Logic circuits from zero forcing

We design logic circuits based on the notion of zero forcing on graphs; each gate of the circuits is a gadget in which zero forcing is performed. We show that such circuits can evaluate every monotone Boolean function. By using two vertices to encode each logical bit, we obtain universal computation. We also highlight a phenomenon of “back forcing” as a property of each function. Such a phenomenon occurs in a circuit when the input of gates which have been already used at a given time step is further modified by a computation actually performed at a later stage. Finally, we show that zero forcing can be also used to implement reversible computation. The model introduced here provides a potentially new tool in the analysis of Boolean functions, with particular attention to monotonicity. Moreover, in the light of applications of zero forcing in quantum mechanics, the link with Boolean functions may suggest a new directions in quantum control theory and in the study of engineered quantum spin systems. It is an open technical problem to verify whether there is a link between zero forcing and computation with contact circuits

Ground State Spin Logic

Designing and optimizing cost functions and energy landscapes is a problem encountered in many fields of science and engineering. These landscapes and cost functions can be embedded and annealed in experimentally controllable spin Hamiltonians. Using an approach based on group theory and symmetries, we examine the embedding of Boolean logic gates into the ground state subspace of such spin systems. We describe parameterized families of diagonal Hamiltonians and symmetry operations which preserve the ground state subspace encoding the truth tables of Boolean formulas. The ground state embeddings of adder circuits are used to illustrate how gates are combined and simplified using symmetry. Our work is relevant for experimental demonstrations of ground state embeddings found in both classical optimization as well as adiabatic quantum optimization.Comment: 6 pages + 3 pages appendix, 7 figures, 1 tabl

The zero forcing polynomial of a graph

Zero forcing is an iterative graph coloring process, where given a set of initially colored vertices, a colored vertex with a single uncolored neighbor causes that neighbor to become colored. A zero forcing set is a set of initially colored vertices which causes the entire graph to eventually become colored. In this paper, we study the counting problem associated with zero forcing. We introduce the zero forcing polynomial of a graph $G$ of order $n$ as the polynomial $\mathcal{Z}(G;x)=\sum_{i=1}^n z(G;i) x^i$, where $z(G;i)$ is the number of zero forcing sets of $G$ of size $i$. We characterize the extremal coefficients of $\mathcal{Z}(G;x)$, derive closed form expressions for the zero forcing polynomials of several families of graphs, and explore various structural properties of $\mathcal{Z}(G;x)$, including multiplicativity, unimodality, and uniqueness.Comment: 23 page

Bounds for the Zero Forcing Number of Graphs with Large Girth

The zero-forcing number, Z(G) is an upper bound for the maximum nullity of all symmetric matrices with a sparsity pattern described by the graph. A simple lower bound is δ ≤ Z(G) where δ is the minimum degree. An improvement of this bound is provided in the case that G has girth of at least 5. In particular, it is shown that 2δ − 2 ≤ Z(G) for graphs with girth of at least 5; this can be further improved when G has a small cut set. Lastly, a conjecture is made regarding a lower bound for Z(G) as a function of the girth, g, and δ; this conjecture is proved in a few cases and numerical evidence is provided

Variants of zero forcing and their applications to the minimum rank problem

The minimum rank problem refers to finding the smallest possible rank, or equivalently the largest possible nullity, among matrices under certain restrictions. These restrictions can be the zero-nonzero pattern, conditions on the inertia, or other properties of a matrix. Zero forcing is a powerful technique for controlling the nullity and plays a significant role in the minimum rank problem. This thesis introduces several zero forcing parameters and their applications on the minimum rank problem. Zero-nonzero patterns can be described by graphs: The edges (including the loops) represent the nonzero entries, while the non-edges correspond to the zero entries. For simple graphs, where no loops are allowed, the diagonal entries can be any real numbers. The maximum nullity of a graph is the maximum nullity among symmetric matrices with the pattern described by the graph. In Chapter 2, the odd cycle zero forcing number Zoc(G) and the enhanced odd cycle zero forcing number Ẑoc(G) are introduced as bounds for the maximum nullities of loop graphs G and simple graphs G, respectively. Also, a relation between loop graphs and simple graphs through graph blowups is developed. The Colin de VerdiÃÂère type parameter ξ(G) is defined as the maximum nullity of real symmetric matrices A with the pattern described by G and with the Strong Arnold Property (SAP), which means X = O is the only symmetric matrix that satisfies A ○ X = I ○ X = AX = O (here ○ is the entrywise product). Chapter 3 introduces zero forcing parameters Zsap(G) and Zvc(G); we show that Zsap(G) = 0 implies every symmetric matrix with the pattern described by G has the SAP and that the inequality M(G) − Zvc(G) ≤ ξ(G) holds for every graph G. Also, the values of ξ(G) are computed for all graphs up to 7 vertices, establishing ξ(G) = ⌊Z⌋(G) for these graphs