SU\G(𝔸)/K·U
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All published works (239)

A posteriori error estimates for the Lindblad master equation

2025-01-16
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 … 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, we establish bounds for these two approximations that can be explicitely 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 significantly reduces computational time and relieves users of the challenge of selecting an appropriate truncation.
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Parameters estimation by fitting correlation functions of continuous quantum measurement

2024-10-15
Pierre Guilmin , Pierre Rouchon , Antoine Tilloy
We propose a simple method to estimate the parameters of a continuously measured quantum system, by fitting correlation functions of the measured signal. We demonstrate the approach in simulation, both … We propose a simple method to estimate the parameters of a continuously measured quantum system, by fitting correlation functions of the measured signal. We demonstrate the approach in simulation, both on toy examples and on a recent superconducting circuits experiment which proved particularly difficult to characterise using conventional methods. The idea is applicable to any system whose evolution is described by a jump or diffusive stochastic master equation. It allows the simultaneous estimation of many parameters, is practical for everyday use, is suitable for large Hilbert space dimensions, and takes into account experimental constraints such as detector imperfections and signal filtering and digitisation. Unlike existing methods, it also provides a direct way to understand how each parameter is estimated from the measured signal. This makes the approach interpretable, facilitates debugging, and enables validating the adequacy of a model with the observed data.
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Monitoring the Energy of a Cavity by Observing the Emission of a Repeatedly Excited Qubit

2024-10-11
Hector Hutin , Antoine Essig , Réouven Assouly +3 more
The number of excitations in a large quantum system (harmonic oscillator or qudit) can be measured in a quantum nondemolition manner using a dispersively coupled qubit. It typically requires a … The number of excitations in a large quantum system (harmonic oscillator or qudit) can be measured in a quantum nondemolition manner using a dispersively coupled qubit. It typically requires a series of qubit pulses that encode various binary questions about the photon number. Recently, a method based on the fluorescence measurement of a qubit driven by a train of identical pulses was introduced to track the photon number in a cavity, hence simplifying its monitoring and raising interesting questions about the measurement backaction of this scheme. A first realization with superconducting circuits demonstrated how the average number of photons could be measured in this way. Here we present an experiment that reaches single-shot photocounting and number tracking owing to a cavity decay rate 4 orders of magnitude smaller than both the dispersive coupling rate and the qubit emission rate. An innovative notch filter and pogo-pin-based galvanic contact makes possible these seemingly incompatible features. The qubit dynamics under the pulse train is characterized. We observe quantum jumps by monitoring the photon number via the qubit fluorescence as photons leave the cavity one at a time. Additionally, we extract the measurement rate and induced dephasing rate and compare them to theoretical models. Our method could be applied to quantum error correction protocols on bosonic codes or qudits.
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Flux-pump induced degradation of $T_1$ for dissipative cat qubits

2024-10-01
Léon Carde , Pierre Rouchon , Joachim Cohen +1 more
Dissipative stabilization of cat qubits autonomously corrects for bit flip errors by ensuring that reservoir-engineered two-photon losses dominate over other mechanisms inducing phase flip errors. To describe the latter, we … Dissipative stabilization of cat qubits autonomously corrects for bit flip errors by ensuring that reservoir-engineered two-photon losses dominate over other mechanisms inducing phase flip errors. To describe the latter, we derive an effective master equation for an asymmetrically threaded SQUID based superconducting circuit used to stabilize a dissipative cat qubit. We analyze the dressing of relaxation processes under drives in time-dependent Schrieffer-Wolff perturbation theory for weakly anharmonic bosonic degrees of freedom, and in numerically exact Floquet theory. We find that spurious single-photon decay rates can increase under the action of the parametric pump that generates the required interactions for cat-qubit stabilization. Our analysis feeds into mitigation strategies that can inform current experiments, and the methods presented here can be extended to other circuit implementations.
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On encoded quantum gate generation by iterative Lyapunov-based methods

2024-09-02
Paulo Sérgio Pereira da Silva , Pierre Rouchon
The problem of encoded quantum gate generation is studied in this paper. The idea is to consider a quantum system of higher dimension $n$ than the dimension $\bar n$ of … The problem of encoded quantum gate generation is studied in this paper. The idea is to consider a quantum system of higher dimension $n$ than the dimension $\bar n$ of the quantum gate to be synthesized. Given two orthonormal subsets $\mathbb{E} = \{e_1, e_2, \ldots, e_{\bar n}\}$ and $\mathbb F = \{f_1, f_2, \ldots, f_{\bar n}\}$ of $\mathbb{C}^n$, the problem of encoded quantum gate generation consists in obtaining an open loop control law defined in an interval $[0, T_f]$ in a way that all initial states $e_i$ are steered to $\exp(\jmath \phi) f_i, i=1,2, \ldots ,\bar n$ up to some desired precision and to some global phase $\phi \in \mathbb{R}$. This problem includes the classical (full) quantum gate generation problem, when $\bar n = n$, the state preparation problem, when $\bar n = 1$, and finally the encoded gate generation when $ 1 < \bar n < n$. Hence, three problems are unified here within a unique common approach. The \emph{Reference Input Generation Algorithm (RIGA)} is generalized in this work for considering the encoded gate generation problem for closed quantum systems. A suitable Lyapunov function is derived from the orthogonal projector on the support of the encoded gate. Three case-studies of physical interest indicate the potential interest of such numerical algorithm: two coupled transmon-qubits, a cavity mode coupled to a transmon-qubit, and a chain of $N$ qubits, including a large dimensional case for which $N=10$.
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Complete positivity violation of the reduced dynamics in higher-order quantum adiabatic elimination

2024-06-03
Masaaki Tokieda , Cyril Elouard , Alain Sarlette +1 more
This paper discusses quantum adiabatic elimination, which is a model reduction technique for a composite Lindblad system consisting of a fast decaying subsystem coupled to another subsystem with a much … This paper discusses quantum adiabatic elimination, which is a model reduction technique for a composite Lindblad system consisting of a fast decaying subsystem coupled to another subsystem with a much slower timescale. Such a system features an invariant manifold that is close to the slow subsystem. This invariant manifold is reached subsequent to the decay of the fast degrees of freedom, after which the slow dynamics follow on it. By parametrizing the invariant manifold, the slow dynamics can be simulated via a reduced model. To find the evolution of the reduced state, we perform an asymptotic expansion with respect to the timescale separation. So far, the second-order expansion has mostly been considered. It has then been revealed that the second-order expansion of the reduced dynamics is generally given by a Lindblad equation, which ensures complete positivity of the time evolution. In this paper, we present two examples where complete positivity of the reduced dynamics is violated with higher-order contributions. In the first example, the violation is detected for the evolution of the partial trace without truncation of the asymptotic expansion. The partial trace is not the only way to parametrize the slow dynamics. Concerning this nonuniqueness, it was conjectured in [Quantum Sci. Technol. 2, 044011 (2017)] that there exists a parameter choice ensuring complete positivity. With the second example, however, we refute this conjecture by showing that complete positivity cannot be restored in any choice of parametrization. We discuss these results in terms of the invariant slow manifold consisting of quantum correlated states.
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Explicit formulas for adiabatic elimination with fast unitary dynamics

2024-04-02
Angela Riva , Alain Sarlette , Pierre Rouchon
The so-called ``adiabatic elimination'' of fast decaying degrees of freedom in open quantum systems can be performed with a series expansion in the timescale separation. The associated computations are significantly … The so-called ``adiabatic elimination'' of fast decaying degrees of freedom in open quantum systems can be performed with a series expansion in the timescale separation. The associated computations are significantly more difficult when the remaining degrees of freedom (center manifold) follow fast unitary dynamics instead of just being slow. This paper highlights how a formulation with Sylvester's equation and with adjoint dynamics leads to systematic, explicit expressions at high orders for settings of physical interest.
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Gate generation for open quantum systems via a monotonic algorithm with time optimization

2024-03-29
Paulo Sérgio Pereira da Silva , Pierre Rouchon
We present a monotonic numerical algorithm including time optimization for generating quantum gates for open systems. Such systems are assumed to be governed by Lindblad master equations for the density … We present a monotonic numerical algorithm including time optimization for generating quantum gates for open systems. Such systems are assumed to be governed by Lindblad master equations for the density operators on a large Hilbert-space whereas the quantum gates are relative to a sub-space of small dimension. Starting from an initial seed of the control input, this algorithm consists in the repetition of the following two steps producing a new control input: (A) backwards integration of adjoint Lindblad-Master equations (in the Heisenberg-picture) from a set of final conditions encoding the quantum gate to generate; (B) forward integration of Lindblad-Master equations in closed-loop where a Lyapunov based control produced the new control input. The numerical stability is ensured by the stability of both the open-loop adjoint backward system and the forward closed-loop system. A clock-control input can be added to the usual control input. The obtained monotonic algorithm allows then to optimise not only the shape of the control imput, but also the gate time. Preliminary numerical implementations indicate that this algorithm is well suited for cat-qubit gates, where Hilbert-space dimensions (2 for the Z-gate and 4 for the CNOT-gate) are much smaller than the dimension of the physical Hilbert-space involving mainly Fock-states (typically 20 or larger for a single cat-qubit). This monotonic algorithm, based on Lyapunov control techniques, is shown to have a straightforward interpretation in terms of optimal control: its stationary conditions coincides with the first-order optimality conditions for a cost depending linearly on the final values of the quantum states.
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Online Parameter Estimation for Continuously Monitored Quantum Systems

2024-03-07
Henrik Glavind Clausen , Pierre Rouchon , Rafał Wiśniewski
In this work, we consider the problem of online (real-time, single-shot) estimation of static or slow-varying parameters along quantum trajectories in quantum dynamical systems. Based on the measurement signal of … In this work, we consider the problem of online (real-time, single-shot) estimation of static or slow-varying parameters along quantum trajectories in quantum dynamical systems. Based on the measurement signal of a continuously-monitored quantum system, we propose a recursive algorithm for computing the maximum likelihood estimate of unknown parameters using an approach based on stochastic gradient ascent on the log-likelihood function. We formulate the algorithm in both discrete-time and continuous-time and illustrate the performance of the algorithm through simulations of a simple two-level system undergoing homodyne measurement from which we are able to track multiple parameters simultaneously.
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Adiabatic elimination for composite open quantum systems: Reduced-model formulation and numerical simulations

2024-03-06
François-Marie Le Régent , Pierre Rouchon
A numerical method is proposed for simulation of composite open quantum systems. It is based on Lindblad master equations and adiabatic elimination. Each subsystem is assumed to converge exponentially towards … A numerical method is proposed for simulation of composite open quantum systems. It is based on Lindblad master equations and adiabatic elimination. Each subsystem is assumed to converge exponentially towards a stationary subspace, slightly impacted by some decoherence channels and weakly coupled to the other subsystems. This numerical method is based on a perturbation analysis with an asymptotic expansion. It exploits the formulation of the slow dynamics with reduced dimension. It relies on the invariant operators of the local and nominal dissipative dynamics attached to each subsystem. Second-order expansion can be computed only with local numerical calculations. It avoids computations on the tensor-product Hilbert space attached to the full system. This numerical method is particularly well suited for autonomous quantum error correction schemes. Simulations of such reduced models agree with complete full model simulations for typical gates acting on one and two cat-qubits (Z, ZZ and CNOT) when the mean photon number of each cat-qubit is less than 8. For larger mean photon numbers and gates with three cat-qubits (ZZZ and CCNOT), full model simulations are almost impossible whereas reduced model simulations remain accessible. In particular, they capture both the dominant phase-flip error-rate and the very small bit-flip error-rate with its exponential suppression versus the mean photon number.
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Monitoring the energy of a cavity by observing the emission of a repeatedly excited qubit

2024-02-07
Hector Hutin , Antoine Essig , Réouven Assouly +3 more
The number of excitations in a large quantum system (harmonic oscillator or qudit) can be measured in a quantum non demolition manner using a dispersively coupled qubit. It typically requires … The number of excitations in a large quantum system (harmonic oscillator or qudit) can be measured in a quantum non demolition manner using a dispersively coupled qubit. It typically requires a series of qubit pulses that encode various binary questions about the photon number. Recently, a method based on the fluorescence measurement of a qubit driven by a train of identical pulses was introduced to track the photon number in a cavity, hence simplifying its monitoring and raising interesting questions about the measurement backaction of this scheme. A first realization with superconducting circuits demonstrated how the average number of photons could be measured in this way. Here we present an experiment that reaches single shot photocounting and number tracking owing to a cavity decay rate 4 orders of magnitude smaller than both the dispersive coupling rate and the qubit emission rate. An innovative notch filter and pogo-pin based galvanic contact makes possible these seemingly incompatible features. The qubit dynamics under the pulse train is characterized. We observe quantum jumps by monitoring the photon number via the qubit fluorescence as photons leave the cavity one at a time. Besides, we extract the measurement rate and induced dephasing rate and compare them to theoretical models. Our method could be applied to quantum error correction protocols on bosonic codes or qudits.
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Online Parameter Estimation for Continuously Monitored Quantum Systems

2024-01-01
Henrik Glavind Clausen , Pierre Rouchon , Rafał Wiśniewski
In this work, we consider the problem of online (real-time, single-shot) estimation of static or slow-varying parameters along quantum trajectories in quantum dynamical systems. Based on the measurement signal of … In this work, we consider the problem of online (real-time, single-shot) estimation of static or slow-varying parameters along quantum trajectories in quantum dynamical systems. Based on the measurement signal of a continuously monitored quantum system, we propose a recursive algorithm for computing the maximum likelihood (ML) estimate of unknown parameters using an approach based on stochastic gradient ascent on the log-likelihood function. We formulate the algorithm in both discrete-time and continuous-time and illustrate the performance of the algorithm through simulations of a simple two-level system undergoing homodyne measurement from which we are able to track multiple parameters simultaneously.
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Gate generation for open quantum systems via a monotonic algorithm with time optimization

2024-01-01
Paulo Sérgio Pereira da Silva , Pierre Rouchon

Heisenberg Formulation of Adiabatic Elimination for Open Quantum Systems with Two Timescales

2023-12-13
François-Marie Le Régent , Pierre Rouchon
Consider an open quantum system governed by a Gorini-Kossakowski-Sudarshan-Lindblad master equation with two times-scales: a fast one, exponentially converging towards a linear subspace of quasi-equilibria; a slow one resulting from … Consider an open quantum system governed by a Gorini-Kossakowski-Sudarshan-Lindblad master equation with two times-scales: a fast one, exponentially converging towards a linear subspace of quasi-equilibria; a slow one resulting from small decoherence and Hamiltonian dynamics. Usually adiabatic elimination is performed in the Schrödinger picture. We propose here a Heisenberg formulation where the invariant operators attached to the fast decay dynamics towards the quasi-equilibria subspace play a key role. Based on geometric singular perturbations, asymptotic expansions of the Heisenberg slow dynamics and of the fast invariant linear subspaces are proposed. They exploit Carr's approximation lemma from center-manifold and bifurcation theory. Second-order expansions are detailed and shown to ensure preservation, up to second-order terms, of the complete positivity for the slow propagator on a slow timescale. Such expansions can be exploited numerically to derive reduced-order dynamical models.
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One Hundred Second Bit-Flip Time in a Two-Photon Dissipative Oscillator

2023-06-23
Camille Berdou , Anil Murani , U. Réglade +7 more
Current implementations of quantum bits (qubits) continue to undergo too many errors to be scaled into useful quantum machines. An emerging strategy is to encode quantum information in the two … Current implementations of quantum bits (qubits) continue to undergo too many errors to be scaled into useful quantum machines. An emerging strategy is to encode quantum information in the two meta-stable pointer states of an oscillator exchanging pairs of photons with its environment, a mechanism shown to provide stability without inducing decoherence. Adding photons in these states increases their separation, and macroscopic bit-flip times are expected even for a handful of photons, a range suitable to implement a qubit. However, previous experimental realizations have saturated in the millisecond range. In this work, we aim for the maximum bit-flip time we could achieve in a two-photon dissipative oscillator. To this end, we design a Josephson circuit in a regime that circumvents all suspected dynamical instabilities, and employ a minimally invasive fluorescence detection tool, at the cost of a two-photon exchange rate dominated by single-photon loss. We attain bit-flip times of the order of 100 seconds for states pinned by two-photon dissipation and containing about 40 photons. This experiment lays a solid foundation from which the two-photon exchange rate can be gradually increased, thus gaining access to the preparation and measurement of quantum superposition states, and pursuing the route towards a logical qubit with built-in bit-flip protection.
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Complete Positivity Violation in Higher-order Quantum Adiabatic Elimination

2023-01-01
Masaaki Tokieda , Cyril Elouard , Alain Sarlette +1 more
When a composite Lindblad system consists of weakly coupled sub-systems with fast and slow timescales, the description of slow dynamics can be simplified by discarding fast degrees of freedom. This … When a composite Lindblad system consists of weakly coupled sub-systems with fast and slow timescales, the description of slow dynamics can be simplified by discarding fast degrees of freedom. This model reduction technique is called adiabatic elimination. While second-order perturbative expansion with respect to the timescale separation has revealed that the evolution of a reduced state is completely positive, this paper presents an example exhibiting complete positivity violation in the fourth-order expansion. Despite the non-uniqueness of slow dynamics parametrization, we prove that complete positivity cannot be ensured in any parametrization. The violation stems from correlation in the initial state.
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Correlation functions for realistic continuous quantum measurement

2023-01-01
Pierre Guilmin , Pierre Rouchon , Antoine Tilloy
We propose a self-contained and accessible derivation of an exact formula for the n-point correlation functions of the signal measured when continuously observing a quantum system. The expression depends on … We propose a self-contained and accessible derivation of an exact formula for the n-point correlation functions of the signal measured when continuously observing a quantum system. The expression depends on the initial quantum state and on the Stochastic Master Equation (SME) governing the dynamics. This derivation applies to both jump and diffusive evolutions and takes into account common imperfections of realistic measurement devices. We show how these correlations can be efficiently computed numerically for commonly filtered and integrated signals available in practice.
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Complete positivity violation of the reduced dynamics in higher-order quantum adiabatic elimination

2023-01-01
Masaaki Tokieda , Cyril Elouard , Alain Sarlette +1 more
This paper discusses quantum adiabatic elimination, which is a model reduction technique for a composite Lindblad system consisting of a fast decaying sub-system coupled to another sub-system with a much … This paper discusses quantum adiabatic elimination, which is a model reduction technique for a composite Lindblad system consisting of a fast decaying sub-system coupled to another sub-system with a much slower timescale. Such a system features an invariant manifold that is close to the slow sub-system. This invariant manifold is reached subsequent to the decay of the fast degrees of freedom, after which the slow dynamics follow on it. By parametrizing invariant manifold, the slow dynamics can be simulated via a reduced model. To find the evolution of the reduced state, we perform the asymptotic expansion with respect to the timescale separation. So far, the second-order expansion has mostly been considered. It has then been revealed that the second-order expansion of the reduced dynamics is generally given by a Lindblad equation, which ensures complete positivity of the time evolution. In this paper, we present two examples where complete positivity of the reduced dynamics is violated with higher-order contributions. In the first example, the violation is detected for the evolution of the partial trace without truncation of the asymptotic expansion. The partial trace is not the only way to parametrize the slow dynamics. Concerning this non-uniqueness, it was conjectured in [R. Azouit, F. Chittaro, A. Sarlette, and P. Rouchon, Quantum Sci. Technol. 2, 044011 (2017)] that there exists a parameter choice ensuring complete positivity. With the second example, however, we refute this conjecture by showing that complete positivity cannot be restored in any choice of parametrization. We discuss these results in terms of unavoidable correlations, in the initial states on the invariant slow manifold, between the fast and the slow degrees of freedom.
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A GKP qubit protected by dissipation in a high-impedance superconducting circuit driven by a microwave frequency comb

2023-01-01
Lev-Arcady Sellem , Alain Sarlette , Zaki Leghtas +3 more
We propose a novel approach to generate, protect and control GKP qubits. It employs a microwave frequency comb parametrically modulating a Josephson circuit to enforce a dissipative dynamics of a … We propose a novel approach to generate, protect and control GKP qubits. It employs a microwave frequency comb parametrically modulating a Josephson circuit to enforce a dissipative dynamics of a high impedance circuit mode, autonomously stabilizing the finite-energy GKP code. The encoded GKP qubit is robustly protected against all dominant decoherence channels plaguing superconducting circuits but quasi-particle poisoning. In particular, noise from ancillary modes leveraged for dissipation engineering does not propagate at the logical level. In a state-of-the-art experimental setup, we estimate that the encoded qubit lifetime could extend two orders of magnitude beyond the break-even point, with substantial margin for improvement through progress in fabrication and control electronics. Qubit initialization, readout and control via Clifford gates can be performed while maintaining the code stabilization, paving the way toward the assembly of GKP qubits in a fault-tolerant quantum computing architecture.
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Stability and decoherence rates of a GKP qubit protected by dissipation

2023-01-01
Lev-Arcady Sellem , Rémi Robin , Philippe Campagne-Ibarcq +1 more
We analyze an experimentally accessible Lindblad master equation for a quantum harmonic oscillator. It approximately stabilizes finite-energy periodic grid states called Gottesman-Kitaev-Preskill (GKP) states, that can be used to encode … We analyze an experimentally accessible Lindblad master equation for a quantum harmonic oscillator. It approximately stabilizes finite-energy periodic grid states called Gottesman-Kitaev-Preskill (GKP) states, that can be used to encode and protect a logical qubit. We give explicit upper bounds for the energy of the solutions of the Lindblad master equation. Using three periodic observables to define the Bloch sphere coordinates of a logical qubit, we show that their dynamics is governed by a diffusion partial differential equation on a 2D-torus with a Witten Laplacian. We show that the evolution of these logical coordinates is exponentially slow even in presence of small diffusive noise processes along the two quadratures of the phase space. Numerical simulations indicate similar results for other physically relevant noise processes.
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Quantum control of a cat-qubit with bit-flip times exceeding ten seconds

2023-01-01
Ulysse Réglade , Adrien Bocquet , Ronan Gautier +7 more
Quantum bits (qubits) are prone to several types of errors due to uncontrolled interactions with their environment. Common strategies to correct these errors are based on architectures of qubits involving … Quantum bits (qubits) are prone to several types of errors due to uncontrolled interactions with their environment. Common strategies to correct these errors are based on architectures of qubits involving daunting hardware overheads. A hopeful path forward is to build qubits that are inherently protected against certain types of errors, so that the overhead required to correct remaining ones is significantly reduced. However, the foreseen benefit rests on a severe condition: quantum manipulations of the qubit must not break the protection that has been so carefully engineered. A recent qubit - the cat-qubit - is encoded in the manifold of metastable states of a quantum dynamical system, thereby acquiring continuous and autonomous protection against bit-flips. Here, in a superconducting circuit experiment, we implement a cat-qubit with bit-flip times exceeding 10 seconds. This is a four order of magnitude improvement over previous cat-qubit implementations. We prepare and image quantum superposition states, and measure phase-flip times above 490 nanoseconds. Most importantly, we control the phase of these quantum superpositions without breaking bit-flip protection. This experiment demonstrates the compatibility of quantum control and inherent bit-flip protection at an unprecedented level, showing the viability of these dynamical qubits for future quantum technologies.
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Convergence of bipartite open quantum systems stabilized by reservoir engineering

2023-01-01
Rémi Robin , Pierre Rouchon , Lev-Arcady Sellem
We study a generic family of Lindblad master equations modeling bipartite open quantum systems, where one tries to stabilize a quantum system by carefully designing its interaction with another, dissipative, … We study a generic family of Lindblad master equations modeling bipartite open quantum systems, where one tries to stabilize a quantum system by carefully designing its interaction with another, dissipative, quantum system - a strategy known as quantum reservoir engineering. We provide sufficient conditions for convergence of the considered Lindblad equations; our setting accommodates the case where steady states are not unique but rather supported on a given subspace of the underlying Hilbert space. We apply our result to a Lindblad master equation proposed for the stabilization of so-called cat qubits, a system that received considerable attention in recent years due to its potential applications in quantum information processing.
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Stability and decoherence rates of a GKP qubit protected by dissipation⋆

2023-01-01
Lev-Arcady Sellem , Rémi Robin , Philippe Campagne-Ibarcq +1 more
We analyze an experimentally accessible Lindblad master equation for a quantum harmonic oscillator. It approximately stabilizes finite-energy periodic grid states called Gottesman-Kitaev-Preskill (GKP) states, that can be used to encode … We analyze an experimentally accessible Lindblad master equation for a quantum harmonic oscillator. It approximately stabilizes finite-energy periodic grid states called Gottesman-Kitaev-Preskill (GKP) states, that can be used to encode and protect a logical qubit. We give explicit upper bounds for the energy of the solutions of the Lindblad master equation. Using three periodic observables to define the Bloch sphere coordinates of a logical qubit, we show that their dynamics is governed by a diffusion partial differential equation on a 2D-torus with a Witten Laplacian. We show that the evolution of these logical coordinates is exponentially slow even in presence of small diffusive noise processes along the two quadratures of the phase space. Numerical simulations indicate similar results for other physically relevant noise processes.
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Structurally Stable Subharmonic Regime of a Driven Quantum Josephson Circuit

2022-12-15
Michiel Burgelman , Pierre Rouchon , Alain Sarlette +1 more
Driven quantum nonlinear oscillators, while essential for quantum technologies, are generally prone to complex chaotic dynamics that fall beyond the reach of perturbative analysis. By focusing on subharmonic bifurcations of … Driven quantum nonlinear oscillators, while essential for quantum technologies, are generally prone to complex chaotic dynamics that fall beyond the reach of perturbative analysis. By focusing on subharmonic bifurcations of a harmonically driven oscillator, we provide a recipe for the choice of the oscillator's parameters that ensures a regular dynamical behavior independently of the driving strength. We show that this suppression of chaotic phenomena is compatible with a strong quantum nonlinear effect reflected by the confinement rate in the degenerate manifold spanned by stable subharmonic orbits.
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Exponential convergence of a dissipative quantum system towards finite-energy grid states of an oscillator

2022-12-06
Lev-Arcady Sellem , Philippe Campagne-Ibarcq , Mazyar Mirrahimi +2 more
Based on the stabilizer formalism underlying Quantum Error Correction (QEC), the design of an original Lindblad master equation for the density operator of a quantum harmonic oscillator is proposed. This … Based on the stabilizer formalism underlying Quantum Error Correction (QEC), the design of an original Lindblad master equation for the density operator of a quantum harmonic oscillator is proposed. This Lindblad dynamics stabilizes exactly the finite-energy grid states introduced in 2001 by Gottesman, Kitaev and Preskill for quantum computation. Stabilization results from an exponential Lyapunov function with an explicit lower-bound on the convergence rate. Numerical simulations indicate the potential interest of such autonomous QEC in presence of non-negligible photon-losses.
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Exponential convergence of a dissipative quantum system towards finite-energy grid states of an oscillator

2022-01-01
Lev-Arcady Sellem , Philippe Campagne-Ibarcq , Mazyar Mirrahimi +2 more
Based on the stabilizer formalism underlying Quantum Error Correction (QEC), the design of an original Lindblad master equation for the density operator of a quantum harmonic oscillator is proposed. This … Based on the stabilizer formalism underlying Quantum Error Correction (QEC), the design of an original Lindblad master equation for the density operator of a quantum harmonic oscillator is proposed. This Lindblad dynamics stabilizes exactly the finite-energy grid states introduced in 2001 by Gottesman, Kitaev and Preskill for quantum computation. Stabilization results from an exponential Lyapunov function with an explicit lower-bound on the convergence rate. Numerical simulations indicate the potential interest of such autonomous QEC in presence of non-negligible photon-losses.
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Structurally stable subharmonic regime of a driven quantum Josephson circuit

2022-01-01
Michiel Burgelman , Pierre Rouchon , Alain Sarlette +1 more
Driven quantum nonlinear oscillators, while essential for quantum technologies, are generally prone to complex chaotic dynamics that fall beyond the reach of perturbative analysis. By focusing on subharmonic bifurcations of … Driven quantum nonlinear oscillators, while essential for quantum technologies, are generally prone to complex chaotic dynamics that fall beyond the reach of perturbative analysis. By focusing on subharmonic bifurcations of a harmonically driven oscillator, we provide a recipe for the choice of the oscillator's parameters that ensures a regular dynamical behavior independently of the driving strength. We show that this suppression of chaotic phenomena is compatible with a strong quantum nonlinear effect reflected by the confinement rate in the degenerate manifold spanned by stable subharmonic orbits.
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A tutorial introduction to quantum stochastic master equations based on the qubit/photon system

2022-01-01
Pierre Rouchon
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A tutorial introduction to quantum stochastic master equations based on the qubit/photon system

2022-01-01
Pierre Rouchon
From the key composite quantum system made of a two-level system (qubit) and a harmonic oscillator (photon) with resonant or dispersive interactions, one derives the corresponding quantum Stochastic Master Equations … From the key composite quantum system made of a two-level system (qubit) and a harmonic oscillator (photon) with resonant or dispersive interactions, one derives the corresponding quantum Stochastic Master Equations (SME) when either the qubits or the photons are measured. Starting with an elementary discrete-time formulation based on explicit formulae for the interaction propagators, one shows how to include measurement imperfections and decoherence. This qubit/photon quantum system illustrates the Kraus-map structure of general discrete-time SME governing the dynamics of an open quantum system subject to measurement back-action and decoherence induced by the environment. Then, on the qubit/photon system, one explains the passage to a continuous-time mathematical model where the measurement signal is either a continuous real-value signal (typically homodyne or heterodyne signal) or a discontinuous and integer-value signal obtained from a counter. During this derivation, the Kraus map formulation is preserved in an infinitesimal way. Such a derivation provides also an equivalent Kraus-map formulation to the continuous-time SME usually expressed as stochastic differential equations driven either by Wiener or Poisson processes. From such Kraus-map formulation, simple linear numerical integration schemes are derived that preserve the positivity and the trace of the density operator, i.e. of the quantum state.
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Complete Positivity Violation in Higher-order Quantum Adiabatic Elimination

2022-01-01
M. Tokieda , Cyril Elouard , Alain Sarlette +1 more
When a composite Lindblad system consists of weakly coupled sub-systems with fast and slow timescales, the description of slow dynamics can be simplified by discarding fast degrees of freedom. This … When a composite Lindblad system consists of weakly coupled sub-systems with fast and slow timescales, the description of slow dynamics can be simplified by discarding fast degrees of freedom. This model reduction technique is called adiabatic elimination. While second-order perturbative expansion with respect to the timescale separation has revealed that the evolution of a reduced state is completely positive, this paper presents an example exhibiting complete positivity violation in the fourth-order expansion. Despite the non-uniqueness of slow dynamics parametrization, we prove that complete positivity cannot be ensured in any parametrization. The violation stems from correlation in the initial state.
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Correlation functions for realistic continuous quantum measurement

2022-01-01
Pierre Guilmin , Pierre Rouchon , Antoine Tilloy
We propose a self-contained and accessible derivation of an exact formula for the $n$-point correlation functions of the signal measured when continuously observing a quantum system. The expression depends on … We propose a self-contained and accessible derivation of an exact formula for the $n$-point correlation functions of the signal measured when continuously observing a quantum system. The expression depends on the initial quantum state and on the Stochastic Master Equation (SME) governing the dynamics. This derivation applies to both jump and diffusive evolutions and takes into account common imperfections of realistic measurement devices. We show how these correlations can be efficiently computed numerically for commonly filtered and integrated signals available in practice.
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Multiplexed Photon Number Measurement

2021-08-27
Antoine Essig , Quentin Ficheux , Théau Peronnin +6 more
By using qubit frequencies to encode information, an experiment demonstrates an approach to quantum measurement that replaces sequential observations with simultaneous ones, greatly improving measurement time. By using qubit frequencies to encode information, an experiment demonstrates an approach to quantum measurement that replaces sequential observations with simultaneous ones, greatly improving measurement time.
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Bilinear Control of Schrödinger PDEs

2021-01-01
Karine Beauchard , Pierre Rouchon
This entry is an introduction to modern issues about controllability of Schrödinger PDEs with bilinear controls. This model is pertinent for a quantum particle, controlled by an electric field. We … This entry is an introduction to modern issues about controllability of Schrödinger PDEs with bilinear controls. This model is pertinent for a quantum particle, controlled by an electric field. We review recent developments in the field, with discrimination between exact and approximate controllabilities, in finite or infinite time. We also underline the variety of mathematical tools used by various teams in the last decade. The results are illustrated on several classical examples.

Multiplexed photon number measurement

2020-12-21
Antoine Essig , Quentin Ficheux , Théau Peronnin +6 more
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Sensorless rotor position estimation by PWM-induced signal injection

2020-10-18
Dilshad Surroop , Pascal Combes , Philippe Martin +1 more
We demonstrate how the rotor position of a PWM-controlled PMSM can be recovered from the measured currents, by suitably using the excitation provided by the PWM itself. This provides the … We demonstrate how the rotor position of a PWM-controlled PMSM can be recovered from the measured currents, by suitably using the excitation provided by the PWM itself. This provides the benefits of signal injection, in particular the ability to operate even at low velocity, without the drawbacks of an external probing signal. We illustrate the relevance of the approach by simulations and experimental results.
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Number-Resolved Photocounter for Propagating Microwave Mode

2020-10-14
Rémy Dassonneville , Réouven Assouly , Théau Peronnin +2 more
Detecting the presence of photons in a propagating microwave mode has been demonstrated only recently, and an important tool still missing is a photocounter able to determine in a single … Detecting the presence of photons in a propagating microwave mode has been demonstrated only recently, and an important tool still missing is a photocounter able to determine in a single shot the number of photons in an incoming mode. The authors create such a photocounter by catching an incoming wave packet in a stationary mode, and then measuring the photon number of that mode bit by bit, using an ancillary qubit. Additionally, this device can measure the envelope of the incoming wave packet $i\phantom{\rule{0}{0ex}}n$ $s\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}u$. Beyond its direct applications in quantum sensing, this photocounter allows development of quantum information protocols that benefit from real-time feedback based on photon number.
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Sensorless rotor position estimation by PWM-induced signal injection

2020-09-10
Dilshad Surroop , Pascal Combes , Philippe Martin +1 more
We demonstrate how the rotor position of a PWM-controlled PMSM can be recovered from the measured currents, by suitably using the excitation provided by the PWM itself. This provides the … We demonstrate how the rotor position of a PWM-controlled PMSM can be recovered from the measured currents, by suitably using the excitation provided by the PWM itself. This provides the benefits of signal injection, in particular the ability to operate even at low velocity, without the drawbacks of an external probing signal. We illustrate the relevance of the approach by simulations and experimental results.

Adding virtual measurements by PWM-induced signal injection

2020-07-01
Dilshad Surroop , Pascal Combes , Philippe Martin +1 more
We show that for PWM-operated devices, it is possible to benefit from signal injection without an external probing signal, by suitably using the excitation provided by the PWM itself. As … We show that for PWM-operated devices, it is possible to benefit from signal injection without an external probing signal, by suitably using the excitation provided by the PWM itself. As in the usual signal injection framework conceptualized in [1], an extra "virtual measurement" can be made available for use in a control law, but without the practical drawbacks caused by an external signal.
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Quantum adiabatic elimination at arbitrary order for photon number measurement

2020-01-01
Alain Sarlette , Pierre Rouchon , Antoine Essig +2 more
Adiabatic elimination is a perturbative model reduction technique based on timescale separation and often used to simplify the description of composite quantum systems. We here analyze a quantum experiment where … Adiabatic elimination is a perturbative model reduction technique based on timescale separation and often used to simplify the description of composite quantum systems. We here analyze a quantum experiment where the perturbative expansion can be carried out to arbitrary order, such that: (i) we can formulate in the end an exact reduced model in quantum form; (ii) as the series provides accuracy for ever larger parameter values, we can discard any condition on the timescale separation, thereby analyzing the intermediate regime where the actual experiment is performing best; (iii) we can clarify the role of some gauge degrees of freedom in this model reduction technique.
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Sensorless rotor position estimation by PWM-induced signal injection

2020-01-01
Dilshad Surroop , Pascal Combes , Philippe Martin +1 more
We demonstrate how the rotor position of a PWM-controlled PMSM can be recovered from the measured currents, by suitably using the excitation provided by the PWM itself. This provides the … We demonstrate how the rotor position of a PWM-controlled PMSM can be recovered from the measured currents, by suitably using the excitation provided by the PWM itself. This provides the benefits of signal injection, in particular the ability to operate even at low velocity, without the drawbacks of an external probing signal. We illustrate the relevance of the approach by simulations and experimental results.
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Adding virtual measurements by PWM-induced signal injection

2020-01-01
Dilshad Surroop , Pascal Combes , Philippe Martin +1 more
We show that for PWM-operated devices, it is possible to benefit from signal injection \emph{without an external probing signal}, by suitably using the excitation provided by the PWM itself. As … We show that for PWM-operated devices, it is possible to benefit from signal injection \emph{without an external probing signal}, by suitably using the excitation provided by the PWM itself. As in the usual signal injection framework conceptualized in [1], an extra "virtual measurement" can be made available for use in a control law, but without the practical drawbacks caused by an external signal.
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Quantum adiabatic elimination at arbitrary order for photon number measurement

2020-01-01
Alain Sarlette , Pierre Rouchon , Antoine Essig +2 more
Adiabatic elimination is a perturbative model reduction technique based on timescale separation and often used to simplify the description of composite quantum systems. We here analyze a quantum experiment where … Adiabatic elimination is a perturbative model reduction technique based on timescale separation and often used to simplify the description of composite quantum systems. We here analyze a quantum experiment where the perturbative expansion can be carried out to arbitrary order, such that: (i) we can formulate in the end an exact reduced model in quantum form; (ii) as the series provides accuracy for ever larger parameter values, we can discard any condition on the timescale separation, thereby analyzing the intermediate regime where the actual experiment is performing best; (iii) we can clarify the role of some gauge degrees of freedom in this model reduction technique.
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Exponential stabilization of quantum systems under continuous non-demolition measurements

2019-11-28
Gerardo Cardona , Alain Sarlette , Pierre Rouchon
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Benchmarking Maximum-Likelihood State Estimation with an Entangled Two-Cavity State

2019-08-09
Valentin Métillon , Stefan Gerlich , M. Brune +3 more
The efficient quantum state reconstruction algorithm described by Six et al. [Phys. Rev. A 93, 012109 (2016)] is experimentally implemented on the nonlocal state of two microwave cavities entangled by … The efficient quantum state reconstruction algorithm described by Six et al. [Phys. Rev. A 93, 012109 (2016)] is experimentally implemented on the nonlocal state of two microwave cavities entangled by a circular Rydberg atom. We use information provided by long sequences of measurements performed by resonant and dispersive probe atoms over timescales involving the system decoherence. Moreover, we benefit from the consolidation, in the same reconstruction, of different measurement protocols providing complementary information. Finally, we obtain realistic error bars for the matrix elements of the reconstructed density operator. These results demonstrate the pertinence and precision of the method, directly applicable to any complex quantum system.
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Exponential stabilization of quantum systems under continuous non-demolition measurements

2019-06-18
Gerardo Cardona , Alain Sarlette , Pierre Rouchon
We present a novel continuous-time control strategy to exponentially stabilize an eigenstate of a Quantum Non-Demolition (QND) measurement operator. In open-loop, the system converges to a random eigenstate of the … We present a novel continuous-time control strategy to exponentially stabilize an eigenstate of a Quantum Non-Demolition (QND) measurement operator. In open-loop, the system converges to a random eigenstate of the measurement operator. The role of the feedback is to prepare a prescribed QND eigenstate with unit probability. To achieve this we introduce the use of Brownian motion to drive the unitary control actions; the feedback loop just adapts the amplitude of this Brownian noise input as a function of the system state. Essentially, it shakes the system away from undesired eigenstates by applying strong noise there, while relying on the open-loop dynamics to progressively reach the target. We prove exponential convergence towards the target eigenstate using standard stochastic Lyapunov methods. The feedback scheme and its stability analysis suggest the use of an approximate filter which only tracks the populations of the eigenstates of the measurement operator. Such reduced filters should play an increasing role towards advanced quantum technologies.

Exact Controllability of a Linear Korteweg--de Vries Equation by the Flatness Approach

2019-01-01
Philippe Martin , Ivonne Rivas , Lionel Rosier +1 more
We consider a linear Korteweg--de Vries equation on a bounded domain with a left Dirichlet boundary control. The controllability to the trajectories of such a system was proved in the … We consider a linear Korteweg--de Vries equation on a bounded domain with a left Dirichlet boundary control. The controllability to the trajectories of such a system was proved in the last decade by using Carleman estimates. Here, we go a step further by establishing the exact controllability in a space of analytic functions with the aid of the flatness approach.
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Continuous-time Quantum Error Correction with Noise-assisted Quantum Feedback

2019-01-01
Gerardo Cardona , Alain Sarlette , Pierre Rouchon
We address the standard quantum error correction using the three-qubit bit-flip code, yet in continuous-time. This entails rendering a target manifold of quantum states globally attractive. Previous feedback designs could … We address the standard quantum error correction using the three-qubit bit-flip code, yet in continuous-time. This entails rendering a target manifold of quantum states globally attractive. Previous feedback designs could feature spurious equilibria, or resort to discrete kicks pushing the system away from these equilibria to ensure global asymptotic stability. We present a new approach that consists of introducing controls driven by Brownian motions. Unlike the previous methods, the resulting closed-loop dynamics can be shown to stabilize the target manifold exponentially. We further present a reduced-order filter formulation with classical probabilities. The exponential property is important to quantify the protection induced by the closed-loop error-correction dynamics against disturbances. We study numerically the performance of this control law and of the reduced filter.
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Exponential stabilization of quantum systems under continuous non-demolition measurements

2019-01-01
Gerardo Cardona , Alain Sarlette , Pierre Rouchon
We present a novel continuous-time control strategy to exponentially stabilize an eigenstate of a Quantum Non-Demolition (QND) measurement operator. In open-loop, the system converges to a random eigenstate of the … We present a novel continuous-time control strategy to exponentially stabilize an eigenstate of a Quantum Non-Demolition (QND) measurement operator. In open-loop, the system converges to a random eigenstate of the measurement operator. The role of the feedback is to prepare a prescribed QND eigenstate with unit probability. To achieve this we introduce the use of Brownian motion to drive the unitary control actions; the feedback loop just adapts the amplitude of this Brownian noise input as a function of the system state. Essentially, it "shakes" the system away from undesired eigenstates by applying strong noise there, while relying on the open-loop dynamics to progressively reach the target. We prove exponential convergence towards the target eigenstate using standard stochastic Lyapunov methods. The feedback scheme and its stability analysis suggest the use of an approximate filter which only tracks the populations of the eigenstates of the measurement operator. Such reduced filters should play an increasing role towards advanced quantum technologies.
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Continuous-time Quantum Error Correction with Noise-assisted Quantum Feedback

2019-01-01
Gerardo Cardona , Alain Sarlette , Pierre Rouchon
We address the standard quantum error correction using the three-qubit bit-flip code, yet in continuous-time. This entails rendering a target manifold of quantum states globally attractive. Previous feedback designs could … We address the standard quantum error correction using the three-qubit bit-flip code, yet in continuous-time. This entails rendering a target manifold of quantum states globally attractive. Previous feedback designs could feature spurious equilibria, or resort to discrete kicks pushing the system away from these equilibria to ensure global asymptotic stability. We present a new approach that consists of introducing controls driven by Brownian motions. Unlike the previous methods, the resulting closed-loop dynamics can be shown to stabilize the target manifold exponentially. We further present a reduced-order filter formulation with classical probabilities. The exponential property is important to quantify the protection induced by the closed-loop error-correction dynamics against disturbances. We study numerically the performance of this control law and of the reduced filter.
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Adiabatic Elimination for Multi-Partite Open Quantum Systems with Non-Trivial Zero-Order Dynamics

2018-12-01
Paolo Forni , Alain Sarlette , Thibault Capelle +3 more
We provide model reduction formulas for open quantum systems consisting of a target component which weakly interacts with a strongly dissipative environment. The time-scale separation between the uncoupled dynamics and … We provide model reduction formulas for open quantum systems consisting of a target component which weakly interacts with a strongly dissipative environment. The time-scale separation between the uncoupled dynamics and the interaction allows to employ tools from center manifold theory and geometric singular perturbation theory to eliminate the variables associated to the environment (adiabatic elimination) with high-order accuracy. An important specificity is to preserve the quantum structure: reduced dynamics in (positive) Lindblad form and coordinate mappings in Kraus form. We provide formulas of the reduced dynamics. The main contributions of this paper are (i) to show how the decomposition of the environment into $K$ components enables its efficient treatment, avoiding the quantum curse of dimension; and (ii) to extend the results to the case where the target component is subject to Hamiltonian evolution at the fast time-scale. We apply our theory to a microwave superconducting quantum resonator subject to material losses, and we show that our reduced-order model can explain the transmission spectrum observed in a recent pump probe experiment.
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Common Coauthors

Coauthor Papers Together
Philippe Martin 73
Alain Sarlette 46
Mazyar Mirrahimi 36
Lionel Rosier 25
François Malrait 21
Paulo Sérgio Pereira da Silva 18
Pascal Combes 17
Benjamin Huard 14
Hadis Amini 13
Karine Beauchard 12
Rémi Azouit 11
Al Kassem Jebai 11
Zaki Leghtas 10
I. Dotsenko 10
Philippe Campagne-Ibarcq 9
Silvère Bonnabel 9
Lev-Arcady Sellem 8
Michel Fliesś 8
Jean Lévine 8
Gerardo Cardona 6
Ram Somaraju 6
Nicolas Petit 6
Pierre Six 6
Antoine Essig 6
C. Sayrin 5
Quentin Ficheux 5
J. M. Raimond 5
Dilshad Surroop 5
S. Jézouin 4
Cyril Elouard 4
Michel Fliess 4
Francesca Chittaro 4
Nathanaël Cottet 4
Landry Bretheau 4
A. Sarlette 4
Hector Bessa Silveira 4
M. Brune 4
Théau Peronnin 4
Rémi Robin 4
Nadège Zarrouati-Vissière 3
Abhinav Somaraju 3
Réouven Assouly 3
Antoine Tilloy 3
Emmanuel Flurin 3
Pierre Guilmin 3
Nadège Zarrouati 3
S. Deléglise 3
Masaaki Tokieda 3
Thibault Capelle 3
Erwan Salaün 3

Commonly Cited References

Quantum feedback by discrete quantum nondemolition measurements: Towards on-demand generation of photon-number states

2009-07-09
I. Dotsenko , Mazyar Mirrahimi , M. Brune +3 more
We propose a quantum feedback scheme for the preparation and protection of photon-number states of light trapped in a high-$Q$ microwave cavity. A quantum nondemolition measurement of the cavity field … We propose a quantum feedback scheme for the preparation and protection of photon-number states of light trapped in a high-$Q$ microwave cavity. A quantum nondemolition measurement of the cavity field provides information on the photon-number distribution. The feedback loop is closed by injecting into the cavity a coherent pulse adjusted to increase the probability of the target photon number. The efficiency and reliability of the closed-loop state stabilization is assessed by quantum Monte Carlo simulations. We show that, in realistic experimental conditions, the Fock states are efficiently produced and protected against decoherence.
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Motion planning for the heat equation

2000-01-01
Béatrice Laroche , Philippe Martin , Pierre Rouchon
The paper gives an explicit open-loop control, able to approximately steer the one-dimensional heat equation with control on the boundary from any state to any other state. The control is … The paper gives an explicit open-loop control, able to approximately steer the one-dimensional heat equation with control on the boundary from any state to any other state. The control is obtained thanks to a parametrization of the solutions of the heat equation by a series involving infinitely many derivatives of the system 'flat output'. Copyright © 2000 John Wiley & Sons, Ltd.

Real-time quantum feedback prepares and stabilizes photon number states

2011-08-30
C. Sayrin , I. Dotsenko , Xingxing Zhou +7 more
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Confining the state of light to a quantum manifold by engineered two-photon loss

2015-02-20
Zaki Leghtas , Steven Touzard , Ioan M. Pop +7 more
A way to induce quantum stability Dynamical systems, whether classical or quantum, usually require a method to stabilize performance and maintain the required state. For instance, communication between computers requires … A way to induce quantum stability Dynamical systems, whether classical or quantum, usually require a method to stabilize performance and maintain the required state. For instance, communication between computers requires error correction codes to ensure that information is transferred correctly. In a quantum system, however, the very act of measuring it can perturb it. Leghtas et al. show that engineering the interaction between a quantum system and its environment can induce stability for the delicate quantum states, a process that could simplify quantum information processing. Science , this issue p. 853
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Dynamically protected cat-qubits: a new paradigm for universal quantum computation

2014-04-22
Mazyar Mirrahimi , Zaki Leghtas , Victor V. Albert +4 more
We present a new hardware-efficient paradigm for universal quantum computation which is based on encoding, protecting and manipulating quantum information in a quantum harmonic oscillator. This proposal exploits multi-photon driven … We present a new hardware-efficient paradigm for universal quantum computation which is based on encoding, protecting and manipulating quantum information in a quantum harmonic oscillator. This proposal exploits multi-photon driven dissipative processes to encode quantum information in logical bases composed of Schrödinger cat states. More precisely, we consider two schemes. In a first scheme, a two-photon driven dissipative process is used to stabilize a logical qubit basis of two-component Schrödinger cat states. While such a scheme ensures a protection of the logical qubit against the photon dephasing errors, the prominent error channel of single-photon loss induces bit-flip type errors that cannot be corrected. Therefore, we consider a second scheme based on a four-photon driven dissipative process which leads to the choice of four-component Schrödinger cat states as the logical qubit. Such a logical qubit can be protected against single-photon loss by continuous photon number parity measurements. Next, applying some specific Hamiltonians, we provide a set of universal quantum gates on the encoded qubits of each of the two schemes. In particular, we illustrate how these operations can be rendered fault-tolerant with respect to various decoherence channels of participating quantum systems. Finally, we also propose experimental schemes based on quantum superconducting circuits and inspired by methods used in Josephson parametric amplification, which should allow one to achieve these driven dissipative processes along with the Hamiltonians ensuring the universal operations in an efficient manner.
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Real and Complex Analysis.

1987-01-01
G. A. Garreau , Walter Rudin
Preface Prologue: The Exponential Function Chapter 1: Abstract Integration Set-theoretic notations and terminology The concept of measurability Simple functions Elementary properties of measures Arithmetic in [0, ] Integration of positive … Preface Prologue: The Exponential Function Chapter 1: Abstract Integration Set-theoretic notations and terminology The concept of measurability Simple functions Elementary properties of measures Arithmetic in [0, ] Integration of positive functions Integration of complex functions The role played by sets of measure zero Exercises Chapter 2: Positive Borel Measures Vector spaces Topological preliminaries The Riesz representation theorem Regularity properties of Borel measures Lebesgue measure Continuity properties of measurable functions Exercises Chapter 3: Lp-Spaces Convex functions and inequalities The Lp-spaces Approximation by continuous functions Exercises Chapter 4: Elementary Hilbert Space Theory Inner products and linear functionals Orthonormal sets Trigonometric series Exercises Chapter 5: Examples of Banach Space Techniques Banach spaces Consequences of Baire's theorem Fourier series of continuous functions Fourier coefficients of L1-functions The Hahn-Banach theorem An abstract approach to the Poisson integral Exercises Chapter 6: Complex Measures Total variation Absolute continuity Consequences of the Radon-Nikodym theorem Bounded linear functionals on Lp The Riesz representation theorem Exercises Chapter 7: Differentiation Derivatives of measures The fundamental theorem of Calculus Differentiable transformations Exercises Chapter 8: Integration on Product Spaces Measurability on cartesian products Product measures The Fubini theorem Completion of product measures Convolutions Distribution functions Exercises Chapter 9: Fourier Transforms Formal properties The inversion theorem The Plancherel theorem The Banach algebra L1 Exercises Chapter 10: Elementary Properties of Holomorphic Functions Complex differentiation Integration over paths The local Cauchy theorem The power series representation The open mapping theorem The global Cauchy theorem The calculus of residues Exercises Chapter 11: Harmonic Functions The Cauchy-Riemann equations The Poisson integral The mean value property Boundary behavior of Poisson integrals Representation theorems Exercises Chapter 12: The Maximum Modulus Principle Introduction The Schwarz lemma The Phragmen-Lindelof method An interpolation theorem A converse of the maximum modulus theorem Exercises Chapter 13: Approximation by Rational Functions Preparation Runge's theorem The Mittag-Leffler theorem Simply connected regions Exercises Chapter 14: Conformal Mapping Preservation of angles Linear fractional transformations Normal families The Riemann mapping theorem The class L Continuity at the boundary Conformal mapping of an annulus Exercises Chapter 15: Zeros of Holomorphic Functions Infinite Products The Weierstrass factorization theorem An interpolation problem Jensen's formula Blaschke products The Muntz-Szas theorem Exercises Chapter 16: Analytic Continuation Regular points and singular points Continuation along curves The monodromy theorem Construction of a modular function The Picard theorem Exercises Chapter 17: Hp-Spaces Subharmonic functions The spaces Hp and N The theorem of F. and M. Riesz Factorization theorems The shift operator Conjugate functions Exercises Chapter 18: Elementary Theory of Banach Algebras Introduction The invertible elements Ideals and homomorphisms Applications Exercises Chapter 19: Holomorphic Fourier Transforms Introduction Two theorems of Paley and Wiener Quasi-analytic classes The Denjoy-Carleman theorem Exercises Chapter 20: Uniform Approximation by Polynomials Introduction Some lemmas Mergelyan's theorem Exercises Appendix: Hausdorff's Maximality Theorem Notes and Comments Bibliography List of Special Symbols Index

Geometric singular perturbation theory for ordinary differential equations

1979-01-01
Neil Fenichel

Quantum stochastic calculus and quantum nonlinear filtering

1992-08-01
V. P. Belavkin
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Stabilizing Feedback Controls for Quantum Systems

2007-01-01
Mazyar Mirrahimi , Ramon van Handel
No quantum measurement can give full information on the state of a quantum system; hence any quantum feedback control problem is necessarily one with partial observations and can generally be … No quantum measurement can give full information on the state of a quantum system; hence any quantum feedback control problem is necessarily one with partial observations and can generally be converted into a completely observed control problem for an appropriate quantum filter as in classical stochastic control theory. Here we study the properties of controlled quantum filtering equations as classical stochastic differential equations. We then develop methods, using a combination of geometric control and classical probabilistic techniques, for global feedback stabilization of a class of quantum filters around a particular eigenstate of the measurement operator.
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Progressive field-state collapse and quantum non-demolition photon counting

2007-08-01
Christine Guerlin , J. Bernu , S. Deléglise +6 more
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Null controllability of the heat equation using flatness

2014-10-25
Philippe Martin , Lionel Rosier , Pierre Rouchon
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A Lie-Backlund approach to equivalence and flatness of nonlinear systems

1999-05-01
Michel Fliess , Jean Lévine , Philippe Martin +1 more
A new system equivalence relation, using the framework of differential geometry of jets and prolongations of infinite order, is studied. In this setting, two systems are said to be equivalent … A new system equivalence relation, using the framework of differential geometry of jets and prolongations of infinite order, is studied. In this setting, two systems are said to be equivalent if any variable of one system may be expressed as a function of the variables of the other system and of a finite number of their time derivatives. This is a Lie-Backlund isomorphism. The authors prove that, although the state dimension is not preserved, the number of input channels is kept fixed. They also prove that a Lie-Backlund isomorphism can be realized by an endogenous feedback. The differentially flat nonlinear systems introduced by the authors (1992) via differential algebraic techniques, are generalized and the new notion of orbitally flat systems is defined. They correspond to systems which are equivalent to a trivial one, with time preservation or not. The endogenous linearizing feedback is explicitly computed in the case of the VTOL aircraft to track given reference trajectories with stability.
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Reconstruction of non-classical cavity field states with snapshots of their decoherence

2008-09-01
S. Deléglise , I. Dotsenko , C. Sayrin +4 more
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Quantum jumps of light recording the birth and death of a photon in a cavity

2007-03-01
Sébastien Gleyzes , Stefan Kuhr , Christine Guerlin +6 more
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Feedback control of quantum state reduction

2005-06-01
Ramon van Handel , John K. Stockton , Hideo Mabuchi
Feedback control of quantum mechanical systems must take into account the probabilistic nature of quantum measurement. We formulate quantum feedback control as a problem of stochastic nonlinear control by considering … Feedback control of quantum mechanical systems must take into account the probabilistic nature of quantum measurement. We formulate quantum feedback control as a problem of stochastic nonlinear control by considering separately a quantum filtering problem and a state feedback control problem for the filter. We explore the use of stochastic Lyapunov techniques for the design of feedback controllers for quantum spin systems and demonstrate the possibility of stabilizing one outcome of a quantum measurement with unit probability.
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Repetition Cat Qubits for Fault-Tolerant Quantum Computation

2019-12-12
Jérémie Guillaud , Mazyar Mirrahimi
We present a 1D repetition code based on the so-called cat qubits as a viable approach toward hardware-efficient universal and fault-tolerant quantum computation. The cat qubits that are stabilized by … We present a 1D repetition code based on the so-called cat qubits as a viable approach toward hardware-efficient universal and fault-tolerant quantum computation. The cat qubits that are stabilized by a two-photon driven-dissipative process exhibit a tunable noise bias where the effective bit-flip errors are exponentially suppressed with the average number of photons. We propose a realization of a set of gates on the cat qubits that preserve such a noise bias. Combining these base qubit operations, we build, at the level of the repetition cat qubit, a universal set of fully protected logical gates. This set includes single-qubit preparations and measurements, not, controlled-not, and controlled-controlled-not (Toffoli) gates. Remarkably, this construction avoids the costly magic state preparation, distillation, and injection. Finally, all required operations on the cat qubits could be performed with slight modifications of existing experimental setups.4 MoreReceived 29 April 2019Revised 11 October 2019DOI:https://doi.org/10.1103/PhysRevX.9.041053Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasQuantum error correctionQuantum gatesQuantum information with solid state qubitsQuantum Information
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Exponential suppression of bit-flips in a qubit encoded in an oscillator

2020-03-16
Raphaël Lescanne , Marius Villiers , Théau Peronnin +6 more
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Wave-function approach to dissipative processes in quantum optics

1992-02-03
Jean Dalibard , Yvan Castin , Klaus Mølmer
A novel treatment of dissipation of energy from a ``small'' quantum system to a reservoir is presented. We replace the usual master equation for the small-system density matrix by a … A novel treatment of dissipation of energy from a ``small'' quantum system to a reservoir is presented. We replace the usual master equation for the small-system density matrix by a wave-function evolution including a stochastic element. This wave-function approach provides new insight and it allows calculations on problems which would otherwise be exceedingly complicated. The approach is applied here to a two- or three-level atom coupled to a laser field and to the vacuum modes of the quantized electromagnetic field.
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Effective operator formalism for open quantum systems

2012-03-09
Florentin Reiter , Anders S. Sørensen
We present an effective operator formalism for open quantum systems. Employing perturbation theory and adiabatic elimination of excited states for a weakly driven system, we derive an effective master equation … We present an effective operator formalism for open quantum systems. Employing perturbation theory and adiabatic elimination of excited states for a weakly driven system, we derive an effective master equation which reduces the evolution to the ground-state dynamics. The effective evolution involves a single effective Hamiltonian and one effective Lindblad operator for each naturally occurring decay process. Simple expressions are derived for the effective operators which can be directly applied to reach effective equations of motion for the ground states. We compare our method with the hitherto existing concepts for effective interactions and present physical examples for the application of our formalism, including dissipative state preparation by engineered decay processes.
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Adiabatic elimination in a lambda system

2007-01-17
E. Brion , L H Pedersen , Klaus Mølmer
This paper deals with different ways to extract the effective two-dimensional lower level dynamics of a lambda system excited by off-resonant laser beams. We present a commonly used procedure for … This paper deals with different ways to extract the effective two-dimensional lower level dynamics of a lambda system excited by off-resonant laser beams. We present a commonly used procedure for elimination of the upper level, and we show that it may lead to ambiguous results. To overcome this problem and better understand the applicability conditions of this scheme, we review two rigorous methods which allow us both to derive an unambiguous effective two-level Hamiltonian of the system and to quantify the accuracy of the approximation achieved: the first relies on the exact solution of the Schrödinger equation, while the second resorts to the Green's function formalism and the Feshbach projection operator technique.
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Generalized Schrieffer-Wolff formalism for dissipative systems

2012-07-30
E. M. Kessler
We present a formalized perturbation theory for Markovian master equations in the language of a generalized Schrieffer-Wolff (SW) transformation. A nonunitary rotation decouples the unperturbed steady states from all fast … We present a formalized perturbation theory for Markovian master equations in the language of a generalized Schrieffer-Wolff (SW) transformation. A nonunitary rotation decouples the unperturbed steady states from all fast degrees of freedom, in order to obtain an effective Liouvillian that reproduces the exact low excitation spectrum of the system. The transformation is derived in a constructive way, yielding a perturbative expansion of the effective Liouville operator. The presented formalism realizes an adiabatic elimination of fast degrees of freedom to arbitrary order. We exemplarily employ the SW formalism to two generic open systems and discuss general properties of the different orders of the perturbation.
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Feedback generation of quantum Fock states by discrete QND measures

2009-12-01
Mazyar Mirrahimi , I. Dotsenko , Pierre Rouchon
A feedback scheme for preparation of photon number states in a microwave cavity is proposed. Quantum Non Demolition (QND) measurement of the cavity field provides information on its actual state. … A feedback scheme for preparation of photon number states in a microwave cavity is proposed. Quantum Non Demolition (QND) measurement of the cavity field provides information on its actual state. The control consists in injecting into the cavity mode a microwave pulse adjusted to maximize the population of the desired target photon number. In the ideal case (perfect cavity and measures), we present the feedback scheme and its detailed convergence proof through stochastic Lyapunov techniques based on super-martingales and other probabilistic arguments. Quantum Monte-Carlo simulations performed with experimental parameters illustrate convergence and robustness of such feedback scheme.
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Completely positive linear maps on complex matrices

1975-06-01
Man-Duen Choi

THE STABILITY OF QUANTUM MARKOV FILTERS

2009-03-01
Ramon van Handel
When are quantum filters asymptotically independent of the initial state? We show that this is the case for absolutely continuous initial states when the quantum stochastic model satisfies an observability … When are quantum filters asymptotically independent of the initial state? We show that this is the case for absolutely continuous initial states when the quantum stochastic model satisfies an observability condition. When the initial system is finite dimensional, this condition can be verified explicitly in terms of a rank condition on the coefficients of the associated quantum stochastic differential equation.
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Feedback linearization and driftless systems

1994-09-01
Philippe Martin , Pierre Rouchon

Feedback stabilization of discrete-time quantum systems subject to non-demolition measurements with imperfections and delays

2013-07-22
Hadis Amini , Ram Somaraju , I. Dotsenko +3 more
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On the generators of quantum dynamical semigroups

1976-06-01
Göran Lindblad
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Adding virtual measurements by signal injection

2016-07-01
Pascal Combes , Al Kassem Jebai , François Malrait +2 more
We propose a method to "create" a new measurement output by exciting the system with a high-frequency oscillation. This new "virtual" measurement may be useful to facilitate the design of … We propose a method to "create" a new measurement output by exciting the system with a high-frequency oscillation. This new "virtual" measurement may be useful to facilitate the design of a suitable control law. The approach is especially interesting when the observability from the actual output degenerates at a steady-state regime of interest. The proposed method is based on second-order averaging and is illustrated by simulations on a simple third-order system.
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An Introduction to Quantum Filtering

2007-01-01
Luc Bouten , Ramon van Handel , M. R. James
This paper provides an introduction to quantum filtering theory. An introduction to quantum probability theory is given, focusing on the spectral theorem and the conditional expectation as a least squares … This paper provides an introduction to quantum filtering theory. An introduction to quantum probability theory is given, focusing on the spectral theorem and the conditional expectation as a least squares estimate, and culminating in the construction of Wiener and Poisson processes on the Fock space. We describe the quantum Itô calculus and its use in the modeling of physical systems. We use both reference probability and innovations methods to obtain quantum filtering equations for system-probe models from quantum optics.
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Towards generic adiabatic elimination for bipartite open quantum systems

2017-09-08
Rémi Azouit , Francesca Chittaro , A. Sarlette +1 more
Abstract We consider a composite open quantum system consisting of a fast subsystem coupled to a slow one. Using the time scale separation, we develop an adiabatic elimination technique to … Abstract We consider a composite open quantum system consisting of a fast subsystem coupled to a slow one. Using the time scale separation, we develop an adiabatic elimination technique to derive at any order the reduced model describing the slow subsystem. The method, based on an asymptotic expansion and geometric singular perturbation theory, ensures the physical interpretation of the reduced second-order model by giving the reduced dynamics in a Lindblad form and the state reduction in Kraus map form. We give explicit second-order formulas for Hamiltonian or cascade coupling between the two subsystems. These formulas can be used to engineer, via a careful choice of the fast subsystem, the Hamiltonian and Lindbald operators governing the dissipative dynamics of the slow subsystem.
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Singular Perturbations and Lindblad-Kossakowski Differential Equations

2009-05-27
Mazyar Mirrahimi , Pierre Rouchon
We consider an ensemble of quantum systems whose average evolution is described by a density matrix, solution of a Lindblad-Kossakowski differential equation. We focus on the special case where the … We consider an ensemble of quantum systems whose average evolution is described by a density matrix, solution of a Lindblad-Kossakowski differential equation. We focus on the special case where the decoherence is only due to a highly unstable excited state and where the spontaneously emitted photons are measured by a photo-detector. We propose a systematic method to eliminate the fast and asymptotically stable dynamics associated to the excited state in order to obtain another differential equation for the slow part. We show that this slow differential equation is still of Lindblad-Kossakowski type, that the decoherence terms and the measured output depend explicitly on the amplitudes of quasi-resonant applied field, i.e., the control. Beside a rigorous proof of the slow/fast (adiabatic) reduction based on singular perturbation theory, we also provide a physical interpretation of the result in the context of coherence population trapping via dark states and decoherence-free subspaces. Numerical simulations illustrate the accuracy of the proposed approximation for a 5-level systems.
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Controllability of the discrete-spectrum Schrödinger equation driven by an external field

2008-06-28
Thomas Chambrion , Paolo Mason , Mario Sigalotti +1 more
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Quantum error correction of a qubit encoded in grid states of an oscillator

2020-08-19
Philippe Campagne-Ibarcq , Alec Eickbusch , Steven Touzard +7 more
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Lyapunov control of a quantum particle in a decaying potential

2009-01-09
Mazyar Mirrahimi
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Null boundary controllability for semilinear heat equations

1995-11-01
Yung-Jen Lin Guo , Walter Littman
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Quantum Back-Action of an Individual Variable-Strength Measurement

2013-01-10
Michael Hatridge , Shyam Shankar , Mazyar Mirrahimi +7 more
Measuring a quantum system can randomly perturb its state. The strength and nature of this back-action depends on the quantity which is measured. In a partial measurement performed by an … Measuring a quantum system can randomly perturb its state. The strength and nature of this back-action depends on the quantity which is measured. In a partial measurement performed by an ideal apparatus, quantum physics predicts that the system remains in a pure state whose evolution can be tracked perfectly from the measurement record. We demonstrate this property using a superconducting qubit dispersively coupled to a cavity traversed by a microwave signal. The back-action on the qubit state of a single measurement of both signal quadratures is observed and shown to produce a stochastic operation whose action is determined by the measurement result. This accurate monitoring of a qubit state is an essential prerequisite for measurement-based feedback control of quantum systems.
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Encoding a qubit in an oscillator

2001-06-11
Daniel Gottesman , Alexei Kitaev , John Preskill
Quantum error-correcting codes are constructed that embed a finite-dimensional code space in the infinite-dimensional Hilbert space of a system described by continuous quantum variables. These codes exploit the noncommutative geometry … Quantum error-correcting codes are constructed that embed a finite-dimensional code space in the infinite-dimensional Hilbert space of a system described by continuous quantum variables. These codes exploit the noncommutative geometry of phase space to protect against errors that shift the values of the canonical variables q and p. In the setting of quantum optics, fault-tolerant universal quantum computation can be executed on the protected code subspace using linear optical operations, squeezing, homodyne detection, and photon counting; however, nonlinear mode coupling is required for the preparation of the encoded states. Finite-dimensional versions of these codes can be constructed that protect encoded quantum information against shifts in the amplitude or phase of a d-state system. Continuous-variable codes can be invoked to establish lower bounds on the quantum capacity of Gaussian quantum channels.
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Observing Interferences between Past and Future Quantum States in Resonance Fluorescence

2014-05-05
P. Campagne-Ibarcq , Landry Bretheau , Emmanuel Flurin +3 more
In quantum physics, measurement results are random but their statistics can be predicted assuming some knowledge about the system in the past. Additional knowledge from a future measurement deeply changes … In quantum physics, measurement results are random but their statistics can be predicted assuming some knowledge about the system in the past. Additional knowledge from a future measurement deeply changes the statistics in the present and leads to purely quantum features. In particular conditioned average outcomes of a weak measurement, so-called weak values, were shown to go beyond the conventional range, give a way to directly measure complex quantities, and can be used to enhance the sensitivity of quantum meters. Recently, these concepts have been considered in the general case of open quantum systems where decoherence occurs. Then, what are the properties of weak values for the unavoidable measurement associated to decoherence, the one performed by the environment? Here, we answer this question in the simplest open quantum system: a quantum bit in presence of a relaxation channel. We continuously monitor the fluorescence emitted by a superconducting qubit driven at resonance. Conditioned on initial preparation and final single shot measurement outcome of the qubit state, we probe weak values displaying all the above properties. The fluorescence signal exhibits interferences between oscillations associated to past and future quantum states. The measured data are in complete agreement with theory.
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Efficient quantum filtering for quantum feedback control

2015-01-28
Pierre Rouchon , Jason F. Ralph
We discuss an efficient numerical scheme for the recursive filtering of diffusive quantum stochastic master equations. We show that the resultant quantum trajectory is robust and may be used for … We discuss an efficient numerical scheme for the recursive filtering of diffusive quantum stochastic master equations. We show that the resultant quantum trajectory is robust and may be used for feedback based on inefficient measurements. The proposed numerical scheme is amenable to approximation, which can be used to further reduce the computational burden associated with calculating quantum trajectories and may allow real-time quantum filtering. We provide a two-qubit example where feedback control of entanglement may be within the scope of current experimental systems.
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Analysis of descriptor systems using numerical algorithms

1981-02-01
Richard F. Sincovec , A. M. Erisman , E. L. Yip +1 more
In this paper we analyze numerical methods for the solution of the large scale dynamical system <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E\dot{y}(t)=Ay(t)+g(t),Y(t_{0})=y_{0}</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</tex> are matrices, … In this paper we analyze numerical methods for the solution of the large scale dynamical system <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E\dot{y}(t)=Ay(t)+g(t),Y(t_{0})=y_{0}</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</tex> are matrices, possibly singular. Systems of this type have been referred to as implicit systems and more recently as descriptor systems since they arise from formulating system equations in physical variables. Special cases of such systems are algebraic-differential systems. We discuss the numerical advantages of this formulation and identify a class of numerical integration algorithms which have accuracy and stability properties appropriate to descriptor systems and which preserve structure, detect nonsolvable systems, resolve initial value consistency problems, and are applicable to "stiff" descriptor systems. We also present an algorithm for the control of the local truncation error on only the state variables.

Parameter estimation from measurements along quantum trajectories

2015-12-01
Pierre Six , Ph. Campagne-Ibarcq , Landry Bretheau +2 more
The dynamics of many open quantum systems are described by stochastic master equations. In the discrete-time case, we recall the structure of the derived quantum filter governing the evolution of … The dynamics of many open quantum systems are described by stochastic master equations. In the discrete-time case, we recall the structure of the derived quantum filter governing the evolution of the density operator conditioned to the measurement outcomes. We then describe the structure of the corresponding particle quantum filters for estimating constant parameter and we prove their stability. In the continuous-time (diffusive) case, we propose a new formulation of these particle quantum filters. The interest of this new formulation is first to prove stability, and also to provide an efficient algorithm preserving, for any discretization step-size, positivity of the quantum states and parameter classical probabilities. This algorithm is tested on experimental data to estimate the detection efficiency for a superconducting qubit whose fluorescence field is measured using a heterodyne detector.
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Differential flatness and absolute equivalence

2002-12-17
Michiel Van Nieuwstadt , Muruhan Rathinam , Richard M. Murray
In this paper we give a formulation of differential flatness-a concept originally introduced by Fliess, Levine, Martin, and Rouchon (1992)-in terms of absolute equivalence between exterior differential systems. Systems which … In this paper we give a formulation of differential flatness-a concept originally introduced by Fliess, Levine, Martin, and Rouchon (1992)-in terms of absolute equivalence between exterior differential systems. Systems which are differentially flat have several useful properties which can be exploited to generate effective control strategies for nonlinear systems. The original definition of flatness was given in the context of differential algebra, and required that all mappings be meromorphic functions. Our formulation of flatness does not require any algebraic structure and allows one to use tools from exterior differential systems to help characterize differentially flat systems. In particular, we show that in the case of single input control systems (i.e., codimension 2 Pfaffian systems), a system is differentially flat if and only if it is feedback linearizable via static state feedback. However, in higher codimensions feedback linearizability and flatness are not equivalent: one must be careful with the role of time as well the use of prolongations which may not be realizable as dynamic feedbacks in a control setting. Applications of differential flatness to nonlinear control systems and open questions are also discussed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>
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Sur l'équivalence absolue de certains systèmes d'équations différentielles et sur certaines familles de courbes

1914-01-01
Élie Cartan
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On dynamic feedback linearization

1989-08-01
B. Charlet , Jean Lévine , R. Marino

Nilpotent bases for a class of nonintegrable distributions with applications to trajectory generation for nonholonomic systems

1994-03-01
Richard M. Murray
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Approximate master equations for atom optics

2003-02-06
Donald Atkins , Howard M. Wiseman , Prahlad Warszawski
In the field of atom optics, the basis of many experiments is a two level atom coupled to a light field. The evolution of this system is governed by a … In the field of atom optics, the basis of many experiments is a two level atom coupled to a light field. The evolution of this system is governed by a master equation. The irreversible components of this master equation describe the spontaneous emission of photons from the atom. For many applications, it is necessary to minimize the effect of this irreversible evolution. This can be achieved by having a far detuned light field. The drawback of this regime is that making the detuning very large makes the timestep required to solve the master equation very small, much smaller than the timescale of any significant evolution. This makes the problem very numerically intensive. For this reason, approximations are used to simulate the master equation which are more numerically tractable to solve. This paper analyses four approximations: The standard adiabatic approximation; a more sophisticated adiabatic approximation (not used before); a secular approximation; and a fully quantum dressed-state approximation. The advantages and disadvantages of each are investigated with respect to accuracy, complexity and the resources required to simulate. In a parameter regime of particular experimental interest, only the sophisticated adiabatic and dressed-state approximations agree well with the exact evolution.
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Observing Quantum State Diffusion by Heterodyne Detection of Fluorescence

2016-01-05
P. Campagne-Ibarcq , Pierre Six , Landry Bretheau +4 more
A qubit can relax by fluorescence, which prompts the release of a photon into its electromagnetic environment. By counting the emitted photons, discrete quantum jumps of the qubit state can … A qubit can relax by fluorescence, which prompts the release of a photon into its electromagnetic environment. By counting the emitted photons, discrete quantum jumps of the qubit state can be observed. The succession of states occupied by the qubit in a single experiment, its quantum trajectory, depends in fact on the kind of detector. How are the quantum trajectories modified if one measures continuously the amplitude of the fluorescence field instead? Using a superconducting parametric amplifier, we perform heterodyne detection of the fluorescence of a superconducting qubit. For each realization of the measurement record, we can reconstruct a different quantum trajectory for the qubit. The observed evolution obeys quantum state diffusion, which is characteristic of quantum measurements subject to zero-point fluctuations. Independent projective measurements of the qubit at various times provide a quantitative verification of the reconstructed trajectories. By exploring the statistics of quantum trajectories, we demonstrate that the qubit states span a deterministic surface in the Bloch sphere at each time in the evolution. Additionally, we show that when monitoring fluorescence field quadratures, coherent superpositions are generated during the decay from excited to ground state. Counterintuitively, measuring light emitted during relaxation can give rise to trajectories with increased excitation probability.Received 28 August 2015DOI:https://doi.org/10.1103/PhysRevX.6.011002This article is available under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasQuantum measurementsQuantum Information
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A differential geometric setting for dynamic equivalence and dynamic linearization

1995-01-01
Jean‐Baptiste Pomet
This paper presents an (infinite-dimensional) geometric framework for control systems, based on infinite jet bundles, where a system is represented by a single vector field and dynamic equivalence (to be … This paper presents an (infinite-dimensional) geometric framework for control systems, based on infinite jet bundles, where a system is represented by a single vector field and dynamic equivalence (to be precise: equivalence by endogenous dynamic feedback
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Measurements continuous in time and a posteriori states in quantum mechanics

1991-04-07
Alberto Barchielli , V. P. Belavkin
Measurements continuous in time have been consistently introduced in quantum mechanics and applications worked out, mainly in quantum optics. In this context a quantum filtering theory has been developed giving … Measurements continuous in time have been consistently introduced in quantum mechanics and applications worked out, mainly in quantum optics. In this context a quantum filtering theory has been developed giving the reduced state after a measurement when a certain trajectory of the measured observables is registered (the a posteriori states). In this paper a new derivation of filtering equations is presented for the cases of counting processes and of measurement processes of diffusive type. It is also shown that the equation for the a posteriori dynamics in the diffusive case can be obtained, by a suitable limit, from that in the counting case. Moreover, the paper is intended to clarify the meaning of the various concepts involved and to discuss the connections among them. As an illustration of the theory, simple models are worked out.
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Monotone metrics on matrix spaces

1996-09-01
Dėnes Petz