Suppressing decoherence and heating with

quantum bang-bang controls

David Vitali and Paolo Tombesi

Dip. di Matematica e Fisica and Unità INFM,Università di Camerino, Italy

• Decoherence is the main limiting factor for quantum information processing Many different routes to decoherence control:

- Coding methods: • quantum error correction codes• error avoiding codes (i.e., decoherence free subspaces)

- Dynamical methods:• closed-loop (feedback) techniques • open-loop techniques, i.e., the application of suitably shaped time-varying control fields acting on the system alone

Open-loop techniques are able to decouple an open quantum system from any environment. They can be seen as the quantum generalization of refocusingtechniques used in NMR to eliminate undesired dephasing due to local magnetic field fluctuations

U1(t) =Texp −i

h dsH1 s( )0

t

∫⎧ ⎨ ⎩

⎫ ⎬ ⎭

=U1(t +Tc)

Decoupling via quantum bang-bang control

• System S interacting with an environment B: H0 = HS + HB + HSB • Cyclic control Hamiltonian H1(t), acting on system only, with period Tc

HTOT(t) = H0+H1(t)

• The stroboscopic evolution at tN = NTc , in the infinitely fast control limit Tc 0, N ∞, tN = NTc = const., is driven by an average Hamiltonian

UTOT tN( ) =exp − i

hH (0)tN⎧ ⎨ ⎩

⎫ ⎬ ⎭ H (0) = 1Tc

dtHeff(t)0

Tc

∫

where Heff t( )=U1+ t( )H0U1 t( )

• quantum bang-bang controls: application of Hamiltonians with very large strength and for very small times (full strength/fast switching scheme) Application of a piecewise constant decoupling operator U1(t) = gj, j∆t ≤ t ≤ (j+1)∆t

• G = {gj} = group of unitary operators with |G| elements

g1 g2 g|G|..... ∆t=Tc/|G|

• Applying the quantum bang-bang control is equivalent to symmetrize the evolution with respect to the group G (Viola, Knill, Lloyd 99, Zanardi 99)

H (0) =ΠC H0( ) = 1|G | gj

+H0gjg j ∈G∑

• The system is perfectly decoupled from the environment when the interaction is symmetrized to zero, c(HSB) = 0

• The effective system evolution is then governed by

lim

N → ∞ρS t =NTc( ) =e−i

hH stρS 0( )e

ih

H st

HS = c(HS) = projection of HS into the commutant of G

If sufficient symmetry conditions are satisfied (for example, G or its commutant are sufficiently large)

this result can be used for implementing universal fault tolerant quantum computation,

using appropriate tricks (Viola, Lloyd, Knill, 99) or encoding into noiseless subsystems (Viola, Knill, Lloyd 2000, Zanardi 2000)

• Example 1: K qubits with linear dissipation HSB = σαi

i,α∑ Bα

i α =x,y,z

• Decoupling is obtained through symmetrization with respect to the tensor power of the Pauli group, G = {1,i(i)

x, i(i)y, i(i)

z}, |G| = 4 (Duan, Guo, 99, Viola, Knill, Lloyd, 99)• Quantum bang-bang cycle = four pulses: x, -z, -x, -z

• Example 2: oscillator (a) with linear dissipation HSB = a†+ a†

• Decoupling is obtained through symmetrization with respect to the group G = Z2 = {1, P (parity)}, |G| = 2 (Vitali, Tombesi, 99)• Bang-bang control = “parity kicks” by pulsing the oscillation frequency

0

0 +

TC

= P = exp{ia†a}

• The most stringent condition for decoupling is on timing: The infinitely fast control limit Tc 0 cannot be met exactly in practice

• How small has Tc to be to achieve significant decoupling ? First estimates (Viola-Lloyd 98, Vitali-Tombesi 99) showed that the significant timescale is determined by environment typical timescale, c ≈ c

-1, inverse of the environment frequency cutoff c , c Tc ≤ 1

• The decoupling technique can be used to eliminate “slow” environments, i.e., typical non-Markovian situations

Open questions: • What is the effect of bath temperature ?• May the scheme be used to decouple from a high temperature bath and for example suppress heating of the vibrational motion in a linear chain of trapped ions ?

Vibrational mode coupled to a thermal bath

H0 =hω0a

+a+ hωii

∑ bi+bi + λi

i∑ abi

+ +a+bi( )

• Environment described as a set of independent bosons (Caldeira-Leggett, 81) It can be decoupled using “parity kicks” (Example 2)

• No Markovian appoximation, no bath elimination numerical solution with a bath of 201 oscillators, with frequencies equally spaced by ∆, in the interval [0, c], c = 20;

ohmic bath i = (∆/2π)1/2, i, = damping rate

• We use the fact that coherent states are preserved either with or without the decoupling parity kicks

α 0( ) βi 0( )i

∏ t ⏐ → ⏐ α t( ) βi t( )i

∏α t( ) =L00 t( )α 0( ) + L0i t( )βi 0( )

i∑

βi t( ) =Li0 t( )α 0( ) + Lij t( )βj 0( )j∑ with the appropriate unitary

matrix Lij (t)

Effect of parity kicks on heating

• The linear chain is first cooled to the ground state and it is then heated by the high temperature bath at temperature T

t =0: 0 0 ⊗ ρT

• The vibrational mode reduced density matrix at time t is a thermal state with mean vibrational number <n(t)> = (t)

ρS t( ) = d2α exp −α 2

ν t( )⎧ ⎨ ⎩

⎫ ⎬ ⎭ α∫ α ν t( )= Nj L0j t( )

j∑ 2

ρT = d2βj exp−βj

2

Nj

⎧ ⎨ ⎩

⎫ ⎬ ⎭

βj βj∫j

∏ Nj = exp hωj

kT⎧ ⎨ ⎩

⎫ ⎬ ⎭ −1⎛ ⎝ ⎜ ⎞

⎠

−1

T = 1K, N(0) = 13144

Vibrational heating is suppressed for c Tc/2 < 1, for any temperature !

0 = 10 Mhz, = 0.1 Mhz,c = 20 Mhz

T = 100 mK, N(0) = 1302.4 T = 10 mK, N(0) = 129.79

Effect of parity kicks on decoherence

Vibrational mode initially in a Schrödinger cat state, which is then heated by the high temperature bath at temperature T

The vibrational mode reduced Wigner function at time t is

t =0: Nϕ2 α0 α0 +−α0 −α0 +eiϕ −α0 α0 +e−iϕ α0 −α0{ }⊗ ρT

W α,t( ) = 2Nϕ2

π 1+2ν t( )[ ]e

−2α −α0L00(t) 2

1+2ν t( ) +e−2α +α0L00(t) 2

1+2ν t( )⎧ ⎨ ⎩

+2exp−2α02η t( )[ ]e

− 2α 2

1+2ν t( ) cosϕ +4Imαα0L00(t)[ ]1+2ν t( )

⎡ ⎣ ⎢

⎤ ⎦ ⎥ ⎫ ⎬ ⎭

(t) describes again heating•L00(t) describes amplitude decay(t): fringe visibility function, describing decoherence

η t( ) =1− L00 t( )2

1+2ν t( )

T = 1K, N(0) = 13144

At variance with heating, decoherenceis well suppressed only at low T !

0 = 10 Mhz, = 0.1 Mhz,c = 20 Mhz

T = 100 mK, N(0) = 1302.4 T = 10 mK, N(0) = 129.79

This is due to the thermal acceleration of decoherence.

ηMark t( )= 1−e−γt( ) 1+2N ω0( )( )1+2N ω0( ) 1−e−γt( ) ⇒ tdec≈

1γ 1+2N ω0( )( )

Decoherence is affected not only by the environment timescale 2/c, but also by the temperature-dependent timescale tdec ≈ [N(0)]-1

Decoherence is suppressed by parity kicks only when Tc < min{2/c , [N(0)]-1}

• On the contrary, heating is affected only by the cutoff frequency and it can be suppressed by parity kicks when Tc < 2/c

In fact, in the Markovian limit

•This quantum bang-bang control can be used to avoid heating of the center-of-mass motion in linear ion traps

Effect of parity kicks on the Wigner function of a Schrödinger cat state with 0 = √5, in a thermal bath with T = 10mK

t = 0

t = 1/

No parity kicksWith parity kicks, with cTc/2 = 1/16