Low rank approximation for the numerical simulation of high dimensional Lindblad equations

Low rank approximation for the numerical simulation of high dimensional Lindblad

equations

pierre.rouchon@mines-paristech.fr

Based on the arXiv preprint

http://arxiv.org/abs/1207.4580 with Claude Le Bris from CERMICS

PRACQSYS, September 2012, Tokyo

Outline

Simulation of high dimensional Lindblad equations

The low rank approximation

Numerical scheme

Numerical tests for oscillation revivals

Concluding remarks

High dimensional Lindblad equations

I

The Lindblad master equation governing open-quantum systems:

d

dtρ=−i[H, ρ]−¹₂(L^†Lρ+ρL^†L) +LρL^†,

where

is the density operator (ρ

1,

0),

is an Hermitian operator and

is any operator on the Hilbert space

of dimension

dim

I

Usually,

j=1n_j large

comes from

H=H₁⊗ H₂⊗. . .⊗ H_c

where each

is of small or intermediate dimension

and

are usually defined as sums with few terms of simple tensor products of operators acting only on some

.

I

Typical situations of composite systems: coherent

feedback scheme, circuit/cavity QED, . . .

Quantum Monte-Carlo (QMC) simulations

The Lindbald equation

is the master equation of the

d|ψ_ti= −iH−¹₂L^†L+hψ_t|L^†L|ψ_ti

|ψ_tidt+

L|ψti

√hψ_t|L^†L|ψ_ti− |ψ_ti

dN_t

with

1},

(Poisson process).

Monte-Carlo simulations: simulate

realizations of such stochastic Schrödinger equation

1, . . .

: for

large (typically

1000)

ρt ≈ _N¹

N

X

k=1

|ψ^k_tihψ^k_t|.

1J. Dalibard, Y. Castion, and K. Mølmer. Wave-function approach to dissipative processes in quantum optics.Phys. Rev. Lett., 68(5):580–583, 1992.

Approximation by projection methods

Based on physical intuition, select an adaptedsub-set of density matrices, i.e. a sub-manifoldDof the vector space of Hermitian matrices equipped with Frobenius Euclidian metric. The approximate evolution is given by theorthogonal projectionΠ^ρ(dρ/dt)ofdρ/dt onto the tangent space atρtoD:

forρ∈ D, d dtρ=

vector field onD

z }| {

Π^ρ

−i[H, ρ]−¹₂(L^†Lρ+ρL^†L) +LρL^†

.

In³, Mabuchi considers a reduced order model for a spin-spring system. The sub-manifoldDwas the (real) 5-dimensional manifold constructed with the tensor products of arbitrary two-level states and pure coherent states.

Computation ofΠ^ρ(dρ/dt)in local coordinatesis not trivial and yields usually to nonlinear ODEs.

2R. van Handel and H. Mabuchi. Quantum projection filter for a highly nonlinear model in cavity qed.Journal of Optics B: Quantum and Semiclassical Optics, 7(10):S226, 2005.

3H. Mabuchi. Derivation of Maxwell-Bloch-type equations by projection of quantum models.Phys. Rev. A, 78:015801, Jul 2008.

Low rank Kalman filters

Fordx=Ax dt+G dω,dy=C dx+H dη, computation of the best estimate ofx attknowing the past values of the outputy relies on the computation of theconditional error covariance matrixPsolution of the Riccati matrix equation

d

dtP=AP+PA⁰+GG⁰−PC⁰(HH⁰)⁻¹CP.

WhenG=0, the Riccati equation is rank preserving. It defines then a vector field on thesub-manifold of rankm<ncovariance matrices (n=dimx here). This sub-manifold admits theover-parameterization

(U,R)7→URU⁰=P!

U

z}|{

R

z}|{

U⁰

z }| {=

P

z }| {

whereUbelongs to the set ofn×morthogonal matrices (U⁰U=Im) andRism×m, positive definite and symmetric.

Lift ofdP/dt (P=URU⁰solution the above Riccati equation):

d

dtU= (In−UU⁰)AU, d

dtR=U⁰AUR+RU⁰AU−RU⁰C(HH⁰)⁻¹CUR

4S. Bonnabel and R. Sepulchre. The geometry of low-rank Kalman filters.

preprint arXiv:1203.4049v1, March 2012.

Projection and lift for rank-m density operators of

The sub-manifoldD_mofdensity matricesρof rankm<nis over-parameterizedvia

ρ=UσU^† !

ρ

z }| {

=

U

z}|{

σ

z}|{

U^†

z }| {

whereσis am×mstrictly positive Hermitian matrix,Uan×m matrix withU^†U=Im.

The family of liftsfordρ/dt =−i[H, ρ]−¹₂(L^†Lρ+ρL^†L) +LρL^†

d

dtU=−iAU+ (In−UU^†) −i(H−A)−¹₂L^†L+LUσU^†L^†Uσ⁻¹U^† U, d

dtσ=−i[U^†(H−A)U, σ]−¹₂(U^†L^†LUσ+σU^†L^†LU) +U^†LUσU^†L^†U +_m¹Tr (L^†(In−UU^†)L UσU^†

Im.

wherethe gage degree of freedomAis any time varyingn×n Hermitian matrix.

The computation of the lifted dynamics

(U, σ)7→UσU^†=ρ!

U

z}|{

σ

z}|{

U^†

z }| {=

ρ

z }| {

with theinfinitesimal variationsδU=ıηUandδσ=ς: (η, ς)7→i[η, ρ] +UςU^†

whereηis anyn×nHermitian matrix,ςis anym×mHermitian matrix with zero trace.

An×nHermitian matrixξin the tangent space atρ=UσU^†toD_m admits the parameterizationξ=i[η, ρ] +UςU^†.

The projectionΠ^ρ_m(_dt^dρ)corresponds to thetangent vectorξ associated toηandς minimizing

Tr

−i[H, ρ]−(L^†Lρ+ρL^†L)/2+LρL^†−i[η, ρ]−UςU^†2 ,

First order stationary conditionsgiveηandς as function ofρ=UσU^†: the lifted evolution is given by_dt^dU=iηUand _dt^dσ=ς where the arbitrary matrixAappears.

Gage

adapted to weak dissipation

In d

dtU=−iAU+ (In−UU^†) −i(H−A)−¹₂L^†L+LUσU^†L^†Uσ⁻¹U^† U, d

dtσ=−i[U^†(H−A)U, σ]−¹₂(U^†L^†LUσ+σU^†L^†LU) +U^†LUσU^†L^†U +_m¹Tr (L^†(In−UU^†)L UσU^†

Im.

setA=H:

d

dtU=−iHU+ (In−UU^†) −¹₂L^†L+LUσU^†L^†Uσ⁻¹U^† U, d

dtσ=−¹₂(U^†L^†LUσ+σU^†L^†LU) +U^†LUσU^†L^†U +_m¹Tr (L^†(In−UU^†)L UσU^†

Im,

Honly appears in the dynamics ofUand not in the dynamics ofσ.

Appropriate whenHdominatesL: aslow evolution ofσas compared to afast evolution ofU(important for the numerical procedure)

A numerical integration scheme adapted to weak dissipation

The update from timekδtto time(k+1)δtis split into3 steps forU and2 steps forσ

U_k+¹

3 =

In−^iδt₂H−^δt₈²H²+i^δt₄₈³H³ Uk

U_k+²

3 =U_k+¹

3 +δt(In−U_k+¹

3U_k+^† 1 3

)

−¹₂L^†LU_k+¹

3 +LU_k+¹

3σ_kU_k+^† 1 3

L^†U_k+¹

3σ_k⁻¹ U_k+1= Υ In−ⁱ₂^δtH−^δt₈²H²+i^δt₄₈³H³

U_k+2 3

(Υortho-normalization)

σ_k+1

2 =σ_k +δt U_k+^† 1 3

LU_k+1 3σ_kU_k+^† 1

3

L^†U_k+1 3

+δt Tr

(U^†

k+¹₃L^†LU_k+1 3 −U^†

k+¹₃L^†U_k+1 3U^†

k+¹₃LU_k+1 3)σ_k

Im

m σ_k+1=

(Im−^δt₂U_k+^† 1 3

L^†LU_k+1 3)σ_k+1

2(Im−^δt₂U_k+^† 1 3

L^†LU_k+1 3) Tr

(Im−^δt₂U^†

k+¹₃L^†LU_k+1 3)σ_k+1

2(Im−^δt₂U^†

k+¹₃L^†LU_k+1 3). This scheme preservesU^†U=Im,σ^†=σ,

σ > 0

Computational cost versus QMC procedure

d|ψ_ti= −iH−¹₂L^†L+hψ_t|L^†L|ψ_ti

|ψ_tidt +

L|ψti

√

hψt|L^†L|ψti− |ψ_ti

dN_t

U_k+¹

3 =

In−^iδt₂H−^δt₈²H²+i^δt₄₈³H³ Uk

U_k+2 3 =U_k+1

3 +δt(In−U_k+1 3U^†

k+¹₃)

−¹₂L^†LU_k+1

3 +LU_k+1 3σ_kU^†

k+¹₃L^†U_k+1 3σ_k⁻¹ Uk+1= Υ In−^iδt₂H−^δt₈²H²+i^δt₄₈³H³

U_k+²

3

Both methods useessentially right multiplications ofH,L,L^†byn×1 orn×mmatrices, as, for example, the productsH|ψi,L|ψiorHU, LU,L^†(LU). No stringn×nmatrices sinceHandLare defined as tensor products of operators of small dimensions. Whennis very large andmis small, this point is crucial for an efficient numerical implementation:evaluations of products likeHUorLUcan be parallelized.

Empirical estimation

of truncation error

I

Based on

for

using:

˙

ρ=−i[H, ρ]−¹₂(L^†Lρ+ρL^†L) +LρL^†

˙

ρ⊥= (In−P_ρ)LρL^†(In−P_ρ)−^Tr(^LρL†(In−Pρ))

m P_ρ

where

.

I

Good approximation when Tr

Tr

.

I

At each time step, Tr

and Tr

may be numerically evaluated with a complexity similar to the complexity of the numerical scheme (no need to explicitly compute

and

as

matrices before taking their Frobenius norms).

5Inspired from R. van Handel and H. Mabuchi. Quantum projection filter for a highly nonlinear model in cavity qed.Journal of Optics B: Quantum and Semiclassical Optics

Initialization procedure

σ0

and

need to be deduced from a given initial condition

:

I When the rank ofρ₀≥m:σ₀

diagonal matrix made of the largest

eigenvalues of

with sum normalized to one;

U₀

the associated normalized eigenvectors.

I

When the rank of

1 and

1:

then natural to take for

a diagonal matrix where the first diagonal element is 1

1) and the over ones are equal to 1. Then

is constructed, up to an

ortho-normalization preserving the first column, with

as the first column,

as the second column, . . . ,

as the last column.

I

When the rank of

in

initialization scheme . . .

Lindblad equation of oscillation revivals

Thecollective symmetric behavior ofN_atwo-level atomsresonantly interacting with aquantized field:

d

dtρ= Ω₀

2 [a^†J⁻−aJ⁺, ρ]−κ(nρ/2+ρn/2−aρa^†) Preliminary tests viatwo different type of simulationsincluding the first complete revival:

I Na=1 atom initially in the excited state, a field initially in a coherent state with¯n=15 photons (truncation to 30 photons):

comparisons between the full-rank and rank-2-4-6 solutionswith κ= Ω₀/500:

I N_a=50 atoms all initially in excited states, a field with¯n=200 (truncation to 300 photons):comparison of the analytic

approximate weak-damping modelproposed in⁶(predicts a reduction of a factorr =2 of the complete first revival between κ=0 andκ=log(r)Ω0/(4π¯n^3/2)) withthe rank-8 approximation given by the above integration scheme withδt =1/(Ω0

√nN¯ a).

6T. Meunier, A. Le Diffon, C. Ruef, P. Degiovanni, and J.-M. Raimond.

Entanglement and decoherence of N atoms and a mesoscopic field in a cavity.Phys. Rev. A, 74:033802, 2006.

Full rank (left) versus rank 2 (right) (N

1,

15,

¯n)

0 2 4 6 8 10

0 0.5 1

population

(a)

0 2 4 6 8 10

0 0.5 1

low rank population

(b)

0 2 4 6 8 10

0 0.5 1

eigenvalues

(c)

0 2 4 6 8 10

0 0.5 1

low rank eigenvalues

(d)

0 2 4 6 8 10

0.9 0.95 1

fidelity

φ (e)

0 2 4 6 8 10

10⁻⁶ 10⁻⁴ 10⁻² 10⁰ 10²

(f)

φ

Frobenius norm

Full rank (left) versus rank 4 (right) (N

1,

15,

¯n)

0 2 4 6 8 10

0 0.5 1

population

(a)

0 2 4 6 8 10

0 0.5 1

low rank population

(b)

0 2 4 6 8 10

0 0.5 1

eigenvalues

(c)

0 2 4 6 8 10

0 0.5 1

low rank eigenvalues

(d)

0 2 4 6 8 10

0.98 0.99 1

fidelity

φ (e)

0 2 4 6 8 10

10⁻⁶ 10⁻⁴ 10⁻² 10⁰ 10²

(f)

φ

Frobenius norm

Full rank (left) versus rank 6 (right) (N

1,

15,

¯n)

0 2 4 6 8 10

0 0.5 1

population

(a)

0 2 4 6 8 10

0 0.5 1

low rank population

(b)

0 2 4 6 8 10

0 0.5 1

eigenvalues

(c)

0 2 4 6 8 10

0 0.5 1

low rank eigenvalues

(d)

0 2 4 6 8 10

0.996 0.998 1

fidelity

φ (e)

0 2 4 6 8 10

10⁻⁶ 10⁻⁴ 10⁻² 10⁰ 10²

(f)

φ

Frobenius norm

Oscillation revival with

0 (N

50,

200,

0 1 2 3 4 5 6 7 8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

population

φ

6.2 6.4 6.6 6.8 7

0 0.05 0.1 0.15 0.2 0.25 0.3

Schrödinger simulation time 1h15 (Dell precision M4440 with Matlab)

Rank-8 solution with

log(2)Ω

(N

50,

200)

0 1 2 3 4 5 6 7 8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

population

φ

6.2 6.4 6.6 6.8 7

0 0.05 0.1 0.15 0.2 0.25 0.3

Rank-8 simulation time 17h00 (Dell precision M4440 with Matlab)

Concluding remarks

A single tuning parameter: the rankmn.

Extension to an arbitrary number of Lindblad operators:

d

dtρ=−i[H, ρ] +X

ν

LνρL^†_ν−¹₂(L^†_νLνρ+ρL^†_νLν) d

dtU=−iHU+ (Iⁿ−UU^†) X

ν

−¹

2L^†_νLν+LνUσU^†L^†_νUσ⁻¹U^†

! U d

dtσ=X

ν

−1

2 (U^†L^†_νLνUσ+σU^†L^†_νLνU) +U^†LνUσU^†L^†_νU +_m¹Tr X

ν

(L^†_ν(In−UU^†)LνUσU^†

! Im.

Similar low-rank approximations could be done forcontinuous-time quantum filters. . .

Implemented in simulation packages such as QuTip⁷?

Adaptation whennis huge⁸?Low-rank quantum tomography ?

7J.R Johansson, P.D. Nation, F.Nori: QuTiP an open-source Python framework for dynamics of open quantum systems. Computers Physics Communications 183 (2012) 1760–1772.

8Ilya Kuprov: Spinach - software library for spin dynamics simulation of large spin systems. PRACQSYS 2010.