Unitary long-time evolution with quantum renormalization groups and artificial neural networks
UUnitary long-time evolution with quantum renormalization groups andartificial neural networks
Heiko Burau ∗ and Markus Heyl Max-Planck-Institut f¨ur Physik komplexer Systeme,N¨othnitzer Straße 38, 01187 Dresden, Germany
In this work we combine quantum renormalization group approaches with deep artificial neuralnetworks for the description of the real-time evolution in strongly disordered quantum matter. Wefind that this allows us to accurately compute the long-time coherent dynamics of large, many-body localized systems in non-perturbative regimes including the effects of many-body resonances.Concretely, we use this approach to describe the spatiotemporal buildup of many-body localizedspin glass order in random Ising chains. We observe a fundamental difference to a non-interactingAnderson insulating Ising chain, where the order only develops over a finite spatial range. We furtherapply the approach to strongly disordered two-dimensional Ising models highlighting that our methodcan be used also for the description of the real-time dynamics of nonergodic quantum matter in ageneral context.
Introduction.
The understanding of emergent behaviorin quantum many-body systems is largely based on the dis-covery of effective descriptions of analytically unsolvablemodels [1]. An essential toolkit to find the former con-stitute renormalization group (RG) methods. They aretraditionally applied on systems in thermal equilibrium,thereby explaining many collective phenomena includingstructured phases, phase transitions, critical scaling anduniversality. In the past decade, real-space RGs havebeen developed that aim to explain analogues of thesewell-known phenomena also in systems where a thermo-dynamic treatment breaks down due to strong quencheddisorder [2–10].Whereas real-space RGs successfully operate in the sta-tionary setting at the level of individual eigenstates [11–16], reaching a quantitative description of the dynamical properties of quantum many-body systems appears evenmore challenging. So far, coherent dynamics of quantummatter far from equilibrium has been mostly simulatedusing tensor networks methods [17–19] or exact diag-onalization [20, 21] with recent developments targetingdynamical descriptions in terms of machine learning meth-ods by utilizing Restricted Boltzmann Machines (RBM)[22] or, more general, Artificial Neural Networks (ANN)[22–24]. Still, accessing quantitatively the long-time dy-namics for large quantum many-body systems, especiallyin spatial dimensions beyond one, represents a majorchallenge [25–27].In this work, we show how ANNs can be utilized ina different way for numerically exactly time-integratingeffective descriptions of generically interacting systems,generated by RG methods. As a concrete example, weexplore the temporal build-up of MBL spin-glass orderout of a simple polarized state for a large, disordered spinchain, see Fig. 1b, among other long-time dynamics in1D- and 2D-lattices. We begin by formulating a prototyp-ical strong-disorder RG (SDRG) for spin-1/2 systems ofarbitrary spatial dimension and map its transformationsinto the time-domain. As a result, we obtain a quantum
Figure 1. (a) Illustration of a random quantum circuit builtup from local unitary RG-transformations. In the courseof the RG, long-distance and higher-order couplings emerge.Adding time-dependence leads to a further broadening withincreasing time. (b) Spatiotemporal build-up of MBL spin-glass order in a random quantum Ising chain with 64 latticesites after quenching a paramagnetic initial condition into thesymmetry-broken phase at J = 5 h, J ( x ) = h/
5. Dashed linesindicate emergence of light-cones except for the non-interactingcase, J ( x ) = 0, where the order stops developing at a finitedistance. The numerical data is obtained from an average over25 disorder realizations. circuit, see Fig. 1a, as an effective description of the time-evolution operator. Hereafter, we show that this circuitcan be encoded efficiently into deep ANNs associated withtypical initial conditions for quantum real-time dynamics.This allows us to quantitatively represent time-evolvedmany-body quantum states not only at short but also longtimes. We note that our method avoids a discretization a r X i v : . [ c ond - m a t . d i s - nn ] S e p of time but relies on a renormalized Hamiltonian that isassumed to effectively describe the relevant physics up tosome finite but nevertheless long time scale.The scheme we introduce in the following can be appliedfor any generic, but strongly disordered, spin-1/2 system.Concretely, we will apply it to a paradigmatic interactingdisordered quantum Ising model [28] of the form H = (cid:88) (cid:104) ij (cid:105) J ij σ zi σ zj + J ( x ) ij σ xi σ xj + (cid:88) i h i σ xj , (1)with next-neighbor couplings J ij ∈ [ − J, J ] and local mag-netic fields h i ∈ [ − h, h ] drawn randomly from uniformdistributions. We use periodic boundary conditions. Forthe 1D case we also add random transverse couplings J ( x ) ij ∈ [ − J ( x ) , J ( x ) ] to obtain a generic and interactingmodel. Solving the time evolution.
Before describing the uti-lized renormalization procedure and the training of theANN in detail, let us start by outlining the general schemefor solving quantum real-time evolution utilizing strong-disorder RGs. Such an RG generates a sequence of localunitary transformations U k in order to iteratively obtain asimplified effective description of the considered quantummany-body system. In the time domain, we will show thatthis leads to the following representation of time-evolvedquantum many-body states: | ψ ( t ) (cid:105) QC = e − iH ( n )0 t U ( t ) · · · U n ( t ) | ψ (cid:105) , (2)where a time-dependence is added to the RG-trans-formations U k through a generalized interaction picture,see the derivation below. The above equation maps quan-tum dynamics onto a quantum circuit generated by thelocal unitaries U k ( t ). As we assume that the effectivedescription in terms of the final Hamiltonian H ( n )0 afterthe end of the RG procedure can be solved exactly, thecomplexity of the quantum circuit emerges solely uni-taries U k ( t ). We find that such quantum circuits canbecome a non-perturbative object, as the spatial supportof the U k ( t ) typically grows over time developing long-distance and higher-order couplings with large overlaps,see Fig 1a. A central contribution of this work is tooutline a numerically exact scheme to encode | ψ ( t ) (cid:105) QC and therefore the RG transformation itself into an ANNusing machine-learning techniques. The numerical learn-ing effort in obtaining | ψ ( t ) (cid:105) QC , as well as its memoryrequirement, scales at most quadratically with systemsize while being independent on the targeted time t orthe spatial dimension. Dynamical strong-disorder Renormalization Group.
Inprinciple, quantum circuits such as in Eq. (2) can begenerated using a variety of standard SDRGs. In thefollowing we introduce a variant of a SDRG, which as wefind improves the quantitative accuracy of the resultingscheme. As other SDRGs, the dynamical variant we introduce isbased on a local separation of energy scales. Consequently,at the beginning of each iteration k we pick the strongestcoupling, also called ”fast mode”, whose correspondingterm in the Hamiltonian we call H . For the first itera-tion this could be either a spin interaction J ij , J ( x ) ij , ortransverse field h i , see the Hamiltonian in Eq. (1). Thoseterms in the Hamiltonian which are not commuting with H we denote by V . These can be eliminated perturba-tively using a Schrieffer-Wolff transformation (SWT) [29]by applying a unitary transformation W k = e S k on theHamiltonian with a generator S k satisfying [ H , S k ] = V and S † k = − S k [12], at the expense of the renormalization H (cid:55)→ H + [ S k , V ] /
2. In general, this modifies existingcouplings and leads to the generation of new terms in theHamiltonian. After the SWT the fast mode is decoupledfrom the remainder and can then be faithfully removedfrom the system as a second-order local integral of motion(LIOM) [10, 30, 31]. After n such iterations, an unper-turbed Hamiltonian H ( n )0 is obtained, formed by the setof LIOMs.The newly generated couplings after each iteration are,of course, not known a-priori, especially if the SDRG isdesigned regardless of details of the model like range ofinteraction, dimensionality etc. We approach this problemby represent at each stage of the RG the Hamiltonian asa sum of arbitrary Pauli-strings σ α l . . . σ α M l M with a realcoefficient λ l ,...,l M each. Certainly, this approach canentail a costly handling of numerous generated higher-order couplings, see below, but it opens the possibilityto take into account many-body resonances, which areneglected using earlier SDRGs [11, 32] and related, so-called flow equation approaches [27, 33, 34].In addition the accuracy of the RG can be further in-creased by splitting the SWT into infinitesimal unitarytransformations, closely resembling in spirit the flow equa-tion framework. This turns out to be particularly helpfulin the vicinity of a critical point, h ≈ J here for the 1Dmodel, where the SWT is least controlled and in orderto capture many-body resonances to an arbitrary degree.For a detailed presentation of the technical details, seethe appendix. To control the exponential number of cou-plings { V i } generated during the RG, we first neglectthose terms where | V i | (cid:28) t ∗− which are much smallerthan the inverse of the targeted time scale t ∗ say, asthey do not influence physics up to t ∗ . Secondly, we per-form the continuous renormalization only w.r.t. those V i whose relative magnitude lies above a fixed threshold, | V i | / | H | > (cid:15) (cid:28)
1. Therefore we have a tradeoff, that iscontrolled by (cid:15) , between exactness and total number ofcouplings within the RG-generators S k and the renormal-ized Hamiltonian H ( n )0 . In our computations, (cid:15) typicallyranges from 10 − . . . − , depending on the closenessto the critical point, h ≈ J , or the ergodic transition, J ( x ) ≈ J . Later we will present a quantitative analysis ofour RG w.r.t. the dynamics of local observables. Time-dependent unitaries.
To derive the time-dependence of U k ( t ) we express the time-evolution opera-tor in the renormalized basis, which yields e − iHt = e S † · · · e S † n e − iH ( n )0 t e S n · · · e S = e − iH ( n )0 t e S † ( t ) · · · e S † n ( t ) e S n · · · e S . (3)We achieve a much more robust learning of the ANN uponsuccessively commuting each factor e S k to the left untilits counterpart e S † k ( t ) is reached. Identifying U k ( t ) = e ˜ S † k ( t ) e ˜ S k gives then the desired form as in (2). Here,˜ S k denotes the total application of all rotations from e S l , l < k on S k , see the appendix for details. Training the artificial neural network.
Utilizing ANNsas a variational ansatz for many-body wavefunctions hasseen an active development recently [22, 35] becomingcompetitive with or partially even superior to otherstate-of-the-art methods. [23, 36, 37]. In contrast tothe commonly used time-dependent variational principle(TDVP), we introduce another way of training anANN.As such, the scattering operators U ( t ) , . . . , U n ( t )are consecutively used during n iterations to trainthe network. As the U k ( t ) are still local operatorswith a finite support in real space, we perform foreach iteration k a supervised learning procedure tofind the set of complex network parameters ˜ W ( k ) that minimize the Fubini-Study metrics, given by L [ ˜ W ( k ) ] = acos (cid:0) | (cid:104) ψ ˜ W ( k ) | U k ( t ) | ψ W ( k ) (cid:105) | (cid:1) . whereas | ψ W (cid:105) = (cid:80) { (cid:126)s } exp[ H ANN ( W , (cid:126)s )] | (cid:126)s (cid:105) refers to a quantumstate defined by the output of an ANN and { (cid:126)s } denotesthe set of all spin configurations (cid:126)s = ( s , s , . . . ) , s i = ± H ANN ( W , (cid:126)s ) can beconsidered as a deep extension of a complex-valuedRBM with up to three hidden layers, see the appendixfor details. After convergence, the ”learned” solution˜ W ( k ) is passed to the next iteration as W ( k +1) . Tocomplete the learning procedure, we write L [ ˜ W ( k ) ] at the k -th iteration, while omitting the index, as L [ ˜ W ( k ) ] = (cid:80) { (cid:126)s } | ψ ( (cid:126)s ) | ˜ ψ ∗ ( (cid:126)s )[ ψ ∗ ( (cid:126)s )] − U loc ( (cid:126)s, t ) , U loc ( (cid:126)s, t ) =[ (cid:104) (cid:126)s | ψ W (cid:105) ] − (cid:104) (cid:126)s | U ( t ) | ψ W (cid:105) , with ψ ( (cid:126)s ) = (cid:104) (cid:126)s | ψ W (cid:105) and U loc ( (cid:126)s, t ) being the equivalent of the local energy knownfrom TDVP methods. We access the above sum with aMarkov chain Monte-Carlo (MCMC) algorithm which,as we can confirm, is sign-problem free in all ourcomputations. In the same way, we calculate the gradient ∂L/∂ ˜ W i with the backpropagation algorithm and passthe result to a stochastic gradient-descent optimizerreferred to as PADAM [38]. Benchmarking.
In order to quantify the overall accuracyof our approach we first benchmark the RG-componentand the machine learning part individually. For the formertask, we calculate Eq. (2) for small system sizes exactlyusing a matrix representation of the quantum circuit. Fig.
Figure 2. (a) Comparing the dynamics of the transversemagnetization to exact diagonalization, averaged over latticesites and 250 disorder realizations, with and without treatmentof many-body resonances. Here, L = 12 , h = { , , / } J fromtop to bottom, and J ( x ) = J/
8. (b) Lower bound F ∗ on themany-body overlap of the trained ANN-state with the stategiven by an hypothetical, exact application of the quantumcircuit. System parameters are the same as in Fig. 1b in theinteracting case. Different sizes of ANNs are compared, where M refers to the number of hidden units in each layer. Shadedareas indicate uncertainties due to a finite disorder ensembleof 25 realizations. The result from a cumulant expansion ofthe quantum circuit is shown for comparison. (c) The sameoverlap on a 12 ×
12 lattice for two different external fieldstrengths. L = 12 spins. The plot reveals that the accuracy of thedynamics depends crucially on the inclusion of many-bodyresonances, which is tuned by the only free RG-parameter (cid:15) , see above. For practical purposes, we set (cid:15) indirectly byimposing a maximum total number n of couplings withinall RG-generators S k . Here, n = 3(10) L corresponds tothe label of excluded (included) many-body resonancesand matches (exceeds) the number of original couplings.Already for n = 10 L we observe a very good agreementeven for the longest times. Importantly, the result can besystematically improved by increasing n .Next, let us benchmark the training of the ANN. Forthis purpose we ideally would like to check the overlap F = | (cid:104) ψ W ( n ) ( t ) | ψ QC ( t ) (cid:105) | of the final ANN-state to theone obtained from an exact application of the quantumcircuit, which is impossible for large system sizes. Nev-ertheless, we can offer a lower bound F ∗ = (cid:81) k F ∗ k < F where F ∗ k = | (cid:104) ψ W ( k +1) ( t ) | U ( k ) ( t ) | ψ W ( k ) ( t ) (cid:105) | denotes thepartial overlaps measured at the end of each iteration k ,which are a by-product of the training procedure. We plot F ∗ in Fig. 2 as a function of time for 1D- and 2D-lattices.It shows a high, macroscopic overlap even for large sys- Figure 3. Quench dynamics for the transverse magnetizationand large system sizes averaged over lattice sites and 25 dis-order realizations. (a) Quench into the MBL-SG phase at h = J/ , J ( x ) = J/ × tem sizes and a systematic improvement on adding moreunits and hidden layers to the ANN. From this findingwe conclude that the quantum circuit can be appliedessentially numerically exactly on the ANN. For compari-son, we also plot the result of a perturbative treatment | ψ QC ( t ) (cid:105) ≈ e − iH ( n )0 t (cid:80) (cid:126)s (cid:81) k exp ( (cid:104) (cid:126)s | S k − S k ( t ) | ψ (cid:105) ) | (cid:126)s (cid:105) ,i.e. a cumulant expansion of the quantum circuit, thatneglects (higher-order) commutators between different S k .It shows a rapid decay and thus confirms the circuit’snon-perturbative nature. In the appendix we show furtherbenchmarks of the whole framework for a large integrablesystem. Numerics.
As an application of our framework wenow explore non-equilibrium dynamics involving globalquenches that has been difficult to access so far in thelarge system size and long-time limit. It is known fromprevious RG-studies that a symmetry-broken state willkeep a non-zero Edwards-Anderson order parameter in thelong-time limit starting from symmetry-broken states ifthe system is in the MBL-spin glass (MBL-SG) phase [32].Here, we aim to address the build-up of spatiotemporalorder starting from a Z -symmetric state upon quenchinginto the MBL-SG phase. We detect the spatiotemporaldynamics of the MBL-SG order via [39], χ ij ( t ) = (cid:88) ν =1 p ( ν ) ij ( t ) (cid:68) (cid:37) ( ν ) ij ( t ) (cid:12)(cid:12)(cid:12) σ zi σ zj (cid:12)(cid:12)(cid:12) (cid:37) ( ν ) ij ( t ) (cid:69) , (4)where (cid:37) ij denotes the reduced density matrix of two latticesites i, j , while ν enumerates its four eigenvectors | (cid:37) ij (cid:105) ( ν ) and eigenvalues (probabilities) p ( ν ) ij . Fixing a distance | i − j | we average χ ij ( t ) across all associated pairs and disorder realizations. This quantity can be interpreted asa local version of the Edwards-Anderson order parameter,which is otherwise mostly used to detect MBL-SG orderin a static context, but which doesn’t exhibit a naturalextension to the dynamical regime considered here.Figure 1b shows χ d ( t ) both for an interacting MBL( J ( x ) = h/
5) and a non-interacting Anderson localized( J ( x ) = 0) case for a 1D chain of 64 spins. At shorttimes tJ ˜ ≤ J/J ( x ) = 25, an almost identical light-cone forthe buildup of MBL-SG correlations is visible, which ap-pears consistent with a logarithmic growth. On longertime scales we observe a fundamental difference betweenthe Anderson and MBL cases. For the non-interactingAnderson-localized limit the growth of MBL-SG orderstops, while for J ( x ) > ∼ /J ( x ) . Interestingly,we find that all light-cones do not become more openas we quench deeper into the MBL-SG phase but ratherthe more close we quench to the critical point. Thisbehavior is reminiscent of the l -bit picture, where LI-OMs become more extended on approaching criticality.We will draw a connection to this picture below. As ex-pected, a quench within the MBL-PM phase does notshow any SG-order. Right at criticality, J = h , evenwithout interaction, we find that the order becomes gen-uinely long-range as it decays algebraically with distancewithin the light-cone. For the interacting case, inside theSG-phase, we observe an exponential decay with distance,but having an essential difference to the non-interactingcase: the order at any fixed distance does not saturate,but increases strictly monotonically for all observed timeswithin the light-cones. This is a drastic non-perturbativeeffect of the interacting model. It is particularly obviousfor next-neighboring spins, see Fig.1b). The importantquestion whether this growing will eventually lead to afinite plateau for | i − j | → ∞ requires access to even muchlater times, which we currently cannot access.When initializing the system in a symmetry-brokenstate, as studied in previous works, the stability of MBL-SG order originates from the large overlap with the LI-OMs. The mechanism for the build-up of long-rangeorder from symmetric states as targeted in this work isof fundamentally different origin, as the initial state isoriented orthogonal to the LIOMs. Here, it is essentialto generate long-distance quantum correlations betweenLIOMs. This is not possible in the Anderson localizedlimit because the LIOMs are independent, as we alsosee from our results in Fig. 1. Only in the interactingMBL limit the MBL-SG order can develop. Quantumcorrelations between two lattice sites i and j can emergeon a time scale [ J ( x ) ] − e | i − j | /ξ where ξ denotes a typicallocalization length. Consequently, at a given time t MBL-SG order can be generated over distances d ∼ ξ log[ J ( x ) t ]explaining the appearance of the logarithmic light-conein Fig. 1.As a closing point, we now turn briefly to quantummany-body dynamics in two dimensions. Whether anonergodic phase due to strong disorder exists there hasremained an outstanding challenge [40]. Its difficultyoriginates from the percolation of many-body resonances[41, 42]. We find that at least for sufficiently small or largeexternal fields, the latter can be effectively captured usingour framework up to an unprecedented long timescale. Fig.3b shows the temporal evolution of the local magnetizationin a quadratic, rectangular lattice, using essentially thesame quench protocol as above. In contrast to the glassydynamics of a chain, see Fig. 3a, the lattice exhibits arapid decay of magnetization at h (cid:28) J , consistent withthermalization. On the other hand, for h (cid:29) J a stablenon-thermal plateau is reached. Our result thereforenumerically confirms a presumed quasi-localization [41,42] in the disordered 2D transverse-field Ising model atinfinite temperature. Conclusion.
We have demonstrated how many-bodyquantum dynamics can be simulated for generic spin-1/2systems up to exponentially long times given that suffi-ciently strong disorder breaks ergodicity at least up tothe targeted timescale. Importantly, this includes an un-biased treatment of many-body resonances, which allowedus to obtain quantitative results in general and to gobeyond one-dimensional systems. We could show that ourproposed framework does not fundamentally rely on anyspecific details of the model and scales up to systems sizesfar beyond of what is possible with exact diagonalization.This opens up for broad investigations e.g. of non-thermalbehavior and quantum aging dynamics in higher dimen-sions [43, 44], long-range interacting systems [45–47] orlocalization in lattice gauge theories [48]. Since this workhas shown that deep ANNs are able to apply the proposedquantum circuit numerically exact, the ansatz could alsobe well suited for random unitary circuit models e.g. tostudy operator spreading [49] [50] [51] or measurementinduced localization transitions [52] [53].
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Comparison to exact solution at large scale
For the one-dimensional and non-interacting case of J ( x ) = 0, χ | i − j | ( t ) is exactly solvable by means of conven-tional free-fermion techniques [54, 55] after performing aJordan-Wigner transformation [56]. We take this exactsolution to compare our numerical result obtained fromthe prescribed framework at large system sizes, see Fig.4. It shows an excellent agreement at the full range of timescales. Figure 4. Comparison of spin-glass order computed with RGand ANN (dashed) to the exact solution (solid) for the non-interacting Ising chain for L = 64 , J = 5 , h = 1, at variousdistances d . Structure of the artificial neural network
We use a complex-valued feed forward network tak-ing a spin-configuration (cid:126)s as input layer and returningthe activation of a single output unit as H ANN ( W , (cid:126)s ) =log[ ψ W ( (cid:126)s )]. In between those, there are one or morehidden layers, each passing the previous, weighted andbiased activations W ( ν ) (cid:126)v ( ν − + (cid:126)b ( ν ) through a non-linear activation function f ( z ) to the next layer, v ( ν ) j = f (cid:16)(cid:80) i W ( ν ) ij v ( ν − i + b ( ν ) j (cid:17) . Building upon the originalansatz in terms of an RBM [22], we take the complexlog cosh( z ) as a natural choice for the activation functionin all hidden layers. In the special case of a single hiddenlayer, both formulations are in fact equivalent, | ψ W (cid:105) = (cid:88) { (cid:126)s,(cid:126)h } e (cid:126)a · (cid:126)s + (cid:126)b · (cid:126)h + (cid:126)s · W · (cid:126)h | (cid:126)s (cid:105) , (cid:126)h = ( h , h , . . . ) , h i = ± (cid:88) { (cid:126)s } e (cid:126)a · (cid:126)s + (cid:80) j log cosh[ θ j ( (cid:126)s )] | (cid:126)s (cid:105) , θ j ( (cid:126)s ) = (cid:88) i W ij s i + b j , (5)which justifies our choice, although we note that a for-mal way of deriving an optimal f ( z ) does not exist inmachine learning. In the above definition, W = ( (cid:126)a,(cid:126)b, W )summarizes all network parameters.Unfortunately, however, by using f ( z ) = log cosh( z )we frequently observe the occurrence of numerical in-stabilities during training, caused by two poles locatedat ± iπ/
2. These instabilities are triggered whenever z comes close to those poles. This manifests itself in suddenjumps of L [ ˜ W ( k ) ], which can ultimately make convergenceimpossible. To fix this problem, we use an approxima-tion ˜ f ( z ) ≈ log cosh( z ) that ”smooths” the poles whilepreserving the asymptotic behavior:log cosh( z ) = − log(2) + z + log (cid:0) e − z (cid:1) (6) ≈ − log(2) + z + P ( z ) Q ( z ) =: ˜ g ( z ) . (7)Here, a Pad´e-(2 ,
4) expansion is done in the second line.The four poles of ˜ g ( z ) are all located within Re( z ) < f ( z ) = (cid:40) ˜ g ( z ) , Re( z ) ≥ g ( − z ) , Re( z ) < f ( z ) an even function likelog cosh( z ). If ˜ f ( z ) is used, no more instabilities occur. Continuous renormalization
As already mentioned in the main text, a SWT witha generator S fulfilling [ H , S ] = V allows to separate˜ H up to second order. We refer to [12] for a formalway to obtain S . Here we recall that H represents asingle coupling of arbitrary type, e.g. σ xi or σ zi σ yj σ yk σ zl .Although the SWT can be extended to any order, there isa subtlety which far more limits the overall accuracy thanits order which is the following. We refer to couplings H (cid:48) which commute with H but still produce new, non-commuting couplings V (cid:48) under the SWT of any order:[ H (cid:48) , H ] = 0 , [ e S H (cid:48) e S † , H ] = [ ˜ H (cid:48) + V (cid:48) , H ] (cid:54) = 0 . (9)These new couplings V (cid:48) need to be damped in subse-quent SWTs before ˜ H is removed, unless they can beconsidered as irrelevant for specific models [32]. Sincewe aim for a general framework and quantitative dynam-ics, our strategy is to keep all emerging couplings whileperforming a continuous unitary transformation (CUT),whereupon ˜ H is removed. To formalize this procedure,we define a continuous scale λ ∈ [0 , ∞ ), where for λ → ∞ ,˜ H = H ( λ → ∞ ) commutes with all other couplings,very much like in the flow equation method [33]. Thereby,like in first order SWT, we require the generator to satisfy[ H ( λ ) , S ( λ )] = (cid:88) | Vi ( λ ) || H λ ) | >(cid:15) V i ( λ ) , (10)but only w.r.t. those non-commuting couplings V i ( λ ),whose relative magnitude lies above a threshold (cid:15) (cid:28) ddλ H ( λ ) = [ S ( λ ) , H ( λ )] ,H ( λ ) = H ( λ ) + H (cid:48) ( λ ) + V ( λ ) , (11) which, under the condition of Eq. (10), converges to H ( λ → ∞ ) = ˜ H + ˜ H (cid:48) + (cid:88) | ˜ V i | / | ˜ H | <(cid:15) ˜ V i . (12)By tuning the threshold (cid:15) , the number of new couplingsemerging during the CUT can be controlled without thetechnical need to restrict their type, i.e. the associatedPauli-string, by any means. In the limit of (cid:15) →
0, theseparation of ˜ H becomes exact.From the CUT we numerically construct a finite se-quence of SW-generators ( S , S , . . . ). The chained se-quences of all RG-steps form the total sequence of { S k } referred to in the main text. Its length can be furtheroptimized by merging commuting consecutive elements. Local rotations of RG-generators
Our definition of time-dependent unitaries U k ( t ) = e ˜ S † k ( t ) e ˜ S k requires RG-generators S k to be successivelyrotated into the frame of all previous ones,˜ S k = e S † · · · e S † k − S k e S k − · · · e S , (13)as shown in the main text. Like the Hamiltonian, we repre-sent all S k as sums of Pauli-strings but with an imaginary coefficient each, to ensure S † k = − S k . In the following, werefer to each element of these sums as coupling. While ingeneral Eq. (13) generates exponentially many couplings,we empirically find that those being smaller than thethreshold (cid:15) can be neglected after each rotation. Such arepetitive cropping does not alter expectation values oflocal observables at arbitrary times, up to a sub-leadingcorrection, see below. We attribute this observation to arelatively low overlap among respective S l . However, westrongly emphasize that this would not hold if the genera-tors were replaced by U l ( t ), i.e. the quantum circuit itself.For intermediate to long times, all U l ( t ) have an extendedspatial support implying a high overlap among them, asillustrated in Fig. 1a. This quickly leads to an explosionof non-negligible couplings during successive rotations aswe confirm in numerical experiments and by the rapiddecay of a cumulant expansion approach as shown in Fig.2b. Instead, these higher-order and long-distant couplingscan be numerically exactly captured using a deep ANN,which is a central result of the present work.As a sub-leading correction we impose the Frobeniusnorm of an original generator S l to the associated croppedgenerator S (cid:48) l ,¯ S (cid:48) l = || S l || F || S (cid:48) l || F S (cid:48) l , S (cid:48) l = (cid:88) | S ( i ) l | >(cid:15) S ( i ) l , (14)where S ( i ) l denotes a coupling within S ll