Tensor Network Contractions for #SAT

Abstract

The computational cost of counting the number of solutions satisfying a Boolean formula, which is a problem instance of #SAT, has proven subtle to quantify. Even when finding individual satisfying solutions is computationally easy (e.g. 2-SAT, which is in $\mathsf{{P}}$), determining the number of solutions can be #$\mathsf{{P}}$-hard. Recently, computational methods simulating quantum systems experienced advancements due to the development of tensor network algorithms and associated quantum physics-inspired techniques. By these methods, we give an algorithm using an axiomatic tensor contraction language for n-variable #SAT instances with complexity $O((g+cd)^{O(1)} 2^c)$ where c is the number of COPY-tensors, g is the number of gates, and d is the maximal degree of any COPY-tensor. Thus, n-variable counting problems can be solved efficiently when their tensor network expression has at most $O(\log n)$ COPY-tensors and polynomial fan-out. This framework also admits an intuitive proof of a variant of the Tovey conjecture (the r,1-SAT instance of the Dubois–Tovey theorem). This study increases the theory, expressiveness and application of tensor based algorithmic tools and provides an alternative insight on these problems which have a long history in statistical physics and computer science.

Appendix 1: Properties of Tensor Contraction

Although the graphical manipulation of tensors (e.g. algebraic rewrite rules) has been axiomatized by several communities, basic theorems and techniques of contraction are less systematically described. Networks have seen little effort toward their systematic formulation. Hence we present the minimal properties of tensor contraction we relied on to provide the main results in the paper.

We assume that the reader has some basic familiarly with tensor networks and thus focus on the properties of contraction properties which we used. This section should hopefully be self contained. For additional background information, see e.g. [46].

The value of a tensor network is obtained by performing all possible partial contractions diagrammed by the network (the order of contraction does not affect the value). If we allow one or more of the tensors in the network to vary, and the network has no dangling wires, we obtain a multilinear function from the direct sum over vertices of the spaces $\bigotimes _{e\in \mathcal {N}(v)}{V(e)}$ to the complex numbers. When there are dangling wires, the multilinear function’s codomain is the tensor product of the vector spaces corresponding to the dangling edges.

We denote such a function by $\mathcal {C}$ when the tensor network defining it is understood. Then multilinearity can be rephrased as follows.

Remark 27

(Linearity of tensor contraction) Tensor contraction is linear in its component tensors provided each appears only once: $\mathcal {C}\{A_{v_1}, \ldots , \alpha A_{v_k}+A_{v_k}', \ldots , A_{v_n}\} = \alpha \mathcal {C}\{A_{v_1}, \ldots , A_{v_k}, \ldots , A_{v_n}\} + \mathcal {C}\{A_{v_1}, \ldots , A_{v_k}', \ldots , A_{v_n}\}.$ and then $A\mapsto kA$ we readily find that the contraction becomes $\mathcal {C}\{A'\}+\mathcal {C}\{B\}$ and $k\cdot \mathcal {C}\{A\}$ respectively.

Several of our results rely on transforming contracted sub-networks, into a sum-over products, and vice versa. In other words, removing a contracted tensor (or part of a contracted tensor) and replacing it by a sum of contracted tensors, where each term contains contractions that are multiplied together. The stage for this can be set by stating several properties of tensor contractions. We will use these properties to reshape networks, such that their transformed geometry is known to be efficiently contractible.

Theorem 28

(COPY-tensors as a resolution of identity) The following sequence of graphical rewrites hold.

Proof

In the above figure, on the left, we abstractly depict a otherwise arbitrary tensor network, by showing only one single wire. The unit for the COPY-tensor is the plus state $|+\rangle =\sum {|n\rangle }$. Contracting $|+\rangle$ against the COPY-tensor therefore gives $\sum _{i,n}{|i\rangle \langle i|\langle i|n\rangle }=\sum _{n}{|n\rangle \langle n|}$ which is precisely the identity. The last equality follows directly from the linearity of tensor contraction. $\square$

This procedure can be iterated over many wires. In particular, if we have a gate in a tensor network, we can remove it from the network by the above method resulting in a nested sum of tensor networks. Of course the number of tensor networks in the summand grows exponentially in the number of wires of the gate removed.

Some counting algorithms are only polynomial time on planar graphs, and this cutting procedure can be used to deal with handles, yielding a running time factor exponential in the genus of the graph [49].

Corollary 29

(Contraction to sum of products transform) Given a tensor $\Gamma ^{\ldots ijk}_{~~~~lmn\ldots }$ in a fully contracted network (e.g. a network without open legs), the following graphical identities, which are equal, transforms the contraction, to a sum over products.

The comprising rectangle above is meant as an abstraction depicting a fully contracted but otherwise general tensor network.

As mentioned, the contraction to sum of products transform in Theorem 29 as well as the tensor network resolution of identity in Theorem 28 will be useful to transform networks into those known to be efficiently contractible.

Theorem 30

(The Cauchy–Schwarz inequality) Given a contracted network and a partition into two halves x, and y. Writing the contraction as $\mathcal {C}\{x,y\}$ the following inequality holds.

$$\begin{aligned} \mathcal {C}\{x,y\}^2\le \mathcal {C}\{x,x\}\cdot \mathcal {C}\{y,y\} \end{aligned}$$

with graphical depiction.

Proof

By the linearity of tensor contraction, we arrive at an abstract form of the Cauchy–Schwarz inequality, with equality in the contraction if and only if $x = \alpha \cdot y$. This leads directly to the concept of an angle between tensors,

$$\begin{aligned} \cos \theta _{xy} = \frac{\mathcal {C}\{x,y\}^2}{\mathcal {C}\{x,x\}\cdot \mathcal {C}\{y,y\}} \end{aligned}$$

where the right side is either real valued, or we take the modulus. $\square$