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(no) direction, the graph is called (un). A spin model is defined by a Hamiltonian H on a lattice E(G)(V (G)) corresponds for the set of edges(vertices) of graph G. The Hamiltonian is of the form the graph G. The size of the two sets are E(G) = m, | | V (G) = n. We can further attach one real value for H = X Jij SiSj + X hiSi, (1) | | − each edge as w(E), such a graph is called weighted graph. hiji∈E(G) i∈V (G) The of a is the number of edges incident where the spin freedoms S live on the vertices of the to the vertex, and a graph with all vertices of degree 3 is i which can take value S = 1. Spin cou- called . A is a graph that can be i plings J are defined on edges of lattice graph± G, while embedded on the plane without any edges crossing, and ij external magnetic fields h are defined on vertices. The the else are nonplanar graph. A plane graph is a pla- i ground state energy of the model is defined as the mini- nar graph that has already been embedded on the plane, mal value of total energy among all spin configurations, there may be more than one plane graph (choice of pla- namely nar embedding) for a planar graph. The of a plane graph G is a graph that has a vertex for each face E0(H) = min H( Si ), (2) of G. The dual graph has an edge whenever two faces of {Si} { } G are separated from each other by an edge. Thus, each where S take values from configuration space with size edge e of G has a corresponding dual edge, whose end- i 2n. { } points are the dual vertices corresponding to the faces on The groud state energy decision problem is defined as: either side of e. A is a partition of the vertices of a graph into two Problem 3 Given the input of the spin model (including disjoint subsets (one of the subsets is defined as a base the lattice graph G and all parameters in the Hamiltonian set S). Any cut S determines a cut-set CUT(S), the set Jij , hi) and a fixed value Ek, is the ground state energy of edges that have one endpoint in each subset of the of the model E0(H) Ek? partition. The size of the cut-set is defined as number of ≤ edges in the cut-set for unweighted graph, which is also This question is formulated as a decision problem and it is obviously in NP since a nondeterministic Turing ma- called simple cut. For weighted graph, the total weight of the cut-set is the sum of all weights of edges from the chine can naturally explores all spin configurations at the same time by different branches and make the compari- cut-set. The simple MAX-CUT problem is defined as: son at the end of each branch. Therefore, the remaining Problem 1 Given the unweighted graph G and an inte- task is to classify all types of spin models into P and ger k, is there any vertices set S, such that CUT(S) NP-complete classes based on their ground state energy k? | |≥ decision problems. We divide spin models in terms of the following con- The simple MAX-CUT problem is NP-complete even on ditions. (1) : the dimension of the model is general cubic graph [15], while simple MAX-CUT prob- related with the underlying lattice graph. For planar lem is in P on planar graph[16]. Similarly we can define graph, the dimension is 2, while for nonplanar graph, the MAX-CUT problems on weighted graphs using the nota- dimension is 3 [17]. In the physics language, D = 3 in tion of w(CUT) = Pe∈CUT w(e) and one can also define terms of lattice graph corresponds to D 3 in real sys- MIN-CUT problems similarly. ≥ tems. (2) Spin-spin coupling sign: If e E(G),Je 0, The MAX-CUT problem separating two given vertices, we call such model ferromagnetic (FM)∀ ∈ models. Other-≥ is defined as: wise for models e E(G), s.t. Je < 0, they are said to ∃ ∈ Problem 2 Given graph G, two vertices s,t on the graph be antiferromagnetic (AFM) models. (3) Randomness: If and the fixed value k, is there any base set S, such that all spin-spin couplings and external fields are constants s S, t / S and w(CUT(S)) k? as e E(G), v V (G),Je = J, hv = H, such models ∈ ∈ ≥ are∀ uniform.∈ Instead,∈ if parameters in the Hamiltonian It can be shown that MAX-CUT problem and MAX- are allowed to taken in several values or a continuum of CUT problem separated by two given vertices are com- parameter windows, the corresponding models are called putationally equivalent. MAX-CUT

0 (hv < 0) on the lattice graph. Moreover, we assign graph of G. The algorithm exploits the famous blossom the newly added edges e = (s, v) (e = (t, v)) of weights algorithm finding max-weight perfect matching on the w(e) = hv = hv (w(e) = hv = hv). The decorated graph[23–25]. The complexity of the state-of-the-art al- weighted| graph| after FM transformation| | − is denoted by gorithm for this task is known as O(mn + n2 log n)[26]. G′. The inverse transformation from any weighted graph Besides, if all the spin-spin couplings are integers with the G′ to a spin model H with underlying lattice graph G integer absolute value bounded by W , the MWPM algo- is obvious. Just define the Hamiltonian on the graph rithm can be further improved to O(m√n log nW )[27]. ′ G without external field and with Je = w(e). Note the For example, a model where short-ranged spin cou- mapping can be easily carried out in polynomial time. plings taking values from 1, 0 can be solved exactly in ± Given a configuration of spins Si , consider the set of O(n√n log n) time where n is the system size. From our vertices S for cut on G′ including{ all} vertices with spin unified picture of max cut analysis, the transformation up and vertex s, i.e. S = v Sv = 1 S s. It is obvious algorithm by Barahona can be understood as a strong to see that the total energy{ of| the system} is given by version of the proof that planar MAX-CUT is in P, i.e. MAX-CUT problem with positive and negative weights H( Si )= E +2w(CUT(S( Si ))), (3) { } − 0 { } on planar graph is in P. Hence the reduction chain can be understood as 2D AFM spin model without external where E0 is a constant independent of the spin config- fields

complete problems with optimal aim of integer values. TABLE I. Summary of the classification on the computational Take the uniform AFM 3D spin models for an example, complexity of spin models. suppose all the couplings are J = 1, the total energy ≥ e Spin Models 2D 3D of the system is integer valued. If there is FPTAS for FM P P ground state energy problem of such models, we can do uniform AFM, no fields P NPC random AFM, no fields P NPC the with precision ǫ =1/(2E0), uniform AFM, external fields NPC NPC such an approximation leads to the approximation solu- tion E E 1/2. Since the energy spectrum is integer | 1− 0|≤ valued, E0 = E1, and thus we obtain the exact results by α β field, we apply the AFM transformation on such model, approximation scheme with the time O(n m ). There- and it gives the reduction from MAX-CUT problem on fore we cannot calculate the approximate ground state general graph to the ground state energy problem in this energy efficiently unless P = NP. class of models. Therefore, it is straightforward to con- From the above analysis, NP-complete physics systems clude that the ground state energy problem of 3D uniform give fundamental limitations to all numerical methods AFM spin models is NP-complete on cubic graph. in general since no efficient approximations are allowed We summarize the full classification on the computa- for such systems. An example is Markov chain Monte tion complexity of spin models by Table I. We omit ran- Carlo (MCMC)[30]. It is known that there exists effi- dom AFM with external fields line, since they are triv- cient update algorithms for 3D FM spin models while ially in NPC as their uniform counterparts are already the slowing down and the exponential increase of auto- in NPC. And all NPC problems in the table are still in correlation time is unavoidable in general 3D AFM spin NPC even restricted to cubic graphs inputs. The uni- systems. This fact in classical MCMC is well explained in fied framework to understand the classification is pro- the framework of computational complexity analysis and vided by the FM/AFM transformation we discussed, shows how the complexity theory plays a vital role in and the investigation on the complexity classes of cor- understanding the behavior of numerical methods. One responding MAX(MIN)-CUT problems on some subsets may try to improve the cluster update algorithm expect- of (un)weighted graph. The general guiding principles in- ing that the slowing down problem can be avoided in clude: on positive weighted graphs, (1) MAX-CUT is in general. Since the intrinsic hardness is in the model it- NPC; (2) MIN-CUT is in P (max flow); (3) MAX-CUT self, this task is as hard as proving P = NP. is in P if restricted to planar graph (MWPM). The classical models also add limitation to numerical There are several observations immediately from the methods for quantum systems. For example, we can use classification results. Firstly, the AFM models are no quantum Monte Carlo (QMC) to solve 3D AFM spin easier than their FM counterparts. This fact is intuitive models by treating the spin as Pauli matrix instead of and can be justified by the more complex energy land- a number. Since this family of models is NP-complete, scape for AFM models. Secondly, randomness brings no QMC must fail in polynomial times and the obstacle is further hardness at least in spin models investigated here. nothing but notorious sign problem[31, 32]. The fact Namely, if a uniform spin model is in P then its counter- shows that sign problem of QMC cannot be systemat- part with random couplings is also in P. And if a random ically solved as it is NP-hard[13]. We can go a step spin model is NP-complete, the uniform counterpart is further, by calculating the same model both in classical also in NPC. This fact may provide new insights into our MCMC and QMC as mentioned above, the algorithms understanding on randomness and spin glasses. Besides, fail due to different reasons: slowing down or sign prob- it is worth noting that the computational hardness has lem. In some sense, the two obstacles of MCMC are two nothing to do with whether the model has a finite tem- sides of the same coin. It is of the same difficulty to sys- perature phase transition. tematically avoid them. Based on the discussion on fun- Discussions: The NP-completeness of ground state en- damental limitations of numerical approaches, we demon- ergy problems imposes strong limitations on the numeri- strate the new understanding on why numerical methods cal methods even for efficient approximations. A reason- become inefficient sometimes. It may be useless to pro- able efficient approximation is the algorithm giving so- pose any fancy improvements on numerical approaches lutions E1 as an approximation of the true ground state to solve certain models because the models are hard by energy E0 with precision ε ( E1 E0 εE0) in poly- their own and the inefficient bottlenecks in each numeri- nomial time with the input size| −n and| ≤ precision 1/ε as cal schemes are just reflections of NP-completeness from O(nαε−β). This efficient approximations is one of the ba- the models. sic requirements for any reasonable numerical approaches The NP-complete problems always come with inputs and the concept is summarized as fully polynomial time encoding scalable infinite freedoms. In this work, the approximation scheme (FPTAS). freedoms are encoded in the input lattice graphs. This is Since all the NP-complete spin models are intrinsically because all problems in NP can be reduced to NPC prob- hard even when all couplings are restricted to 1, they lems, and the encoded large freedoms correspond to each are denoted by NP-complete in the strong sense± [29]. It NP problems. In other words, it is hard to believe one is easy to prove that there is no FPTAS for strong NP- can solve all problems in NP by a problem with one inte- 5 ger input. So we can not talk about the NP-completeness ficiently (BQP cannot cover NPC). Therefore, the whole of very specific models. For example, the input for AFM discussion in this work may still hold even if quantum model with uniform fields and uniform couplings on some computer are used to carry out numerical simulations fixed lattice has only three integers: size n, coupling J on NP-complete physical system. As for the complex- and external field H. Ground state energy problem of ity classification on quantum models, it is more suitable such specific model is hard to describe in the context to use the QMA-complete[33] class to define hard prob- of NP-completeness, since it is of little hope that every lems. QMA is the quantum counterpart of NP class, problem in NP can be easily reduced to a problem with and thus QMA-complete is no easier than NP-complete three natural freedoms. There is no good way to char- problems. Various quantum Hamiltonians[34] have been acterize the complexity of such problems which is too recognized as QMA-complete in terms of ground state en- specific to be NP-complete but might also hard at the ergy promise problems since the initial work for 5-local same time. Hamiltonians[33]. All the general ideas and insights in On the other hand, the subset of NP-complete prob- the above discussions has no substantive change consid- lems are not necessarily hard. They can be in P as well. ering QMA-complete problems. For example, the funda- For example, 3D uniform AFM model without external mental limitations of density functional theory(DFT) is field on arbitrary graph is NP-complete. But for its sub- shown by QMA-complete quantum models[14]. Perhaps set where the given graph is 3D grids, the ground state the most amazing conjecture from NP(QMA)-complete configuration and energy value is obvious in P. This fact models is that the ground state energy is inaccessible in leaves another possibility how we can study NP-complete polynomial time even in experiments. It is the spirit of systems. Although we have no efficient numerical ap- the strong version of Church-Turing thesis which states proach on NPC models, we can still compute the subset that no nature process can be faster than a Turing ma- we are interested by showing the subset of the models we chine in terms of computation complexity[35][36]. care about is actually in P. Namely, one can always try to In conclusion, we give a full classification on the com- make a finer classification on the computation complexity putational complexity of classical spin models by a unify- of models. Not only a subset of problems from NP might ing framework of FM/AFM transformations. The ground be in P, a collection of NPC problems may also become state energy problems for spin models are related with easy. For example, although the ground state energy for some kinds of MAX(MIN)-CUT problems under the each disorder configuration is NP-complete problem, the transformations. Furthermore, we discuss in detail on disorder averaged results are not necessarily NP-complete how to correctly understand NP-complete problems in due to the possible emergent structure of disorder aver- physics and how NP-completeness impose fundamental age. limitations on numerical approximations. The effective- Finally, we give a brief discussion on the picture when ness of our understanding remains when quantum com- quantum computation and quantum complexity theory plexity is considered. are included. Firstly, it is widely believed that a quan- Acknowledgment: We thank Hong Yao and Zi-Xiang tum computer also can’t solve NP-complete problems ef- Li for useful discussions.

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