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Register a new userTensor Network Rewriting Strategies for Satisfiability and Counting
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Niel de Beaudrap
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We provide a graphical treatment of SAT and #SAT on equal footing. Instances of #SAT can be represented as tensor networks in a standard way. These tensor networks are interpreted by diagrams of the ZH-calculus: a system to reason about tensors over C in terms of diagrams built from simple generators, in which computation may be carried out by transformations of diagrams alone. In general, nodes of ZH diagrams take parameters over C which determine the tensor coefficients; for the standard representation of #SAT instances, the coefficients take the value 0 or 1. Then, by choosing the coefficients of a diagram to range over B, we represent the corresponding instance of SAT. Thus, by interpreting a diagram either over the boolean semiring or the complex numbers, we instantiate either the decision or counting version of the problem. We find that for classes known to be in P, such as 2SAT and #XORSAT, the existence of appropriate rewrite rules allows for efficient simplification of the diagram, producing the solution in polynomial time. In contrast, for classes known to be NP-complete, such as 3SAT, or #P-complete, such as #2SAT, the corresponding rewrite rules introduce hyperedges to the diagrams, in numbers which are not easily bounded above by a polynomial. This diagrammatic approach unifies the diagnosis of the complexity of CSPs and #CSPs and shows promise in aiding tensor network contraction-based algorithms.
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It has been known for almost three decades that many $mathrm{NP}$-hard optimization problems can be solved in polynomial time when restricted to structures of constant treewidth. In this work we provide the first extension of such results to the quantum setting. We show that given a quantum circuit $C$ with $n$ uninitialized inputs, $mathit{poly}(n)$ gates, and treewidth $t$, one can compute in time $(frac{n}{delta})^{exp(O(t))}$ a classical assignment $yin {0,1}^n$ that maximizes the acceptance probability of $C$ up to a $delta$ additive factor. In particular, our algorithm runs in polynomial time if $t$ is constant and $1/poly(n) < delta < 1$. For unrestricted values of $t$, this problem is known to be complete for the complexity class $mathrm{QCMA}$, a quantum generalization of MA. In contrast, we show that the same problem is $mathrm{NP}$-complete if $t=O(log n)$ even when $delta$ is constant. On the other hand, we show that given a $n$-input quantum circuit $C$ of treewidth $t=O(log n)$, and a constant $delta<1/2$, it is $mathrm{QMA}$-complete to determine whether there exists a quantum state $mid!varphirangle in (mathbb{C}^d)^{otimes n}$ such that the acceptance probability of $Cmid!varphirangle$ is greater than $1-delta$, or whether for every such state $mid!varphirangle$, the acceptance probability of $Cmid!varphirangle$ is less than $delta$. As a consequence, under the widely believed assumption that $mathrm{QMA} eq mathrm{NP}$, we have that quantum witnesses are strictly more powerful than classical witnesses with respect to Merlin-Arthur protocols in which the verifier is a quantum circuit of logarithmic treewidth.
We prove a query complexity lower bound for $mathsf{QMA}$ protocols that solve approximate counting: estimating the size of a set given a membership oracle. This gives rise to an oracle $A$ such that $mathsf{SBP}^A otsubset mathsf{QMA}^A$, resolving an open problem of Aaronson [2]. Our proof uses the polynomial method to derive a lower bound for the $mathsf{SBQP}$ query complexity of the $mathsf{AND}$ of two approximate counting instances. We use Laurent polynomials as a tool in our proof, showing that the Laurent polynomial method can be useful even for problems involving ordinary polynomials.
We discuss in detail algorithms for implementing tensor network renormalization (TNR) for the study of classical statistical and quantum many-body systems. Firstly, we recall established techniques for how the partition function of a 2D classical many-body system or the Euclidean path integral of a 1D quantum system can be represented as a network of tensors, before describing how TNR can be implemented to efficiently contract the network via a sequence of coarse-graining transformations. The efficacy of the TNR approach is then benchmarked for the 2D classical statistical and 1D quantum Ising models; in particular the ability of TNR to maintain a high level of accuracy over sustained coarse-graining transformations, even at a critical point, is demonstrated.
For Boolean satisfiability problems, the structure of the solution space is characterized by the solution graph, where the vertices are the solutions, and two solutions are connected iff they differ in exactly one variable. In 2006, Gopalan et al. studied connectivity properties of the solution graph and related complexity issues for CSPs, motivated mainly by research on satisfiability algorithms and the satisfiability threshold. They proved dichotomies for the diameter of connected components and for the complexity of the st-connectivity question, and conjectured a trichotomy for the connectivity question. Recently, we were able to establish the trichotomy [arXiv:1312.4524]. Here, we consider connectivity issues of satisfiability problems defined by Boolean circuits and propositional formulas that use gates, resp. connectives, from a fixed set of Boolean functions. We obtain dichotomies for the diameter and the two connectivity problems: on one side, the diameter is linear in the number of variables, and both problems are in P, while on the other side, the diameter can be exponential, and the problems are PSPACE-complete. For partially quantified formulas, we show an analogous dichotomy.
For Boolean satisfiability problems, the structure of the solution space is characterized by the solution graph, where the vertices are the solutions, and two solutions are connected iff they differ in exactly one variable. For this implicitly defined graph, we here study the st-connectivity and connectivity problems. Building on the work of Gopalan et al. (The Connectivity of Boolean Satisfiability: Computational and Structural Dichotomies, 2006/2009), we first investigate satisfiability problems given by CSPs, more exactly CNF(S)-formulas with constants (as considered in Schaefers famous 1978 dichotomy theorem); we prove a computational dichotomy for the st-connectivity problem, asserting that it is either solvable in polynomial time or PSPACE-complete, and an aligned structural dichotomy, asserting that the maximal diameter of connected components is either linear in the number of variables, or can be exponential; further, we show a trichotomy for the connectivity problem, asserting that it is either in P, coNP-complete, or PSPACE-complete. Next we investigate two important variants: CNF(S)-formulas without constants, and partially quantified formulas; in both cases, we prove analogous dichotomies for st-connectivity and the diameter; for for the connectivity problem, we show a trichotomy in the case of quantified formulas, while in the case of formulas without constants, we identify fragments of a possible trichotomy. Finally, we consider the connectivity issues for B-formulas, which are arbitrarily nested formulas built from some fixed set B of connectives, and for B-circuits, which are Boolean circuits where the gates are from some finite set B; we prove a common dichotomy for both connectivity problems and the diameter; for partially quantified B-formulas, we show an analogous dichotomy.
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