Simulation methods for open quantum many-body systems

Hendrik Weimer∗ Institut f¨urTheoretische Physik, Leibniz Universit¨atHannover, Appelstraße 2, 30167 Hannover, Germany Augustine Kshetrimayum Dahlem Center for Complex Quantum Systems, Physics Department, Freie Universit¨atBerlin, 14195 Berlin, Germany Rom´anOr´us Donostia International Physics Center, Paseo Manuel de Lardizabal 4, E-20018 San Sebasti´an, Spain Ikerbasque Foundation for Science, Maria Diaz de Haro 3, E-48013 Bilbao, Spain

Coupling a quantum many-body system to an external environment dramatically changes its dynamics and offers novel possibilities not found in closed systems. Of special interest are the properties of the steady state of such open quantum many-body systems, as well as the relaxation dynamics towards the steady state. However, new com- putational tools are required to simulate open quantum many-body systems, as methods developed for closed systems cannot be readily applied. We review several approaches to simulate open many-body systems and point out the advances made in recent years towards the simulation of large system sizes.

CONTENTS Acknowledgments 20

I. Introduction 1 References 20 A. The open quantum many-body problem 1 B. The Markovian quantum master equation 2 C. Steady state solution versus time evolution 3 I. INTRODUCTION D. Differences to equilibrium problems 3 E. Paradigmatic models 4 A. The open quantum many-body problem II. Stochastic methods 4 Open quantum many-body systems have witnessed a III. Tensor network methods 6 surge of interest in recent years, chiefly for two reasons. A. One spatial dimension 6 B. Extensions to higher dimensions 8 On the one hand, these systems offer the exciting possi- bility to use controlled dissipation channels to engineer IV. Variational methods 12 interesting quantum many-body states as the stationary

arXiv:1907.07079v2 [quant-ph] 26 Mar 2020 A. The variational principle for open quantum systems 12 B. Comparison with mean-field methods 13 state of their dynamics (Diehl et al., 2008; Verstraete C. Variational tensor network methods 15 et al., 2009; Weimer et al., 2010). On the other hand, D. Variational quantum Monte-Carlo methods 16 open quantum many-body systems are attractive from a fundamental perspective, as their dynamics exhibits a V. Phase space and related methods 16 A. Truncated Wigner approximation 17 wide range of features not found in equilibrium systems. B. BBGKY hierarchy equations 17 As in the case of closed quantum systems, the complex- ity of the problem scales exponentially with the size of VI. Linked cluster expansion methods 18 the system, requiring the use of sophisticated simulation VII. Summary and outlook 19 methods to obtain useful results. Interestingly, open quantum many-body systems are even harder to simulate on classical computers than closed systems, while at the same time the station- ∗ [email protected] ary state of an open quantum system is much eas- 2 ier to experimentally prepare than the ground state of form a closed system. These properties make open quan- d tum systems one of the prime candidates to show a ρ = [ρ] quantum advantage of quantum simulators over clas- dt L 1 1 sical methods within noisy intermediate-scale quantum = i [H, ρ] + L ρL† L† L ρ ρL† L , − µ µ − 2 µ µ − 2 µ µ devices (Preskill, 2018). However, this requires a thor- µ X   ough assessment of the capabilities of classical simulation (2) methods, which we will provide in this review. In our review, we first provide a general introduction where H is the Hamiltonian of the system and L ,L† { µ µ} to open quantum many-body systems, laying particular the Lindblad operators responsible for the incoherent dy- emphasis on the key differences compared to simulating namics arising from the coupling to an external envi- closed quantum systems and on the paradigmatic models ronment, which are also known as the jump operators that have emerged to benchmark simulation methods for (Gorini et al., 1976; Lindblad, 1976). open systems. In the main part, we first review stochas- The validity of the Lindblad master equation Eq. (2) tic methods commonly known as wave-function Monte- for a concrete physical system crucially depends on the Carlo techniques, which are based on a numerical exact separation of several timescales. Considering a system treatment of the total Hilbert space of the problem. We of interest coupled to a larger environment, one first as- then turn to tensor network simulation techniques aim- sumes a weak coupling between system and environment, ing to describe the “physical corner” of the Hilbert space, such that the entanglement between system and environ- i.e., the quantum states that are most relevant to de- ment remains low. Furthermore, the environment must scribe the dynamical evolution and steady states of open not retain any memory of the system degrees of free- quantum many-body systems. Subsequently, we review dom. The approximations related to these conditions are variational methods that employ very similar strategies, commonly refered to as the Born-Markov approximation including variational methods that are based on a tensor (Breuer and Petruccione, 2002) and require that the cor- network description. We also cover phase space methods relation time of the environment τE is much smaller than and closely related counterparts. Finally, we have added the relaxation time of the system τR. Finally, the differ- a section on linked cluster expansion. Within our review, ences in eigenfrequencies in the system ωs has to be large 1 we will not cover methods derived from a field-theoretical compared to the inverse relaxation time τR− . description of open quantum systems within the Keldysh These approximations are well justified in quantum op- formalism, as this has already been extensively covered in tical systems, in particular atoms coupled to electroni- a previous review article (Sieberer et al., 2016). We will cally excited states (Barreiro et al., 2011; Baumann et al., also not cover integrable models (Foss-Feig et al., 2017; 2010; Krauter et al., 2011; Malossi et al., 2014; Raitzsch Guo and Poletti, 2018; Medvedyeva et al., 2016; Prosen, et al., 2009). There, the optical frequencies of the tran- 2011a,b, 2008) for which analytical techniques such as sition leave a large time scale to observe the complete the Bethe ansatz can be employed. relaxation to its equilibrium state. Additionally, the re- laxation of the electronic excitation into the vacuum of the radiation field as the correlation time of the radia- tion field is related to the photon frequency (Breuer and B. The Markovian quantum master equation Petruccione, 2002), which again is much larger than the 1 relaxation rate τR− . Artificial atomic systems such as the The state of an open system is described by its den- nitrogen-vacancy center in diamond (Dutt et al., 2007; sity operator ρ, which can be described as a statistical Jelezko et al., 2004; Robledo et al., 2011) offer similar ensembles of pure states, benefits. Another advantage of quantum optical systems for ρ = p ψ ψ , (1) studying open quantum many-body systems is the pos- i| iih i| i sibility to drive them with time-dependent laser fields. X Importantly, if all the jump operators in the master equa- where pi denotes the probability to find the system in the tion describe transitions between the eigenstates of the state ψi . Note that the decomposition into pure states Hamiltonian, the resulting steady state of the system is is not| unique.i In our review, we will limit ourselves to the guaranteed to be a thermal state (Breuer and Petruc- discussion of Markovian systems, i.e., dynamical systems cione, 2002). However, if an oscillatory driving term is in which the generator of the dynamics [ρ] (commonly added to the system Hamiltonian, it is possible to observe called the Liouvillian) depends only onL the state at the non-equilibrium steady states in the rotating frame of the present time t and not on the state at earlier times. Such driving. Optically excited atoms can also exhibit strong Markovian systems form a dynamical semigroup and can interactions when excited to Rydberg states (Saffman be described by a quantum master equation in Lindblad et al., 2010), which can be used to realize a rich variety 3 of driven-dissipative quantum many-body systems (Ates efficient to compute the full time evolution of the sys- et al., 2012; Carr and Saffman, 2013; Glaetzle et al., 2012; tem. This is comparable to imaginary time evolution Lee et al., 2011; Lemeshko and Weimer, 2013; Rao and algorithms to find the ground state of a closed many- Mølmer, 2013). body system. In our review, we will contrast the two There are also interesting solid state platforms to study approaches and address this distinction when discussing strong interaction and dissipation. One example is that individual simulation methods in the main part of our of semiconductor polaritonic systems see, e.g., (Caru- review. sotto and Ciuti, 2013)), where semiconductor microstruc- Investigating the full time evolution also offers the pos- tures are used to embed quantum wells or quantum dots, sibility to investigate interesting many-body effects dur- becoming a photonic resonator where strong interactions ing the relaxation dynamics. For instance, it is possible can be induced. Another example is that of circuit-QED for open many-body systems exhibiting a quite trivial systems (Fitzpatrick et al., 2017; Ma et al., 2019), where steady state, while the relaxation behavior is dominated superconducting circuits can be used to construct Bose- by complex glassy quantum dynamics (Olmos et al., Hubbard lattices of microwave photons, and where dissi- 2012). pation can be engineered, so that one can have a tailored reservoir. It is important to remark that the Lindblad opera- tors are usually considered to be local, but this approx- imation holds only in the weak-coupling limit. To be more precise, a Markovian master equation with quasi- D. Differences to equilibrium problems local Lindblad operators holds as long as the coupling between the system and the environment is weak, which To find the steady state of an open quantum many- in practice amounts to (1) a slow development of corre- body system, it might first be tempting to take well es- lations between system and environment, (2) fast decay tablished methods for ground state calculations for closed of excitations of the environment, and (3) neglect of fast- systems and try to adapt it to the open case. Unfortu- oscillating terms when compared to the typical system nately, this approach fails in many cases. For example, timescale. One should be careful, however, since when quantum Monte-Carlo methods that are highly successful dealing with strongly correlated systems, strong interac- for ground state calculations, require to rewrite the par- tions within the system of interest may lead to a break- tition function of the quantum system to a correspond- down of the local Lindblad dissipation (Beaudoin et al., ing classical system. However, for the steady state of 2011; Wichterich et al., 2007). In these cases, it may an open system it is unclear a priori (and often incor- be necessary to consider additional steps to derive the rect (Sieberer et al., 2013)) whether the steady state of correct Lindblad operators (Reiter and Sørensen, 2012). the system is a thermal state that can be described in For the purpose of our review, we assume that the correct terms of a partition function. The same argument holds Lindblad form has already been derived. for density functional theory approaches trying to mini- mize the ground state energy; usually, the steady state of an open system is completely different from the ground C. Steady state solution versus time evolution state of the Hamiltonian. This is even true in the limit of infinitely weak dissipation, as the strength of the dissipa- Typically, there are two different aspects that are of in- tion will predominantly control the relaxation rate rather terest when studying open quantum many-body systems. than the properties of the steady state. First, one wishes to understand the properties of one or Some methods from the study of closed quantum sys- several steady states that the system reaches in the long tems out of equilibrium can be adopted to open systems; time limit. This is similar to understanding the ground we will discuss these cases in detail. In general, the sim- state properties of a closed many-body system. Second, ulation of an open quantum system is computationally one is interested in the dynamical evolution of the system much harder than for a closed system due to the statis- towards the steady state. The latter is particularly in- tical nature of the state. teresting when the system exhibits several steady states that can be reached depending on the initial condition of Additionally, one can benefit to some extent from the the system. vast body of works commited to the study of classical While the requirements for the appearance of a unique non-equilibrium dynamics. For example, the importance steady state are well understood for finite systems of the symmetries of the open quantum many-body dy- (Spohn, 1976), many-body systems add the additional namics is equally important as in the classical case (Ho- complication that the long time limit and the thermody- henberg and Halperin, 1977) and allows for the classifi- namic limit do not necessarily commute. In some cases, cation of dissipative phase transitions in terms of their even when chiefly interested in the steady state, it is more universality classes. 4

E. Paradigmatic models is given by

U 2 Within the analysis of ground state many-body prob- H = J b†bj + n ∆ωni + F bi + b† . − i 2 i − i lems, there is a number of particular models that have i,j i   hXi X   found especially wide interest and are often used as a first (4) example to benchmark a numerical method. These mod- In this model, J describes the hopping of bosons between els include, e.g., the Ising model in a transverse field, sites, while the on-site interaction U involves the square the Heisenberg model, and the Hubbard model (both of the density operator ni = bi†bi. Furthermore, ∆ω is bosonic and fermionic). A similar observation can be the chemical potential for the bosons, and F describes made about open quantum many-body problems, where the aforementioned coherent driving. Finally, the quan- these paradigmatic models are often derived from the cor- tum jump operators capturing the loss of a single bo- responding ground state counterparts, i.e., the Hamilto- son are given by ci = √γbi. While the dissipation term nian dynamics is the same. However, adding dissipation also breaks the U(1) symmetry of the conventional Bose- to a closed many-body model can be done in different Hubbard model, here, the symmetry is already broken ways and can lead to drastically different results. In the on the level of the Hamiltonian by the inclusion of the following, we present and briefly discuss the two most driving term F . promiment dissipative many-body models; we will pro- As with the dissipative Ising model, the driven- vide a more detailed discussion in later sections when re- dissipative Bose-Hubbard model has a very intriguing ferring to particular numerical strategies to tackle them. mean-field phase diagram, where several islands of mul- tistability occur in a way that is somewhat reminiscent One of the most widely studied open many-body mod- of Mott lobes (Le Boit´e et al., 2013), see Fig. 1. The els in recent years is the transverse field Ising model with stability of the mean-field solutions has been evaluated longitudinal dissipation (Lee et al., 2011). Its Hamilto- by considering density matrices of the form nian is of the form of the conventional Ising model, given in terms of Pauli matrices σα by MF ρ = (ρi + δρi), (5) i Y h V H = σ(i) + σ(i)σ(j), (3) with ρMF being the mean-field solution for the steady 2 x 4 z z i i ij state. Expanding the quantum master equation up to X Xh i first order in δρi allows to evaluate the stability by check- where h is the strength of the transverse field and V ing whether none of eigenvalues of the Liouvillian has a accounts for the Ising interaction. The dissipation is positive real part. incorporated in terms of jump operators of the form c = γσ , with γ being the rate of dissipative flips from i √ II. STOCHASTIC METHODS the spin up− to the spin down state. An important aspect is that the dissipation breaks the Z Ising symmetry of 2 Upon first glance, the computational complexity of an the Hamiltonian, i.e., the quantum master equation does open quantum system in terms of the Hilbert space di- not exhibit such a symmetry. The model is also relevant mension d appears to be at least O(d2), as there are to ongoing experiments in the field of interacting Ryd- O(d2) independent entries in the density matrix ρ. How- berg atoms (Carr et al., 2013; Malossi et al., 2014). ever, the density matrix at an initial time t0 can be Within a mean-field calculation (Lee et al., 2011), the written as a statistical ensemble of pure states, ρ(t0) = model is predicted to support a large range of h values i pi ψi(t0) ψi(t0) . Instead of propagating the entire for which the system exhibits two stable steady states. density| matrix,ih the| key strategy is to propagate the in- P We will discuss in later sections of our review how dif- dividual pure states ψi to the time t and then calculate ferent numerical approaches address the question on the observables according| toi existence of such a bistable thermodynamic phase. Ac- cording to mean-field theory, the bistable region ends in O = Tr Oρ = pi ψi O ψi . (6) h i { } h | | i i a critical point that belongs to the Ising universality class X (Marcuzzi et al., 2014). The probability distribution pi can then be sampled Another important dissipative model is the driven- using standard Monte-Carlo techniques, which is why dissipative Bose-Hubbard model. While there are differ- the approach is often called the wave function Monte- ent ways to generalize the famous Bose-Hubbard model Carlo method. In practice, the most common strategy (Fisher et al., 1989) to the dissipative case, the most com- is to start from an initial pure state ψ and perform | 0i monly studied one involves a dissipative particle loss that M = 1/pi numerical simulations. Since the trajecto- can be countered by a coherent driving term (Carusotto ries ψi are independent from each other, the statisti- and Ciuti, 2013; Le Boit´e et al., 2013). Its Hamiltonian cal error| i associated with the observable will behave as 5

This stochastic Schr¨odingerequation conserves the norm of the state vector and can be solved by standard tech- niques for stochastic differential equations. 5 An alternative strategy to propagate a single trajec- A A’ tory is the quantum jump method (Dalibard et al., 1992; 4 0 Dum et al., 1992; Mølmer et al., 1993; Plenio and Knight, 1 1998). This approach has been recently reviewed exten- sively in (Daley, 2014), so we will only cover the basic ∆ω 3 1 2 strategy. Within the quantum jump method, the dynam-

U/ B B’ ics is split into to parts. First, the state ψi is propagated | i 2 under an effective non-Hermitian Hamiltonian HNH , i H = H c†c . (10) NH − 2 j j 1 j X 1 Once the norm of the state drops below a previously 0 1 2 3 4 drawn random number r, a quantum jump occurs. Which J/∆ω quantum jump occurs is drawn from the probability dis- tribution FIG. 1 Mean-field phase diagram of the driven-dissipative p = ψ c†c ψ , (11) Bose-Hubbard model. The numbers inside the plot represent j N h i| j j| ii the number of stable mean-field solutions. The yellow region with being a normalization factor. While the high exhibits two mean-field solutions, one of which is unstable. N From (Le Boit´e et al., 2013). order integration of HNH is straightforward, a high or- der simulation of the quantum jumps requires a more subtle identification of the time the jump operator needs ∆O 1/√M. The entire computational cost will be to be applied. For instance, the popular QuTiP library ∼ 2 O(Md), which is considerably lower than d already for (Johansson et al., 2012, 2013) uses a logarithmic secant quite modest system sizes. Importantly, the requirement method to numerically solve the equation ψi(t) ψi(t) = to repeat the simulation M times results in the simula- r for the time t. h | i tion time being significantly longer than for a comparable No matter which approach is used to propagate a single closed quantum system. Depending on the observable, trajectory, the computations can be highly parallelized M 1000 is a reasonable choice to get the statistical ≈ since the trajectories are independent from each other error down to a few percent. For spin 1/2 systems, this by construction. Doing so, it is possible to simulate open essentially means that the system sizes that can be stud- many-body spin 1/2 models with up to 20 spins (Raghu- ied in an open system consist of log M 10 particles et al. 2 ≈ nandan , 2018). The relatively small system sizes less than in a closed system. when compared to equilibrium problems demand the de- The central question is now how can a single trajectory velopment of new data analysis techniques, e.g., concern- ψi be propagated such that the ensemble of all trajecto- ing finite size scaling methods. One possiblity is to use |riesi satisfies ρ(t) = p ψ (t) ψ (t) . One possiblity is i i| i ih i | anisotropic system sizes to obtain more data points for a to describe the evolution of the density operator in terms reliable finite size scaling extrapolation. Close to a phase P of a quantum state diffusion approach (Gisin and Perci- transition, the susceptibility χ of a system may be ex- val, 1992; Percival, 1998), in which the incoherent dy- pressed as namics from the Lindblad operators is captured in terms α of a stochastic Schr¨odingerequation, χ = N χ˜(λ), (12) dψ (t) = iH ψ (t) dt + M ψ (t) dW , (7) where N is the number of particles and α is an exponent i − eff | i i j| i i j j associated with the underlying phase transition (Binder X and Wang, 1989). The reduced susceptibilityχ ˜ is only where the dW refer to Wiener increments. The effective j a function of the anisotropy λ of the system and can be Hamiltonian H describes the drift of the state vector eff determined by symmetry considerations as well as nu- in the Hilbert space, merical data (Raghunandan et al., 2018). H = H + 2 c† c c†c c† c . (8) The wave-function Monte-Carlo method has been used eff h ji j − j j − h jih ji j to analyze the one-dimensional dissipative Ising model of X Eq. (3) (Ates et al., 2012; Hu et al., 2013). While these The diffusion operators M describe the random fluctu- j works have not found a bistable phase as predicted by ations arising from each associated jump operator c , j mean-field theory, a significant increase in the spin cor- M = c c . (9) relations has been reported in the same region (Hu et al., j j − h ji 6

2013). Additionally, finite size scaling of a similar two- (a) (b) ⇢ dimensional model believed to lie in the same universality | i class as the dissipative Ising model has found evidence for Ak Ak⇤ a first order transition (Raghunandan et al., 2018). Ak

III. TENSOR NETWORK METHODS FIG. 2 (a) Writing a wave function |ψi as an MPS for 6 sites. Each site has a physical dimension d. (b) A density A. One spatial dimension matrix ρ can be written as an MPDO, an extension of the MPS formalism. Such a construction automatically ensures We would, first, like to describe the important nu- positivity of the density matrix. merical techniques that has been developed for studying open quantum many-body systems using Matrix Prod- uct States (MPS) which is the one-dimensional ansatz Hilbert space. An MPDO ρ of N d-level particles with of the tensor network (TN) family. MPS, by far, is the (D1,D2,...,DN )-dimensional bonds is then defined as most successful and widely used ansatz in comparison to other ansatz of the Tensor network family, thanks to d s1,s10 sN ,sN0 the success of the density matrix renormalization group ρ = (M ...M ) s1, . . . , sN 1 N | i (DMRG) (White, 1992, 1993) and related techniques (Vi- s1,s0 ,...,sN ,s0 =1 1 X N dal, 2004; Vidal, 2003). Not only are its properties very s10 , . . . , sN0 , (14) well understood, contraction of MPS tensors can be done × h | efficiently and exactly unlike the case for its higher di- sk,sk0 2 2 where Mk are Dk Dk+1 matrices that can be de- mensional counterparts (Haferkamp et al., 2018; Schuch composed as × et al., 2007). For these reasons, MPS have been used dk extensively producing extremely accurate results, how- s,s0 s,a s0,a M = A (A )∗. (15) ever, mostly in the context of ground state calculations k k ⊗ k a=1 of many-body systems (Schollw¨ock, 2005). For a detailed X review on MPS and other tensor networks in general, s,k where dk is at most dDkDk+1 and the matrices Ak are we ask the readers to refer to (Biamonte and Bergholm, of size Dk Dk+1. Such a construction of MPDOs au- 2017; Cirac and Verstraete, 2009; Eisert, 2013; Orus, tomatically× ensures the positivity of the reduced density 2014; Orus, 2018; Schollw¨ock, 2011; Verstraete et al., matrix ρ. This is shown in Fig. 2. This MPDO can be 2008). The application to open quantum systems, mean- expressed in terms of a pure state MPS by defining it over while, is more rare and there are only a few known ap- a larger Hilbert space and using the concept of purifica- proaches one can take for such systems. Not only are tion (Nielsen and Chuang, 2000). This can be done by open systems more computationally challenging (since associating an ancilla with a Hilbert space of dimension we need to deal with matrices in place of vectors for the dk with each physical system. One can then choose an pure states), there are also several intrinsic bottlenecks orthonormal basis sk, ak for these physical and ancilla such as the positivity, hermiticity in the numerical op- indices. The corresponding| i MPS for this system can be timization of the density matrix. Nevertheless, many of written as the ideas in the pure state formalism have been success- Ψ = As1,a1 ...AsN ,aN s a , . . . , s a fully applied in the context of open systems using the | i 1 N | 1 1 N N i s ,...,s a ,...,a concept of Matrix Product Operators (MPOs) or Matrix 1X N 1X N Product Density Operators (MPDOs) (Cirac et al., 2017; (16) Pirvu et al., 2010). These appraches have also been used The MPDO ρ can be obtained by tracing over the ancil- las i.e. ρ = Tr ( Ψ Ψ ). This process is illustrated in to study thermodynamic properties of 1d systems. We a | ih | discuss them below. Fig. 3. The original Ak matrices can be recovered from In 2004, (Verstraete et al., 2004) introduced the con- Mk by doing some eigenvalue decomposition. To deter- cept of MPDO which extended the MPS formalism from mine the evolution of a Hamiltonian of a mixed state pure to mixed states. Let us recall that an MPS can be in real and imaginary time, they simply simulated the written in the following form evolution of the purification by updating the Ak matri- ces using an iterative procedure similar to the standard d DMRG in this technique. The purification could then ψ = As1 ...AsN s , . . . , s (13) | i 1 N | 1 N i be used to reconstruct the density operator at any time s ,...,s =1 1 XN and compute the expectation values of the observables. where the A’s are matrices whose dimension is bounded Such a purification scheme can be used for mixed state by some fixed number D (also called the bond di- evolution under dissipation as well as for thermal equi- mension χ) and d is the physical dimension of the librium and can be implemented irrespective of periodic 7

(a) ⇢ ⇢ | i | i] (c) ⇢ =Tr ( ) a | ih |

(b) | ih |

FIG. 4 Choi isomorphism: vectorizing a density matrix writ- ten in terms of an MPO. In TN diagram, it is simply reshaping one of the indices and gluing it with the other thereby giving us an MPS. FIG. 3 (a) Defining an MPS |Ψi over the enlarged Hilbert space using ancilas (in red) (b) Taking the projector of the MPS with ancillas (c) Tracing out the ancillas from the pro- jector to obtain the MPDO ρ. where il ] is an orthonormal basis of Cd2 for site l. Further| assumingi that the Liouvillian superoperator can be decomposed into terms involving at most nearest-L

neighbors i.e. [ρ] = l l,l+1[ρ], one could in principle or open boundary conditions, finite or infinite systems. use the usual TEBDL algorithmL to solve Eq. 2 by start- The main source of errors in this procedure, like most ing from some initialP Matrix Product Operator (MPO) other TN techniques are (i) Trotter error and (ii) trunca- (shown in in the left side of Fig. 4). This was the ba- tion error. Such an approach with ancillas was also ap- sic idea behind the technique in Ref. (Zwolak and Vidal, plied to study the thermodynamic properties of several 2004). One of the first applications of this technique was spin chains in Ref. (Feiguin and White, 2005). Although the study of the driven-dissipative Bose-Hubbard model the MPDOs in Ref. (Verstraete et al., 2004) are positive in the context of optical resonators (Hartmann, 2010). by construction, it was shown in Ref. (las Cuevas et al., More detailed explanation of this vectorization process 2013) that such a MPDO descriptions of mixed states will be explained later when we discuss the case for higher are not exactly equivalent to the one obtained using lo- dimensional systems. Although the technique proved to cal purification schemes. In particular, it was shown that be extremely simple and efficient, the issue of positivity the bond dimension of the locally purified MPS D0 is not still remained at large. In fact, checking the positivity upper bounded by the bond dimension of the MPDO D. of a reduced density matrix is known to be a very hard In fact, the local purification techniques can be much problem in physics (Kliesch et al., 2014). more costly than the MPDO form itself. Thus, the au- Another approach was taken in Ref. (Werner et al., thors concluded that a description of mixed states which 2016) to solve the problem of positivity. In this approach, is both efficient and locally positive semi definite does instead of expressing ρ directly as an MPO, at every stage not exist and that one can only make approximations. of the algorithm, ρ was kept in its locally purified ρ = Around the same time, (Zwolak and Vidal, 2004) pro- XX†, where the purification operator X is decomposed posed another technique to study the mixed state dy- as a variational tensor network. namics in one dimensional lattice systems. Their tech- s1,...,sN [1]s1,r1 [2]s2,r2 [N]sN ,rN nique, which is also based on MPS, used the Time Evolv- [X]r1,...,rN = Am1 Am1,m2 ...AmN 1 − m1,...,mN 1 ing Block Decimation (TEBD) to simulate the real time X − Markovian dynamics given by a master equation with (18) where 1 s d , 1 r K and 1 m D. A[l] nearest-neighbor couplings. At the heart of this algo- ≤ l ≤ ≤ l ≤ ≤ l ≤ rithm, lies the concept of ‘Choi isomorphism’. It is more are rank-four tensors with physical dimension d, bond di- of a mathematical trick and it states that one can rewrite mension D and Kraus dimension K. Then, a technique the coefficients of a matrix as those of a vector. In other similar to the usual TEBD was used to update the ten- words, this is simply turning a bra index into a ket index sors. Such an approach never required to contract the for a density matrix (understanding the coefficients of ρ two TN layers (X and X†) together, thereby ensuring positivity at all times during the evolution. The tech- as those of a vectorized density matrix denoted by ρ ]). And in the language of TN diagrams, it can be regarded| i nique also provided more control of the approximation as reshaping one of the legs and gluing it with the other error with respect to the trace norm. In Ref. (Cui et al., 2015), a very interesting and dif- (Fig. 4). Once vectorized, ρ ] now lives in the n-fold | i ferent approach based on MPO was taken for finding the tensor product of Cd2 and the master equation can be steady states of dissipative 1D systems governed by the written in the vector form. The mixed state will now dρ look like as follows master equation of the Lindbladian form, dt = [ρ], where is the Liouvillian superoperator. In this tech-L L d2 1 d2 1 nique, instead of doing the full real time evolution of − − the Liouvillian, they proposed a variational method that ρ ] = ci1 iN i1 ] iN ]. (17) | i ··· ··· | i ⊗ · · · ⊗ | i i =0 i =0 searches for the null eigenvector of which is, by def- X1 XN L 8 inition, the steady state of the master equation in the matrx ρ0 Lindbladian form. Their results were based on the prin- | i τ e−H ρ0 ciple that if ρs is the steady state of the Lindbladian mas- ρG lim τ | i (20) ter equation satisfying ˆ ρ = 0, then ρ will also be | i ≈ τ e−H ρ0 L| si] | si] →∞ || | i|| the ground state of the non-local Hamiltonian ˆ† ˆ (since L L (ii) Real time evolution of the Liouvillian superopera- it is Hermitian and positive semi-definite) where ρs ] is | i tor starting from ρ the vectorized form of the steady state density matrix. | Gi Then using a variational algorithm, they directly tar- e T ρ ˆ ˆ L G geted the ground state of † to find the steady state of ρS lim T | i (21) L L | i ≈ T eL ρG the Lindbladian master equation for a finite chain. One →∞ || | i|| of the reasons why directly targeting the ground state where ρS is the desired steady state of the Liouvillian of ˆ† ˆ might be advantageous is that unlike imaginary | i L L master equation. Imaginary time evolution in step(i) en- time evolution, where the sequence of states visited by sures that one does not pass through highly entangled the algorithm is unimportant, the simulation of a master transient regime. Step(ii) increases the accuracy of the equation requires us to follow real time evolution. There- stationary state since ρG is the ground state of which fore, if there are errors in the intermediate states visited is a truncated approximation| i of the non local Hamilto-H by the algorithm, it may lead to problems in the con- nian ˆ† ˆ. vergence of our steady state. For example, some of the It isL worthL mentioning that in one spatial dimension, intermediate states may require large bond dimensions many of the above techniques and their combinations of the MPO although it is known that the final steady have been used for studying not only other important state can be well-represented by an MPO of small bond disspative models (Carollo et al., 2019; H¨oning et al., dimensions (Bonnes et al., 2014; Cai and Barthel, 2013). 2012; Mascarenhas et al., 2015; Piˇzorn,2013), including Also, one doesn’t need to worry about the large entan- the dissipative Ising model of Eq. (3) (H¨oning et al., 2013; glement growth of real time evolution. A very similar Mendoza-Arenas et al., 2016), but also in dissipative approach was taken in Ref. (Mascarenhas et al., 2015) preparation of topologically ordered materials (Iemini where the algorithm instead of doing a time-evolution, et al., 2016) as well as in the energy transport (Guo et al., searched for the null eigenvalue of the Liouvillian super- 2015). Very recently, MPO based techniques have been operator by sweeping along the system. Their method L applied to study vibronic states which extends the ap- claimed to work even in the weakly dissipative regime plication to quantum biology and organic photo voltaics by slowly tuning the dissipation rates along the sweeps. (Somoza et al., 2019). and also to study the dynam- However, it needs to be noted that such techniques, while ics of photonic circuits with time delays and quantum advantageous numerically, cannot be used for obtaining feedback (Pichler and Zoller, 2016). We do not discuss the transient states. the later two works due to the non-Markovian nature of In another paper (Gangat et al., 2017), this idea was the problem which is beyond the scope of this review. applied to infinite 1D systems (i.e. the thermodynamic Similar MPS based techniques that go beyond the Lind- limit) using a hybrid technique of both imaginary and blad master equation (Xu et al., 2019) or the markovian real time evolution. They took a local auxiliary Hamil- approximation (Guo et al., 2018) are also not discussed tonian whose ground state is a good approximation H here. to the ground state of the nonlocal Hamiltonian ˆ† ˆ by taking its kth root as L L B. Extensions to higher dimensions 1/k = ( ˆ† ˆ ) (19) H LrLr r Unlike the case for 1d, the generalization of MPS in X∈Z higher dimensions, also known as Projected Entangled where ˆ = ˆ since ˆ is a translationally invari- Pair States (PEPS) or Tensor Product States (TPS) L r Z Lr L ant local operator.∈ The kth root was taken in order to comes with some serious limitations and there are still P yield faster convergence. The idea is that if the gap be- many open problems (Cirac et al., 2019). Not only do ˆ ˆ tween the two lowest eigenvalues of r† r is less than the PEPS algorithm require serious programming effort, L L ˆ ˆ one, then k > 1 will increase the gap since r† r is pos- exact contratcion of PEPS is known to be a mathemati- itive semi definite, thereby achieving fasterL convergenceL cally hard problem (Haferkamp et al., 2018; Schuch et al., to the ground state. The authors then performed a real 2007). To achieve this, one requires additional PEPS time evolution to obtain a more accurate steady state. contraction algorithms (Jordan et al., 2008; Or´us,2012; In quick summary, the main steps of the algorithm is: Or´usand Vidal, 2009) that are nevertheless known to give very accurate results, in particular, for gapped sys- (i) Imaginary time evolution of the auxiliary Hamilto- tems. Even for critical systems with algebraically de- nian starting from some vectorized initial density caying correlations, the PEPS contraction schemes are H 9 known to provide reasonably accurate results with suf- (a) ⇢A ⇢A = pA A A ficiently high bond dimension of the environment (Or´us Diagonalize i | i ih i | and Vidal, 2009). In fact, recently, techniques have been i B X introduced to capture the infinite correlation length of 2D ⇢ Diagonalize ⇢B = pB B B critical systems using iPEPS based on finite correlation i | i ih i | i length scaling (Corboz et al., 2018; Rader and L¨auchli, X (b) ⇢A ⇢B ⇢() 2018). Thus, despite the higher requirement of numerical ⌦ dedications and limitations, PEPS algorithms are becom- Truncate ing state of the art numerical tools for strongly correlated ⇢()= A B , A B ,..., A B two dimensional systems. Recently, PEPS provided the {| i1i| i01i | i2i| i02i | ii| i0i} best variational energy for the 2D Hubbard model (Cor- boz, 2016a), have offered several new insights on paradig- FIG. 5 (a) Steady state density matrices of two systems A and B are first obtained using brute force. They are then matic models and real materials in the lab (Corboz and expressed in their respective diagonal forms. (b) We then Mila, 2014; Kshetrimayum et al., 2019; Liao et al., 2017; merge the two systems and keep only the χ most probable Matsuda et al., 2013). The successes of PEPS so far, how- pair of states. The process is repeated for different χs until ever, is mostly confined to ground state calculations and we get some convergence. Larger systems can be simulated partially to thermal states (Czarnik et al., 2012; Czarnik by merging more systems in step (b). and Dziarmaga, 2015; Czarnik et al., 2016; Dai et al., 2017; Kshetrimayum et al., 2019) using the concept of Projected Entangled Pair Operators (PEPOs) or Ten- sor Product Operators (TPOs), which we will discuss in the density matrix in this corner space can be determined more detail later, and, more recently, to time evolution either by direct numerical integration in time (for small (Czarnik et al., 2019; Hubig and Cirac, 2019; Kshetri- χ) or by using a stochastic wave function Monte Carlo mayum et al., 2019). For the context of open dissipative algorithms for large χ, see Sec. II. One can then increase quantum system, so far there is only one known approach the size of the corner χ until convergence in some ob- using PEPS (Kshetrimayum et al., 2017) and another servables is reached. Larger systems can be simulated by one using a Corner Space Renormalization method (Fi- merging more systems as we discussed in the initial steps. nazzi et al., 2015). We describe them below. We will A simplified summary of the steps involved is shown in also discuss briefly other potential implementation tech- Fig. 5. niques and possible issues while using PEPS formalism The proposed CSR method was used to study the in particular for such open systems. driven-dissipative Bose-Hubbard model in 2D in both pe- The Corner Space Renormalization method (Finazzi riodic and open boundary conditions for system sizes up et al., 2015) solves the master equation in a corner of the to 16 16 lattice sites. The technique has also been Hilbert space through an iterative procedure. It starts by used to× study the critical Heisenberg model (Rota et al., finding the steady state density matrix for small lattice 2017) for system size up to 6 6 lattice sites and more systems (say ρA and ρB for systems A and B respec- recently the critical regime in× the Bose Hubbard model tively). This can be done by a brute force integration of (Rota et al., 2019) for up to 8 8 lattices. The size of the master equation since the system size is very small. the lattice that can be simulated× using this technique de- The steady state density matrices can be diagonalized pends on the entanglement of the steady state. Even if and written as not obvious at first sight, the structure of the density op- erator generated by the Corner Space Renormalization ρA = pA φA φA , i | i ih i | method amounts to that of a Tree Tensor Network (Shi i X (22) et al., 2006). As such, this particular method, even if un- ρB = pB φB φB , i | i ih i | derstood in terms of TNs, is tailored to driven-dissipative i X systems of finite size. For generalizing it to the thermo- where the states φA form an orthonormal basis for dynamic limit or for non-driven non-dissipative systems, | i i A (the Hilbert space corresponding to system A) and one needs to use more general TN techniques. We will HA pi are the corresponding probabilities. Similar nota- discuss one such technique which we developed recently tion follows for system B. The two systems are then below. merged and the χ most probable product states span- In (Kshetrimayum et al., 2017), we make use of the ning the so-called corner space are selected i.e. we only concept of PEPO by vectorizing them. PEPOs are sim- keep the subspace generated by the orthonormal basis ply the operator version of PEPS, in the same way that φA φB , φA φB ,..., φA φB where the prod- an MPO is the operator version of MPS for the 1d case. i1 i01 i2 i02 iχ i0χ uct{| ofi| the probabilitiesi | i| i of the| twoi| systemsi} are arranged in Hence, PEPOs are used to represent mixed states ρ in decreasing order of magnitude. In this way, we only keep 2D, even beyond dissipative systems, e.g., for thermal the χ most probable pair of states. The steady state of states (Czarnik et al., 2012; Czarnik and Dziarmaga, 10

ground states of local Hamiltonians by imaginary-time ⇢ D ⇢ ] D | i evolution, which we detail in Table I. Given the paral-

Ground states Steady states P [i,j] P [i,j] H = hi,ji h L] = hi,ji L] { d 2 −Ht L t d e e ] |e0i |ρsi]

FIG. 6 TN diagram for the PEPO of ρ on a 2d square lattice, he0|H|e0i = e0 ]h|ρsiL]|ρsi] = 0 with bond dimension D and physical dimension d. When vec- Imaginary time Real time torized, it can be understood as a PEPS for |ρi] with physical 2 dimension d TABLE I Ground state calculation in a closed quantum sys- tem (left) and Steady state calculation in an open quantum system (right). The former one requires an imaginary time 2015; Czarnik et al., 2016; Kshetrimayum et al., 2019). evolution while the latter follows a real time evolution. Both the Hamiltonian H and the vectorized Liouvillian L can be As mentioned before, such a construction of density ma- ] decomposed as a sum of local terms. |e0i is the ground state trices using PEPOs does not automatically guarantee the of the many-body Hamiltonian with e0 as its ground state. positivity of the density matrix. However, for simulations |ρsi] is the non-equilibrium steady state of the Liouvillian in targeting the steady states, this lack of exact positivity is their vectorized forms. not a bottleneck if the fixed point is not very highly en- tangled. For the moment, we will restrict our discussion lelism above, it is clear that one can adapt, at least in to this case. Once we have our PEPO, we vectorize it principle, the methods to compute imaginary time evo- i.e. rewrite the coefficients of the PEPO as a PEPS (also lution of a pure state as generated by local Hamiltoni- called Choi’s isomorphism). Once vectorized, the PEPO ans, to compute also the real time evolution of a mixed ρ can be understood as a PEPS of physical dimension d2 state as generated by local Liouvillians. This was, in fact, and bond dimension D (now called ρ ]), as shown also the approach taken in Ref.(Zwolak and Vidal, 2004) for in Fig. 6. The vectorized form of the| i Lindblad master finite-size 1d systems, using Matrix Product Operators equation Eq. (2) can be written as (MPO) to describe the 1d reduced density matrix, and proceeding as in the Time-Evolving Block Decimation d ρ ] = ] ρ ] (23) (TEBD) algorithm for ground states of 1d local Hamil- dt| i L | i tonians (Or´usand Vidal, 2008; Vidal, 2004; Vidal, 2007, where the vectorized Liouvillian operator is given by 2003) as we have discussed previously. In (Kshetrimayum T et al., 2017), we extended this implementation for the ] i H I I H L ≡ − ⊗ − ⊗ case of 2D systems using the concept of PEPO with phys- 1
1 T (24) ical dimension d and bond dimension D, see Fig.6. For + Lµ L∗ L† Lµ I I L∗ L . ⊗ µ − 2 µ ⊗ − 2 ⊗ µ µ µ   the case of an infinite-size 2d system, this setting is ac- X tually equivalent to that of the infinite-PEPS algorithm H is the Hamiltonian of the system and I corresond to (iPEPS) to compute ground states of local Hamiltonians the identity operator. Lµ and Lµ† correspond to the in 2d in the thermodynamic limit. Thus, in principle, on-site Lindblad/jump operators, responsible for dissi- one can use the full machinery of iPEPS to tackle as well pation. The tensor product separates the operator ⊗ the problem of 2d dissipation and steady states. acting on the ket and bra index of ρ before the vectoriza- There seems to be, however, one problem with this tion. When the vectorized Liouvillian superoperator L] idea: unlike in imaginary-time evolution, we are now is independent of time, Eq. 23 can be integrated as dealing with real time. In the master equation, part of ]t ρ(t) ] = eL ρ(0) ], (25) the evolution is generated by a Hamiltonian H, and part | i | i by the Lindblad operators Lµ. The Hamiltonian part cor- where ρ(0) is some vectorized initial density matrix, | i] responds actually to a unitary “Schr¨odinger-like” evolu- written as a PEPS. In the limit of t , we obtain the tion in real time, which typically increases the “operator- non-equilibirum steady state (NESS)→ as ∞ the fixed point entanglement” in ρ ], up to a point where it may be too of the master equation which we denote by ρ . From | i | si] large to handle for a TN representation (e.g., 1d MPO Eq. 23, it is also obvious that ρs ] is the right eigen or 2d PEPO) with a reasonable bond dimension. In 1d vector of with zero eigen value| soi that L] this is the reason why the simulations of master equa- tions are only valid for a finite amount of time. In 2d, ] ρs ] = 0. (26) L | i simple numerical experiments indicate that in a typical For a Liouvillian consisting of local terms say [ρ] = simulation the growth of entanglement is even faster than L L [i,j][ρ], the vectorized form of the Lindblad equa- in 1d. Luckily, this is not a dead-end: if the dissipation i,j L tionh Eq.i (2) yields a parallelism with the calculation of is strong compared to the rate of entanglement growth, P 11

101

0 10 γ = 0.01 γ = 0.05 γ = 0.1 γ = 0.15 γ = 0.2 ) ) 4 4 ρ -1 ( ⇢ 10 ( op S op S

10-2

10-3 0 500 1000 1500 2000 timetime steps hx/

FIG. 7 Operator entanglement entropy Sop for a block of 2×2 unit cell for real time evolution of the master equation for FIG. 8 Our study, based on iPEPO found bistability in the different values of dissipation strength. Stronger dissipation phase diagram of the dissipative Ising model for low bond di- implies lower entanglement growth and faster convergence to mensions D = 1, 2. The bistability is replaced by a first order the NESS. Figure taken from (Kshetrimayum et al., 2017) for transition for higher Ds. Figure taken from (Kshetrimayum the dissipative Ising model. et al., 2017) for the dissipative Ising model.

then the evolution drives the system into the steady state sults of the dissipative Ising model have been reproduced before hitting a large-entanglement region. In fact, even independently using a different update scheme (Czarnik if there is too much entanglement for the TN at inter- et al., 2019) compared to the one used in (Kshetrimayum mediate times, the dissipation may still drive the evolu- et al., 2017). While the technique also employed vector- tion towards a good approximation of the correct steady ization along with iPEPS, the update scheme is based on state. In short, dissipation limits the growth of entangle- maximizing the fidelity between two consecutive steps of ment if the fixed point attractor is strong enough. This the update of the iPEPS tensors. For the case of the can be verified numerically by plotting the operator en- dissipative Heisenberg model (Lee et al., 2013) with the tanglement entropy for different dissipation strengths as Hamiltonian it flows into the NESS. This is shown in Fig. 7. De- i j i j i j tails on how to compute this quantity can be found in H = Jxσxσx + Jyσyσy + Jzσzσz (27) (Kshetrimayum, 2017; Kshetrimayum et al., 2017). i,j hXi
Hence, one can apply the iPEPS machinery to com- and the same Lindblad operators as before, our studies pute the time evolution in 2d with a local Liouvillian found no phenomenon of re-emergence in the phase di- and some initial state. This procedure was used to in-L agram, confirming a prediction by studies using cluster vestigate the dissipative Ising and the XYZ model, con- mean-field approaches (Jin et al., 2016). firming and offering several insights that were inaccesible To the best of our knowledge, we have discussed most before using mean field and other techniques. For exam- of the state-of-the-art numerical techniques based on TN ple, for the dissipative Ising model of Eq. (3), given by for the study of open quantum many-body systems in V [i] [j] hx [i] the Hamiltonian H = 4 i,j σz σz + 2 i σx and both one and two spatial dimensions. We would, now, h i[µ] Lindblad operators Lµ =P√γσ . The phaseP diagram like to discuss some of the possible ideas that could be is controversial with some papers− suggesting the exis- helpful in improving the existing algorithm and possible tence of a bistable steady state (Lee et al., 2011; Marcuzzi new implementation techniques specially in 2d. First of et al., 2014) and others supporting a first order transition all, we remark that the 2d algorithm suggested above (Maghrebi and Gorshkov, 2016; Weimer, 2015a,b). Our does not guarantee the positivity of the density matri- technique found bistability for low bond dimensions of ces. This problem can be solved by starting from an the PEPO (D = 1, 2) which was replaced by a first order initial state that is positive by construction, for example transition for higher Ds, thus confirming that the bista- taking the product of two PEPOs which are the conju- bility is an artifact of mean field. This is shown in Figure gate of each other (A and A∗). One can then think about 8. Furthermore, some studies suggested the existence of using a positivity preserving algorithm such as the one in an antiferromagnetic region in the presence of the trans- (Werner et al., 2016). Such an algorithm will ensure the verse field hx (Lee et al., 2011; Weimer, 2015a). Once positivity of the density matrix at all times of the evolu- again, while our technique found evidence of such an AF tion. We can call this initial density matrix as Projected region, it eventually shrank with increasing bond dimen- Entangled Pair Density Operator (PEPDO) as shown in sion until it finally disappears for large enough D. Re- Figure 9. While such an approach may avoid the problem 12

A. The variational principle for open quantum systems

A⇤

Variational methods generically consist of two steps. The first step is a paramterization of the state of the sys- tem in terms of a set of variational parameters α . For { i} A open quantum system, it is convenient to parameterize the density matrix, i.e., ρ = ρ( αi ), although parame- terizations based on statistical{ ensembles} of pure states are also possible (Transchel et al., 2014). The second FIG. 9 TN diagram for the PEPDO of ρ on a 2d square step is to identify a suitable functional that can be op- lattice, with bond dimension D and physical dimension d. timized by tuning the variational parameters. For open When vectorized, it can be understood as a PEPS for |ρi] 2 quantum systems, it is very natural to apply a variational with physical dimension d principle to find the steady state of the quantum master equation, which can be found by solving the equation ρ˙ = 0. Importantly, the exact steady state can no longer be determined after the variational parametrization, as of negative eigenvalues of the density matrix, in practice, the steady state will generically lie outside the variational it may require very high bond dimension of the PEPDO manifold. Hence, the best possible option is to find the and one, therefore, needs to consider the practical aspect variational parameters that will minimize the functional of the implementation. ρ for a suitable norm (Weimer, 2015b). The other possibility would be to target the ground ||L || state of the Hermitian and positive semidefinite operator The correct norm for the variational optimization can † ]. This ground state could be computed, e.g., by an be identified as the trace norm ρ = Tr ρ˙ , i.e., L] L imaginary time evolution. However, there are two major the sum of the absolute values of||L the|| eigenvalues{| |} ofρ ˙ hurdles associated with this approach. First, the crossed (Weimer, 2015b). This choice can be motivated on dif- products in † ] are non-local, and therefore the usual ferent grounds. First, the trace distance being the nat- L] L algorithms for time evolution are difficult to implement ural distance measure for density matrices (Nielsen and unless one introduces extra approximations in the range Chuang, 2000) is highly suggestive of the trace norm be- of the crossed terms. Another option is to approximate ing the natural norm for the tangent spaceρ ˙. This can the ground state variationally, e.g., via the Density Ma- be formalized in the sense that the trace norm describes trix Renormalization Group (White, 1992, 1993) in 1d, an optimal measurement to distinguishρ ˙ from the zero or variational PEPS in 2d (Corboz, 2016b). In the ther- matrix (Gilchrist et al., 2005). A second way to mo- modynamic limit, however, this approach does not look tivate the trace norm is to consider classes of possible very promising because of the non-locality of † ] men- alternatives. It can be shown that all Schatten p-norms L] L tioned before. In any case, one could always represent of the form ( ρ˙ p)1/p are inherently biased towards the this operator as a PEPO (in 2d), which would simplify maximally mixed| | state for all values of p > 1 (Weimer, some of the calculations, but at the cost of introducing a 2015b). Since functionals with p < 1 do not constitute very large bond dimension in the representation of † ]. proper norms, this leaves the trace norm as the only valid L] L For instance, if a typical PEPO bond dimension for ] choice. One can also understand the variational princi- L is 4, then for ]† ] it is 16, which in 2d implies ex- ple as a direct solution of the overdetermined steady state tremely∼ slow calculations.L L ∼ Another option would be to equation ρ = 0 in terms of a trace norm minimization. target the variational minimization of the real part for L the expectation value of . In general, the evaluation of the variational functional L is still an exponentially hard problem, as the computa- tion of the trace norm requires the diagonalization of the matrixρ ˙. However, it is possible to construct upper IV. VARIATIONAL METHODS bounds to the variational norm that retain the variational character (Weimer, 2015b) and appear to introduce only Variational techniques are often very powerful tools to small quantitative deviations even close to phase tran- analyze quantum many-body systems, as demonstrated sitions (Weimer, 2015a). The upper bound depends on by the successes of density functional theory (Kohn, the variational manifold and its tangent space, i.e., the 1999) and matrix product state approaches (Schollw¨ock, degree of additional correlations that can be build up by 2011) for ground state problems. As we will discuss in applying the Liouvillian to states within the variational this section, variational methods can also be successfully manifold. For example, for a variational class of product applied to open quantum many-body systems. states of the form ρ = i ρi, the upper bound D can be Q 13 given as variational principle (Kraus and Osborne, 2012; Tran- schel et al., 2014). There, the variational functional is D = Tr ρ˙ , (28) replaced by a variational integration of the quantum mas- {| ij|} ij ter equation for small time steps τ. For example, in the X∈T lowest order Euler approximation, it is given by where contains pairs of sites that are connected to each T other by the Liouvillian (Weimer, 2015b). D = Tr ρ(t + τ) ρ(t) τ ρ(t) , (31) The variational principle has been applied to find the {| − − L |} steady states of the dissipative Ising and Bose-Hubbard where ρ(t + τ) is the density matrix containing the vari- models introduced in Eqs. (3) and (4), respectively ational parameters. Higher-order schemes exist as well, (Weimer, 2015a,b), as well as dissipative Ising models but constructing an upper bound similar to Eq. (28) re- including a Z2 symmetry (Overbeck et al., 2017), purely quires to consider higher-order correlations due to multi- dissipative Heisenberg models (Weimer, 2017), dissipa- ple applications of the Liouvillian to the density matrix. tive Rydberg gases (Weimer, 2015a), dissipative ensem- A good compromise is the implicit midpoint method, bles of nitrogen-vacancy centers (Raghunandan et al., which is exact up to second order in τ while only re- 2018), entanglement generation in cavity QED arrays quiring a single application of the Liouvillian (Overbeck (Lammers et al., 2016), and dissipative Fermi-Hubbard and Weimer, 2016). models (Kaczmarczyk et al., 2016). In the latter case, the study of fermionic models was realized by employing a two-dimensional Jordan-Wigner transformation, where B. Comparison with mean-field methods the appearance of nonlocal Wigner strings was ruled out by the choice of the variational manifold. For equilibrium problems, the variational method In the case where the steady state of the system is based on product states is exactly equivalent to a mean- close to criticality, it is possible to construct a dissipa- field decoupling of the interaction terms. Remarkably, tive Ginzburg-Landau theory based on the variational this is not the case for open quantum systems. Within principle (Overbeck et al., 2017). The essential step is the mean-field approach to open systems (Diehl et al., to perform a series expansion of the variational norm of 2010a; Tomadin et al., 2010), a set of effective single site Eq. 28 in terms of an order parameter field φ(x) and its master equations is considered that is obtained by tracing spatial gradient φ(x), leading to out the rest of the system. For the ith site, the mean-field ∇ master equation reads D[φ] = dx v [ φ(x)]m + u [φ(x)]n. (29) m n d d MF ∇ ρi = Tri ρ = i[Hi , ρi] + i(ρi), (32) Z m n dt 6 dt − D X X   All the coefficients v and u can be calculated from the n n where HMF and are the mean-field Hamiltonian and microscopic quantum master equation. The series can be i i the mean-field dissipators,D respectively. This set of equa- truncated at low orders of m and n, as higher order terms tion is then solved self-consistently, while for translation- are irrelevant close to criticality. In the case of steady ally invariant systems it is often sufficient to consider an states with thermal statistics due to the presence of a effective single site problem. dynamical symmetry (Sieberer et al., 2013), it is possible Due to the nonlinear structure of the mean-field equa- to construct a Ginzburg-Landau-Wilson functional inte- tions of motion, it is possible to have two or more inde- gral for an effective partition function (Hohenberg and pendent solutions for the steady state (Lee et al., 2011), Krekhov, 2015), given by see Fig. 10. This also occurs within mean-field theory for equilibrium systems close to first order transitions. Z = φ exp ( β D[φ]) . (30) eff D − eff However, there one can always resort to the free energy, Z which has to be minimal in thermal equilbrium. Unless Here, the effective inverse temperature βeff can be de- one invokes the variational principle, one cannot decide rived from the u0 coeffecient, as this coefficient captures which of the solutions of mean-field theory are stable and the strength of fluctuations beyond a spatially homoge- which ones are not. Interestingly, the solution according neous order parameter field (Overbeck et al., 2017). The to the variational principle and mean-field theory are only subsequent statistical field theory of Eq. (30) can then identical in the limit of infinite dimensions, where both be analyzed using standard techniques such as the per- approaches become exact (Weimer, 2015b). turbative renormalization group. Mean-field theory predicts bistability for a wide range Finally, the variational principle can also be extended of models, including the dissipative Ising model (Lee and towards the full time evolution of open quantum sys- Cross, 2012; Lee et al., 2011; Marcuzzi et al., 2014) or ex- tems (Overbeck and Weimer, 2016), following very simi- tended spin models (Parmee and Cooper, 2018), as well lar ideas discussed in the context of the time-dependent as driven-dissipative Bose-Hubbard models (Jin et al., 14

that such mean-field results need to be taken with cau- 0.5 tion. Classical models exhibiting extended coexistence regions (Mu˜noz et al., 2005) might still exhibit bistabil- 0.4 ity after including quantum fluctuations. The situation

r 0.3 is similar when it comes to limit cycles of open quan- n tum many-body systems (Chan et al., 2015), which have 0.2 been predicted to exist in sufficiently high-dimensional 0.1 systems (Owen et al., 2018). 0 One systematic extension of mean-field theory is clus- 0 2 4 6 8 10 ter mean-field theory, where the trace in Eq. (32) is not Ω/γ carried out over all but one site but results in a larger cluster that has again to be solved self-consistently (Jin FIG. 10 Comparison of the solutions according to the vari- et al., 2016). This strategy is in close analogy to the ational principle (solid), the mean-field decoupling (dashed), cluster mean-field theory for statistical mechanics and and wave-function Monte-Carlo simulations for 4 × 4 lattices ground state problems (Bethe, 1935; Oguchi, 1955). Es- for the up-spin density nr of the dissipative Ising model. The mean-field solution displays a region of bistability, while the sentially, cluster mean-field approaches treat the short- variational solution correctly predicts a first order transition. range physics more accurately than bare mean-field the- From (Weimer, 2015a). ory, leading to better quantitative estimates for phase transitions. However, the qualitative limitations of bare mean-field theory remains, as these are resulting of long- 2013; Le Boit´e et al., 2013, 2014; Mertz et al., 2016). range fluctuations in the system. For open quantum So far, mean-field bistability has been found in the ab- many-body models, cluster mean-field theory has been sence of symmetries in the underlying master equation, used to calculate the phase diagram of the dissipative i.e., the two solutions are not connected by a symmetry Heisenberg model given by Eq. 27 (Jin et al., 2016) and transformation. These properties have led to speculation dissipative Ising models with and without a Z2 symmetry that bistability could be a genuine nonequilibrium phase, (Jin et al., 2018). which has stimulated several investigations whether this Finally, it is also possible to systematically go beyond could indeed be the case. However, the results of these the mean-field approximation using open system dynam- investigation have all been negative so far. Specifically, ical mean-field theory (DMFT). DMFT is a mapping of a the variational principle predicts that bistability is re- many-body lattice model onto a single impurity problem placed by a first order transition both in the dissipa- that has to be solved in a self-consistent way (Georges tive Ising and in the driven-dissipative Bose-Hubbard et al., 1996). Within DMFT, the approach is to start model (Weimer, 2015b). For the dissipative Ising model, with an effective dynamical Green’s function 0, which the existence of the first order transition has been con- serves as a time-dependent version of a mean-fieldG cou- firmed in tensor network simulations, where bistability is pling. Considering the Fermi-Hubbard model as an ex- found for low bond dimensions, but a first order transi- ample, 0 can used to express the effective action of a tion appears for higher bond dimensions (Kshetrimayum single siteG as et al., 2017), see Sec. III.B. In the case of the driven- dissipative Bose-Hubbard model, the first order transi- tion has also been found in a field-theoretic treatment β β based on the Keldysh formalism (Maghrebi and Gor- 1 S = dτ dτ 0 f †(τ) − f (τ 0) shkov, 2016), again confirming the variational prediction. eff − σ G0 σ σ These results underscore that the conventional argument Z0 Z0 X of mean field theory becoming qualitiatively correct if the β spatial dimension becomes large enough appears to be in- + U f †f f †f , (33) ↑ ↑ ↓ ↓ correct for open quantum systems. On the other hand, Z0 this argument seems to be much more justified when ap- plied to the variational principle (especially, when con- sidering the connection to equilibrium statistical physics where fσ annihilates a fermion with spin σ, β is the in- through the existence of the dissipative Ginzburg-Landau verse temperature, and U is the on-site interaction. The theory of Eq. (29)), however, even there one may have central idea of DMFT is to consider a self-consistent so- possible counterexamples (Mesterh´azyand Hebenstreit, lution that repoduces the dynamical Green’s function 0. 2017), which are not yet fully understood. This constraint is satisfied by the solution to the DMFTG Nevertheless, these findings do not rule out genuine equations for the local Green’s function G0, the dynami- bistability in open quantum systems per se, but only cal Green’s function , and the self-energy Σ evaluated G0 15 at the Matsubara frequencies ωn = (2n + 1)π/β, been reported, which for sufficiently large spatial dimen- sions also survives under inclusion of the terms beyond G0(iωn) = cσ(iωn)cσ∗ (iωn) Seff (34) mean-field. Consequently, it would be interesting to learn h i 1 1 whether the projection operator approach is also capable G0(iωn) = 0(iωn)− Σ(iωn) − (35) G − to correctly identify the replacement of mean-field bista- N() G (iω ) =  d  , (36) bility by a first order transition in the dissipative Ising 0 n iω + µ Σ(iω )  Z n − n − model. where µ is the chemical potential and N() is the den- sity of states (Kollar, 2011). The first step in bringing C. Variational tensor network methods DMFT to open systems has been to use effective Lindblad master equations have to describe quantum transport in Given the successes of tensor network methods dis- closed quantum systems using DMFT (Arrigoni et al., cussed in Sec. III, it appears natural to combine them 2013; Titvinidze et al., 2015, 2016). Recently, this ap- with variational methods for the study of open quan- proach has been extended to the case where already the tum many-body systems. However, the main challenge initial many-body problem describes an open quantum is that the natural trace norm for constructing the varia- system (Panas et al., 2019). tional principle cannot be calculated efficiently in a ten- A different method to systematically extend mean-field sor network respresentation. This has led to the use of theory is to use projection operator methods. The cen- different norms as possible alternatives (Cui et al., 2015; tral idea is to consider a single site of the many-body Mascarenhas et al., 2015), see Sec. III.A. problem, with the rest of the system forming a non- On the one hand, the choice of the norm is not really Markovian environment. This non-Markovian master relevant if the value of the norm is very low (i.e., compa- equation is then solved using standard projection oper- rable to the machine precision of the numerical simula- ator techniques such as the Nakajima-Zwanzig method tion), as then the solution is almost exact from any point or the time-convolutionless master equation (Breuer and of view. On the other hand, choosing a non-natural norm Petruccione, 2002). Initially, this approach has been is a potential source of errors that is not under control of used to describe the relaxation dynamics of local observ- the variational algorithm. In practice, this difficulty will ables in a closed quantum system (Weimer et al., 2008), mostly manifest itself for higher-dimensional problems, which has later been extended to the Lindblad dynamics as there the bond dimensions that can be reached are sev- of open systems (Degenfeld-Schonburg and Hartmann, erly constrained by the compuational resources (Kshetri- 2014). There, the initial step is to introduce corrections mayum et al., 2017). But even for one-dimensional sys- ∆ to the mean-field Liouvillian are introduced ac- L LMF tems, there are computationally challenging problems in- cording to volving long relaxation times (Carollo et al., 2019), where an arbitrarily low variational norm might not be reach- = + ∆ . (37) L LMF L able. The projection removes all correlations and projects A way out of this problem can be realized by represent- the system ontoP a product state, i.e., ing the density matrix in terms of an ensemble of pure states and use a variational tensor network formulation ρ = ρ . (38) for these pure states (Transchel et al., 2014). In this case, P i i the density matrix is parametrized according to Y If the initial state at time t is also a product state, 0 ρ = p(α, α¯) ψ(α) ψ(α) dαdα,¯ (40) the projected Lindblad master equation may be formally | ih | written as Z where ψ(α) is a variational wave function with varia- t | i d tional parameters α and p(α, α¯) is the associated proba- ρ(t) = MF ρ(t) + ∆ dt0 (t, t0) ρ(t0), (39) bility distribution. Crucially, the variational norm asso- P dt L P P L K P Z0 ciated with the effective Hamiltonian of the master equa- tion Heff = H i/2 i ci†ci can now be calculated as where the generator has been introduced (Degenfeld- − K P 2 Schonburg and Hartmann, 2014). The generator may DH = Heff ψ(α) . (41) then be expanded in terms of a power series of the beyondK | | i| mean-field corrections ∆ . This projection operator ap- This expression can both be computed efficiently using proach has been used toL investigate both dissipative XY tensor network methods and corresponds to the natu- models (Degenfeld-Schonburg and Hartmann, 2014) and ral trace norm when evaluated over the full ensemble. the dissipative Heisenberg model (Owen et al., 2018). Re- The quantum jump terms of the master equation can be markably, in the latter case a limit cycle behavior has treated in a similar fashion (Transchel et al., 2014). 16

D. Variational quantum Monte-Carlo methods

The central idea behind quantum Monte-Carlo meth- ods is to rewrite a quantum many-body problem in terms of a sampling over a classical probability distribution (Batrouni and Scalettar, 2011). However, the existence of destructive interference in quantum mechanics can lead to corresponding classical probabilities that are negative, which is the root of the famous sign problem. One com- mon workaround is to sample over the absolute value of the probability distribution instead, but this comes at the price of the complexity of the computation increas- ing exponentially with the system size (Troyer and Wiese, FIG. 11 Node structure of a restricted Boltzmann machine 2005). Open quantum many-body systems are especially for open quantum systems. The vectorized density matrix is prone to the sign problem since the eigenvalues of the Li- realized in terms of a physical layer σi, corresponding to a ouvillian can even be complex (Nagy and Savona, 2018). set of spin 1/2 variables. These are coupled to the nodes of Nevertheless, Monte-Carlo sampling can be useful even a hidden layer hi, which are again coupled to the third layer in the presence of the sign problem, if the required re- τi, which represents the adjoint of the physical layer. sources for the Monte-Carlo sampling are lower than for a full solution of the problem. For the variational Monte-Carlo samplings, different The first quantum Monte-Carlo simulation of an open norms have been put forward. One possibility is to con- quantum many-body problem has been based on a non- sider the Hilbert-Schmidt norm of the time evolution variational full-configuration-interaction Monte Carlo al- (Hartmann and Carleo, 2019) or the steady state (Vi- gorithm (Nagy and Savona, 2018), which is better centini et al., 2019a). Interestingly, in the latter case, equipped to deal with the sign problem without com- the variational norm D has been normalized according pletely negating it. For the magnetization of a dissipative to the purity Tr ρ2 , i.e., XYZ model on small lattices, the quantum Monte-Carlo { } simulation is in excellent agreement with wave-function Tr ρ˙2 Monte-Carlo results. D = . (43) Tr ρ2 Recently, variational Monte-Carlo methods have been { } applied to open quantum systems (Hartmann and Car- This norm is not biased towards the maximally mixed leo, 2019; Nagy and Savona, 2019; Vicentini et al., 2019a; state as mentioned above. An alternative approach to Yoshioka and Hamazaki, 2019). These approaches are in- construct a suitable norm is to mimize the Hermitian spired by using variational wave function corresponding † in close analogy to a ground state problem (Yoshioka to restricted Boltzmann machines (RBMs) (Carleo and andL L Hamazaki, 2019). Finally, it is possible to consider Troyer, 2017), which were first introduced in the context the equivalent of an expectation value for vectorized den- of neural network simulations. The main idea behind sity matrices according to ] ρ ] ρ ]/] ρ ρ ] (Nagy and RBM wave functions is shown in Fig. 11, where an ad- Savona, 2019). With the respecth |L | i to theh | i more natural ditional hidden layer introduces variational parameters trace norm for density matrices, the RBM approaches associated with the quantum correlations of the many- behave similarly to the tensor network simulations dis- body state. The entries of the vectorized density matrix cussed in Sec. IV.C. However, since RBMs can be applied are then given by to two-dimensional models in a straightforward way, it will be very interesting to see how these methods perform 1 for the investigation of dissipative phase transitions, in σ, τ ρ = exp W σ h + W ∗ τ h ]h | i] Z  ij i j ij i j particular in critical systems. hj ij {X} X  

exp a σ + a∗τ + b h , (42) V. PHASE SPACE AND RELATED METHODS ×  i i i i j j i j X X   Other methods have also been used with relative suc- where the Wij, ai, and hj are variational parameters. cess in the study of open quantum systems, such as phase Interestingly, there is a close connection between RBM space methods, as well as methods based on hierarchy wave functions and matrix product states (Chen et al., equations. In this section we explain two of such ex- 2018; Deng et al., 2017), however, RBMs are potentially amples, namely, truncated Wigner approximations and also capable to describe long-range entangled quantum Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hier- states. archies. The resulting methods are very general in pur- 17 pose, and can be applied to a wide variety of systems, κX,C the exciton and photon decay rates, and keeping up yet in what follow we discuss concrete examples. to second-order derivatives only, one obtains the stochas- tic differential equation

A. Truncated Wigner approximation ψX ψX 0 √κX dWX id = HHF0 + dt+i . ψC ! ψC ! F !! √κC dWC ! In the context of phase-space related methods, trun- (46) cated Wigner approximations were first used in (Caru- In this equation, dWl=X,C are Wiener noise terms, and sotto and Ciuti, 2005) to driven-dissipative microcavity H0 is given by polariton system coherently driven into the optical para- MF 2 metric oscillator regime, also reviewed in (Carusotto and 2 1 ΩR −∇ + g ψ 2 iκ 2mX X X a X 2 Ciuti, 2013) and revisited in (Dagvadorj et al., 2015) as H0 = | | − − 2 MF ΩR −∇ iκC ! an example of a 2d driven-dissipative non-equilibrium 2
2mC − phase transition. The Hamiltonian for the system is given (47) by The resulting stochastic differential equation can then be solved using standard methods and softeare packages for 2 gX 2 ΩR this purpose. −∇ + ψ 2mX 2 X 2 ψX H = d~r ψ† ψ† | | 2 , S X C ΩR Let us add that recently, this method has also been −∇ ! ψC ! Z   2 2mC used to study critical slowing down in photonic lattices (44) (Vicentini et al., 2018), as well as extended to disor- with cavity and photon field operators ψX,C (~r, t), spa- dered quantum many-body system (the so-called optical tial coordinate ~r = (x, y), mX,C the exciton and photon stochastic unraveling for disordered systems) (Vicentini masses, gX the exciton-exciton interaction strength, and et al., 2019b). ΩR the Rabi splitting. One introduces the effect of an ex- ternal drive (pump) as well as incoherent decay by adding a system-bath Hamiltonian given by B. BBGKY hierarchy equations

HSB = d~r F (~r, t)ψC† (~r, t) + h.c. It is also possible to study open quantum systems via Z the so-called Bogoliubov-Born-Kirkwood-Yvon hierarchy  l  + ξ ψ† (t)B + h.c. + ω B† B , (Liboff, 2003). In a nutshell, this is a hierarchy of equa- ~k l,~k l,~k l,~k l,~k l,~k l=X,C tions aimed to describe a system of a large number of X~k X     (45) interacting particles. As such, the idea is very generic. But as shown in (Navez and Sch¨utzhold,2010), it can with ψl,~k(t) the Fourier transform of the field operators also be applied directly in the context of open dissipative systems in order to obtain a hierarchy of equations for in real space, B ~ and B† the bath’s bosonic anihilation l,k l,~k the different reduced density matrices. and creation operators with energy ωl,~k, which describes the decay for both excitons and cavity photons. The de- cay is compensated by an external homogeneous coherent i(~kp ~r ωpt) pump F (~r, t) = fpe · − , injecting polaritons with momentum ~kp and energy ωp. By using standard quantum optical methods, one can trace out the bath within the Markovian approximation and obtain a Master equation for the system. There is, however, an alternative approach by means of phase- space techniques. In particular, one can represent the quantum fields as quasiprobability distribution functions. The Fokker-Planck partial differential equation that gov- erns the dynamics of such distributions can be mapped to a stochastic differential equation, which can be solved using different techniques. For the example that we are discussing, one solves the equation on a finite grid with lattice spacing a. The most suitable quasiprobability dis- tribution for this example is the Wigner representation, which is also the most suitable one for numerical im- plementation. By truncating the corresponding Fokker- Planck equation in the limit (g /κ a2) 1, with X X,C  18

The way this approach works is quite intuitive. Con- generating functional (αµ) = log Tr(ρ (Iµ + αµ)), F µ sider the reduced density matrices for one lattice site with αµ an arbitrary operator acting on site µ. Using Q ρµ, for two lattice sites ρµν , and so on. We separate such a functional one has ρµ = ∂ /∂αµ α=0, as well c c 2 F | the correlated parts as ρµν = ρµν + ρµρν , as well as as ρµν = ∂ /∂αµ∂αν α=0, and so on. Next, the Li- ρ = ρc ρ + ρc ρ + ρc ρ + ρ ρ ρ , and so on. The ouville operatorsF and| acting on one and two µνλ µν λ µλ ν νλ µ µ ν λ Lµ Lµν method that will be discussed in what follows in based sites are introduce via the dissipation equation i∂tρ = on the scaling hierarchy of correlations [H, ρ] + µ µρ + µν µν ρ/Z, with Z the coordination number of theL HamiltonianL (e.g., the number of tunneling c 1 P P ρ = O Z −|S| , (48) neighbours at any given site for a Hubbard-like Hamilto- S   nian). Following these equations, the time evolution of with the number of lattice sites in set . The different is given by F reduced|S| density matrices can also be computedS using the

∂ ∂ 1 ∂2 ∂ ∂ i (α) = Tr α F + Tr (α + α + α α ) F + F F . (49) ∂tF µ µLµ ∂α Z µν µ ν µ ν Lµν ∂α ∂α ∂α ∂α µ µ µν µ ν µ ν X   X   

Using this equation, one can take derivatives and obtain a set of equations for the correlated density matrices,

¯= µ P∪P S\{ } ∂ c c 1 c 1 S c S c c i ρ = µρ + µν ρ + Trk µkρ k + µkρ µ ρ k ¯ ∂t S L S Z L S Z L S∪ L { }∪P { }∪P  µ µν k / µ µ X∈S X∈S X∈S X∈S P⊆S\{X } ¯= µ,ν  ¯= ¯  P∪P S\{ } 1 c c S c Q∪Q P c c c + µν ρ µ ρ ν ¯ Trν µν ρ µ,ν ¯ + ρ µ ρ ν ¯ ρ ν (50) Z L { }∪P { }∪P − L  { }∪P { }∪Q { }∪Q { }∪P  µν µ,ν ¯ X∈S P⊆S\{X } Q⊆XP     

S with µν = µν + νµ. This hierarchy of equations for For instance, in (Navez and Sch¨utzhold,2010) it was ap- the reducedL densityL L matrices is preserved in time. More- plied to a lattice Bose-Hubbard model. The method can over, it allows us to write explicit equations for the one- be used to obtain analytical expansions, as well as to and two-site density matrices. For the one-site matrix facilitate efficient numerical simulations. one gets

∂ 1 i ρ = + Tr S ρc + ρ ρ , (51) ∂t µ Lµ Z k Lµk µk µ k VI. LINKED CLUSTER EXPANSION METHODS k X

and for the two-site matrix one has Methods based on linked-cluster expansions have also been recently put forward in the study of open quantum ∂ 1 many-body systems, so far focusing on the study of two- i ρ = ρc ρc + ρ ρ ∂t µν Lµ µν Z Lµν µν µ ν dimensional spin systems with incoherent spin relaxation 1
(Biella et al., 2018). The method numerically targets + Tr S ρc + ρc ρ + ρc ρ Z k Lµk µνk µν k νk µ expectation values of observables in the steady state (at k=µ,ν X6

long times) of the master equation. ρµ Tr S ρc + ρ ρ + (µ ν). (52) Mathematically, the procedure is as follows: let us as- − Z µ Lµν µν µ ν ↔ sume (without loss of generality) that the Liouvillian can

By combining the above expressions with Eq.(48), one be expanded as a sum of two-body terms, i.e., can expand in powers of 1/Z and obtain different ap- = α , (53) proximations for the one- and two-particle behaviour. L ijLij i,j This approach can be implemented for a variety of sys- hXi tems (spins, bosons, fermions...) and has the advantage with αij some local coupling strength. For the sake of of being independent of the dimensionality of the system. simplicity let us define k (i, j) as a combined index. ≡ 19

The expectation value O of an observable Oˆ can be ex- enabled us to review a large variety of numerical meth- panded in terms of powers of αk, i.e., ods. To be more specific, in this review we considered methods for the Markovian quantum master equation nk O( αk ) = O nk αk , (54) (assuming a weak-coupling limit), including mean-field, { } { } nk k stochastic methods, tensor networks, variational meth- {X} Y ods, quantum Monte-Carlo, truncated Wigner approxi- with nk running over all non-negative integers for all k. mation, BBGKY hierarchy equations, and linked cluster It is clear that all possible polynomials in αk are included expansions. While so far, no method has emerged that in the above expression, which can be reorganized in clus- is universally optimal for all cases, there have been sev- ters as follows: eral very promising developments with different methods for different regimes. Even with such major technical O = W (c), (55) [O] advances discussed in this review, there are still many c X open problems which are inacessible with these state-of- with c a non-empty set of k-indexes identifying the sites the-art numerical techniques. To give concrete exam- belonging to the cluster. The cluster weight W[O](c) con- ples of actual physical problems, one may consider a very tains all the terms in the expansion with at least one common setting in the context of Rydberg atoms where power of αk, for all k in c, and no powers of αk of k the interaction is often long-ranged and cannot be ap- does not belong to c. These terms obey the recurrence proximated with just a nearest-neighbour Hamiltonian relation (Browaeys et al., 2016; Labuhn et al., 2016; Schachen- mayer et al., 2015). Even TN techniques will face a diffi- W (c) = O(c) W (s), (56) cult challenge specially in 2D while encountering such [O] − [O] s c problems although there has been promising develop- X⊂ ments even in this direction recently (O’Rourke and Kin- with Lic Chan, 2019). Other challenging problems include the existence of AF order in 3D dissipative Ising model which O(c) = Tr(Oρˆ s(c)) (57) is an open question that appears hard to answer. This being the expectation value of the observable in the is again relevant to ongoing experiments with Rydberg steady-state ρs(c) for the finite cluster c. Taking into atoms, which one cannot reliably simulate at the mo- account symmetries in the system, the expectation value ment (Carr et al., 2013; Helmrich et al., 2018; Malossi per site in the thermodynamic limit can be written as et al., 2014). Phase transitions and universality classes of dissipative models is another class of problem which O ∞ has proven to be quite difficult for numerical techniques = l(c )W (c ) , (58) L n [O] n (Biondi et al., 2017; Carmichael, 2015; Diehl et al., 2010b; n=1 c ! X Xn Fink et al., 2017). with L the size of the system, the outer sum run- Certainly, the largest confidence in a simulation re- ning over→ all ∞ possible cluster sizes n, and the inner sum sult can be achieved if it is reproducible using a comple- mentary simulation approach. Despite these caveats, one over all topologically different clusters cn of size n, with can make several key observations about the particular l(cn) their multiplicity. This series expansion can be truncated up to a cluster size R, thus giving rise to a methods covered in this review. The first observation is plausible approximation method also valid for open sys- that mean-field methods are considerably less reliable for tems. open system than their counterparts for closed systems, The linked cluster expansion works very well for the although the reason for this discrepancy is still an open dissipative Heisenberg model (Biella et al., 2018), where question. Furthermore, tensor network methods have an exact product state solution can be used as a starting demonstrated their ability to successfully tackle many point of the expansion. In this case, it is even possible hard problems surrounding open many-body systems and to calculate phase boundaries and critical exponents of resolve long-standing open questions. A particularly in- a dissipative phase transition between a paramagnet and teresting and promising case is that of open 2d systems, a ferromagnet. The situation is quite different for the which is unexplored territory to a great extent. As for dissipative Ising model, where the expansion series failed the variational methods discussed in this review, there to converge even for a 10th order expansion (Jin et al., appears to be a tradeoff between the formal suitability 2018). of the norm and its efficient computability. It will be in- teresting to see if and how this tradeoff will be resolved VII. SUMMARY AND OUTLOOK in future work. We provide a summary in Tab. II com- paring the different techniques we have discussed above. The enormous effort to develop novel simulation meth- ods to investigate open quantum many-body systems has The progress in recent years in simulating open quan- 20

WFMC TN Variational Principle VQMC CMF TWA System size 20 TDL TDL 16 TDL 400 Dimensions 1D,2D 1D,2D anya any anyb any Local Hilbert space small small large large small large Fermionic systems Yes Yes partially No partially unknown Inhomogeneous systems good good bad good good good Critical exponents good good goodc unknown bad unknown a Works better in higher dimensions b Works better in higher dimensions c For states with thermal statistics

TABLE II Table comparing the different simulation methods discussed in this review. We differentiate the methods by the system sizes that can be simulated, the spatial dimensions, contraints on the local Hilbert space dimension, whether fermionic systems can be treated, the simulation performance for inhomogeneous systems, and whether the correct critical exponents of phase transitions can be obtained. tum systems has brought the field to a level where one cia Cugliandolo, Artur Ekert, and Kok Khoo Phua (Oxford has a wide range of tools at hand to systematically com- University Press) pp. 356–394. pare to experimental results, in particular in the context Baumann, K, C. Guerlin, F. Brennecke, and T. Esslinger of quantum simulations. Combined with the experimen- (2010), “Dicke quantum phase transition with a super- fluid gas in an optical cavity,” Nature 464, 1301–1306, tal ease of preparing the steady state of an open quantum arXiv:0912.3261 [quant-ph]. system, these are good reasons to believe that the study Beaudoin, F´elix,Jay M. Gambetta, and A. Blais (2011), of strongly-correlated open quantum many-body systems “Dissipation and ultrastrong coupling in circuit qed,” Phys. will become a research topic with impact in other areas Rev. A 84, 043832. of science, such as material design and quantum compu- Bethe, H A (1935), “Statistical theory of superlattices,” tation. Proc. R. Soc. London, Ser. A 150 (871), 552–575, http://rspa.royalsocietypublishing.org/content/150/871/552.full.pdf. Biamonte, Jacob, and Ville Bergholm (2017), “Ten- sor Networks in a Nutshell,” arXiv e-prints , ACKNOWLEDGMENTS arXiv:1708.00006arXiv:1708.00006 [quant-ph]. Biella, Alberto, Jiasen Jin, Oscar Viyuela, Cristiano Ciuti, We thank C. Ciuti for valuable feedback on our Rosario Fazio, and Davide Rossini (2018), “Linked cluster manuscript. This work was funded by the Volkswagen expansions for open quantum systems on a lattice,” Phys. Foundation, by the Deutsche Forschungsgemeinschaft Rev. B 97, 035103. Binder, Kurt, and Jian Sheng Wang (1989), “Finite-size ef- (DFG, German Research Foundation) within SFB 1227 fects at critical points with anisotropic correlations: Phe- (DQ-mat, project A04), SPP 1929 (GiRyd), and under nomenological scaling theory and monte carlo simulations,” Germanys Excellence Strategy – EXC-2123 Quantum- J. Stat. Phys. 55 (1), 87–126. Frontiers – 390837967. Biondi, Matteo, Gianni Blatter, Hakan E. T¨ureci,and Sebas- tian Schmidt (2017), “Nonequilibrium gas-liquid transition in the driven-dissipative photonic lattice,” Phys. Rev. A REFERENCES 96, 043809. Bonnes, Lars, Daniel Charrier, and Andreas M. L¨auchli (2014), “Dynamical and steady-state properties of a bose- Arrigoni, Enrico, Michael Knap, and Wolfgang von der Lin- hubbard chain with bond dissipation: A study based on den (2013), “Nonequilibrium dynamical mean-field theory: matrix product operators,” Phys. Rev. A 90, 033612. An auxiliary quantum master equation approach,” Phys. Breuer, Heinz-Peter, and Francesco Petruccione (2002), Rev. Lett. 110, 086403. The Theory of Open Quantum Systems (Oxford University Ates, Cenap, Beatriz Olmos, Juan P. Garrahan, and Igor Press, Oxford). Lesanovsky (2012), “Dynamical phases and intermittency Browaeys, Antoine, Daniel Barredo, and Thierry Lahaye of the dissipative quantum ising model,” Phys. Rev. A 85, (2016), “Experimental investigations of dipole–dipole inter- 043620. actions between a few rydberg atoms,” Journal of Physics Barreiro, Julio T, Markus M¨uller,Philipp Schindler, Daniel B: Atomic, Molecular and Optical Physics 49 (15), 152001. Nigg, Thomas Monz, Michael Chwalla, Markus Hennrich, Cai, Zi, and Thomas Barthel (2013), “Algebraic versus expo- Christian F. Roos, Peter Zoller, and Rainer Blatt (2011), nential decoherence in dissipative many-particle systems,” “An open-system quantum simulator with trapped ions,” Phys. Rev. Lett. 111, 150403. Nature 470, 486. Carleo, Giuseppe, and Matthias Troyer (2017), “Solv- Batrouni, G G, and R. T. Scalettar (2011), “Quantum phase ing the quantum many-body problem with artifi- transitions,” in Ultracold Gases and Quantum Information, cial neural networks,” Science 355 (6325), 602–606, edited by Christian Miniatura, Leong-Chuan Kwek, Mar- http://science.sciencemag.org/content/355/6325/602.full.pdf. tial Ducloy, Benot Grmaud, Berthold-Georg Englert, Leti- 21

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