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Feedback Issues Underlying Quantum Error Correction

IFAC Mechatronics & Nolcos Conference 2019 September 4–6, Vienna, Austria

Pierre Rouchon

Centre Automatique et Systèmes, Mines ParisTech, PSL Research University Quantic Research Team, Inria

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Nobel Prize in Physics 2012 (second quantum revolution)

Serge Haroche David J. Wineland

" This year’s Nobel Prize in Physics honours the experimental inventions and discoveries that have allowed themeasurement and control of individual quantum systems. They belong to two separate but related technologies: ions in a harmonic trap and photons in a cavity"

From the Scientific Background on the Nobel Prize in Physics 2012 compiled by the Class for Physics of the Royal Swedish Academy of Sciences, 9 October 2012.

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Investigated technologies for quantum computation1

© OBrien

Superconduc�ng circuits

Photons

© S. Kuhr

Ultra-cold neutral/Rydberg

atoms

© Bla� & Wineland

Trapped ions

© Pe�a

Quantum dots

© IBM

Requirements:

• scalable modular architecture;

• control software from the very beginning.

1Courtesy of Walter Riess, IBM Research - Zurich.

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Issues underlying this talk

Quantum Error Corrections (QEC) is based on an elementary discrete-time feedback loop: a static-output feedback neglecting the finite bandwidth of the measurement and actuation processes.

I Current experiments: 10−2is the typical error probability during elementary gates (manipulations) involving few physical qubits.

I High-order error-correcting codes with an important overhead;more than 1000 physical qubits to encode a controllable logical qubit2.

I Today, no such controllable logical qubit has been built.

I Key issue: reduction by several magnitude orders such error rates, far below the threshold required by actual QEC, to build a controllable logical qubit encoded in a reasonable number of physical qubits and protected by QEC.

Control engineering can play a crucial roleto built a controllable logical qubit protected by muchmore elaborated feedback schemes increasing precision and stability.

2A.G. Fowler, M. Mariantoni, J.M. Martinis, A.N. Cleland (2012): Surface codes: Towards practical large-scale quantum computation. Phys. Rev.

A,86(3):032324.

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Outline

Continuous-time dynamics of open quantum system Stochastic Master Equation (SME)

Key characteristics of SME

Quantum filtering and state estimation

Feedback schemes

Measurement-based feedback and classical controller Coherent feedback and quantum controller

Merging measurement-based and coherent feedbacks

Quantum Error Correction (QEC) and feedback QEC from scratch

Storing a logical qubit in a high-quality harmonic oscillator Simplified feedback scheme for cat-qubit experiment

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Dynamics of open quantum systems based on three quantum features3

1. Schrödinger (~=1): wave funct. |ψi ∈ H,density op. ρ∼ |ψihψ| d

dt|ψi=−iH|ψi, H=H0+uH1=H, d

dtρ=−i[H,ρ].

2. Origin of dissipation: collapse of the wave packetinduced by the measurement ofO=O with spectral decomp. P

yλyPy:

I measurement outcomey with proba.

Py =hψ|Py|ψi= Tr(ρPy)depending on|ψi,ρjust before the measurement

I measurement back-action if outcomey:

|ψi 7→ |ψi+= Py|ψi

phψ|Py|ψi, ρ7→ρ+= PyρPy

Tr(ρPy) 3. Tensor product for the description of composite systems (S,C):

I Hilbert spaceH=Hs⊗ Hc

I HamiltonianH=Hs⊗Ic+Hsc+Is⊗Hc I observable on sub-systemC only: O=Is⊗Oc.

3S. Haroche and J.M. Raimond (2006). Exploring the Quantum: Atoms, Cavities and Photons. Oxford Graduate Texts.

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Outline

Continuous-time dynamics of open quantum system Stochastic Master Equation (SME)

Key characteristics of SME

Quantum filtering and state estimation

Feedback schemes

Measurement-based feedback and classical controller Coherent feedback and quantum controller

Merging measurement-based and coherent feedbacks

Quantum Error Correction (QEC) and feedback QEC from scratch

Storing a logical qubit in a high-quality harmonic oscillator Simplified feedback scheme for cat-qubit experiment

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Classical I/O dynamics based on Stochastic Master Equation (SME)4

QUANTUM WORLD decoherence

Hilbert space (dissipation) CLASSICAL WORLD

Continuous-time models: stochastic differential systems (It¯o formulation) density operatorρ(ρ=ρ,ρ≥0, Tr(ρ) =1) as state (~≡1 here):

t =

−i[H0+utH1t] + X

ν=d,m

LνρtLν1

2(LνLνρttLνLν) dt +√

ηm

LmρttLm− Tr

(Lm+Lmt ρt

dWt

driven by the Wiener processWt, with measurementyt,

dyt=√ ηmTr

(Lm+Lmt

dt+dWt detection efficienciesηm∈[0,1].

Measurement backaction: dρanddy share the same noisesdW. Very different from the usual Kalman I/O state-space description.

4A. Barchielli, M. Gregoratti (2009): Quantum Trajectories and Measurements in Continuous Time: the Diffusive Case. Springer Verlag.

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SME well adapted to super-conducting Josephson circuits

Classical model (C+CCl

l =1):

d

dtΦ=C1Q+2u2 q`

c Φ

L sin 1

ΦΦ

d

dtQ=ΦLsin

1 ΦΦ

withy=u q`

c Φ

L sin

1 ΦΦ

. Hs(Φ,Q) =2C1Q2ΦL2cos

1 ΦΦ

with nonlinearity (Φ<(L/C)1/4):

I anharmonic spectrum: frequency transition between the ground and first excited states larger than frequency transition between first and second excited states, . . .

I qubit model based on restriction to these two slowest energy levels,|gi and|ei, with pulsationωq∼1/√

LC. Two weak coupling regimes of the transmon qubit5:

I resonant, in/out wave pulsationωq;

I off-resonant, in/out wave pulsationωq+ ∆with|∆| ωq.

5J. Koch et al. (2007): Charge-insensitive qubit design derived from the Cooper pair box. Phys. Rev. A, 76:042319.

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A key physical example with super-conducting Josephson circuits

6

Superconducting qubit dispersively coupled to a cavity traversed by a microwave signal (input/output theory).

The back-action on the qubit state of a single measurement of one output field quadrature y is described by a simple SME for the qubit density operatorρ, 2×2 Hermitian0 matrix.

t=

2iqZ,ρt] +γ(ZρZ−ρt) dt +√ηγ

ttZ−2 Tr(Zρtt dWt

withyt given bydyt=2√ηγ Tr(Zρt)dt+dWt whereγ≥0 is related to the measurementstrengthandη∈[0,1] is the detection efficiency.

6M. Hatridge et al. (2013): Quantum Back-Action of an Individual Variable-Strength Measurement. Science, 339, 178-181.

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Outline

Continuous-time dynamics of open quantum system Stochastic Master Equation (SME)

Key characteristics of SME

Quantum filtering and state estimation

Feedback schemes

Measurement-based feedback and classical controller Coherent feedback and quantum controller

Merging measurement-based and coherent feedbacks

Quantum Error Correction (QEC) and feedback QEC from scratch

Storing a logical qubit in a high-quality harmonic oscillator Simplified feedback scheme for cat-qubit experiment

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Another formulation of diffusive SME

7

With a single imperfect measurementdyt=√

ηTr (L+Lt

dt+dWt and detection efficiencyη∈[0,1], the quantum stateρt obeys to

t =

−i[H0+utH1t] +LρtL1

2(LttLL) dt +√

η

ttL− Tr

(L+Lt ρt

dWt

driven by the Wiener processdWt

WithIt¯o rules, it can be written as the following "discrete-time" Markov model

ρt+dtt+dρt= Mut,dytρtMu

t,dyt+ (1−η)LρtLdt Tr

Mut,dytρtMu

t,dyt+ (1−η)LρtLdt

withMut,dyt=I− i(H0+utH1) +12 LL dt+√

ηLdyt.

7PR (2014): Models and Feedback Stabilization of Open Quantum Systems. Proc.

of Int. Congress of Mathematicians, vol. IV, pp 921–946, Seoul.

(http://arxiv.org/abs/1407.7810).

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Key characteristics of quantum SME (1)

Measured output mapdyt =√η Tr

(L+Lt

dt+dWt and measurement backaction described by

ρt+dtt+dρt = Mut,dytρtMu

t,dyt+ (1−η)LρtLdt

Tr

Mut,dytρtMu

t,dyt+ (1−η)LρtLdt

I if ρ0 density operator, then, for all t>0,ρt remains a density operator

The dynamics preserve the cone of non-negative Hermitian operators.

I Positivity and trace preserving numerical scheme for quantum Monte-Carlo simulations.

I Whenη=1, rank(ρt)≤rank(ρ0)for allt ≥0. In particular ifρ0is a rank one projector, thenρt remains a rank one projector (pure state).

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Key characteristics of quantum SME (2)

t =

−i[H0+uH1t] +LρtL1

2(LttLL) dt +√

η

ttL− Tr

(L+Lt ρt

dWt

with measured output mapdyt=√

ηTr (L+Lt

dt+dWt

I Invariance of the SME structure under unitary transformations.

A time-varying change of frameeρ=UtρUt withUt unitary.

The new density operatorρeobeys to a similar SME where He0+uHe1=Ut(H0+uH0)Ut+iUt dtdUt

andeL=UtLUt. I "L1-contraction" properties. Whenη=0, SME becomes deterministic

d

dtρ=−i[H0+uH1t] +LρtL1

2(LttLL) generating a contraction semi-group for many distances (nuclear distance8, Hilbert metric on the cone of non negative operators9).

I If the non-negative Hermitian operatorAsatisfies the operator inequality i[H0+uH1,A] +LAL−1

2(LLA+ALL)≤0 thenV(ρ) = Tr(Aρ)is asuper-martingale (Lyapunov function).

8D.Petz (1996). Monotone metrics on matrix spaces. Linear Algebra and its Applications, 244, 81-96.

9R. Sepulchre, A. Sarlette, PR (2010). Consensus in non-commutative spaces. IEEE-CDC. 14 / 37

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Outline

Continuous-time dynamics of open quantum system Stochastic Master Equation (SME)

Key characteristics of SME

Quantum filtering and state estimation

Feedback schemes

Measurement-based feedback and classical controller Coherent feedback and quantum controller

Merging measurement-based and coherent feedbacks

Quantum Error Correction (QEC) and feedback QEC from scratch

Storing a logical qubit in a high-quality harmonic oscillator Simplified feedback scheme for cat-qubit experiment

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Belavkin quantum filter: stability and convergence issues

Quantum stateρt producingdyt=√

ηTr LρttL

dt+dWt and its estimateρbt:

t =

−i[H0+uH1t] +LρtL1

2(LttLL) dt +√

η

ttL− Tr

(L+Lt ρt

dWt

dρbt =

−i[H0+uH1,ρbt] +LρbtL1

2(LLρbt+ρbtLL) dt +√

η

Lρbt+ρbtL− Tr

(L+L)ρbt ρbt

dyt−√

ηTr

Lρbt+ρbtL dt

. I Stability10: the fidelityF(ρt,ρbt) = Tr2 p√

ρtρbt√ ρt

is always a sub-martingale:

∀t1≤t2, E

F ρt2,ρbt2 ρt1,ρbt1

≥F ρt1,ρbt1 .

Fidelity: 0≤F(ρ,ρ)b ≤1 andF(ρ,ρ) =b 1 iffρ=ρ.b

I Convergence11ofρbt towardsρt whent7→+∞is an open problem

10H. Amini, C. Pellegrini, C., PR (2014). Stability of continuous-time quantum filters with measurement imperfections. Russian Journal of Mathematical Physics, 21(3), 297-315.

11Partial result: R. van Handel (2009): The stability of quantum Markov filters. Infin. Dimens.

Anal. Quantum Probab. Relat. Top., 12, 153-172.

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Outline

Continuous-time dynamics of open quantum system Stochastic Master Equation (SME)

Key characteristics of SME

Quantum filtering and state estimation

Feedback schemes

Measurement-based feedback and classical controller Coherent feedback and quantum controller

Merging measurement-based and coherent feedbacks

Quantum Error Correction (QEC) and feedback QEC from scratch

Storing a logical qubit in a high-quality harmonic oscillator Simplified feedback scheme for cat-qubit experiment

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Measurement-based feedback

QUANTUM WORLD CLASSICAL WORLD

classical controller

decoherence classical

input

classical output

quantum measurement classical

reference

SME

I P-controller (Markovian feedback12) forut dt=k dyt, the ensemble average closed-loop dynamics ofρremains governed by a linear Lindblad master equation.

I PID controller: no Lindblad master equation in closed-loop for dynamics output feedback

I Nonlinear hidden-state stochastic systems: Lyapunov state-feedback13; many open issues on convergence rates, delays, robustness, . . . I Short sampling times limit feedback complexity

12H. Wiseman, G. Milburn (2009). Quantum Measurement and Control. Cambridge University Press.

13See e.g.: C. Ahn et. al (2002): Continuous quantum error correction via quantum feedback control. Phys. Rev. A 65;

M. Mirrahimi, R. Handel (2007): Stabilizing feedback controls for quantum systems. SIAM Journal on Control and Optimization, 46(2), 445-467;

G. Cardona, A. Sarlette, PR (2019): Continuous-time quantum error correction with noise-assisted quantum feedback. IFAC Mechatronics & Nolcos Conf.

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First MIMO measurement-based feedback for a superconducting qubit14

FM Rabi

b)

~ ~

FM

~ ~

~

Rabi

JPC out in

Drift

a) ,

c) 3 inputs

2 outputs

14P. Campagne-Ibarcq, . . . , PR, B. Huard (2016): Using Spontaneous Emission of a Qubit as a Resource for Feedback Control. Phys. Rev. Lett.

117(6).

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Outline

Continuous-time dynamics of open quantum system Stochastic Master Equation (SME)

Key characteristics of SME

Quantum filtering and state estimation

Feedback schemes

Measurement-based feedback and classical controller Coherent feedback and quantum controller

Merging measurement-based and coherent feedbacks

Quantum Error Correction (QEC) and feedback QEC from scratch

Storing a logical qubit in a high-quality harmonic oscillator Simplified feedback scheme for cat-qubit experiment

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Coherent (autonomous) feedback (dissipation engineering)

Quantum analogue of Watt speed governor: adissipativemechanical system controls another mechanical system15

QUANTUM WORLD CLASSICAL WORLD

Hilbert space

quantum controller Hilbert space Hilbert space

system S

decoherence

decoherence

quantum interaction

Optical pumping (Kastler1950), coherent population trapping (Arimondo1996) Dissipation engineering, autonomous feedback: (Zoller, Cirac, Wolf, Verstraete, Devoret, Schoelkopf, Siddiqi, Martinis, Raimond, Brune,. . . ,Lloyd, Viola, Ticozzi, Leghtas, Mirrahimi, Sarlette, PR, ...) (S,L,H) theoryandlinear quantum systems: quantum feedback networks based on stochastic Schrödinger equation, Heisenberg picture (Gardiner, Yurke, Mabuchi, Genoni, Serafini, Milburn, Wiseman, Doherty, . . . ,Gough, James, Petersen, Nurdin, Yamamoto, Zhang, Dong, . . . )

Stability analysis: Kraus maps and Lindblad propagators are always contractions (non commutative diffusion and consensus).

15J.C. Maxwell (1868): On governors. Proc. of the Royal Society, No.100.

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Coherent feedback involves tensor products and many time-scales

The closed-loop Lindblad master equation onH=Hs⊗ Hc: d

dtρ=−ih

Hs⊗Ic+Is⊗Hc+Hsc , ρi +X

ν

DLs,ν⊗Ic(ρ) +X

ν0

DIs⊗Lc,ν0(ρ)

withDL(ρ) =LρL12 LLρ+ρLL

and operators made oftensor products.

•Consider a convex subsetDsof steady-states for original systemS: each density operatorρs onHs belonging toDssatisfyi[Hss] =P

νDLs,νs).

•Designing arealisticquantum controllerC (Hc,Lc,ν0) and coupling HamiltonianHsc stabilizingDsis non trivial. Realisticmeans in particular relying onphysical time-scalesand constraints:

I Fastest time-scales attached toHsandHc (Bohr frequencies) and averaging approximations: kHsk,kHck kHsck,

I High-quality oscillations: kHsk kLs,νLs,νkandkHck kLc,ν0Lc,ν0k.

I Decoherence rates ofSmuch slower than those ofC: kLs,νLs,νk kLc,ν0Lc,ν0k: model reduction byquasi-static approximations(adiabatic elimination, singular perturbations).

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Outline

Continuous-time dynamics of open quantum system Stochastic Master Equation (SME)

Key characteristics of SME

Quantum filtering and state estimation

Feedback schemes

Measurement-based feedback and classical controller Coherent feedback and quantum controller

Merging measurement-based and coherent feedbacks

Quantum Error Correction (QEC) and feedback QEC from scratch

Storing a logical qubit in a high-quality harmonic oscillator Simplified feedback scheme for cat-qubit experiment

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Quantum feedback engineering

QUANTUM WORLD CLASSICAL WORLD

Hilbert space

classical controller

quantum controller Hilbert space Hilbert space

system S

decoherence decoherence

classical input

classical output

quantum measurement classical

reference

quantum interaction

To stabilize the quantum information localized in system S:

I fast decoherence addressed preferentially byquantum controllers (coherent feedback);

I slowdecoherence, perturbations and parameter drifts tackled mainly byclassical controllers (measurement-based feedback).

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Outline

Continuous-time dynamics of open quantum system Stochastic Master Equation (SME)

Key characteristics of SME

Quantum filtering and state estimation

Feedback schemes

Measurement-based feedback and classical controller Coherent feedback and quantum controller

Merging measurement-based and coherent feedbacks

Quantum Error Correction (QEC) and feedback QEC from scratch

Storing a logical qubit in a high-quality harmonic oscillator Simplified feedback scheme for cat-qubit experiment

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The simplest classical error correction code

•Single bit error model: the bitb∈ {0,1} flips with probabilityp<1/2 during∆t (for usual DRAM:p/∆t ≤10−14 s−1).

•Multi-bit error model: each bitbk ∈ {0,1} flips with probability p<1/2 during∆t;no correlation between the bit flips.

•Useredundancyto construct with several physical bitsbk of flip probabilityp, a logical bitbL with a flip probabilitypL<p.

•The simplest solution, the3-bit code(sampling time∆t):

t=0: bL= [bbb]withb∈ {0,1}

t = ∆t: measure the three physical bits ofbL= [b1b2b3] (instantaneous) :

1. if all 3 bits coincide, nothing to do.

2. if one bit differs from the two other ones, flip this bit (instantaneous);

•Since the flip probability laws of the physical bits are independent, the probability that the logical bitbL (protected with the above error correction code) flips during∆t ispL =3p2−2p3<p sincep<1/2.

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The 3-qubit bit-flip code (Peter Shor (1995))

17

Local bit-flip errors: each physical qubit|ψi=α|0i+β|1ibecomes X|ψi=α|1i+β|0i16with probabilityp<1/2 during∆t.

(for actual super-conducting qubitp/∆t>103s−1).

t=0: Li=α|0Li+β|1Li ∈C2C2C2C8with|0Li=|000iand

|1Li=|111i.

t= ∆t: Libecomes with

1 flip:

α|100i+β|011i α|010i+β|101i α|001i+β|110i

; 2 flips:

α|110i+β|001i α|101i+β|010i α|011i+β|100i

; 3 flips:α|111i+β|000i.

Key fact: 4 orthogonal planesPc=span(|000i,|111i),P1=span(|100i,|011i), P2=span(|010i,|101i)andP3=span(|001i,|110i).

Error syndromes: 3 commuting observablesS1=IZZ,S2=ZIZ and S3=ZZI with spectrum{−1,+1}and outcomes(s1,s2,s3)∈ {−1,+1}.

-1-s1=s2=s3:Pc3 |ψLi=

α|000i+β|111i0 flip

β|000i+α|111i3 flips ; no correction -2-s16=s2=s3:P13 |ψLi=

α|100i+β|011i1 flip

β|100i+α|011i2 flips ;(XII)|ψLi ∈ Pc. -3-s26=s3=s1:P23 |ψLi=

α|010i+β|101i1 flip

β|010i+α|101i2 flips ;(IXI)|ψLi ∈ Pc. -4-s36=s1=s2:P33 |ψLi=

α|001i+β|110i1 flip

β|001i+α|110i2 flips ;(IIX)|ψLi ∈ Pc.

16X=|1ih0|+|0ih1|andZ=|0ih0| − |1ih1|.

17M.A Nielsen, I.L. Chuang (2000): Quantum Computation and Quantum Information.Cambridge

University Press. 27 / 37

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The 3-qubit phase-flip code

Local phase-flip error: each physical qubit|ψi=α|0i+β|1ibecomes Z|ψi=α|0i −β|1i18with probabilityp<1/2 during∆t.

SinceX=HZ HandZ=HX H(H2=I), use the3-qubit bit flip code in the frame defined byH:

|0i 7→ |0i+|1i

2 ,|+i, |1i 7→|0i − |1i

2 ,|−i, X7→HX H=Z=|+ih+|+|−ih−|.

t= +:Li=α|+Li+β|−Liwith|+Li=|+ ++iand|−Li=| − −−i.

t= ∆t:Libecomes with

1 flip:

α| −++i+β|+−−i α|+−+i+β| −+−i α|+ +−i+β| − −+i

;2 flips:

α| − −+i+β|+ +−i α| −+−i+β|+−+i α|+−−i+β| −++i

;3 flips:α| − −−i+β|+ ++i.

Key fact:4 orthogonal planesPc=span(|+ ++i,| − −−i),P1=span(| −++i,|+−−i, P2=span(|+−+i,| −+−i)andP3=span(|+ +−i,| − −+i).

Error syndromes: 3 commuting observablesS1=IXX,S2=XIXandS3=XXI with spectrum{−1,+1}and outcomes(s1,s2,s3)∈ {−1,+1}.

-1-s1=s2=s3:Pc3 |ψLi=

α|+ ++i+β| − −−i0 flip

β|+ ++i+α| − −−i3 flips ; no correction -2-s16=s2=s3:P13 |ψLi=

α| −++i+β|+−−i1 flip

β| −++i+α|+−−i2 flips ;(ZII)|ψLi ∈ Pc. -3-s26=s3=s1:P23 |ψLi=

α|+−+i+β| −+−i1 flip

β|+−+i+α| −+−i2 flips ;(IZI)|ψLi ∈ Pc. -4-s36=s1=s2:P33 |ψLi=

α|+ +−i+β| − −+i1 flip

β|+ +−i+α| − −+i2 flips ;(IIZ)|ψLi ∈ Pc. 18X=|1ih0|+|0ih1|,Z=|0ih0| − |1ih1|andH=|0i+|1i

2

h0|+|0i−|1i

2

h1|.

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The 9-qubit bit-flip and phase-flip code (Shor code (1995))

Take the phase flip code|+ ++iand| − −−i. Replace each|+i(resp.|−i) by

|000i+|111i

2 (resp.|000i−|111i

2 ).

New logical qubit Li=α|0Li+β|1Li ∈C2

9 C512:

|0Li= |000i+|111i

|000i+|111i

|000i+|111i

2

2 ,|1Li= |000i−|111i

|000i−|111i

|000i−|111i

2 2

Local errors: each of the 9 physical qubits can have a bit-flipX, a phase flipZ or a bit flip followed by a phase flipZ X=iY 19with probabilitypduring∆t.

Denote byXk (resp.Yk andZk), the local operatorX (resp.Y andZ) acting on physical qubit nok∈ {1, . . . ,9}. Denote byPc=span(|0Li,|1Li)the code space.

One get a family of the 1+3×9=28 orthogonal planes:

Pc, XkPc

k=1,...,9, YkPc

k=1,...,9, ZkPc

k=1,...,9.

One can always constructerror syndromesto obtain, when there is only one error among the 9 qubits during∆t,the numberkof the qubit and the error type it has undergone(X,Y orZ). These 28 planes are then eigen-planes by the syndromes.

If the physical qubitk is subject toany kind of local errorsassociated to arbitrary operatorMk=gI+aXk+bYk+cZk(g,a,b,cC),Li 7→q MkLi

L|MkMkLi

, the syndrome measurements will project the corrupted logical qubit on one of the 4 planes Pc,XkPc,YkPc orZkPc.It is then simple by using eitherI,Xk,YkorZk, to recover up to a global phase the original logical qubitLi.

19X=|1ih0|+|0ih1|,Z=|0ih0| − |1ih1|andY=i|1i|0i −i|0i|1i.

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Practical open issues with usual QEC

•For a logical qubit relying onnphysical qubits, the dimension of the Hilbert has to be larger than 2(1+3n)to recover a single but arbitrary qubit error: 2n≥2(1+3n)imposingn≥5 (H=C2

5 =C32)

•Efficient constructions of quantum error-correcting codes: stabilizer codes, surface codes where the physical qubits are located on a 2D-lattice, topological codes, . . .

•Fault tolerant computations: computing on encoded quantum states;

fault-tolerant operations to avoid propagations of errors during encoding, gates and measurement; concatenation and threshold theorem, . . .

•Actual experiments: 10−2 is the typical error probability during elementary gates involving few physical qubits.

•High-order error-correcting codes with an important overhead;more than 1000 physical qubits to encode a logical qubit20 H ∼C2

1000.

20A.G. Fowler, M. Mariantoni, J.M. Martinis, A.N. Cleland (2012): Surface codes: Towards practical large-scale quantum computation. Phys. Rev.

A,86(3):032324.

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Outline

Continuous-time dynamics of open quantum system Stochastic Master Equation (SME)

Key characteristics of SME

Quantum filtering and state estimation

Feedback schemes

Measurement-based feedback and classical controller Coherent feedback and quantum controller

Merging measurement-based and coherent feedbacks

Quantum Error Correction (QEC) and feedback QEC from scratch

Storing a logical qubit in a high-quality harmonic oscillator Simplified feedback scheme for cat-qubit experiment

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Cat-qubit: storing a logical qubit in a high-quality harmonic oscillator21

•SystemS(LC circuit): a high-quality harmonic oscillator (1ωss∼106):

d

dteρs=−iωs[Ns,ρes] +κsDas(ρes)

withHs=span{|0i,|1i, . . . ,|ni, . . .}, photon-number operatorNs and annihilation operatoras (Ns=asas,Ns|ni=n|ni,as|ni=√

n|n−1i).

•In the rotation frame (eρs=e−iωstNsρse+iωstNs):

d

dtρssDass) =κs

asρas12(asasρ+ρasas) .

•Goal: engineer alogical qubit with superpositions of coherent states

|βi=e|β|

2

2 X

n≥0

βn

√ n!|ni

of complex amplitudesβ∈ {α,iα,-α,-iα}withα1 (typicallyα≥3).

•Coherent states are robust to decoherence: as|βi=β|βi: ifρs(0) =|β0ihβ0|, thenρs(t) =|βtihβt|withβt0e−κst/2.

21M.Mirrahimi, Z. Leghtas . . . , M. Devoret. (2014). Dynamically protected cat-qubits: a new paradigm for universal quantum computation. New Journal of Physics, 16, 045014.

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Outline

Continuous-time dynamics of open quantum system Stochastic Master Equation (SME)

Key characteristics of SME

Quantum filtering and state estimation

Feedback schemes

Measurement-based feedback and classical controller Coherent feedback and quantum controller

Merging measurement-based and coherent feedbacks

Quantum Error Correction (QEC) and feedback QEC from scratch

Storing a logical qubit in a high-quality harmonic oscillator Simplified feedback scheme for cat-qubit experiment

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Simplified feedback scheme for a possible cat-qubit experiment22

(measurement-based feedback)

high Q mode (logical qubit)

low Q mode

(coherent feedback)

Re Im

dispersive coupling non-linear mixer

low Q mode

Stochastic master equation onHs⊗ H1⊗ H2:

=κsDas(ρ)dti[H,ρ]dt +κ1Da1(ρ)dt+κ2Da2(ρ)dt +

ηκ2

ia2ρiρa2 Tr

ia2ρiρa2 ρ

dWt

Engineered Hamiltonian

H=i(u1a1u1a1) +i(u2a2u2a2) +g1

a4sa1+ (a4s)a1

+g2a2a2eiπasas,

Classical control inputsu1,u2C Measurement output classical signal:

dyt = ηκ2Tr

ia2ρiρa2

dt+dWt

Many time-scales,κsg1,g2κ1, κ2and κsgκ12

1,κg22

2, providing a reduced slow SME on span{|αi,|iαi,|-αi,|-iαi}withα=p4

u1/g1 22Inspired by

N. Ofek, . . . , M. Mirrahimi, M. Devoret, R. Schoelkopf (2016). Extending the lifetime of a quantum bit with error correction in superconducting circuits. Nature, 536(7617),441-445.

R. Lescanne, . . . , M. Mirrahimi, Z. Leghtas (2019): Exponential suppression of bit-flips in a qubit encoded in an oscillator. arXiv:1907.11729 [quant-ph].

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Quantum feedback engineering for robust quantum information processing

QUANTUM WORLD

CLASSICAL WORLD Hilbert space

classical controller

quantum controller Hilbert space Hilbert space

system S

decoherence decoherence

classical input

classical output

quantum measurement classical

reference

quantum interaction

To protect quantum information stored in system S (alternative to usual QEC):

I fast stabilization and protection mainly achieved by aquantum controller (coherent feedback stabilizing decoherence-free sub-spaces);

I slow decoherence and perturbations mainly tackled by aclassical controller (measurement-based feedback "finishing the job")

Underlyingmathematical methodsfor high-precision dynamical modeling and control based onstochastic master equations(SME):

I High-order averaging methods and geometric singular perturbations for coherent feedback.

I Stochastic control Lyapunov methods for exponential stabilization via measurement-based feedback.

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Qubit (2-level system, half-spin)

23

I Hilbert space:

HM =C2=n

cg|gi+ce|ei, cg,ce ∈C o

.

I Quantum state space:

D={ρ∈ L(HM),ρ=ρ, Tr(ρ) =1,ρ≥0}.

I Operators and commutations:

σ-=|gihe|,σ+-=|eihg| X ≡σx-+ =|gihe|+|eihg|; Y ≡σy =iσ-−iσ+=i|gihe| −i|eihg|; Z ≡σz+σ-−σ-σ+=|eihe| − |gihg|; σx2=I,σxσy =iσz, [σxy] =2iσz, . . .

I Hamiltonian: HMqσz/2+uqσx.

I Bloch sphere representation:

D=n

1

2 I+xσx +yσy +zσz (x,y,z)∈R3, x2+y2+z2≤1o

| g

| e ω

q

u

q

23See S. M. Barnett, P.M. Radmore (2003): Methods in Theoretical Quantum Optics. Oxford University Press.

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Quantum harmonic oscillator (spring system)

23

I Hilbert space:

HS =n P

n≥0ψn|ni, (ψn)n≥0∈l2(C)o

≡L2(R,C)

I Quantum state space:

D={ρ∈ L(HS),ρ=ρ, Tr(ρ) =1,ρ≥0}.

I Operators and commutations:

a|ni=√n|n-1i,a|ni=√

n+1|n+1i; N=aa,N|ni=n|ni;

[a,a] =I,af(N) =f(N+I)a;

Dα=eαa−αa. a=X+iP = 1

2 x+∂x

, [X,P] =ıI/2.

I Hamiltonian: HScaa+uc(a+a).

(associated classical dynamics:

dx

dtcp, dpdt =−ωcx−√ 2uc).

I Classical pure state≡coherent state|αi α∈C: |αi=P

n≥0

e−|α|2/2αn

n!

|ni;|αi ≡ π1/41 eı

2x=αe(x−

2<α)2 2

a|αi=α|αi,Dα|0i=|αi.

| 0

|1

|2 ω

c

|n

ω

c

u

c

... ...

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