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## 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 steadystate 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 eigenstate, indexed by ` 2 {1, . . . , d} for all realizations. More precisely, we will design a real-valued continuous stochastic control process v, depending on the state , such that limt!1 E[p`()] = 1 with exponential convergence rate, for any initial condition 0 2 S.

Feedback actions are incorporated as a unitary actuators of the form t = 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), e add unitary actuators of form t+dt = X U(t)(t + dt)U(t)†.

### Main contribution: noise-assisted feedback stabilization

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 exponentially 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 discourage the system to converge towards any undesired situation. The challenge is to show that this procedure induces exponential convergence towards the target state.

• Identify opportunities towards the implementation of efficiently computable 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 preclude 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 = (t)dB, where Bt a standard Brownian motion independent of Wt. We will construct controls continuously differentiable (). Our approach for m measurements and c controls translates to the closed-loop SDE: d = mX μ=1 DLμ()dt+p MLμ()dW+ Xc =1 ()2DH ()dt−i()[H, ]dB.

#### Static output feedback on a qubit

As we exposed in the introduction, there is an obstruction for static output feedback to stabilize a QND eigenstate. One could ask about defining the static output feedback gains as to approach arbitrarily a target QND eigenstate. The next Proposition indeed shows that, at least for a two-level system, any other pure state can be exponentially stabilized by a static output feedback with a fixed measurement operator and detection efficiency = 1. We use the standard Pauli matrix and Bloch sphere notation for the qubit system.

**Noise assisted stabilization of Qubit eigenstates**

It was already highlighted in [71] the difficulties associated to designing a globally stabilizing control law even in the case of a qubit. It was noted there that in principle it suffices that the control drives a continuous field that only vanishes on the target. This is the approach followed in the literature, e.g. [44, 69, 71, 50]. The point of view adopted here is, in a sense, weaker: the control field on (3.13) will be for a large parte of the state space, turnedoff. It is only when the state is closed to an undesired situation that the controller will be turned on. It suffices to show that this procedure induces exponential convergence. We can readily illustrate on the qubit how the use of noise allows us to exponentially stabilize a target eigenstate. From (3.13) with L = Z, > 0, and H = Y , the closed loop dynamics read d = ZZ − dt − p Z + Z − 2Tr (Z) dW

**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