Continuous-time Quantum Error Correction with Noise-assisted Quantum Feedback

Continuous-time Quantum Error Correction with Noise-assisted Quantum Feedback

11th 11th IFAC IFAC Symposium Symposium on on Nonlinear Nonlinear Control Control Systems Systems 11th IFAC Symposium on Nonlinear Control Systems Vie...

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11th 11th IFAC IFAC Symposium Symposium on on Nonlinear Nonlinear Control Control Systems Systems 11th IFAC Symposium on Nonlinear Control Systems Vienna, Austria, Sept. 4-6, 2019 11th Symposium on Control Systems 11th IFAC IFAC Symposium on Nonlinear Nonlinear Control Systems Vienna, Austria, Sept. 4-6, 2019 Available online at www.sciencedirect.com Vienna, Austria, Sept. 4-6, 2019 11th IFAC Symposium on Nonlinear Control Systems Vienna, Austria, Sept. 4-6, 2019 Vienna, Austria, Sept. 4-6, 2019

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IFAC PapersOnLine 52-16 (2019) 198–203

Continuous-time Quantum Error Correction Continuous-time Continuous-time Quantum Quantum Error Error Correction Correction Continuous-time Quantum Error Correction with Noise-assisted Quantum Feedback Continuous-time Quantum Error Correction with Noise-assisted Quantum Feedback with Noise-assisted Quantum Feedback with Noise-assisted Quantum Feedback with Cardona Noise-assisted Quantum Feedback ∗,∗∗ ∗,∗∗ ∗∗,∗∗∗ ∗,∗∗ Alain Sarlette ∗∗,∗∗∗ ∗∗,∗∗∗ Pierre Rouchon ∗,∗∗ ∗,∗∗ Gerardo ∗,∗∗

Gerardo Cardona Cardona ∗,∗∗ Alain Alain Sarlette Sarlette Pierre Rouchon Rouchon ∗,∗∗ Gerardo Pierre ∗∗,∗∗∗ Pierre Rouchon ∗,∗∗ Gerardo Alain Sarlette ∗∗,∗∗∗ Gerardo Cardona Cardona ∗,∗∗ ∗,∗∗ Alain Sarlette ∗∗,∗∗∗ Pierre Rouchon ∗,∗∗ ∗ Gerardo Cardona etAlain Sarlette Pierre PSL Rouchon ∗ Centre Automatique Systèmes, Mines-ParisTech, Research ∗ et Systèmes, Mines-ParisTech, PSL Research Centre Automatique Automatique et Systèmes, Mines-ParisTech, PSL Research ∗ Centre ∗ Centre Automatique et Systèmes, Mines-ParisTech, PSL Research University. 60 Bd Saint-Michel, 75006 Paris, France. Centre Automatique et Systèmes, Mines-ParisTech, PSL Research University. 60 Bd Saint-Michel, 75006 Paris, France. University. 60 Bd Saint-Michel, 75006 Paris, France. ∗∗∗ Automatique et Systèmes, Mines-ParisTech, PSL Research University. 60 Bd Saint-Michel, 75006 Paris, France. ∗∗Centre QUANTIC lab, INRIA, rue Simone Iff 2, 75012 Paris, France. ∗∗ University. 60 Bd Saint-Michel, 75006 Paris, France. QUANTIC lab, lab, INRIA, INRIA, rue rue Simone Simone Iff Iff 2, 2, 75012 75012 Paris, Paris, France. France. QUANTIC ∗∗∗∗∗ University. 60 Bd Saint-Michel, 75006 Paris, France. ∗∗∗∗∗ QUANTIC lab, INRIA, rue Simone Iff 2, 75012 Paris, France. ∗∗∗ Electronics and Information Systems, Ghent Univ., Belgium. QUANTIC lab, INRIA, rue Simone Iff 2, 75012 Paris, France. Electronics and Information Systems, Ghent Univ., Belgium. ∗∗∗∗∗ Electronics and Information Systems, Ghent Univ., Belgium. QUANTIC INRIA, rue Simone Iff 2, 75012Univ., Paris,Belgium. France. ∗∗∗ Electronicslab, and Information Systems, Ghent and Information Systems, Ghent Univ., Belgium. ∗∗∗ Electronics e-mail: e-mail: [email protected] Electronics [email protected] Information Systems, Ghent Univ., Belgium. e-mail: [email protected] e-mail: e-mail: [email protected] [email protected] e-mail: [email protected] Abstract: We We address address the the standard standard quantum quantum error correction correction using using the the three-qubit three-qubit bit-flip bit-flip code, code, Abstract: Abstract: We address the standard quantum error error correction using the three-qubit bit-flip code, Abstract: We address the standard error correction using the three-qubit bit-flip code, yet in in continuous-time. continuous-time. This entailsquantum rendering a target target manifold of quantum quantum states globally Abstract: We address the standard quantum error correction using the three-qubit bit-flip code, yet This entails rendering a manifold of states globally yet in continuous-time. This entails rendering aa target manifold of quantum states globally Abstract: We address the standard quantum correction using the three-qubit bit-flip code, yet in This entails rendering manifold of states attractive. Previous feedback designs could error feature spurious equilibria, or resort resort to globally discrete yet in continuous-time. continuous-time. This entails rendering a target target manifold of quantum quantum states globally attractive. Previous feedback designs could feature spurious equilibria, or to discrete attractive. Previous feedback designs could feature spurious equilibria, or resort to discrete 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 attractive. Previous feedback designs could feature spurious equilibria, or resort to discrete kicks pushing pushing the the system system away away from from these these equilibria equilibria to to ensure ensure global global asymptotic asymptotic stability. stability. We We kicks attractive. Previous feedback designs could feature to spurious equilibria, resortstability. to motions. discrete kicks pushing the system away from these equilibria ensure global We present a new new approach that consists of introducing introducing controls drivenasymptotic byorBrownian Brownian motions. kicks pushing the system away from these equilibria to ensure global asymptotic stability. We present a approach that consists of controls driven by present a newthe approach that of introducing driven by Brownian motions. kicks system away consists fromresulting these equilibria to controls ensure global asymptotic We present a approach that consists of controls driven by Unlikepushing the previous methods, the resulting closed-loop dynamics can be Brownian shownstability. to motions. stabilize present a new new approach that consists of introducing introducing controls driven by Brownian motions. Unlike the previous methods, the closed-loop dynamics can be shown to stabilize Unlike the previous methods, the resulting closed-loop dynamics can be shown to stabilize 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 Unlike the manifold previous exponentially. methods, the We resulting dynamics can filter be shown to stabilize the target target manifold exponentially. We furtherclosed-loop present aa reduced-order reduced-order filter formulation with the further present formulation with Unlike the previous exponentially. methods, the We resulting dynamics can filter be shown to stabilize the target manifold exponentially. We furtherclosed-loop present reduced-order filter formulation with classical probabilities. The exponential exponential property is important important to quantify quantify the protection induced the target manifold further present aa reduced-order formulation with classical probabilities. The property is to the protection induced classical probabilities. The exponential property is important to quantify the protection induced the target manifolderror-correction exponentially. We furtheragainst present a reduced-order formulation with classical probabilities. The exponential property is important to quantify the protection induced by the the closed-loop error-correction dynamics against disturbances. We filter study numerically the classical probabilities. The exponential property is important to quantify the protection induced by closed-loop dynamics disturbances. We study numerically the by the closed-loop error-correction dynamics against disturbances. We study numerically the 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. by the closed-loop error-correction dynamics against disturbances. We study numerically the performance of this control law and of the reduced filter. performance of control law and dynamics of the reduced filter. by the closed-loop against disturbances. We study numerically the performance of this this control law of the filter. performance thiserror-correction controlFederation law and and of of Automatic the reduced reduced filter. © 2019, IFAC of (International Control) Hosting by Elsevier Ltd. All rights reserved. performance of this control law and of the reduced filter.stochastic Keywords: Quantum control, quantum error correction, systems, Lyapunov Lyapunov method method Keywords: Quantum control, quantum error correction, stochastic systems, Keywords: Quantum control, quantum error correction, stochastic systems, Lyapunov method Keywords: Quantum control, quantum error correction, stochastic systems, Lyapunov method Keywords: Quantum control, quantum error correction, stochastic systems, Lyapunov method Keywords: Quantum control, quantum error correction, stochastic systems, Lyapunov method 1. INTRODUCTION INTRODUCTION in 1. in Murch Murch et et al. al. [2012], [2012], Cohen Cohen et et al. al. [2014], [2014], Guillaud Guillaud et et al. al. 1. INTRODUCTION in Murch et al. [2012], Cohen et al. [2014], Guillaud et al. 1. INTRODUCTION in Murch et al. [2012], Cohen et al. [2014], Guillaud [2019]. An advantage of this approach is that there is no 1. INTRODUCTION in Murch et al. [2012], Cohen et al. [2014], Guillaud et al. [2019]. An An advantage advantage of of this this approach approach is is that that there there et is al. no [2019]. is no 1. INTRODUCTION Murch etadvantage al. [2012], Cohen etHowever, al. [2014], Guillaud et al. [2019]. An of this approach is that there is no need for external control logic. the challenge is Developing methods methods to protect protect quantum quantum information information in in in [2019]. An advantage of this approach is that there is no need for external control logic. However, the challenge is Developing to need for external control logic. However, the challenge is Developing methods to protect quantum information in [2019]. An advantage of this approach is that there is no need for external control logic. However, the challenge is Developing methods to protect quantum information in to implement the specific ancillary system and coupling the presence of disturbances is essential to improve existfor external control logic. However, the challenge is Developing methods to protect quantum information in need to implement the specific ancillary system and coupling the presence of disturbances is essential to improve existto implement the specific ancillary system and coupling need for external control logic. However, the challenge is Developing to protect quantum in to to implement the specific ancillary system and coupling the presencemethods of disturbances is essential essential to information improve existwithin experimental constraints. ing quantum technologies (Reed et al. [2012], Ofek et al. implement the specific ancillary system and coupling the presence of disturbances is to improve existwithin experimental experimental constraints. constraints. ing quantum technologies (Reed et al. [2012], Ofek et al. within to implement the specific ancillary system and coupling the presence of disturbances is essential to improve existexperimental constraints. ing quantum technologies (Reed et (QEC) al. [2012], [2012], Ofekencode et al. al. within [2016]). Quantum error correction correction (QEC) codes, encode within experimental constraints. ing quantum technologies (Reed et al. Ofek et [2016]). Quantum error codes, [2016]). Quantum error correction codes, Experimental progress on Experimental progress on performing performing high-fidelity high-fidelity quanquanexperimental constraints. ing quantum (Reed et (QEC) al.states. [2012], Ofekencode et al. [2016]). Quantum error correction (QEC) codes, encode a logical logical statetechnologies intoerror multiple physical states. Similarly to within Experimental progress on performing high-fidelity quan[2016]). Quantum correction (QEC) codes, encode a state into multiple physical Similarly to Experimental progress on performing high-fidelity quana logical state into multiple physical states. Similarly to tum measurements now allows to consider measurementExperimental progress on performing high-fidelity quantum measurements now allows to consider measurement[2016]). Quantum error correction (QEC) codes, encode a logical state into multiple physical states. Similarly to classical error correction, this redundancy allows to protect tum measurements now allows to consider measurementa logical error statecorrection, into multiple physical states. Similarly to Experimental progress on performing high-fidelity quantum measurements now allows to consider measurementclassical this redundancy allows to protect based feedback in continuous-time. In the context of QEC, tum measurements now allows to consider measurementbased feedback in continuous-time. In the context of QEC, a logical state into multiple physical states. Similarly to classical error correction, thisdisturbances redundancy allows allows to protect protect quantumerror information from by stabilizing stabilizing a based feedback in continuous-time. In the context of QEC, classical correction, this redundancy to tum measurements now allows to consider measurementbased feedback in continuous-time. In the context of QEC, quantum information from disturbances by a this has been addressed in Ahn et al. [2002, 2003], Sarovar based feedback in continuous-time. In the context of QEC, this has been addressed in Ahn et al. [2002, 2003], Sarovar classical error correction, this redundancy allows to protect quantum information from disturbances by stabilizing a submanifold of steady steady states, states, which represent represent the nominal nominala this has been addressed in Ahn et al. [2002, 2003], Sarovar quantum information from disturbances by stabilizing submanifold of which the based feedback in continuous-time. In the context of QEC, this has been addressed in Ahn et al. [2002, 2003], Sarovar submanifold of steady states, which represent the nominal et al. [2004], Mabuchi [2009], essentially as proposals ilthis has been addressed in Ahn essentially et al. [2002,as 2003], Sarovar et al. [2004], Mabuchi [2009], essentially as proposals ilquantum information from disturbances by stabilizing a submanifold of steady states, which represent the nominal logical states Lidar and Brun [2013], Nielsen and Chuang et al. [2004], Mabuchi [2009], proposals ilsubmanifold ofLidar steady states, which represent the nominal logical states and Brun [2013], Nielsen and Chuang this has been addressed in Ahn et al. [2002, 2003], Sarovar et al. [2004], Mabuchi [2009], essentially as proposals illogical states Lidar and Brun [2013], Nielsen and Chuang lustrated by simulation. The short dynamical timescales et al. [2004], Mabuchi [2009], essentially as proposals illustrated by simulation. The short dynamical timescales submanifold of steady states, which represent the nominal logical states Lidar and Brun [2013], Nielsen and Chuang [2002]. As long as a disturbance does not drive the system lustrated by simulation. The short dynamical timescales logical states Lidar and Brun [2013], Nielsen and Chuang et al. [2004], Mabuchi [2009], essentially as proposals illustrated by simulation. The short dynamical timescales [2002]. As long as a disturbance does not drive the system of experimental setups is a main difficulty towards imlustrated by simulation. The short dynamical timescales of experimental setups is a main difficulty towards imlogical states Lidar and Brun [2013], Nielsen and Chuang [2002]. As basin long as as disturbance does not drive drive the out of of the the of aaattraction attraction of the the original nominal state, of experimental setups is aa main difficulty towards im[2002]. As long disturbance does not the system system lustrated by simulation. The short dynamical timescales of experimental setups is main difficulty towards imout basin of of original nominal state, plementing complex feedback laws. Furthermore, data acexperimental setups is a main difficulty towards implementing complex feedback laws. Furthermore, data ac[2002]. As basin long as disturbance does not drive thestabilize system out of the basin of aattraction attraction of the the original nominal state, of the of logical information remains unperturbed. To stabilize plementing complex feedback laws. Furthermore, data acout the of of original nominal state, the logical information remains unperturbed. To of experimental setups is leads a main difficulty towards implementing complex feedback laws. Furthermore, data acthe logical information remains unperturbed. To stabilize quisition and processing to latencies in the plementing complex feedback laws. Furthermore, datafeedacquisition and processing leads to latencies in the feedout of the basin of attraction of the original nominal state, the logical information remains unperturbed. To stabilize nominal submanifold in a quantum system, a syndrome quisition and processing leads to latencies in the feedthe logical information remains unperturbed. To stabilize nominal submanifold in a quantum system, a syndrome plementing complex feedback laws. Furthermore, data acquisition and processing leads to latencies in the feedthe nominal submanifold in a quantum system, a syndrome back loop. This motivates the development of efficiently quisition and processing leads to latencies in the feedback loop. This motivates the development of efficiently the logical information remains unperturbed. To stabilize nominal submanifold in a quantum system, a syndrome diagnosis stage performs quantum non-destructive (QND) back loop. This motivates the development of efficiently the nominal submanifold in a quantum system, a syndrome quisition and processing leads to latencies in the feedback loop. This motivates the development of efficiently diagnosis stage performs quantum non-destructive (QND) computable control techniques that are robust against loop. This motivates the development of efficiently computable control techniques that are robust against the nominal submanifold a quantum system, a syndrome diagnosis stage performs in quantum non-destructive (QND) measurements extracting information about code code distur- back computable control techniques that are robust against diagnosis stage performs quantum non-destructive (QND) back loop. This motivates the development of efficiently computable control techniques that are robust against measurements extracting information about disturunmodeled dynamics. computable control techniques that are robust against unmodeled dynamics. diagnosis stage performs quantum non-destructive (QND) measurements extracting information about code disturbances without perturbing the encoded data. Based on unmodeled dynamics. measurements extracting information about code disturbances without perturbing the encoded data. Based on computable control techniques that are robust against unmodeled dynamics. bances without perturbing the encoded data. Based on unmodeled dynamics. measurements extracting information about code disturbances without perturbing the encoded data. Based on this information, a recovery feedback action restores the In this paper we establish analytical results about the bances without perturbing the encoded data. Based the on unmodeled In this paper we establish analytical results about the dynamics. this information, a recovery feedback action restores In this paper we establish analytical results about the bances without perturbing the encoded data. Based on this information, a recovery feedback action restores the In this paper we establish analytical results about the corrupted state. QEC is most often presented as discreteconvergence rate of QEC systems towards the nominal this information, a recovery feedback action restores the In this paper we establish analytical results about the convergence rate of QEC systems towards the nominal corrupted state. QEC is most often presented as discreteconvergence rate of QEC systems towards the nominal this information, a recovery feedback action restores the corrupted state. QEC is most often presented as discreteIn this paper we establish analytical results about the convergence rate of QEC systems towards the nominal time operations towards digital quantum computing, see submanifold, a prerequisite for analytically quantifying corrupted state. QEC is most often presented as discreteconvergence of QEC for systems towards the nominal submanifold, rate a prerequisite prerequisite for analytically quantifying the time operations towards digital quantum computing, see submanifold, a analytically quantifying the corrupted state. is most often presented discretetime operations towards digital quantum computing, see convergence rate of QEC systems towards the nominal a prerequisite for analytically quantifying the e.g. Nielsen Nielsen and QEC Chuang [2002]. Not onlycomputing, theasdesign design of submanifold, protection of quantum information. To obtain exponential time operations towards digital quantum see submanifold, a prerequisite for analytically quantifying the e.g. and Chuang [2002]. Not only the of protection of of quantum quantum information. information. To To obtain obtain exponential exponential e.g. Nielsen and Chuang [2002]. Not only the design of protection time operations towards digital quantum computing, see e.g. Nielsen and Chuang [2002]. Not only the design of submanifold, a prerequisite for analytically quantifying the protection of quantum information. To obtain exponential the underlying control layer, but also the proposal convergence in a compact space, it is necessary to suppress e.g. underlying Nielsen andcontrol Chuanglayer, [2002]. Not only the design of protection of quantum information. To obtain exponential convergence in a compact space, it is necessary to suppress the but also the proposal of convergence in aa compact space, it is necessary to suppress e.g. Nielsen and Chuang [2002]. Not only the design of the underlying control layer, but also the proposal protection of quantum information. To obtain exponential convergence in compact space, it is necessary to suppress analog quantum technologies, like solving optimization any spurious unstable equilibria that might remain in the the underlying control layer, but also the optimization proposal of convergence inunstable a compact space, it is necessary to suppress any spurious equilibria that might remain in the analog quantum technologies, like solving spurious unstable equilibria that might remain in the the underlying control layer, but also the proposal of any analog quantum technologies, like solving optimization convergence in a compact space, it is necessary to suppress any spurious unstable equilibria that might remain in problems by quantum annealing, motivate a study of QEC closed-loop dynamics. As we noted in Cardona et al. [2018], analog quantum technologies, like solving optimization any spuriousdynamics. unstable As equilibria that might remain in the the closed-loop dynamics. As we noted noted in Cardona Cardona et al. al. [2018], [2018], problems by quantum annealing, motivate a study of QEC closed-loop we in et analog quantum technologies, like solving optimization problems by quantum annealing, motivate a study of QEC any spurious unstable equilibria that might remain in the closed-loop dynamics. As we noted in Cardona et al. [2018], in continuous-time, among them reservoir engineering and this is simplified problems by quantum annealing, motivate engineering a study of QEC closed-loop we notedby Cardona et stochastic al. [2018], in continuous-time, among them reservoir and this problem problemdynamics. is greatly greatlyAs simplified byinconsidering considering stochastic in continuous-time, among them reservoir engineering and this problem is greatly simplified by considering stochastic problems by quantum annealing, motivate a study of QEC in continuous-time, among them reservoir engineering and closed-loop dynamics. As we noted in Cardona et al. [2018], this problem is greatly simplified by considering stochastic measurement-based feedback. processes to drive the controls (see also Zhang et al. in continuous-time, among them reservoir engineering and processes this problem isdrive greatly simplified by considering stochastic to the controls (see also Zhang et al. measurement-based feedback. processes to drive the controls (see also Zhang et al. in continuous-time, among them reservoir engineering and measurement-based feedback. this problem is greatly simplified by considering stochastic processes to drive the controls (see also Zhang et al. [2018] for feedback laws with similar stochastic terms). measurement-based feedback. processes to drive the controls (see also Zhang et al. [2018] for feedback laws with similar stochastic terms). Reservoir engineering couples the target system to a dis[2018] for feedback laws with similar stochastic terms). Reservoir engineering couples the target system to a dismeasurement-based feedback. processes to drive the controls (see also Zhang et al. [2018] for feedback laws with similar stochastic terms). Reservoir engineering couples the target system to a disTherefore in the present paper, in the context of QEC, we [2018] for feedback laws with similar stochastic terms). Therefore in the present paper, in the context of QEC, we Reservoir engineering couples the target system to a dissipative ancillary quantum system, such that the entropy Therefore in the present paper, in the context of QEC, we Reservoirancillary engineering couples the target system toentropy a dis- Therefore [2018] foraa in feedback lawspaper, with similar stochastic terms). the present in the context of QEC, we sipative quantum system, such that the propose noise-assisted quantum feedback, acting with Therefore in the present paper, in the context of QEC, we propose noise-assisted quantum feedback, acting with Reservoir engineering couples the target system toentropy a dis- propose a noise-assisted quantum feedback, acting with sipative ancillary quantum system, such that the entropy introduced by errors errors on the target system isthe evacuated sipative ancillary quantum system, such that Therefore the whose presentgain paper, in the context of QEC, we propose aa in noise-assisted quantum feedback, acting with introduced by on the target system is evacuated noise is adjusted in real-time. We propose noise-assisted quantum feedback, acting with Brownian noise whose gain is adjusted in real-time. We sipative quantum system, such that entropy Brownian introduced bydissipation errors on of the target system isthe evacuated through ancillary theby dissipation of the ancillary one. Reservoir Brownian noise whose gain is adjusted in real-time. We introduced errors on the target system is evacuated through the the ancillary one. Reservoir propose a noise-assisted quantum feedback, acting with Brownian noise whose gain is adjusted in real-time. We through the dissipation of the ancillary one. Reservoir show via standard stochastic Lyapunov arguments that noise whose gain is Lyapunov adjusted inarguments real-time.that We show via via standard standard stochastic Lyapunov arguments that introduced errors on of thethe target system isinvestigated evacuated through theby dissipation of the ancillary one. Reservoir Brownian engineering for autonomous QEC has been investigated show stochastic through the dissipation ancillary one. Reservoir engineering for autonomous QEC has been Brownian noise whose gain is target adjusted inarguments real-time. We show viaapproach standard stochastic Lyapunov arguments that engineering for autonomous QEC has been investigated this new renders the subspace, containing show via standard stochastic Lyapunov that this new approach renders the target subspace, containing through the dissipation of the ancillary one. Reservoir engineering for autonomous QEC has been investigated this new approach renders the target subspace, containing  engineering for autonomous QEC has been investigated This work has been supported by the ANR project HAMROQS. show via standard stochastic Lyapunov arguments that  this new approach renders the target subspace, containing  This work work has has been supported by by the ANR ANR project HAMROQS. this new approach renders the target subspace, containing engineering forbeen autonomous QEC has project been investigated This supported the HAMROQS.  This work work has has been been supported supported by by the the ANR ANR project project HAMROQS. HAMROQS.  This this new approach renders the target subspace, containing

 This work has been supported by the ANR project HAMROQS. 2405-8963 © © 2019 2019, IFAC IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright 276 Copyright 2019 276 Copyright © under 2019 IFAC IFAC 276 Control. Peer review© responsibility of International Federation of Automatic Copyright © 2019 IFAC 276 Copyright © 2019 IFAC 276 10.1016/j.ifacol.2019.11.778 Copyright © 2019 IFAC 276

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the nominal encoding of quantum information, globally exponentially stable thanks to feedback from syndrome measurements. Furthermore, our strategy allows to work with a reduced state estimator: while other feedback schemes require to keep track of quantum coherences, the proposed feedback scheme allows for the implementation of a reduced filter that only tracks the populations on the various joint eigenspaces of the measurement operators. The paper is organized as follows. Section 2 presents the dynamical model of the three-qubit bit-flip code, which is the most basic model in QEC. In section 3 we introduce our approach to feedback using noise and we prove exponential stabilization of the target manifold of the three-qubit bitflip code. It presents as well the reduced order filter that follows from the feedback scheme. Section 4 examines the performance of this feedback and reduced filter to protect quantum information from bit-flip errors. Remark 1. (Stochastic Calculus): We will consider concrete instances of It¯ o stochastic differential equations (SDEs) on Rn of the form dx = µ(x)dt + σ(x)dW, (1) where W is a standard Brownian motion on Rk , and µ, σ are regular functions of x with image in Rn and Rn×k respectively, satisfying the usual conditions for existence and uniqueness of solutions ([Khasminskii, 2011, Chapter 3]) on S, a compact and positively invariant subset of Rn .

We will use results on stochastic stability (Khasminskii [2011]). Consider (1) with µ(x) = σ(x) = 0 for x ∈ S0 ⊂ S, thus S0 is a compact set of equilibria. Let V (x), a nonnegative real-valued twice continuously differentiable function with respect to every x ∈ S \ S0 . Its Markov generator associated with (1) is  1 ∂ ∂2 µi V + σi σj V, (2) AV = ∂xi 2 i,j ∂xi xj i and

E[V (xt )] = V (x0 ) + E



t AV 0

 (xs )ds .

Theorem 2. (Khasminskii [2011]). If there exists r > 0 such that AV (x) ≤ −rV (x), ∀x ∈ S \ S0 , then V (xt ) is a supermartingale on S with exponential decay: E[V (xt )] ≤ V (x0 ) exp(−r t) . If V is a meaningful way to quantify the distance to a target set { x : V (x) = 0 } ⊇ S0 , then this theorem establishes an exponential convergence result in the sense of expectation of V . Analysis in the rest of this paper consists in defining a function V and constructing controls that ensure exponential convergence in the above sense.

   ρL† − Tr ρ(L + L† ) ρ , where L† denotes the complex conjugate transpose of L. The state ρ belongs to the set of density matrices S = {ρ ∈ Cn×n : ρ = ρ† , ρ positive semidefinite , Tr (ρ) = 1} on the Hilbert space of the system H  Cn×n ; the {Wk } are independent standard Brownian motions and the {dYk } correspond to the measurement processes of each measurement channel. The ηk ∈ [0, 1] express the corresponding measurement efficiencies, i.e. the ratio of the corresponding channel linking the system to the outside world which is effectively captured by the measurement device; channels k with ηk = 0 represent pure loss channels. The simplest way to model the feedback stage consists in applying an infinitesimal unitary operation to the openloop evolution, ρt+dt = Ut (ρt + dρt )Ut† , where Ut =  exp(−i j Hj ut,j dt) with Hj hermitian operators denoting the control Hamiltonians that can be applied, and each ut,j dt a real control input. The fact that ut,j dt may contain stochastic processes requires to treat this feedback action with care, we will come back to this in the next section. 2.1 Dynamics of the three-qubit bit-flip code The three-qubit bit-flip code corresponds to a Hilbert space H = (C2 )⊗3  C8 , where ⊗ denotes tensor product (Kronecker product, in matrix representation). We denote In the identity operator on Cn and we write Xk , Yk and Zk the local Pauli operators acting on qubit k, e.g. X2 = I2 ⊗ σx ⊗ I2 . We denote {|0 , |1 } the usual basis states, i.e. the -1 and +1 eigenstates of the σz operator on each individual qubit (Nielsen and Chuang [2002]). The encoding on this 3-qubit system is meant to counter bit-flip errors, which map a ±1 eigenstate of Zk to the ∓1 eigenstate for each k = 1, 2, 3. The nominal encoding for a logical information 0 (resp. 1) is on the state |000 (resp. |111 ). A single bit-flip on e.g. the first qubit brings this to X1 |000 = |100 (resp. |011 ), which by majority vote can be brought back to the nominal encoding. In the continuous-time model (3), bit-flip errors occurring with a probability γk dt 1 during a time interval [t, t + dt] are modeled by disturbance channels, with Lk+3 = √ γk Xk and ηk+3 = 0, k = 1, 2, 3. The measurements needed to implement “majority vote” corrections, so-called syndromes, continuously compare the σz value of pairs of qubits.√The associated measurements correspond in (3) to Lk = Γk Sk for k = 1, 2, 3, with S1 = Z2 Z3 , S2 = Z1 Z3 , S3 = Z1 Z2 and Γk representing the measurement strength. This yields the following open-loop model: 3 3    dρ =

2. CONTINUOUS-TIME DYNAMICS OF THE THREE-QUBIT BIT-FLIP CODE

199

k=1

Γk DSk (ρ)dt+

ηk Γk MSk (ρ)dWk +

s=1

γs DXs (ρ)dt. (4)

We further define the operators:

The general model for a quantum system subject to several measurement channels (see, e.g., Barchielli and Gregoratti [2009]) is an It¯ o stochastic differential equation of the type  √ dρt = k DLk (ρ)dt + ηk MLk (ρ)dWk , (3)   √ † dYk = η k Tr (Lk + Lk )ρ dt + dWk .

We have used the standard super-operator notation   DL (ρ) = LρL† − 12 (L† Lρ + ρL† L) , ML (ρ) = Lρ +

277

ΠC =

1 4



I8 +

3 

k=1

 Sk , Πj := Xj ΠC Xj , j ∈ {1, 2, 3},

(5)

corresponding to orthogonal projectors onto the eigenspaces of the measurement syndromes. ΠC projects onto the nominal code C := span(|000 , |111 ) (+1 eigenspace of all the Sk ), whereas Πj projects onto the subspace where qubit j is flipped with respect to the two others. For each k ∈ {C, 1, 2, 3}, we write

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pt,k := Tr (Πk ρt ) ≥ 0 the so-called population of subspace k, i.e. the probability that a projective measurement of the syndromes would give the output corresponding to subspace k. By the law  of total probabilities, k∈{C,1,2,3} pt,k = 1 for all t. 2.2 Behavior under measurement only

We have the following behavior in absence of feedback actions and disturbances. Lemma 3. Consider (4) with γs = 0 for s ∈ 1, 2, 3.

(i) For each k ∈ {C, 1, 2, 3}, the subspace population pt,k is a martingale i.e. E(pt,k |p0,k ) = p0,k for all t ≥ 0. (ii) For a given ρ0 , if there exists k¯ ∈ {C, 1, 2, 3} such ¯ then ρ0 is a that p0,k¯ = 1 and p0,k = 0 for all k = k, steady state of (4). (iii) The Lyapunov function  √ V (ρ) = pk pk  k∈{C,1,2,3} k =k

decreases exponentially as E[V (ρt )] ≤ e−rt V (ρ0 ) for all t ≥ 0, with rate r = 4 mink∈{1,2,3} ηk Γk . In this sense the system exponentially approaches the set of invariant states described in point (ii).

Proof. The first two statements are easily verified, we √ prove the last one. The variables ξj = pj , j ∈ {1, 2, 3, C} satisfy the following SDE’s:    ηk Γk (1 − ξC2 − ξk2 )2 dt dξC = −2ξC k∈{1,2,3}

+ 2ξC 





k∈{1,2,3}

  ηk Γk (1 − ξC2 − ξk2 ) dWk ,

• Drive any initial state ρ0 towards a state with support only on the nominal codespace C = span{|000, |111}. This comes down to making Tr (ΠC ρt ) converge to 1. • For Tr (ΠC ρ0 ) = 1 and in the presence of disturbances γs = 0, minimize the distance between ρt and ρ0 for all t ≥ 0.

We now directly address the first point, the second one will be discussed in the sequel.

As mentioned in the introduction, this problem has already been considered before, yet without proof of exponential convergence. Towards establishing such proof, we introduce a key novelty into the feedback signal: we drive it by a stochastic process. Indeed, noise can be as efficient as a deterministic action to exponentially destabilize a spurious equilibrium where k¯ = C; in turn, using noise simplifies the study of the average dynamics, both in the analysis via Theorem 2 and towards implementing a quantum filter to estimate ρ. We thus introduce noise-assisted quantum feedback, where the control input consists of pure noise with state-dependent gain; i.e. we take uj dt = σj (ρ)dBj , with Bj (t) a Brownian motion independent of any Wk (t). As control Hamiltonians we take Hj = Xj , thus rotating back the bit-flip actions. The closed-loop dynamics in It¯ o sense then writes: 3 3    Γk DSk (ρ)dt + +

3  j=1



ηj Γj (1 − ξC2 − ξj2 ) dWj    − ηk Γk (ξC2 + ξk2 ) dWk , k∈{1,2,3}\j

 while V = k∈{C,1,2,3} k =k ξk ξk . Noting that 2(1−ξC2 − ξk2 ) and 2(ξC2 + ξk2 ) just correspond to 1 ± Tr (ρSk ), we only have to keep track of ± signs to compute    ξj ξk

j,k,l ηl Γl AV = −2 k∈{C,1,2,3} j∈{C,1,2,,3}\k

Error correction requires to design a control law satisfying two properties:

k=1

k∈{1,2,3}\j



3.1 Controller design

dρ =

dξj=C = −2ξj ηj Γj (1 − ξC2 − ξj2 )2   + ηk Γk (ξC2 + ξk2 )2 dt + 2ξj

3. ERROR CORRECTION VIA NOISE-ASSISTED FEEDBACK STABILIZATION

l∈{1,2,3}

where, for each pair (j, k), the selector j,k,l ∈ {0, 1} equals 1 for two l values, namely C,k,l = k,C,l = 1 if l = k ∈ {1, 2, 3} and j,k,j = j,k,k = 1 for j, k ∈ {1, 2, 3}. This readily leads to AV ≤ −4 mink∈{1,2,3} (ηk Γk ) V . We conclude by Theorem 2 and noting that V = 0 necessarily corresponds to a state as described in point (ii).  The above Lyapunov function describes the convergence of the state towards Tr (Πk¯ ρ) = 1, for a random subspace k¯ ∈ {C, 1, 2, , 3} chosen with probability p0,k¯ . We now address how to render a particular subspace globally attractive, namely the one associated to ΠC and nominal codewords.

278

ηk Γk MSk (ρ)dWk +

s=1

γs DXs (ρ)dt

−iσj (ρ)[Xj , ρ]dBj + σj (ρ)2 DXj (ρ)dt .

(6)

The last term can be viewed as “encouraging” a bit-flip with a rate depending on the value of σj and thus on ρ. The remaining task is to design the gains σj . For this many options will work — its only essential role is to “shake” the state when it is close to Tr (ΠC ρ) = 0, since the open loop already ensures stochastic convergence to either Tr (ΠC ρ) = 0 or Tr (ΠC ρ) = 1. The following hysteresisbased control law, illustrated by Fig. 1, depends only on the pt,k and should not be too hard to implement. Select real parameters αj and βj such that 12 < βj < αj < 1 for j ∈ {1, 2, 3}, and take a constant c > 0.  6cη Γ (1) If pj ≥ αj then take σj = 2αjj−1j ; (2) If pj ≤ βj then take σj = 0; (3) In the hysteresis region, i.e. for values of pj ∈]βj , αj [: keep the previous value of σj . 3.2 Closed-loop exponential convergence We propose the closed-loop Lyapunov function: V (ρ) = V1 (ρ) + V2 (ρ) + V3 (ρ) √ with Vk (ρ) = pk + p1 + p2 + p3 for k = 1, 2, 3.

(7)

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201

Via circular permutation and summation, we get 3  √ g(ρ)V (ρ) σj2 (ρ)gj (ρ) − 34ηΓ AV (ρ) ≤ 2

(8)

j=1

where

1−f gj (ρ) = √ j +

1−2(pj +pj  ) 2

fj



fj 

{j, j , j } = {1, 2, 3} and g(ρ) =  2  



(p2 +p3 )(1−f1 )

+

1−2(pj +pj  ) 2



with

fj 

2 

+ p1 +(p1 +p3 )(1−f1 ) (2p1 +p2 +p3 )2

+ p1 +(p1 +p2 )(1−f1 )

2

 2  2  2 (p3 +p1 )(1−f2 ) + p2 +(p2 +p1 )(1−f2 ) + p2 +(p2 +p3 )(1−f2 ) + (2p2 +p3 +p1 )2  2  2  2 (p1 +p2 )(1−f3 ) + p3 +(p3 +p2 )(1−f3 ) + p3 +(p3 +p1 )(1−f3 ) + . (2p3 +p1 +p2 )2

When ρ ∈ Q, we have pj ≥ αj > 1/2 for a unique j ∈ {1, 2, 3}, since  p1 + p2 + p3 ≤ 1. Assume first that

Fig. 1. for αj ≡ α and   βj ≡ β, the 6 active feedback zones in the simplex (p1 , p2 , p3 )  p1 , p2 , p3 ≥ 0, p1 +p2 +p3 ≤ 1 .

6cηΓ and σ2 (ρ) = σ3 (ρ) = 0. Since p1 ≥ α1 , thus σ1 = 2α 1 −1 Theorem 4. Consider (6) with all γs = 0 and feedback g(ρ) ≥ 0, inequality (8) implies   gains (σj ) as specified just before section 3.2. Then 1−2(p1 +p2 ) 1−2(p1 +p3 ) 6cηΓ 1−f 1 √ √ AV ≤ 2α1 −1 √ + + . E[V (ρt )] ≤ V (ρ0 )e−rt , ∀t ≥ 0, f1 2 f2 2 f3 √ with the exponential convergence rate estimated as:     Since f1 ≥ 2α1 , 1 − 2p1 ≤ 0, f1 ≤ 2 and V ≤ 3 2 we get 6cηΓ 1−2α √ 1 = − 6cηΓ √ V ≤ −cηΓV. AV ≤ 2α 4 r= min ηj Γj min c , √ min g(s, x1 , x2 , x3 ) 1 −1 f1 V f1 3 2 j∈{1,2,3}

(s,x1 ,x2 ,x3 )∈K

We get a similar inequality when p2 ≥ α2 or p3 ≥ α3 . Thus ∀ρ ∈ Q, AV (ρ) ≤ −cηΓV (ρ). Consider now ρ ∈ S \ Q. Then, pj < αj for all j. Since σj (ρ) = 0 when pj ≤ 1/2 we have σj2 (ρ)gj (ρ) ≤ 0. √ g(ρ)V (ρ). Let us prove From (8), we have AV (ρ) ≤ − 34ηΓ 2 that g(ρ) ≥ r for any ρ ∈ S \ Q. With s = p1 + p2 + p3 and xj = pj /s, g can be seen as a function of (s, x1 , x2 , x3 ),

where g(s, x1 , x2 , x3 ) is given in (9) below and     K = (s, x1 , x2 , x3 ) ∈ [0, 1]4  x1 + x2 + x3 = 1; sxj ≤ αj

Proof. By design of the hysteresis, well-posedness of the solution then follows from standard arguments on the construction of solutions of SDE’s. The proof then consists in showing that V (ρt ) on S is an exponential supermartingale satisfying A(V ) ≤ −rV . Towards this we  partition the state-space into Q := ∪3j=1 ρ ∈ S | pj ≥ αj and S \ Q on which we compute the Markov generator A(V ) separately.

g(ρ) = g(s, x1 , x2 , x3 ) 



+

Let  us write  from (6) the expression of AV (ρ) = E dVt | ρt = ρ /dt for any value of the control gain vector σ. We exploit here the following formula based on It¯ o rules and valid for any non-negative operator F :  Tr (F dρ) (Tr (F dρ))2 d Tr (F ρ) =  − .  2

Tr (F ρ)

4 Tr (F ρ)

+

AV1 (ρ) =

− 4ηΓ



Since



4ηΓV



2 σ1

f1 ≥

AV1 (ρ) ≤ −

1−f1

(p2 +p3 )(1−f1 )



2 +σ2

Tr

2 

2

1−2(p1 +p2) 2



f1

+ p1 +(p1 +p3 )(1−f1 ) f1

2 ([X1 ,ρ]F1 )+σ2



Tr (F ρ)

f1

, we have  

2 

+ p1 +(p1 +p2 )(1−f1 )

1 √ V 3 2





f1



2 2 2 2σ1 1−f1 +σ2 1−2(p1 +p2) +σ3 1−2(p1 +p3 )

(p2 +p3 )(1−f1 )

2 

2



f1

+ p1 +(p1 +p3 )(1−f1 ) √ 3 2f12

2 



(x3 +x1 )(1−f2 )

(x1 +x2 )(1−f3 )

2 

(1+x1 )2

2 

2

2 

2

+ x1 +(x1 +x2 )(1−f1 )

+ x2 +(x2 +x1 )(1−f2 )

2 

(1+x2 )2

+ x3 +(x3 +x2 )(1−f3 ) (1+x3 )2

2 

+ x2 +(x2 +x3 )(1−f2 )

+ x3 +(x3 +x1 )(1−f3 )

2 (9)

For a heuristic estimate of r, take s = αj with xj = 1 for    some j to get r ∼ minj∈{1,2,3} ηj Γj min c, √82 (1 − α ¯ )2 , with α ¯ = maxj∈{1,2,3} αj . Typically one would take c = 1 and α1 = α2 = α3 = α close to 1. When η√ j Γj are all equal, such a rough estimate simplifies to r = 4 2(1 − α)2 ηΓ .

2 .

+ p1 +(p1 +p2 )(1−f1 )

+ x1 +(x1 +x3 )(1−f1 )

Taking all things together, we have proved that AV (ρ) ≤ −rV (ρ) always holds. We conclude with Theorem 2. 

1−2(p1 +p3 )

2 Tr2 ([X2 ,ρ]F1 )+σ3 Tr2 ([X3 ,ρ]F1 )

4f1



2 +σ3





2 

with fj = 1 − s − sxj . Here (s, x1 , x2 , x3 ) belongs to the compact set s ∈ [0, 1], xj ≥ 0, j xj = 1 and sxj ≤ αj for all j. On this compact set, g is a smooth function. Moreover it is strictly positive since g = 0 implies that s = 1 and xj = 1 for some j ∈ {1, 2, 3} which would not satisfy sxj ≤ αj . This means that minρ∈S\Q g(ρ) > 0.

We detail below the computations when ηj ≡ η and Γj ≡ Γ (the formulas in the general case are slightly more complicated). With F1 = 2Π1 + Π2 + Π3 and V1 (ρ) =  √ f1 = Tr (F1 ρ), we get       2 2σ1

(x2 +x3 )(1−f1 )

3.3 Reduced quantum filter 2

.

279

Towards implementing the control law we have to reconstruct in real-time the quantum state estimate ρ via a quantum filter. For (6), this filter reads:

2019 IFAC NOLCOS 202 Vienna, Austria, Sept. 4-6, 2019

dρ =

3 

k=1

Γk DSk (ρ)dt +

Gerardo Cardona et al. / IFAC PapersOnLine 52-16 (2019) 198–203

3  s=1

shown on Figure 2 where we consider that the quantum filter perfectly follows (6). Figure 3 corresponds to a more realistic situation where the same feedback law relies on the reduced order quantum filter (12) corrupted by errors and feedback latency: we observe a small change of performance but still a clear improvement compared to a single qubit.

γs DXs (ρ)dt

3      ηk Γk MSk (ρ) dYk − 2 ηk Γk Tr (Sk ρ) dt + k=1

+

3  j=1

−iσj (ρ)[Xj , ρ]dBj + σj (ρ)2 DXj (ρ)dt .

(10)

√ where dYk = 2 ηk Γk Tr (Sk ρ) dt + dWk is the measurement outcome of syndrome Sk , and the random dBj applied to the system are accessible too a posteriori. Instead, we can replace the state ρt in the feedback law, by ρt corresponding to the Bayesian estimate of ρt knowing its initial condition ρ0 and the syndrome measurements dYk between 0 and the current time t > 0, but not the dBj . Then ρt obeys to the SME: d ρ=

3 

k=1

+

3 

k=1



Γk DSk ( ρ)dt +

3 

(γs + σj2 ( ρ))DXj ( ρ)dt

0.95 0.9 0.85 0.8

Tr(;;0 ) closed loop Tr(;& c) closed loop

0.75

Tr(;;0 ) open loop qubit

0.7

Tr((;+X1 ; X 1 +X2 ; X 2 +X3 ; X 3 );0 )

0.65 0.6

j=1

   ηk Γk MSk ( ρ) dYk − 2 ηk Γk Tr (Sk ρ) dt

Average fidelity over 1000 realizations

1

(11)

√ where dYk = 2 ηk Γk Tr (Sk ρ) dt + dWk with ρ governed by (6) where σj (ρ) is replaced by σj ( ρ). Denote pˆj = Tr (Πj ρ) and sˆk = Tr (Sk ρ). Then we have

s1 dt dˆ s1 = −2(γ2 + σ22 + γ3 + σ33 )ˆ     2 + 2 η1 Γ1 (1 − sˆ1 ) dY1 − 2 η1 Γ1 sˆ1 dt     + 2 η2 Γ2 (ˆ s3 − sˆ1 sˆ2 ) dY2 − 2 η2 Γ2 sˆ2 dt     + 2 η3 Γ3 (ˆ s2 − sˆ1 sˆ3 ) dY3 − 2 η3 Γ3 sˆ3 dt (12) s2,3 with pˆ1 = (1 + sˆ1 − sˆ2 − sˆ3 )/4. The formulas for dˆ and pˆ2,3 are obtained via circular permutation in {1, 2, 3}. Since the feedback law depends only on the populations pˆj , it can be implemented with the exact quantum filter reduced to (ˆ s1 , sˆ2 , sˆ3 ) ∈ R3 . Contrarily to the full quantum filter (10), here the syndrome dynamics sˆk are independent of any coherences among the different subspaces and we get a closed system on classical probabilities, driven by the measurement signals. 4. ON THE PROTECTION OF QUANTUM INFORMATION It is well-known in control theory that exponential stability gives an indication of robustness against unmodeled dynamics. In the present case, this concerns the first control goal, namely stabilization of ρt close to the nominal subspace C in the presence of bit-flip errors γs = 0. About the second control goal, namely keeping the dynamics on C close to zero such that logical information remains protected, the analysis of the previous section is less telling. We can illustrate both control goals by simulation. As in Ahn et al. [2002] we set as initial condition ρ0 = |000000| and simulate 1000 closed-loop trajectories under the feedback law of section 3.1. We compare the average evolution of this encoded qubit with a single physical qubit subject to a σx decoherence of the same strength, since this is the situation that the bit-flip code is meant to improve. Parameter values and simulation results are 280

0.55 0.5 0

10

20

30

40

50

60

70

80

90

100

t

Fig. 2. Ideal situation where the feedback of subsection 3.1 is based on ρ governed by (6). Solid red: mean overlap of the state with the code space. Solid black: mean fidelity of the logical qubit versus ρ0 . Solid blue: mean correctable fidelity under active quantum feedback. For the three solid curves, the initial state is chosen as ρ0 = |000000| and closed-loop simulation parameters based on (6) are Γj = 1, γj = 1/64, ηj = 0.8, and for the feedback law βj = 0.6, αj = 0.95, c = 3/2. Dashed line, for comparison: mean fidelity towards |00| for a single physical qubit without measurement nor control and subject to bitflip disturbances with γ = 1/64. Regarding the first control goal, we observe that the controller indeed confines the mean evolution to a small neighborhood of C, for all times, as expected from our analysis. Regarding the second criterion, the distance between ρt and ρ0 cannot be confined to a small value for all times. Indeed, majority vote can decrease the rate of information corruption but not totally suppress it; as corrupted information is irremediably lost, ρt progressively converges towards an equal distribution of logical 0 and logical 1. However, for the protected 3-qubit code, this information loss is much slower than for the single qubit; this indicates that the 3-qubit code with our feedback law indeed improves on its components. In our feedback design, making αj closer to 1/2 would improve the convergence rate estimate in Theorem 4; however, this also has a negative effect on the logical information, since it means that we turn on the noisy drives more often. Analytically computing the optimal tradeoff is the subject of ongoing work. Similary, making c larger would accelerate the recovery action but increase the level of noise, and we want to keep the induced motion slower than the measurement timescale. Simulations (not

2019 IFAC NOLCOS Vienna, Austria, Sept. 4-6, 2019

Gerardo Cardona et al. / IFAC PapersOnLine 52-16 (2019) 198–203

word fidelity, despite leading to a slower convergence rate estimate. A theoretical analysis of information protection capabilities is the subject of ongoing work.

Average fidelity over 1000 realizations with errors and delays

1

203

0.95 0.9

The authors would like to thank K. Birgitta Whaley and Leigh S. Martin for discussions on continuous-time QEC.

0.85 0.8

REFERENCES

0.75 0.7 0.65

Tr(;;0 ) closed loop Tr(;& c) closed loop

0.6

Tr(;;0 ) open loop qubit

0.55

Tr((;+X1 ; X 1 +X2 ; X 2 +X3 ; X 3 );0 )

0.5 0

10

20

30

40

50

60

70

80

90

100

t

Fig. 3. Simulation similar to Fig. 2 for a more realistic case where feedback is based on the reduced order filter (12) and includes modeling/measurement errors and feedback latency. Marked with subscript ∗, the parameter values used in (12) are as follows: γ∗ = 0.8γ, Γ∗ = 0.9Γ, η∗ = 0.9η; constant measurement √ ηΓ bias according to dY∗,1 = dY1 + 10 dt, dY∗,2 = dY2 − √ √ ηΓ ηΓ 10 dt and dY∗,3 = dY3 + 20 dt, , and feedback latency of 1/(2Γ); measurement signals Yk are based on (6) with nominal values identical to simulation of Fig. 2 and control values σj ( ρ). reproduced here) clearly show that intermediate values of the control parameters deliver better overall results. 5. CONCLUSION We have approached continuous-time quantum error correction in the same spirit as Ahn et al. [2002], and showed how introducing Brownian motion to drive control fields yields exponential stabilization of the nominal codeword manifold. The main idea relies on the fact that the SDE in open loop stochastically converges to one of a few steadystate situations, but on the average does not move closer to any particular one. It is then sufficient to activate noise only when the state is close to a bad equilibrium, in order to induce globale convergence to the target ones. This general idea can be extended to other systems with this property, and in particular to more advanced errorcorrecting schemes. In the same line, while we have proposed particular controls with hysteresis, proving a similar property with smoother control gains should not be too different. The convergence rate obtained is dependent on our choice of Lyapunov function and on the values of αj ; from parallel investigation it seems possible to get a closedloop convergence rate arbitrarily close to the measurement rate. However, unlike in classical control problems, the key performance indicator is not how fast we approach the target manifold. Instead, what matters is how well, in presence of disturbances, we preserve the encoded information. Towards this goal, we should refrain from disturbing the system with feedback actions; accordingly, we have noticed that taking αj closer to 1 can improve the code281

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