Classification on the Computational Complexity of Spin Models Shi-Xin Zhang∗ Institute for Advanced Study, Tsinghua University, Beijing 100084, China (Dated: November 12, 2019) In this note, we provide a unifying framework to investigate the computational complexity of classical spin models and give the full classification on spin models in terms of system dimensions, randomness, external magnetic fields and types of spin coupling. We further discuss about the implications of NP-complete Hamiltonian models in physics and the fundamental limitations of all numerical methods imposed by such models. We conclude by a brief discussion on the picture when quantum computation and quantum complexity theory are included. Introduction: Computational complexity classes are hardness of specific numerical methods with the help of very important tools in computer science to character- models in NP(QMA)-complete complexity classes[13, 14]. ize the hardness of problems [1]. After the original work In this work, we elaborate on this idea and show that the of Cook[2], NP-completeness (NPC)[3] and in general the fundamental limitations of numerical approximation are notion of complete problems of corresponding complex- universal to all numerical simulation schemes due to the ity classes have become the dominant approach for ad- existence of NP-hard models. This fact brings invaluable dressing how hard a problem is. For decision problems insight into the numerical study on physics systems in in nondeterministic polynomial time (NP), namely deci- general. Efficient numerical schemes often fail in some sion problems that can be solved in polynomial time on a models (namely the time to get reasonable approxima- nondeterministic Turing machine, they can be classified tion results scales exponentially with the system size), as polynomial time (P), NP-complete or NP-intermediate and in such cases we often attribute the failure to the al- (neither in NPC nor in P). In this work, we take the con- gorithm itself and try to update the algorithm or replace jecture P = NP throughout[4], implying NP-complete it with other schemes. However, when various numerical problems are6 intractable. Surprisingly, most of the com- schemes fail on the same model due to seemingly differ- mon NP problems are either NP-complete or in P and ent reasons, it is highly possible that the hardness is not there are fewer NP-intermediate problem candidates than due to the drawbacks of individual numerical methods, na¨ıve thought [5]. This fact shows the deep structure of but from the model itself, indicating that there is no effi- NP problems and the universality of NP-completeness cient numerical simulation method at all. We also discuss language. how the above picture remains the same when quantum Barahona[6] introduced the NP-completeness notion computation is allowed. into statistical mechanics by considering the ground state Notations: A decision problem is a problem that can decision problems and gave the proof of NP-completeness be posed as a yes-no question of the input values. The set on two types of classical spin models. Furthermore, of decision problems can be solved in polynomial time on following previous works on exactly solving Ising mod- a deterministic Turing machine in terms of the input size els and dimer models[7–11], Barahona gave a general is P. As for NP class, there are two equivalent definitions. polynomial-time algorithm which can exactly solve the The sets of decision problems can be solved in polynomial ground state energy for all 2D spin models without ex- time on a nondeterministic Turing machine is NP, or the ternal magnetic field on arbitrary lattice. Therefore, it sets of problems that the yes answer instances can be is clear the classical spin models are totally different in verified given a proof in polynomial times is NP. It is terms of computational complexity. In this work, we will obvious P NP , as P problems require null proofs. give a full classification on the hardness of spin mod- Problem reduction⊆ is defined as a function f, such that els: some of them are claimed in P by directly providing for any legal input x of problem A, we have f(x) as input polynomial-time algorithms and the others are shown as of problem B and A(x)=1 B(f(x)) = 1. If such ⇐⇒ arXiv:1911.04122v1 [cond-mat.dis-nn] 11 Nov 2019 NP-complete by the reduction proofs from 3SAT. The a problem reduction f can be carried out in polynomial framework for NPC proof in this work is inspired by the time by a Turing machine, we have A<P B. Intuitively, results on the computational complexity of random field problem A is no harder than B, since solution to B also Ising model[12]. solves problem A. If A <P B and B <P A, problem A Notion of NP-completeness in physics, as an intrinsic and B are said to be computationally equivalent, written property of some statistical models, is not only of aca- as A =P B. If Q NP,Q <P R, we call such problems demic interest, but also plays an vital role when such R NP-hard. A∀ problem∈ is NP-complete if it is both in NP physics systems are numerically investigated in practice. and NP-hard. It is conjectured that P = NP. For more For example, there are various works dealing with the formal definitions and discussions on the6 computational complexity classes, please refer to [1]. We use standard conventions on the definition of a graph. A graph G = (V, E) is composed of vertices V ∗ [email protected] and edges E connecting two vertices. If the edge has 2 (no) direction, the graph is called (un)directed graph. 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 degree of a vertex 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 lattice graph which can take value S = 1. Spin cou- called cubic graph. A planar graph 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 dual graph 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 cut 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) Dimension: 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.