Search

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

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

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., Comment: 7 pages, 3 figures. Minor modifications of the text and added references

This paper deals with the problem of string stability in a chain of acceleration-controlled vehicles. It is known that string stability cannot be achieved, with any linear controller, when the vehicles' control inputs are based on relative distances to a fixed number of predecessors. We extend the set of impossible settings by including elements like dynamic sensor parts and local inter-vehicular communication, as in cooperative adaptive cruise control. It is also known that a weaker form of string stability is achievable by adding absolute velocity measurements (e.g. "time-headway" policy). We show that a stronger property can also be achieved, provided steady-state control gain is infinite e.g. by using integral control.

This paper deals with the problem of string stability in a chain of acceleration-controlled vehicles, i.e. how input disturbances affect the distributed system for very long chains. There exist variants of string stability, like avoiding that a local disturbance gets amplified along the chain, or more strongly ensuring that the output vector's p-norm remains bounded for any bounded vector of input disturbances independently of the string length. They are all impossible to achieve with any linear controller if the vehicles only use relative information of few vehicles in front. Previous work has shown that adding absolute velocity into the controller, allows to at least avoid amplification of a local disturbance. In this paper, we consider the stronger definitions of string stability, under this same relaxation of using absolute velocity in the controller. We prove that the influence from input vector to output vector cannot be bounded independently of chain length in the most popular 2-norm sense, with any bounded stabilizing linear controller; while a proportional derivative (PD) controller can guarantee it in the practically relevant infinity-norm sense. Moreover, we identify the disturbance acting on the leader as the main issue for string stability., Comment: presented at IEEE-CDC 2018, Miami; more complete version and correcting a few typos

We provide a comprehensive impossibility result towards achieving string stability, i.e. keeping local relative errors in check with local controllers independently of the size of a chain of subsystems. We significantly extend existing results, from the LTI setting to any homogeneous controllers that can be nonlinear, time-varying, and locally communicating. We prove this impossibility for a set of definitions with various norm choices, including the $L_2$-type which is more standard in the literature and a BIBO type criterion. All results hold for a general discrete-time controller which should cover most applications.

Lifted Markov chains are Markov chains on graphs with added local "memory" and can be used to mix towards a target distribution faster than their memoryless counterparts. Upper and lower bounds on the achievable performance have been provided under specific assumptions. In this paper, we analyze which assumptions and constraints are relevant for mixing, and how changing these assumptions affects those bounds. Explicitly, we find that requesting mixing on either the original or the full lifted graph, and allowing for reducible lifted chains or not, have no essential influence on mixing time bounds. On the other hand, allowing for suitable initialization of the lifted dynamics and/or imposing invariance of the target distribution for any initialization do significantly affect the convergence performance. The achievable convergence speed for a lifted chain goes from diameter-time to no acceleration over a standard Markov chain, with conductance bounds limiting the effectiveness of the intermediate cases. In addition, we show that the relevance of imposing ergodic flows depends on the other criteria. The presented analysis allows us to clarify in which scenarios designing lifted dynamics can lead to better mixing, and provide a flexible framework to compare lifted walks with other acceleration methods., Comment: 15 pages, 5 figures. Comments welcome

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., Comment: Submitted

This paper applies the recently developed framework for integral control on nonlinear spaces to two non-standard cases. First, we show that the property of perfect target stabilization in presence of actuation bias holds also if this bias is state dependent. This might not be surprising, but for practical purposes it provides an easy way to robustly cancel nonlinear dynamics of the uncontrolled plant. We specifically illustrate this for robust stabilization of a pendulum at arbitrary angle, a problem posed as non-trivial by some colleagues. Second, as previous work has been restricted to systems with as many control inputs as configuration dimensions, we here provide results for integral control of a non-holonomic system. More precisely, we design robust steering control of a rigid body under velocity bias., Comment: submitted to NOLCOS conference 2016

This paper deals with the problem of string stability of interconnected systems with double-integrator open loop dynamics (e.g.~acceleration-controlled vehicles). We analyze an asymmetric bidirectional linear controller, where each vehicle is coupled solely to its immediate predecessor and to its immediate follower with different gains in these two directions. We show that in this setting, unlike with unidirectional or symmetric bidirectional controllers, string stability can be recovered when disturbances act only on a small ($N$-independent) set of leading vehicles. This improves existing results from the literature with this assumption. We also indicate that string stability with respect to arbitrarily distributed disturbances cannot be achieved with this controller., Comment: Version 2 corrects a typo in the proof, and adds the proof of stability before string stability. Slightly longer than published version

We consider an open quantum system described by a Lindblad-type master equation with two times-scales. The fast time-scale is strongly dissipative and drives the system towards a low-dimensional decoherence-free space. To perform the adiabatic elimination of this fast relaxation, we propose a geometric asymptotic expansion based on the small positive parameter describing the time-scale separation. This expansion exploits geometric singular perturbation theory and center-manifold techniques. We conjecture that, at any order, it provides an effective slow Lindblad master equation and a completely positive parameterization of the slow invariant sub-manifold associated to the low-dimensional decoherence-free space. By preserving complete positivity and trace, two important structural properties attached to open quantum dynamics, we obtain a reduced-order model that directly conveys a physical interpretation since it relies on effective Lindbladian descriptions of the slow evolution. At the first order, we derive simple formulae for the effective Lindblad master equation. For a specific type of fast dissipation, we show how any Hamiltonian perturbation yields Lindbladian second-order corrections to the first-order slow evolution governed by the Zeno-Hamiltonian. These results are illustrated on a composite system made of a strongly dissipative harmonic oscillator, the ancilla, weakly coupled to another quantum system., Comment: 9 pages, one figure

Previous papers have proposed to add memory registers to the dynamics of discrete-time linear systems in order to accelerate their convergence. In particular, it has been proved that adding one memory slot per agent allows faster convergence towards average consensus. We here prove that this situation \emph{cannot} be improved by adding more memory slots, when the knowledge about the self-adjoint linear map to be accelerated reduces to bounds on its extreme eigenvalues.

We investigate performance bounds for feedback control of distributed plants where the controller can be centralized (i.e. it has access to measurements from the whole plant), but sensors only measure differences between neighboring subsystem outputs. Such "distributed sensing" can be a technological necessity in applications where system size exceeds accuracy requirements by many orders of magnitude. We formulate how distributed sensing generally limits feedback performance robust to measurement noise and to model uncertainty, without assuming any controller restrictions (among others, no "distributed control" restriction). A major practical consequence is the necessity to cut down integral action on some modes. We particularize the results to spatially invariant systems and finally illustrate implications of our developments for stabilizing the segmented primary mirror of the European Extremely Large Telescope., Comment: submitted to Automatica

This paper extends the consensus framework, widely studied in the literature on distributed computing and control algorithms, to networks of quantum systems. We define consensus situations on the basis of invariance and symmetry properties, finding four different probabilistic generalizations of classical consensus states. We then extend the gossip consensus algorithm to the quantum setting and prove its convergence properties, showing how it converges to symmetric states while preserving the expectation of permutation-invariant global observables., Comment: 14 pages, 2 figures

For discrete-time systems, governed by Kraus maps, the work of D. Petz has characterized the set of universal contraction metrics. In the present paper, we use this characterization to derive a set of quadratic Lyapunov functions for continuous-time systems, governed by Lindblad differential equations, that have a steady-state with full rank. An extremity of this set is given by the Bures metric, for which the quadratic Lyapunov function is obtained by inverting a Sylvester equation. We illustrate the method by providing a strict Lyapunov function for a Lindblad equation designed to stabilize a quantum electrodynamic "cat" state by reservoir engineering. In fact we prove that any Lindblad equation on the Hilbert space of the (truncated) harmonic oscillator, which has a full-rank equilibrium and which has, among its decoherence channels, a channel corresponding to the photon loss operator, globally converges to that equilibrium., Comment: Submitted (10 pages, 1 figure)

This paper studies autonomous synchronization of k agents whose states evolve on SO(n), but which are only coupled through the action of their states on one "reference vector" in Rn for each link. Thus each link conveys only partial state information at each time, and to reach synchronization agents must combine this information over time or throughout the network. A natural gradient coupling law for synchronization is proposed. Extensive convergence analysis of the coupled agents is provided, both for fixed and time-varying reference vectors. The case of SO(3) with fixed reference vectors is discussed in more detail. For comparison, we also treat the equivalent setting in Rn, i.e. with states in Rn and connected agents comparing scalar product of their states with a reference vector., Comment: to be submitted to SIAM Journal on Control and Optimization

Convergence analysis of consensus algorithms is revisited in the light of the Hilbert distance. Tsitsiklis Lyapunov function is shown to be the Hilbert distance to consensus in log coordinates. Birkhoff theorem, which proves contraction of the Hilbert metric for any positive homogeneous monotone map, provides an early yet general convergence result for consensus algorithms. Because Birkhoff theorem holds in arbitrary cones, we extend consensus algorithms to the cone of positive definite matrices. The proposed generalization finds applications in the convergence analysis of quantum stochastic maps, which are a generalization of stochastic maps to non-commutative probability spaces., Comment: Submitted

The goal of the present paper is to highlight the fundamental differences of so-called synchronization or consensus algorithms when the agents to synchronize evolve on a compact homogeneous manifold (like the circle, sphere or the group of rotation matrices), instead of a vector space. For the benefit of understanding, the discussion is restricted to the circle. First, a fundamental consensus algorithm on R^n is reviewed, from which continuous- and discrete-time synchronization algorithms on the circle are deduced by analogy. It is shown how they are connected to Kuramoto and Vicsek models from the literature. Their convergence properties are similar to vector spaces only for specific graphs or if the agents are all located within a semicircle. Examples are proposed to illustrate several other possible behaviors. Finally, new algorithms are proposed to recover (almost-)global synchronization properties., Comment: submitted 01/2009 for possible book contribution

The present paper considers distributed consensus algorithms that involve N agents evolving on a connected compact homogeneous manifold. The agents track no external reference and communicate their relative state according to a communication graph. The consensus problem is formulated in terms of the extrema of a cost function. This leads to efficient gradient algorithms to synchronize (i.e. maximizing the consensus) or balance (i.e. minimizing the consensus) the agents; a convenient adaptation of the gradient algorithms is used when the communication graph is directed and time-varying. The cost function is linked to a specific centroid definition on manifolds, introduced here as the induced arithmetic mean, that is easily computable in closed form and may be of independent interest for a number of manifolds. The special orthogonal group SO(n) and the Grassmann manifold Gr(p,n) are treated as original examples. A link is also drawn with the many existing results on the circle., Comment: Preprint submitted to SIAM/SICON September 2006; revised November 2007; accepted for publication April 2008; publication date unknown before 2009

The present paper proposes a unified geometric framework for coordinated motion on Lie groups. It first gives a general problem formulation and analyzes ensuing conditions for coordinated motion. Then, it introduces a precise method to design control laws in fully actuated and underactuated settings with simple integrator dynamics. It thereby shows that coordination can be studied in a systematic way once the Lie group geometry of the configuration space is well characterized. This allows among others to retrieve control laws in the literature for particular examples. A link with Brockett's double bracket flows is also made. The concepts are illustrated on SO(3), SE(2) and SE(3)., Comment: Submitted