj Quantum Information​61Jan 2020​‌, 8URL: https://doi.org/10.1038/s41534-019-0231-2​​DOIback to text​​​‌
• 133 articleW. C.​William C Smith,​‌ M.Marius Villiers,​​ A.Antoine Marquet,​​​‌ J.Jose Palomo,​ M.Matthieu Delbecq,​‌ T.Takis Kontos,​​ P.Philippe Campagne-Ibarcq,​​​‌ B.Benôit Douçot and​ Z.Zaki Leghtas.​‌ Magnifying quantum phase fluctuations​​ with Cooper-pair pairing.​​​‌Physical Review X12​211 pages, 6​‌ figuresApril 2022,​​ 021002HALDOIback​​​‌ to text
• 134 inproceedings​A.A. Somaraju,​‌ I.I. Dotsenko,​​ C.C. Sayrin and​​​‌ P.P. Rouchon.​ Design and Stability of​‌ Discrete-Time Quantum Filters with​​ Measurement Imperfections.American​​​‌ Control Conference2012,​ 5084--5089back to text​‌back to text
• 135​​ articleA.A. Somaraju​​​‌, M.M. Mirrahimi​ and P.P. Rouchon​‌. Approximate stabilization of​​ infinite dimensional quantum stochastic​​​‌ system.Reviews in​ Mathematical Physics251350001​‌2013back to text​​back to text
• 136​​​‌ articleA.A. Steane​. Error Correcting Codes​‌ in Quantum Theory.​​Phys. Rev. Lett77​​​‌51996back to​ textback to text​‌
• 137 articleL.L.​​ Sun, A.A.​​​‌ Petrenko, Z.Z.​ Leghtas, B.B.​‌ Vlastakis, G.G.​​ Kirchmair, K.K.M.​​​‌ Sliwa, A.A.​ Narla, M.M.​‌ Hatridge, S.S.​​ Shankar, J.J.​​​‌ Blumoff, L.L.​ Frunzio, M.M.​‌ Mirrahimi, M. H.​​M. H. Devoret and​​ R. J.R. J.​​​‌ Schoelkopf. Tracking photon‌ jumps with repeated quantum‌​‌ non-demolition parity measurements.​​Nature5112014,​​​‌ 444--448back to text‌
• 138 articleT.T‌​‌ Thorbeck, S.S​​ Zhu, E.E​​​‌ Leonard Jr, R.‌R Barends, J.‌​‌J Kelly, J.​​ M.John M Martinis​​​‌ and R.R McDermott‌. Reverse isolation and‌​‌ backaction of the SLUG​​ microwave amplifier.Physical​​​‌ Review Applied85‌2017, 054007back‌​‌ to text
• 139 article​​J.J.N. Tsitsiklis.​​​‌ Problems in decentralized decision‌ making and computation.‌​‌PhD Thesis, MIT1984​​back to text
• 140​​​‌ articleR.R. Vijay‌, C.C. Macklin‌​‌, D.D.H. Slichter​​, S.S.J. Weber​​​‌, K.K.W. Murch‌, R.R. Naik‌​‌, A.A.N. Korotkov​​ and I.I. Siddiqi​​​‌. Stabilizing Rabi oscillations‌ in a superconducting qubit‌​‌ using quantum feedback.​​Nature4902012,​​​‌ 77--80back to text‌back to text
• 141‌​‌ articleM.Marius Villiers​​, W. C.William​​​‌ C Smith, A.‌Alexandru Petrescu, A.‌​‌Alvise Borgognoni, M.​​Matthieu Delbecq, A.​​​‌Alain Sarlette, M.‌Mazyar Mirrahimi, P.‌​‌Philippe Campagne-Ibarcq, T.​​Takis Kontos and Z.​​​‌Zaki Leghtas. Dynamically‌ Enhancing Qubit-Photon Interactions with‌​‌ Antisqueezing.PRX Quantum​​522024,​​​‌ 020306HALDOIback‌ to text
• 142 article‌​‌G.Giovanni Viola and​​ D. P.David P​​​‌ DiVincenzo. Hall effect‌ gyrators and circulators.‌​‌Physical Review X4​​22014, 021019​​​‌back to text
• 143‌ articleL.L. Viola‌​‌, E.E. Knill​​ and S.S. Lloyd​​​‌. Dynamical decoupling of‌ open quantum system.‌​‌Phys. Rev. Lett.82​​1999, 2417-2421back​​​‌ to text
• 144 article‌B.B. Vlastakis,‌​‌ G.G. Kirchmair,​​ Z.Z. Leghtas,​​​‌ S.S.E. Nigg,‌ L.L. Frunzio,‌​‌ S.S.M. Girvin,​​ M.M. Mirrahimi,​​​‌ M. H.M. H.‌ Devoret and R. J.‌​‌R. J. Schoelkopf.​​ Deterministically encoding quantum information​​​‌ using 100-photon Schrödinger cat‌ states.Science342‌​‌2013, 607--610back​​ to text
• 145 article​​​‌X.X. Zhou,‌ I.I. Dotsenko,‌​‌ B.B. Peaudecerf,​​ T.T. Rybarczyk,​​​‌ C.C. Sayrin,‌ S.S. Gleyzes,‌​‌ J.-M.J.-M. Raimond,​​ M.M. Brune and​​​‌ S.S. Haroche.‌ Field locked to Fock‌​‌ state by quantum feedback​​ with single photon corrections​​​‌.Physical Review Letter‌1082436022012back‌​‌ to text