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Quantum non-demolition measurements and feedback
Dans ce chapitre, nous introduisons les modèles mathématiques correspon-dant aux mesures QND en temps continu. Nous presentons le comportement dynamique des systèmes QND à boucle ouverte. Il est montré via une fonc-tion de Lyapunov originale que, pour toute condition initiale, l’état converge de manière exponentielle vers l’ensemble des états propres QND, l’ensemble de vecteurs propres de la matrice hermitienne de la mesure. Cette propriété de «multistabilité stochastique» signifie que le système converge vers l’un des états QND possibles, mais en moyenne, le système ne bouge pas vers un état QND en particulier. Nous présentons ensuite le problème de con-trôle, qui consiste à stabiliser de manière exponentielle un état propre QND parmi l’ensemble existant en boucle ouverte. Nous rappelerons aussi les ar-chitectures de contrôle existantes. Enfin, nous presenterons la stratégie de commande constituant le coeur de cette thèse.
The concept of measurement and feedback, which are so natural in engi-neering applications, becomes more subtle in the quantum domain. In order to control the system, a measurement must be performed, but due to the measurement back-action, this would invariably perturb the system. The formalism of stochastic master equations [25, 11] provides a mathematical framework to model this measurement process. It is then possible to use the information obtained from the measurements to influence the behavior of a quantum system.
Next, we need to make sure that the measurement iteslf does not per-turb the target state. A special case of stochastic master equations are those that model quantum non-demolition (QND) measurements, which are a continuous-time versions of the projection postulate. The eigenstates of the measurement operator, called QND eigenstates, are equilibria of the measurement dynamics and thus naturally protected from the measurement backaction. Then, a QND measurement can be chosen advantageously such that its eigenstates/eigenspaces contain states/subspaces suitable for quan-tum computation. Thus, an appropiate QND measurement can itself be considered as an open-loop preparation tool for quantum states [33], al-though the resulting state will be random. Then, added to a fixed QND measurement, feedback control can be used for preparing a target QND eigenstate with probability one.
Throughout this chapter, we overview some properties of stochastic mas-ter equations and in paricular of QND systems, which are considered as our open-loop models. Afterwards we will present the feedback stabilization problem, the previous work around this problem, and lastly address the control methodology that will be pursued in this Thesis.
Open loop-models: continuous QND measure-ments
The mathematical setup for quantum measurements in continuous time is that of stochastic master equations, for a more complete introduction, the reader is encouraged to follow the book of A. Barchielli & M. Gregoratti [11] for a mathematical introduction, or the lecture notes of H. Carmichael
[25]. We do not consider here discrete-time models, for which the reader is invited to read the previous works [12, 4, 3, 6], addressing discrete-time quantum measurements and feedback control.
We work in a finite dimensional complex Hilbert space H isomorphic to Cn. The kets |xi are complex vectors on Cn and hx| its complex conjugate transpose called bra, Tr ( • ) denotes the trace operator, [ •, •] denotes the commutator and k • k is the norm induced by the inner product h •||• i.
The state space is set of density matrices as S = {ρ ∈ Cn×n : ρ ≥ 0, ρ = ρ†, Tr (ρ) = 1}
Control problem and existing feedback designs
QND measurements are a common element in measurement based feedback. Part of this consideration is the practical interest of such measurements for engineering quantum systems: from property ii) of Lemma 3.1.1, steady states remain unperturbed from the measurement process. On the other hand, QND measurements pose interesting challenges: as seen in Lemma 3.1.1, the open-loop system stochastically converges to one of a few steady-state situations, but on the average it does not move closer to any particular one. It is then our goal to bias this average to a target eigenstate.
The control objective is to ensure convergence to a target QND eigen-state, indexed by ` ∈ {1, . . . , d} for all realizations. More precisely, we will design a real-valued continuous stochastic control process v, depending on the state ρ, such that limt→∞ E[p`(ρ)] = 1 with exponential convergence rate, for any initial condition ρ0 ∈ S.
Feedback actions are incorporated as a unitary actuators of the form Ut = exp (−iHdv), where H = H† is the actuation Hamiltonian, and dv is the control process that drives the actuator. To the QND dynamics (3.1), we add unitary actuators of form ρt+dt = Uν (t)(ρt + dρt)Uν (t)† (3.5) with m measurements channels {Yµ}1≤µ≤m.
We aim to showing exponential convergence via global Lyapunov func-tions V (ρ) such that V (ρt) is a supermartingale with exponential decay for all t ≥ 0 and for all ρ0 ∈ S.
Static output feedback
We consider the simplest possible feedback control scheme, a proportional output feedback of the form dv = f dt + σdY,
where f, σ are constants. Since the control signal depends on a stochastic process, care is needed in order to derive the closed-loop dynamics.
Since dv depends on a stochastic proces, care is needed to derive the closed-loop dynamics. The derivation of the closed loop dynamics was first done in [73], here we only provide a simpler formulation. We relate Eq. (3.1) with (3.5) via the Baker-Campbell-Hausdorff (BCH) formula exH P e−xH = X Tj xj , the terms Tj of the series expansion on the right hand side of are defined recursively as T0 = P and Tj+1 = [H, Tj ] for j ≥ 0. We identify the right hand side up-to second order of the BCH formula with It¯o’s formula. Take P = ρt + dρt and consider It¯o rules (dW 2 = dt, dW dt = dt2 = 0). The closed-loop equation reads dρ = −if[H, ρ]dt − iσ√η[H, Lρ + ρL]dt + DL(ρ)dt + σ2DH (ρ)dt + √ηDL(ρ)dW − iσ[F, ρ]dW.
The simplicity of the feedback scheme makes it attractive for experi-mental implementations, since there is no control processing overhead in the feedback loop. This feedback equation has been derived as well using the rules of the quantum stochastic calculus [35]. The next Lemma shows that the controller (3.6) cannot be tuned to achieve global asymptotic sta-bility towards a prescribed QND eigenstate Π`.
Main contribution: noise-assisted feedback sta-bilization
There are two main issues that we want to address for stabilization of a QND eigenstate:
• Achieve exponential stabilization of a prescribed QND eigenstate. Lemma 3.1.1 indicates that the open-loop system (3.2) approaches exponen-tially the set of QND eigenstates, the resulting state being at random. Then a point of view on the role of feedback is that it has to dis-courage the system to converge towards any undesired situation. The challenge is to show that this procedure induces exponential conver-gence towards the target state.
• Identify opportunities towards the implementation of efficiently com-putable controls on an experimental setup. Global stabilization of a QND eigenstate can be achieved by using a quantum filter, but the explicit dependence of known control laws on quantum coherences pre-clude a simpler implementation. Developing feedback controls that are dependent only on observed quantities, like monitoring the population on a target eigenstate, and the formulation of reduced models that avoid the computation of the full quantum state.
The main idea to address this two problems is Use an external noise to drive the actuator.
Here the control law dv present on the unitary actuation e−iHdv will have attached a gain computed via a quantum filter, but now we let the state feedback signal dv to be driven by Brownian noise, i.e. dv = σ(ρt)dB, where Bt a standard Brownian motion independent of Wt. We will construct controls continuously differentiable σ(ρ). Our approach for m measurements and c controls translates to the closed-loop SDE: Where dBν dBν0 = δν,ν0 dt, dWµ dWµ0 = δµµ0 dt, dBν dWµ = 0.
Theorem 3.3.1. Let σ(ρ) be a smooth function of ρ. Then the closed-loop system (3.13) admits a unique solution on the set S.
Proof. From Eq. (3.13), the smooth control σ(ρ) is bounded since it is de-fined on a compact set and the terms DH (ρ) = −12 [H, [H, ρ]] and −i[H, ρ] are Lipschitz in ρ. Terms corresponding to the QND measurement fullfills the existence and solution properties by Theorem (3.1.1). Putting all to-gether, existence and uniqueness of solutions on (3.13) follows from standard arguments of SDE’s.
The way we address the two problems at the beginning of this section is
• Exponential stabilization via noise-assisted feedback. The main idea relies on the fact that the open loop system (3.2) stochastically con-verges to one of a few steady-state situations, i.e. to a state supported on one of the eigenspaces of the measurement operator, but on the av-erage does not move closer to any particular state. Thus it suffices that the controller tracks the eigenspace populations {pk(ρ)}1≤k≤d, that is, the diagonal elements on the eigenbasis fixed by the measurement.
Control design consists then on activating noise only when the state is close to a bad equilibrium, in order to « shake it » away from it.
Table of contents :
1 Résumé
1.1 Contexte
1.2 Énoncé du problème et idée principale
1.3 Plan de la thèse
2 Introduction
2.1 Background
2.2 Problem statement and main idea
2.3 Thesis outline
3 Quantum non-demolition measurements and feedback
3.1 QND measurements
3.2 Control problem and existing feedback designs
3.2.1 Static output feedback
3.2.2 State feedback
3.3 Main contribution: noise-assisted feedback stabilization
4 Exponential stabilization of a qubit
4.1 Introduction
4.2 Qubit system
4.3 Static output feedback on a qubit
4.4 Noise assisted stabilization of Qubit eigenstates
4.5 Reduced order filtering on the qubit
4.6 Moving beyond a qubit: generation of GHZ states
4.7 Conclusions
5 Exponential stabilization of a QND eigenstate
5.1 Introduction
5.2 Connectivity graph and Laplacian matrix
5.3 Exponential stabilization via noise-assisted feedback
5.4 Approximated quantum filtering
5.5 Simulations
5.6 Conclusions
6 On continuous-time quantum error correction
6.1 Introduction
6.2 Dynamics of the three-qubit bit-flip code
6.3 Some open issues on continuous-time QEC
6.4 Error correction as noise-assisted feedback stabilization
6.4.1 Controller design
6.4.2 Closed-loop exponential stabilization
6.4.3 Reduced order filtering
6.4.4 On the protection of quantum information
6.5 Conclusions
7 Concluding remarks and perspectives
7.1 Towards robust control methods for quantum information processing
7.2 Towards dynamical output feedback controllers
A Lyapunov’s second method for stochastic stability
A.1 Lyapunov functions for QND systems
