Quadratization of Symmetric Pseudo-Boolean Functions

A pseudo-Boolean function is a real-valued function $f(x)=f(x_1,x_2,\ldots,x_n)$ of $n$ binary variables; that is, a mapping from $\{0,1\}^n$ to $\mathbb{R}$. For a pseudo-Boolean function $f(x)$ on $\{0,1\}^n$, we say that $g(x,y)$ is a quadratization of $f$ if $g(x,y)$ is a quadratic polynomial depending on $x$ and on $m$ auxiliary binary variables $y_1,y_2,\ldots,y_m$ such that $f(x)= \min \{g(x,y) : y \in \{0,1\}^m \}$ for all $x \in \{0,1\}^n$. By means of quadratizations, minimization of $f$ is reduced to minimization (over its extended set of variables) of the quadratic function $g(x,y)$. This is of some practical interest because minimization of quadratic functions has been thoroughly studied for the last few decades, and much progress has been made in solving such problems exactly or heuristically. A related paper \cite{ABCG} initiated a systematic study of the minimum number of auxiliary $y$-variables required in a quadratization of an arbitrary function $f$ (a natural question, since the complexity of minimizing the quadratic function $g(x,y)$ depends, among other factors, on the number of binary variables). In this paper, we determine more precisely the number of auxiliary variables required by quadratizations of symmetric pseudo-Boolean functions $f(x)$, those functions whose value depends only on the Hamming weight of the input $x$ (the number of variables equal to $1$).Comment: 17 page

On Symmetric Pseudo-Boolean Functions: Factorization, Kernels and Applications

A symmetric pseudo-Boolean function is a map from Boolean tuples to real numbers which is invariant under input variable interchange. We prove that any such function can be equivalently expressed as a power series or factorized. The kernel of a pseudo-Boolean function is the set of all inputs that cause the function to vanish identically. Any $n$-variable symmetric pseudo-Boolean function $f(x_1, x_2, \dots, x_n)$ has a kernel corresponding to at least one $n$-affine hyperplane, each hyperplane is given by a constraint $\sum_{l=1}^n x_l = \lambda$ for $\lambda\in \mathbb{C}$ constant. We use these results to analyze symmetric pseudo-Boolean functions appearing in the literature of spin glass energy functions (Ising models), quantum information and tensor networks.Comment: 10 page

On Commutative Penalty Functions in Parent-Hamiltonian Constructions

There are several known techniques to construct a Hamiltonian with an expected value that is minimized uniquely by a given quantum state. Common approaches include the parent Hamiltonian construction from matrix product states, building approximate ground state projectors, and, in a common case, developing penalty functions from the generalized Ising model. Here we consider the framework that enables one to engineer exact parent Hamiltonians from commuting polynomials. We derive elementary classification results of quadratic Ising parent Hamiltonians and to generally derive a non-injective parent Hamiltonian construction. We also consider that any $n$-qubit stabilizer state has a commutative parent Hamiltonian with $n+1$ terms and we develop an approach that allows the derivation of parent Hamiltonians by composition of network elements that embed the truth tables of discrete functions into a kernel space. This work presents a unifying framework that captures components of what is known about exact parent Hamiltonians and bridges a few techniques across the domains that are concerned with such constructions.Comment: 23 page

Quadratic reformulations of nonlinear binary optimization problems

Very large nonlinear unconstrained binary optimization problems arise in a broad array of applications. Several exact or heuristic techniques have proved quite successful for solving many of these problems when the objective function is a quadratic polynomial. However, no similarly efficient methods are available for the higher degree case. Since high degree objectives are becoming increasingly important in certain application areas, such as computer vision, various techniques have been recently developed to reduce the general case to the quadratic one, at the cost of increasing the number of variables. In this paper we initiate a systematic study of these quadratization approaches. We provide tight lower and upper bounds on the number of auxiliary variables needed in the worst-case for general objective functions, for bounded-degree functions, and for a restricted class of quadratizations. Our upper bounds are constructive, thus yielding new quadratization procedures. Finally, we completely characterize all ``minimal'' quadratizations of negative monomials

Quadratization of symmetric pseudo-Boolean functions

A pseudo-Boolean function is a real-valued function $f(x)=f(x_1,x_2,\ldots,x_n)$ of $n$ binary variables; that is, a mapping from $\{0,1\}^n$ to {\bbr}. For a pseudo-Boolean function $f(x)$ on $\{0,1\}^n$, we say that $g(x,y)$ is a quadratization of $f$ if $g(x,y)$ is a Quadratic polynomial depending on $x$ and on $m$ auxiliary binary variables $y_1,y_2,\ldots,y_m$ such that $f(x)= \min \{ g(x,y) : y \in \{0,1\}^m \}$ for all $x \in \{0,1\}^n$. By means of quadratizations, minimization of $f$ is reduced to minimization (over its extended set of variables) of the quadratic function $g(x,y)$. This is of some practical interest because minimization of quadratic functions has been thoroughly studied for the last few decades, and much progress has been made in solving such problems exactly or heuristically. A related paper initiated a systematic study of the minimum number of auxiliary $y$-variables required in a quadratization of an arbitrary function $f$ (a natural question, since the complexity of minimizing the quadratic function $g(x,y)$ depends, among other factors, on the number of binary variables). In this paper, we determine more precisely the number of auxiliary variables required by quadratizations of \emph{symmetric} pseudo-Boolean functions $f(x)$, those functions whose value depends only on the Hamming weight of the input $x$ (the number of variables equal to 1).PAI COME

Quadratization of symmetric pseudo-Boolean functions

We consider the problem of minimizing an arbitrary pseudo-Boolean function f(x), that is, a real-valued function of 0-1 variables. In recent years, several authors have proposed to reduce this problem to the quadratic case by expressing f(x) as min{g(x,y):y∈{0,1}^m}, where g(x,y) is a quadratic pseudo-Boolean function of x and of additional binary variables y. We say that g(x,y) is a quadratization of f. In this talk, we investigate the number of additional variables needed in a quadratization when f is a symmetric function of the x-variables. The cases where f is either a positive or a negative monomial are of particular interest, but some of our techniques also extend to more complex functions, like k-out-of-n or parity functions. Joint work with Martin Anthony, Endre Boros and Aritanan Grube