A complexity classification of spin systems with an external field

Abstract

We study the computational complexity of approximating the partition function of a q-state spin system with an external field. There are just three possible levels of computational difficulty, depending on the interaction strengths between adjacent spins: (i) efficiently exactly computable, (ii) equivalent to the ferromagnetic Ising model, and (iii) equivalent to the antiferromagnetic Ising model. Thus, every nontrivial q-state spin system, irrespective of the number q of spins, is computationally equivalent to one of two fundamental two-state spin systems.computational complexity | partition function | spin system L et Q = f1,2, . . . , qg denote a set of spins. A q-state spin system is specified by a nonnegative real symmetric interaction matrix A ∈ ðR ≥0 Þ q×q . The entries ða ij Þ of A represent "interaction strengths" between spins in Q. An instance of such a spin system is a graph G = ðV , EÞ, where V is a set of vertices (sites) and E is a set of edges (bonds) together with a collection h = fh w : Q → R ≥0 jw ∈ V g of functions, representing the action of an external field. The function h w represents the effect of the field on vertex w. The partition function of the system is then a weighted sum over configurations σ : V → Q.The above setting encompasses all spin systems with uniform interactions between pairs of sites and includes many familiar models. For example, the interaction matricescapture the Ising, independent set (hard-core), three-state Potts, and Widom-Rowlinson models, respectively. In the case of the Ising and Potts models, λ > 1 corresponds to a ferromagnetic system and λ < 1 to an antiferromagnetic one. The Problem and What Is KnownWe study the following computational problem. Fix an interaction matrix A. Given an instance consisting of a graph G and field h, approximately evaluate the partition function Z A ðG, hÞ. An important point to note is that the interaction matrix A does not form part of the problem instance. By fixing A, we fix a particular model, say the hardcore model or the q-state ferromagnetic Potts model. We then ask, for that model, What is the computational complexity of approximately evaluating the partition function given the input pair ðG, hÞ?We are interested in determining how that complexity depends on A. The purpose of this paper is to map the space of interaction matrices, delineating which case arises for each interaction matrix A. Before stating our result, we fill in some details about the three possibilities above (and approximation complexity in general), we say a little bit more about the result of ref. 1, and we present some matrix preliminaries.The first possibility, that the partition function can be evaluated exactly in polynomial time, is straightforward. Unfortunately, as we shall see, it rarely arises. Let us turn to the second possibility. It will be helpful to introduce the problem #BIS, which plays an important role in the complexity theory of approximate counting problems (2). #BIS is the problem of counting the independent sets of a bipartite g...

Exact Recursive Calculation of Circulant Permanents: A Band of Different Diagonals inside a Uniform Matrix

“…(A standard PC with Wolfram Mathematica fails already at n ∼ 25.) A fully polynomial randomized approximation scheme [21,22] does not work for matrices with sign indefinite or complex entries and requires a rather long polynomial computing time that scales as ∼n 11 for a general matrix with nonnegative entries or ∼n 4 (log n) in a special case of a very dense matrix. Figure 1 illustrates a strong dependence of the permanent on the entries c 0 , c 1 , c 2 .…”

“…The permanents have been studied in mathematics for more than a century (for a review, see [13][14][15][16][17][18]), the most actively after discovery of the Ryser's algorithm [19], the publication of the comprehensive book "Permanents" [13], proof of the famous Valiant's theorem stating that their computing is a P-hard problem within the computational complexity theory [20] and a recent development of a fully polynomial randomized approximation scheme [21,22] for their computing. In fact, the permanents are intimately related to many fields of mathematics, including matrix algebra, combinatorics, number theory, theory of symmetric polynomials, discrete Fourier transform, q-analysis, dynamical systems, generalized harmonic and wavelet analysis [23][24][25] and computational complexity theory.…”

“…Importantly, the permanents play a significant role in the algebraic complexity theory as universal polynomials [28]. Nevertheless, despite amazing recent activity and interest (see, e.g., [1,[5][6][7][8]15,18,22,[29][30][31][32][33][34][35][36][37][38] and references therein), the methods for analyzing and calculating the permanents and their asymptotics for most of large matrices remain illusive.…”

“…As is known from a phenomelogical renormalization-group approach [9,50], they are very different, since the correlation matrix abruptly tends to zero in the disordered phase, but remains nonzero for a very long distance due to long-range correlations in the ordered phase. Note that we can choose any (except the truly special, trivial zero) constant nonzero value, c = 1 or c = 1, for the entries of the uniform background matrix (see Equation (22)) in the considered model of the circulant matrix with a band of k any-value diagonals since it amounts to a simple rescaling of the permanent, as is explained in Remark 3. Computing the permanent of such matrices is very important in a number of applications in mathematics, physics, computing, information systems, cryptography, etc.…”

Unification of the Nature’s Complexities via a Matrix Permanent—Critical Phenomena, Fractals, Quantum Computing, ♯P-Complexity

“…In particular, an ad hoc counting of all matchings of a bipartite graph representing a monomer-dimer model of phase transitions allows one to express its partition function via the permanent of a 0-1 matrix adjacent to the bipartite graph [35]. Importantly, the graph theory and the Markov chain Monte Carlo method provide a fully polynomial randomized approximation scheme (FPRAS) for numerical computation of the permanent of nonnegative matrices and a ferromagnetic Ising model [36][37][38][39]; for a discussion of a different scheme, see Reference [40].…”