Simulation complexity of open quantum dynamics: Connection with tensor networks

I. A. Luchnikov,1, 2 S. V. Vintskevich,2, 3 H. Ouerdane,1 and S. N. Filippov2, 4, 5

1Center for Energy Science and Technology, Skolkovo Institute of Science and Technology,3 Nobel Street, Skolkovo, Moscow Region 121205, Russia

2Moscow Institute of Physics and Technology, Institutskii Per. 9, Dolgoprudny, Moscow Region 141700, Russia3A.M. Prokhorov General Physics Institute, Russian Academy of Sciences, Moscow, Russia

4Valiev Institute of Physics and Technology of Russian Academy of Sciences, Nakhimovskii Pr. 34, Moscow 117218, Russia5Steklov Mathematical Institute of Russian Academy of Sciences, Gubkina St. 8, Moscow 119991, Russia

The difficulty to simulate the dynamics of open quantum systems resides in their coupling tomany-body reservoirs with exponentially large Hilbert space. Applying a tensor network approachin the time domain, we demonstrate that effective small reservoirs can be defined and used formodeling open quantum dynamics. The key element of our technique is the timeline reservoirnetwork (TRN), which contains all the information on the reservoir’s characteristics, in particular,the memory effects timescale. The TRN has a one-dimensional tensor network structure, whichcan be effectively approximated in full analogy with the matrix product approximation of spin-chain states. We derive the sufficient bond dimension in the approximated TRN with a reduced setof physical parameters: coupling strength, reservoir correlation time, minimal timescale, and thesystem’s number of degrees of freedom interacting with the environment. The bond dimension canbe viewed as a measure of the open dynamics complexity. Simulation is based on the semigroupdynamics of the system and effective reservoir of finite dimension. We provide an illustrative exampleshowing scope for new numerical and machine learning-based methods for open quantum systems.

PACS numbers: 02.10.Xm,02.10.Yn,03.65.Yz,03.65.Aa,03.65.Ca

Introduction. One of the most challenging and impor-tant problems of modern theoretical physics is the accu-rate simulation of an interacting many-body system. Asthe dimension of its Hilbert space grows exponentiallywith the system size, direct simulations become impos-sible. Exactly solvable models exist nonetheless [1]; theyprovide some insights into the properties of actual phys-ical systems. Perturbation theory can be used only forproblems that can be split into an exactly solvable partand a perturbative one provided that a relevant small pa-rameter (e.g., weak interaction strength with respect toother energy scales) can be defined. For strongly interact-ing many-body systems, a range of techniques including,e.g., the Bethe ansatz [2], the dynamical mean field the-ory [3, 4], or the slave boson techniques [5, 6] have beendeveloped and applied to problems like the diagonaliza-tion of the Kondo Hamiltonian or the Anderson impuritymodel [7–11]. Numerical approaches, which may signif-icantly go beyond the range of applicability of analyti-cal methods, have been also developed and proved quitesuccessful, though they suffer from limits. For example,methods based on tensor networks [12–14] and the densitymatrix renormalization group [15–17] work well mainly forone-dimensional models. Quantum Monte Carlo (QMC)methods provide reliable ways to study the many-bodyproblem [18], but for interacting fermion systems theseapproaches are plagued by the sign problem [19].

Unitary evolution of a many-body system is completelyout of reach: dynamical versions of QMC calculationsor efficient methods like the time-evolving block decima-tion algorithm cannot predict long-term time dynamicsbecause of the Lieb-Robinson bound [20–23]. Having ex-perimental access only to a part of a many-body system,one in fact deals with an open quantum dynamics of thesubsystem (S), whereas the rest of the particles (modes)

FIG. 1: Schematic of reservoir truncation.

play the role of environment also referred to as reser-voir (R), see Fig. 1(a). The subsystem is described bya density operator ρS(t) = trR[U(t)ρ(0)U†(t)], the evo-lution of which is still challenging to determine thoughthe subsystem is relatively small compared to the envi-ronment [24, 25]: the partial trace, trR, disregards theenvironment degrees of freedom but ρ(0) is the initialstate of the whole many-body system and its evolutionoperator is U(t) = e−itH . There exist particular exactlysolvable models of open quantum dynamics [26–28]; how-ever, without the Markov approximation the problem ofopen dynamics is typically impossible to solve directly be-cause of the exponentially large dimension of the reser-voir’s Hilbert space [25, 29, 30]. Examples of complexopen dynamics in structured reservoirs, where it is nec-essary to go beyond the Markov approximation, are pre-sented in [31–42]. Therefore, new, numerically tractableapproaches permitting significant progress in the field ofopen quantum dynamics simulation are highly desirable,and their development constitutes a timely challenge, es-pecially in the study of quantum control and dynamicaldecoupling [40–42], and quantum dynamics induced bymany-body reservoirs [43–46].

In this work, we show that the actual infinite envi-ronment can be replaced by a finite-dimensional effec-

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tive reservoir (ER) in such a way that the aggregate“S+ER” experiences semigroup dynamics, see Fig. 1(b).This approach resembles the idea of Markovian embed-ding of non-Markovian dynamics [47–50] and the pseu-domode method [51–53]. Our main result is the estima-tion of the minimal (sufficient) dimension dER of the ef-fective reservoir expressed through a reduced set of pa-rameters. Knowledge of dER enables one to efficientlysimulate the complex dynamics of a subsystem of di-mension dS via ρS(t) = trER[eLtρS+ER(0)], where theGorini-Kossakowski-Sudarshan-Lindblad (GKSL) genera-tor L [54, 55] is easy to parameterize in this case: L acts ondSdER×dSdER matrices and can be numerically found viamachine learning techniques provided a sequence of mea-surements on the subsystem is performed [56]. Anothermachine learning algorithm [57] estimates dER within thetraining range dER = 1, 2, 8, 16 based on interventions inthe open qubit evolution at 4 time moments.

Open quantum systems properties. Let HS and HR beHilbert spaces of the subsystem and the reservoir, respec-tively. Typically dim(HS) dim(HR) as HS could beassociated with, e.g., a qubit or other small system, andHR with a many-body quantum environment of a hugedimension. The total Hilbert space is H = HS ⊗ HR.As the environment is assumed to be in the thermody-namic limit, the dynamics of ρS(t) is irreversible, i.e. thePoincare recurrence time is infinite. When the subsys-tem and reservoir exchange energy, thermalization is ex-pected on a long timescale [58], though the dynamics canbe strongly non-Markovian at finite times [59].

The total Hamiltonian reads H = H0 + Hint, whereH0 = HS ⊗ 1+ 1⊗HR involves individual Hamiltoniansof the subsystem and the reservoir, Hint = γ

∑ni=1Ai ⊗

Bi is the interaction part with characteristic interactionstrength γ; n is the effective subsystem’s number of de-grees of freedom interacting with the reservoir. DenoteBi(t) = U†(t)BiU(t), then gij(t, t − s) = tr[B†i (t)Bj(t −s)ρ(0)]− tr[B†i (t)ρ(0)]tr[Bj(t−s)ρ(0)] is the reservoir cor-relation function [60]. Suppose gij(t, t − s) decays expo-nentially with the growth of s over a characteristic timesij (see examples in [61, 62]), then T = maxij sij is thereservoir correlation time. Suppose the Fourier transformof the reservoir correlation function decays significantly atthe characteristic frequency Ωij , then τ = (maxij Ωij)

−1

defines the minimal time scale in the dynamics. In thecase of bosonic bath, τ = ω−1

c , where ωc is the cutofffrequency of the spectral function [63]. Our approach todetermine the dimension dER of truncated environment isbased on tensor network formalism in time domain, wherethe building blocks are responsible for evolution duringtime τ and the ancillary space is capable of transferringtemporal correlations for the period T . Therefore, dERdepends only on the following few physical parameters:γ, n, τ , and T .

Tensor network representation of open quantum dynam-ics. For simplicity, we resort to the vector representation

of a density operator:

ρ =∑jk

ρjk |j〉 〈k| → |ρ〉 =∑jk

ρjk |j〉 ⊗ |k〉 , (1)

which implies QρP → Q ⊗ PT |ρ〉. The dynamics of thewhole system reads

|ρ(t)〉 = exp[−itH]⊗ exp[itHT ] |ρ(0)〉 . (2)

The initial state ρ(0) can be correlated in general, i.e.

ρ(0) =∑l σ

(l)S ⊗σ

(l)R . The subsystem state ρS(t) = trRρ(t)

in terms of vectors reads |ρS(t)〉 = 〈ψ+| |ρ(t)〉, where

〈ψ+| =∑dRj=1 1S ⊗ 〈j| ⊗ 1S ⊗ 〈j|. For further con-

venience we introduce a new order of Hilbert spaces:HS ⊗HR ⊗HS† ⊗HR† → HS ⊗HS† ⊗HR ⊗HR†.

The minimal timescale τ is a time step in discretizedevolution; τ can always be reduced in such a way thatγτ 1, which permits application of the Trotter decom-position [64]. Note that we do not restrict our frameworkto a weak coupling between a subsystem and reservoir(γ ‖H0‖); we rather adjust the minimal timescale τ inaccordance with the interaction strength, which yields

|ρ(t)〉 = Φ0(τ)Φint(τ) · · ·Φ0(τ)Φint(τ)︸ ︷︷ ︸t/τ times

|ρ(0)〉+O(γτ),

(3)where Φ0(τ) and Φint(τ) are responsible for the nonin-teractive and interactive evolutions of the subsystem andenvironment:

Φ0(τ) = exp [−iτHS ]⊗ exp [iτHTS ]

⊗ exp [−iτHR]⊗ exp [iτHTR ], (4)

Φint(τ) =

2n∑i=0

Ai(τ)⊗ Bi(τ), (5)

Ai(τ) =

1⊗ 1 if i = 0,√γτ Ai ⊗ 1 if 1 ≤ i ≤ n,√γτ 1⊗ATi−n if i ≥ n+ 1,

(6)

Bi(τ) =

1⊗ 1 if i = 0,−i√γτ Bi ⊗ 1 if 1 ≤ i ≤ n,i√γτ 1⊗BTi−n if i ≥ n+ 1.

(7)

The tensor network representation to calculate |ρS(t)〉is presented in Fig. 2. It is a particular case of the gen-eral quantum circuits [65–69]. Each building block with marms corresponds to a tensor of rank m. Connecting linksdenote contractions over the same indices. The vector|ρ(t)〉 has two multi-indices j = (jS , jR) and k = (kS , kR),so it is represented as a tensor of rank 4. The upper (bot-tom) row corresponds to the degrees of freedom of sub-system S (reservoir R). The operator Φ0(τ) is depictedby solid squares. The dashed squares with a link betweenthem denote the operator Φint(τ), with the link being re-

sponsible for summation∑2ni=0 in formula (5). Concatena-

tion with the building block ψ+ in right bottom of Fig. 2corresponds to the partial trace over R and will be furtherdenoted by connecting link ⊃.

3

S

R +

l iR

jS

kS

j

kR

0

y

t 2t Nt

FIG. 2: Tensor network for open system dynamics.

A key object in our study is the tensor in Fig. 3(c),which we call a timeline reservoir network (TRN). TheTRN contains all the information on the reservoir andcontrols all features of open dynamics, including dissipa-tion, Lamb shift, memory effects like revivals [25]. Fromthe computational viewpoint, for a fixed time t, TRN isa tensor with t/τ indices. Since the physical reservoirhas a finite memory depth T , the considered tensor musthave vanishing correlations between apart indices. Ten-sors of such a type can be effectively approximated byone-dimensional tensor networks of matrix-product (MP)form.

The TRN is closely related to the recent reformula-tions of the Feynman-Vernon path integrals [63, 70–73]and the process tensor [68, 74–76], see Figs. 3(a)–3(b).In fact, the influence functional in Fig. 3(b) can be ex-plicitly calculated in the case of a bosonic bath linearlycoupled to the system [77] but it remains difficult to con-tract with the system initial state and unitary evolutiontensors, so in Refs. [70–72] the contraction calculation isapproximated by fixing a finite memory depth K = T/τ .References [63, 73] further use MP approximation of theinfluence functional (with rank λmax and accuracy λc),which allows to deal with longer memory depths. Sinceλmax is d2

ER in our model, we actually estimate the com-plexity (λmax) of the algorithm in Ref. [63].

MP approximation of the TRN. From a mathematicalviewpoint, the constructed TRN can be treated as a puremultipartite quantum state |ψ〉, where summation indexim = 0, . . . , 2n at time moment tm = mτ plays the role ofthe physical index assigned to the mth particle:

|ψ〉 =∑

l,i1,i2,...,iN

ψli1i2...iN |l〉⊗|i1〉⊗|i2〉⊗ . . .⊗|iN 〉 . (8)

The only difference between the TRN and |ψ〉 is the nor-

S

R

S

R

R

(a)

(b)

(c)l iii1i 2 3 N

FIG. 3: (a) Process tensor T (t1; t3; tn) [68, 74–76]. (b) Influ-ence functional [63, 70–73]. (c) Timeline reservoir network.

MP approximation of states MP approximation of TRN

position of m’th particle in space time moment tm = mτ on timeline

dimension of a particle’s Hilbert twice the number of subsystem’s

space degrees of freedom plus one, 2n+ 1

rank, bond dimension r square of dimension

(dimension of ancillary space) of effective reservoir, d2ER

correlation length, L reservoir correlation time, T

TABLE I: Correspondence between physical descriptions ofpure quantum states and TRN in MP approximation.

malization: TRN∝ |ψ〉, 〈ψ|ψ〉 = 1.A multipartite quantum state |ψ〉 with a finite correla-

tion length L can be effectively described via MP approxi-mation [12]. The benefit of such an approximation is thatit is able to reproduce spatial correlations among particleswithin the characteristic length L. The effectiveness of ap-proximation means that the bond dimension r of the an-cillary space (rank of MP state) is rather small. Similarly,the TRN is able to effectively reproduce temporal corre-lations within the period T with a rather small dimensiondER of effective reservoir. Since we deal with matrices,d2ER is equivalent to the rank of corresponding MP state.

The dimension of the physical space in the MP state is2n+ 1, where n is the number of the subsystem’s degreesof freedom involved in the interaction with environment.Note that n ≤ d2

S . The physics of MP approximations forstates and TRN is summarized in Table I.

Suppose |ψ(r)〉 is a rank-r MP approximation of thepure state |ψ〉. The approximation error ε(r) equals theFrobenius distance between |ψ(r)〉 and |ψ〉. Consider |ψ〉as a bipartite state, with parties being separated by a cutbetween the mth and (m + 1)th particles. MP approxi-mation effectively disregards low-weight contributions inthe Schmidt decomposition of |ψ〉 with respect to such acut. The approximation error ε(r) is related to the Renyientropy of order α (0 < α < 1), Sα of a single reduceddensity operator as follows [78]:

ln[ε(r)] ≤ 1− αα

[Sα − ln

(r

1− α

)], (9)

from which one readily obtains the sufficient bond dimen-sion guaranteeing the arbitrary desired accuracy ε:

rsuff(ε) = min0<α<1

(1− α)ε−α/(1−α) exp(Sα). (10)

In the language of TRN, the sufficient rank is thesquare of minimal dimension of effective reservoir, d2

ER,which can reproduce all temporal correlations with errorε. Therefore, to find dER one needs to estimate the Renyientropy Sα of tensor in Fig. 4 considered as a matrix Mwith 2 multi-indices (l, i1, . . . , im) and (l′, i′1, . . . , i

′m):

M(l,i1,...,im),(l′,i′1,...,i′m)

=∑

im+1,...,iN

ψl,i1,...,im,im+1,...,iNψ∗l′,i′1,...,i

′m,im+1,...,iN

. (11)

4

i1

im+1 iNil

l ’ i ’1 i ’mm

FIG. 4: Reduced matrix of TRN.

The Renyi entropy reads [79]:

Sα .1

1− αln

[1 + 2n(γτ)α]T/τ

(1 + 2nγτ)αT/τ≈ 2nγT

(γτ)α−1 − α1− α

.

(12)The entropy Sα is a measure of the time correlations inTRN. Substituting (12) in (10), we obtain the sufficientrank rsuff of MP approximation of TRN with desired phys-ical properties (parameters γ, n, T , τ) and accuracy ε.On the other hand, the rank of MP approximation is thesquare of the dimension dER of the effective reservoir thatcan reproduce all the features of open dynamics (includingmemory effects) with accuracy ε. Therefore, it is possibleto simulate the complex open system dynamics by usingthe effective reservoir of dimension

dER(ε) = min0<α<1

√1− α

εα/2(1−α)exp

[nγT

(γτ)α−1 − α1− α

].

(13)Once the effective reservoir is constructed, the aggre-

gate “S+ER” experiences the semigroup dynamics. Thisfollows from the tensor network representation in Fig. 2.The TRN has a homogeneous structure and so does itsMP approximation. The regular structure of buildingblocks in the time scale means that the same transfor-mation 1 + τL acts on “S + ER” between the successivetimes mτ and (m + 1)τ . The GKSL generator L [54, 55]acts on dSdER × dSdER density matrices and guaranteescomplete positivity of evolution.

Discussion. Although the actual environment consistsof infinitely many modes, the developed theory facilitatesthe simulation of complex open system dynamics with afinite dimensional effective reservoir. The sufficient di-mension dER depends on two combinations of physicalparameters: nγT and γτ . Figure 5 shows that one cansimulate the open dynamics on a classical computer for awide range of parameters: nγT and γτ , accounting for allpotential initial correlations between the system and itsenvironment.

The first illustrative example is a decay of the two-levelsystem (qubit) in multimode environment [79]. We com-pare the exactly solvable qubit dynamics, the Markov ap-proximation (d′ER = 1), and the approximation obtainedwith the reservoir of fixed dimension (d′′ER = 2). Fig-ure 6 shows that the best Markovian approximation can-not reproduce oscillations in the exact dynamics, whereasthe approximation with the fixed dimension d′′ER = 2 fitswell the exact dynamics when the simulation complexitydER ∼ d′′ER. However, if dER is several orders of magni-tude larger than d′′ER, then the approximation is not able

FIG. 5: Number of qubits log2(dER) in effective reservoir,which is sufficient for simulation of open dynamics with ac-curacy ε = 0.05. nγT is dimensionless memory time and γτ isdimensionless minimal timescale.

to reproduce memory effects present in the exact solution.Thus, dER does quantify the complexity of dynamics.

The second example is the double quantum dot chargequbit coupled to piezoelectric acoustic phonons [62]. Here,n = 1, T ≈ 4τ , τ = ω−1

c , γ = 0.05ωc, and ωc = 83 GHzis the cutoff frequency of the spectral function. Eq. (13)yields log2dER = 4 for ε = 0.05, i.e. the non-Markovianqubit dynamics can be embedded into a Markovian evo-lution of the very qubit and 4 auxiliary ones.

The third example is the non-Markovian evolution ofthe qubit due to interaction with the dissipative pseudo-mode [52, 53]: dρ

dt = −i[H0, ρ]+Γ(aρa†− 12a†a, ρ), where

H0 = ω0σ+σ−+ωa†a+ Ω0σx(a†+ a) and ρ is the densityoperator for the qubit and the pseudomode. Physical pa-rameters are n = 1, T = Γ−1, τ = ω−1, γ = Ω0

√n0 + 1,

where n0 is the effective number of photons in the pseudo-mode. Our result, Eq. (13), estimates where the pseudo-mode oscillator can be truncated (Fock states with num-ber of photons less than dER) to reproduce the systemdynamics with precision ε at any time despite the mem-ory effects and the counter-rotating terms in H0. In thisexample, construction of the effective reservoir reduces tothe subspace spanned by dER lowest energy states of thepseudomode because the particular dissipator forces thepseudomode to the ground state.

There are physical scenarios in which the structure ofthe effective reservoir follows from the model. For in-

FIG. 6: Typical evolutions of parameter tr[σzρ(t)] of openqubit system ρ(t) for different values of simulation complex-ity dER(0.05) [79]. Dotted lines depict the exact dynam-ics. Dashed lines depict the best Markov approximations(d′ER = 1). Solid lines are the approximations obtained withthe reservoir of fixed dimension d′′ER = 2.

5

stance, in a nitrogen-vacancy center in diamond, the in-herent nitrogen (14N) nuclear spin (I = 1) serves as aneffective reservoir for the electronic spin qubit [42]. In thiscase, dER = 3. Similarly, in a composite bipartite colli-sion model [83], dER is given by the size of an ancillarysystem. In general, however, the structure of the effectivereservoir is to be determined from the experimental data.Reference [56] proposes a machine learning algorithm toreconstruct the generator L based on a series of repeatedmeasurements on the open system.

Finally, our result is applicable to the influence func-tional tensor networks in Refs. [63, 73], where the ana-lytical solution for open dynamics is not accessible, andprovides the upper bound on the maximum bond dimen-sion, λmax < d2

ER. Conversely, for a fixed computationallytractable size of bond dimension, e.g., λmax ∼ 103, our re-sult provides the region of physical parameters γτ andnγT , for which the algorithm in Ref. [63] definitely workswell.

Importantly, the TRN is a multidimensional tensor, soit can be approximated with MP form but also with other

constructions like multiscale entanglement renormaliza-tion ansatz [84, 85] or artificial neural networks [86, 87].The benefit of such networks is that time correlations inthe environment do not have to decay exponentially as forthe MP approximation.

Conclusion. We gave a definition of simulation com-plexity of open quantum dynamics in terms of a reservoir’seffective dimension. We showed that the tensor networksapproach can be utilized to analyze memory effects inopen dynamics. We provided an estimation of simula-tion complexity using a set of physical parameters. Ourestimation is universal and fits well the arbitrary openquantum dynamics with finite memory.

Acknowledgments The authors thank Alexey Akimov,Mikhail Krechetov, Eugene Polyakov, Alexey Rubtsov,and Mario Ziman for fruitful discussions. The studyis supported by Russian Foundation for Basic Researchunder Project No. 18-37-20073. I.A.L. thanks RussianFoundation for Basic Research for partial support underProject No. 18-37-00282.

Supplemental Material

Upper bound on Renyi entropy

Here we discuss the upper bound on Renyi entropy to clarify the derivation of formula (12) in the main text.Consider an arbitrary density matrix M that has the form M =

∑q vqv

†q, where the set of vectors vqq is not

orthogonal nor normalized. Let us prove that the Renyi entropy Sα(M) = 11−α ln trMα satisfies the inequality Sα(M) ≤

11−α ln

∑q

(v†qvq

)αif 0 < α < 1.

In fact, the classical Renyi entropy Hα is Schur-concave for all α > 0 (see, e.g., [80]), i.e. Hα(R) ≤ Hα(P ) whenever

the probability distribution P is majorized by the probability distribution R (∑ki=1 P

↓i ≤

∑ki=1R

↓i for all k = 1, 2, . . .).

By Theorem 10 of Ref. [81] the equality M =∑q vqv

†q implies that the distribution V = v†qvqq is majorized by the

vector Λ of eigenvalues of M . Therefore,

Sα(M) = Hα(Λ) ≤ Hα(V ) =1

1− αln∑q

(v†qvq

)α. (14)

Diagrammatic language for tensor networks

In this section we demonstrate key physical objects using tensor network representation (language).

Fig. 7(a) illustrates the timeline reservoir network (TRN) as an analogue of multipartite quantum state |ψ〉 . It alsodepicts the TRN matrix in analogy with the density matrix |ψ〉 〈ψ| for states.

In Fig. 7(b) we simplify the notation and coarse grain the elements of the tensor network. We replace double linesby single thick lines. The tracing element in the right hand side of network is depicted as a semicircle.

Fig. 7(c) depicts TRN matrix in terms of new notation.

In Fig. 7(d) we depict the reduced TRN matrix, which is constructed in analogy with the reduced density matrix(ρs = trrρs+r). Connected indices between upper and lower parts of network mean summation, i.e. reduction overnon-relevant part with time t ≥ mτ .

Given a reduced matrix M of TRN, namely,

M(l,i1,...,im),(l′,i′1,...,i′m) =

∑im+1,...,iN

ψl,i1,...,im,im+1,...,iNψ∗l′,i′1,...,i

′m,im+1,...,iN

. (15)

one needs to estimate its Renyi entropy Sα. The result of section A is that Sα(M) ≤ (1− α)−1 ln∑q

(v†qvq

)α. In our

6

FIG. 7: (a) TRN and TRN matrix. (b) Change of notation to simplify tensor network representation. (c) TRN matrix insimplified notation. (d) Reduced matrix of TRN.

FIG. 8: (a) Reduced matrix of TRN and the reservoir correlation time T . (b) The absence of time correlations between blocksseparated by time T . (c) The right hand side is a rank-0 tensor. (d) The marked tensor of length T is exactly the element =|⊃in Fig. 8(c).

FIG. 9: Estimation of the von Neumann entropy of the reduced TRN within the framework of diagrammatic representation.

7

case q = (im+1, . . . , iN ) and vq is a vector with components ψl,i1,...,im,q, so we have

Sα ≤ (1− α)−1 ln∑

im+1,...,iN

∑l,i1,...,im

|ψl,i1,...,im,im+1,...,iN |2α . (16)

Let us consider the physical structure of M , see Fig. 8. In Fig. 8(a), we explicitly mark out the reservoir correlationtime T . Since temporal correlations decay within the time period T , we assume that time correlations between nodesseparated by time t ≥ T can be neglected. This means that the action of tensor network of length T can be replacedby a simpler tensor of the form =|⊃ ⊂|=, see Fig. 8(b). Thus,

=( tensor of length t ≥ T )= ⇐⇒ =|⊃ ⊂|= . (17)

Concatenation of the tensor ⊂|= with the rest of right-hand side tensor is merely a number (rank-0 tensor), Fig. 8(c).In other words, the contribution of indexes im+T/τ+1, im+T/τ+2, . . . , iN in formula (16) is trivial: the summation overindexes im+T/τ+1, im+T/τ+2, . . . , iN does not change the structure of the matrix M as it only leads to a multiplicationfactor. The reduced matrix M of TRN is not affected by time moments happening after the reservoir correlationtimescale has passed.

That observation explains the fact why in formula (16) one can replace the tensor ψl,i1,...,im,im+1,...,iN by a tensor

ψl,i1,...,im,im+1,...,im+T/τof finite length and appropriate normalization

∑l,i1,...,im+T/τ

|ψl,i1,...,im+T/τ|2 = 1.

Therefore, the Renyi entropy of reduced TRN matrix equals the Renyi entropy of a tensor depicted in the left-handside of Fig. 8(c). The element =|⊃ in the left-hand side of Fig. 8(c) can be replaced by a tensor of finite length T , andwe end up with a tensor network illustrated in Fig. 8(d). Following the notation of the main text,

Fig. 8(d) ⇐⇒∑

im+1,...,im+T/τ

ψl,i1,...,im,im+1,...,im+T/τψ∗l′,i′1,...,i′m,im+1,...,im+T/τ

. (18)

and we get

Sα . (1− α)−1 ln∑

im+1,...,im+T/τ

∑l,i1,...,im

|ψl,i1,...,im,im+1,...,im+T/τ|2α . (19)

Recalling the explicit structure of TRN given by formulas (5)–(7) in the main text, we find

∑l,i1,...,im

|ψl,i1,...,im,im+1,...,im+T/τ|2 ≈ (1 + 2nγτ)−T/τ

T/τ∏p=1

1, im+p = 0,

γτ, im+p = 1, . . . , 2n .(20)

Substituting this result in (19), we obtain

Sα .1

1− αln

[1 + 2n(γτ)α]T/τ

(1 + 2nγτ)αT/τ≈ 2nγT

(γτ)α−1 − α1− α

. (21)

To finalize the explanation of diagrammatic language in the estimation of entropy Sα, in Fig. 9 we depict the caseα→ 1 when Sα tends to the von Neumann entropy.

Exactly solvable model

In this section, we consider a particular example of an exactly solvable open quantum dynamics. Let S be a two-levelsystem with energy separation Ω and R be a bosonic bath at zero temperature. The total Hamiltonian reads (hereafterthe Planck constant ~ = 1)

H = Ωσ+σ− +∑m

ωmb†mbm +

∑m

gm(b†mσ− + bmσ+

), (22)

where σ+ = |1〉 〈0| and σ− = |0〉 〈1|, |0〉 and |1〉 are the ground and excited states of the two-level system, respectively,index m enumerates bosonic modes, bm and b†m are the annihilation and creation operators for bosons in m’th mode,gm is the coupling strength given by formula

gm =

g if ωmin ≤ ωm ≤ ωmax,

0 otherwise.(23)

8

Consider the equidistant set of mode frequencies ωm with ωm+1−ωm = ∆ω, then the bath consists of N = ωmax−ωmin∆ω

modes with freqencies ωm = ωmin +m∆ω, m = 1, . . . , N .The initial state of “S+R” is pure and has the form

|ψ(0)〉 = |1, vac〉 , (24)

which implies that the two-level system is in the excited state and that the bath has no excitation. The Hamiltonian (22)preserves the number of excitations, so the solution of the Schrodinger equation i ddt |ψ(t)〉 = H |ψ(t)〉 has the form

|ψ(t)〉 = α(t) |1, vac〉+∑m

βm(t) |0, 1m〉 , (25)

where the coefficients α(t) and βm(t) satisfy the following system of differential equations

id

dt

α(t)

β1(t)

β2(t)...

βN (t)

=

Ω g g . . . g

g ∆ω 0 . . . 0

g 0 2∆ω . . . 0...

......

. . ....

g 0 0 . . . N∆ω

α(t)

β1(t)

β2(t)...

βN (t)

(26)

and meet the initial requirements α(0) = 1 and βm(0) = 0.The parameter α(t) is a solution of:

d

dtα(t) = −iΩα(t)−

N∑k=1

g2

∫ t

0

exp[−ik∆ω(t− t′)

]α(t′)dt′. (27)

Assume ∆ω → 0, we replace the summation∑k by the integration:

N∑k=1

→ 1

∆ω

∫ ωmax

ωmin

dω. (28)

Therefore, we obtain the model with reservoir containing an infinite number of modes. Nonetheless, such a model is still solvableas the parameter α(t) is a solution of the following integrodifferential equation:

d

dtα(t) = −iΩα(t)− g2

∆ω

∫ t

0

G(t− t′)α(t′)dt′, G(t− t′) =

∫ ωmax

ωmin

exp[−iω(t− t′)

]dω. (29)

Formally α(t) reads

α(t) =1

2πi

∫ ∞+ε

−∞+ε

dsest1

s+ iΩ + g2

∆ω

∫ ωmax

ωmaxdω 1

s+iω

, (30)

and the quantity tr[σzρ(t)] illustrated in Fig. 6 in the main text equals tr[σzρ(t)] = 2|α(t)|2 − 1.Now, let us define the parameters T , τ , γ, n needed for the estimation of simulation complexity (the dimension of effective

reservoir, dER). The memory kernel G decays significantly within the reservoir correlation time T , which in our case equalsT = 1

ωmax−ωmin. The minimal timescale of evolution is τ = 1

ωmax. Comparing the general form of interaction Hamiltonian

Hint = γ∑ni=1 Ai ⊗ Bi and the interaction Hamiltonian in our example g

∑ωmax−ωmin∆ω

m=1 (b†mσ− + bmσ+), we conclude that n = 2

and γ = g ωmax−ωmin∆ω

. Fixing the accuracy ε = 0.05 and substituting parameters T , τ , γ, n in formula (15) in the main text, wecalculate the sufficient dimension of effective reservoir dER for three different evolutions depicted in Fig. 6 in the main text.

Semigroup evolution of system (d′ER = 1)

Direct application of the weak-coupling and Born–Markov approximations [25] to the model in section C results in the ex-ponential decay tr[σ+σ−ρ(t)] = exp(−Γt) because the reservoir is assumed to be time-independent (always in a vacuum state).Such a behavior is in strong contrast to the exact solution in Fig. 6 in the main text, where the steady state has a non-zeropopulation tr[σ+σ−ρ(∞)].

A general semigroup evolution of the very qubit corresponds to the situation, when one forcibly fixes the dimension d′ER = 1.Let us consider the semigroup evolution of the qubit with a GKSL generator, which takes into account the finite population

of excited state:dρ

dt= −iΩ [σ+σ−, ρ] + γ↓

(σ−ρσ+ −

1

2σ+σ−, ρ

)+ γ↑

(σ+ρσ− −

1

2σ−σ+, ρ

). (31)

Note that this equation is nothing else but the generalized amplitude damping process [82]. Adjusting the relaxation ratesγ↓ and γ↑, we get the best Markov approximations shown in Fig. 6 in the main text. This figure also illustrates that the bestMarkov approximation cannot reproduce non-Markovian memory effects, the correct effective dimension dER being much greaterthan d′ER = 1.

9

Semigroup evolution of system and reservoir of fixed dimension d′′ER = 2

Let us consider a model with the small reservoir of forcibly fixed dimension d′′ER = 2 such that the system and the smallreservoir altogether experience the semigroup dynamics. Such a model would be a good approximation of the exact dynamicsfound in section C if the simulation complexity dER ∼ d′′ER.

We consider the following parameterization of the GKSL generator L governing the evolution equation dρdt

= Lρ for the densityoperator ρ(t) of the two level system and small reservoir:

Lρ(t) = −i [h, ρ(t)] +D(ρ(t)),

h = Ω1σ+σ− ⊗ 1+ Ω21⊗ σ+σ− + g (σ+ ⊗ σ− + σ− ⊗ σ+) ,

D(ρ(t)) = γ1,↓

(σ− ⊗ 1ρ(t)σ+ ⊗ 1−

1

2σ+σ− ⊗ 1, ρ(t)

)+γ1,↑

(σ+ ⊗ 1ρ(t)σ− ⊗ 1−

1

2σ−σ+ ⊗ 1, ρ(t)

)+γ2,↓

(1⊗ σ−ρ(t)1⊗ σ+ −

1

21⊗ σ+σ−, ρ(t)

)+γ2,↑

(1⊗ σ+ρ(t)1⊗ σ− −

1

21⊗ σ−σ+, ρ(t)

). (32)

The parameter Ω1 is the energy difference between excited and ground states of the system S with a potential contribution ofthe Lamb shift; Ω2 is the energy separation between levels of the effective reservoir; g is the coupling constant; γ1,↓ and γ1,↑are decay rates of the system; and γ2,↓ and γ2,↑ are decay rates of the effective reservoir. All these parameters are adjusted toapproximate the exact dynamics described in section C.

Figure 6 in the main text shows that the simple model with reservoir of fixed dimension d′′ER = 2 adequately describesnon-Markovian dynamics with simulation complexity dER(0.05) ≈ 7. However, such a simple model fails to reproduce memoryeffects associated with dynamics for which dER(0.05) ∼ 103. This fact justifies that in order to reproduce memory effects withsimulation complexity dER one has to deal with a reservoir of comparable dimension.

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