Feedback Issues Underlying Quantum Error Correction

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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

Superconduc�ng circuits

Photons

Ultra-cold neutral/Rydberg

atoms

Trapped ions

Quantum dots

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=H^s⊗ H^c

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

Feedback schemes

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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):

dρ_t =

−i[H0+utH1,ρ_t] + X

ν=d,m

Lνρ_tL^†_ν−¹

2(L^†_νLνρ_t+ρ_tL^†_νLν) dt +√

ηm

Lmρ_t+ρ_tL^†_m− Tr

(Lm+L^†_m)ρ_t ρ_t

dWt

driven by the Wiener processWt, with measurementyt,

dyt=√ ηmTr

(Lm+L^†_m)ρ_t

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+C^C^l

l =1):

d

dtΦ=_C¹Q+2u−² q`

c Φ∗

L sin 1

Φ∗Φ

d

dtQ=−^Φ_L^∗sin

1 Φ∗Φ

withy=u− q`

c Φ∗

L sin

1 Φ∗Φ

. Hs(Φ,Q) =_2C¹Q²−^Φ_L²^∗cos

1 Φ∗Φ

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 Hermitian≥0 matrix.

dρ_t=

−2ⁱ[ω_qZ,ρ_t] +γ(ZρZ−ρ_t) dt +√ηγ

Zρ_t+ρ_tZ−2 Tr(Zρ_t)ρ_t dWt

withyt given bydyt=2√ηγ Tr(Zρ_t)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

Feedback schemes

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Another formulation of diffusive SME

⁷

With a single imperfect measurementdyt=√

ηTr (L+L^†)ρ_t

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

dρ_t =

−i[H0+utH1,ρ_t] +Lρ_tL^†−¹

2(L^†Lρ_t+ρ_tL^†L) dt +√

η

Lρ_t+ρ_tL^†− Tr

(L+L^†)ρ_t ρ_t

dWt

driven by the Wiener processdWt

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

ρ_t+dt,ρ_t+dρ_t= Mu_t,dy_tρ_tM^†_u

t,dy_t+ (1−η)Lρ_tL^†dt Tr

Mu_t,dy_tρ_tM^†_u

t,dy_t+ (1−η)Lρ_tL^†dt

withMu_t,dy_t=I− i(H0+utH1) +¹₂ L^†L 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+L^†)ρ_t

dt+dWt and measurement backaction described by

ρ_t+dt,ρ_t+dρ_t = Mu_t,dy_tρ_tM^†_u

Tr

Mut,dytρ_tM^†_u

I if ρ₀ density operator, then, for all t>0,ρ_t 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(ρ_t)≤rank(ρ₀)for allt ≥0. In particular ifρ₀is a rank one projector, thenρ_t remains a rank one projector (pure state).

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Key characteristics of quantum SME (2)

dρ_t =

−i[H0+uH1,ρ_t] +Lρ_tL^†−¹

η

(L+L^†)ρ_t ρ_t

dWt

with measured output mapdyt=√

ηTr (L+L^†)ρ_t

dt+dWt

I Invariance of the SME structure under unitary transformations.

A time-varying change of frameeρ=U^†_tρUt withUt unitary.

The new density operatorρeobeys to a similar SME where He0+uHe1=U^†_t(H0+uH0)Ut+iU^†_t _dt^dUt

andeL=U^†_tLUt. I "L¹-contraction" properties. Whenη=0, SME becomes deterministic

d

dtρ=−i[H0+uH1,ρ_t] +Lρ_tL^†−¹

2(L^†Lρ_t+ρ_tL^†L) 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] +L^†AL−¹

2(L^†LA+AL^†L)≤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

Feedback schemes

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Belavkin quantum filter: stability and convergence issues

Quantum stateρ_t producingdyt=√

ηTr Lρ_t+ρ_tL^†

dt+dWt and its estimateρb_t:

dρ_t =

−i[H0+uH1,ρ_t] +Lρ_tL^†−¹

η

(L+L^†)ρ_t ρ_t

dWt

dρb_t =

−i[H0+uH1,ρb_t] +Lρb_tL^†−¹

2(L^†Lρb_t+ρb_tL^†L) dt +√

η

Lρb_t+ρb_tL^†− Tr

(L+L^†)ρb_t ρb_t

dyt−√

ηTr

Lρb_t+ρb_tL^† dt

. I Stability¹⁰: the fidelityF(ρ_t,ρb_t) = Tr² p√

ρ_tρb_t√ ρ_t

is always a sub-martingale:

∀t1≤t2, E

F ρ_t₂,ρb_t₂ ρ_t₁,ρb_t₁

≥F ρ_t₁,ρb_t₁ .

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

I Convergence¹¹ofρb_t towardsρ_t 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

Feedback schemes

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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

Feedback schemes

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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

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

D^Ls,ν⊗I_c(ρ) +X

ν⁰

D^Is⊗L_c,ν0(ρ)

withD^L(ρ) =LρL^†−¹₂ L^†Lρ+ρL^†L

and operators made oftensor products.

•Consider a convex subsetDsof steady-states for original systemS: each density operatorρ_s onHs belonging toDssatisfyi[Hs,ρ_s] =P

νDLs,ν(ρ_s).

•Designing arealisticquantum controllerC (Hc,L_c,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 kL^†s,νLs,νkandkHck kL^†_c,ν0L_c,ν0k.

I Decoherence rates ofSmuch slower than those ofC: kL^†s,νLs,νk kL^†_c,ν0L_c,ν0k: model reduction byquasi-static approximations(adiabatic elimination, singular perturbations).

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Outline

Feedback schemes

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Quantum feedback engineering

QUANTUM WORLD CLASSICAL WORLD

Hilbert space

system S

decoherence decoherence

classical input

classical output

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

Feedback schemes

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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 bitb_k ∈ {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 isp_L =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: |ψ_Li=α|0_Li+β|1_Li ∈C²⊗C²⊗C²≡C⁸with|0_Li=|000iand

|1_Li=|111i.

•t= ∆t: |ψ_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 planesP_c=span(|000i,|111i),P1=span(|100i,|011i), P₂=span(|010i,|101i)andP₃=span(|001i,|110i).

•Error syndromes: 3 commuting observablesS1=I⊗Z⊗Z,S2=Z⊗I⊗Z and S3=Z⊗Z⊗I with spectrum{−1,+1}and outcomes(s1,s2,s3)∈ {−1,+1}.

-1-s1=s2=s3:Pc3 |ψ_Li=

α|000i+β|111i0 flip

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

β|100i+α|011i2 flips ;(X⊗I⊗I)|ψ_Li ∈ Pc. -3-s26=s3=s1:P23 |ψ_Li=

β|010i+α|101i2 flips ;(I⊗X⊗I)|ψ_Li ∈ Pc. -4-s36=s1=s2:P33 |ψ_Li=

β|001i+α|110i2 flips ;(I⊗I⊗X)|ψ_Li ∈ P_c.

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= +:|ψ_Li=α|+_Li+β|−_Liwith|+_Li=|+ ++iand|−_Li=| − −−i.

•t= ∆t:|ψ_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 planesP_c=span(|+ ++i,| − −−i),P₁=span(| −++i,|+−−i, P₂=span(|+−+i,| −+−i)andP₃=span(|+ +−i,| − −+i).

•Error syndromes: 3 commuting observablesS1=I⊗X⊗X,S2=X⊗I⊗XandS3=X⊗X⊗I with spectrum{−1,+1}and outcomes(s₁,s₂,s₃)∈ {−1,+1}.

-1-s1=s2=s3:P_c3 |ψ_Li=

α|+ ++i+β| − −−i0 flip

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

α| −++i+β|+−−i1 flip

β| −++i+α|+−−i2 flips ;(Z⊗I⊗I)|ψ_Li ∈ P_c. -3-s26=s3=s1:P23 |ψ_Li=

α|+−+i+β| −+−i1 flip

β|+−+i+α| −+−i2 flips ;(I⊗Z⊗I)|ψ_Li ∈ P_c. -4-s36=s1=s2:P33 |ψ_Li=

α|+ +−i+β| − −+i1 flip

β|+ +−i+α| − −+i2 flips ;(I⊗I⊗Z)|ψ_Li ∈ P_c. 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 |ψ_Li=α|0_Li+β|1_Li ∈C²

9 ≡C⁵¹²:

|0Li= |000i+|111i

|000i+|111i

2√

2 ,|1Li= |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(|0_Li,|1_Li)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 operatorM_k=gI+aX_k+bY_k+cZ_k(g,a,b,c∈C),|ψ_Li 7→q ^M^k^|ψ^Lⁱ

hψ_L|M^†_kM_k|ψ_Li

, 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 qubit|ψ_Li.

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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Feedback schemes

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

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

d

dteρ_s=−iωs[Ns,ρe_s] +κsD^as(ρe_s)

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

n|n−1i).

•In the rotation frame (eρ_s=e^−iω^s^tN^sρ_se⁺ⁱ^ω^s^tN^s):

d

dtρ_s=κsDas(ρ_s) =κs

asρa^†_s−¹₂(a^†_sasρ+ρa^†_sas) .

•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).

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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Feedback schemes

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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:

dρ=κsDa_s(ρ)dt−i[H,ρ]dt +κ1Da1(ρ)dt+κ2Da2(ρ)dt +√

ηκ2

ia2ρ−iρa^†₂− Tr

ia2ρ−iρa^†₂ ρ

dWt

Engineered Hamiltonian

H=i(u1a^†₁−u^∗₁a1) +i(u2a^†₂−u^∗₂a2) +g1

a⁴_sa^†₁+ (a⁴_s)^†a1

+g2a^†₂a2e^iπa^†^s^a^s,

Classical control inputsu1,u2∈C Measurement output classical signal:

dyt =√ ηκ2Tr

ia2ρ−iρa^†₂

dt+dWt

Many time-scales,κsg1,g2κ1, κ2and κs^g_κ¹²

1,_κ^g²²

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

system S

decoherence decoherence

classical input

classical output

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:

H^M =C²=n

c_g|gi+c_e|ei, c_g,c_e ∈C o

.

I Quantum state space:

D={ρ∈ L(H^M),ρ^†=ρ, Tr(ρ) =1,ρ≥0}.

I Operators and commutations:

I Hamiltonian: HM =ωqσ_z/2+uqσ_x.

I Bloch sphere representation:

D=n

1

2 I+xσ_x +yσ_y +zσ_z (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:

H^S =n P

n≥0ψn|ni, (ψn)_n≥0∈l²(C)o

≡L²(R,C)

I Quantum state space:

D={ρ∈ L(H^S),ρ^†=ρ, Tr(ρ) =1,ρ≥0}.

I Operators and commutations:

a|ni=√n|n-1i,a^†|ni=√

n+1|n+1i; N=a^†a,N|ni=n|ni;

[a,a^†] =I,af(N) =f(N+I)a;

Dα=e^αa^†^−α^†^a. a=X+iP = ^√¹

2 x+_∂x^∂

, [X,P] =ıI/2.

I Hamiltonian: HS =ωca^†a+u_c(a+a^†).

(associated classical dynamics:

dx

dt =ω_cp, ^dp_dt =−ω_cx−√ 2uc).

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

n≥0

e^−|α|²^/2^√^αⁿ

n!

|ni;|αi ≡ π^1/4¹ e^ı

√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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