A Matrix Product State Representation of Boolean Functions

Abstract: We introduce a novel normal form representation of Boolean functions in terms of products of binary matrices, hereafter referred to as the Binary Matrix Product (BMP) representation. BMPs are analogous to the Tensor-Trains (TT) and Matrix Product States (MPS) used, respectively, in applied mathematics and in quantum many-body physics to accelerate computations that are usually inaccessible by more traditional approaches. BMPs turn out to be closely related to Binary Decision Diagrams (BDDs), a powerful compressed representation of Boolean functions invented in the late 80s by Bryant that has found a broad range of applications in many areas of computer science and engineering. We present a direct and natural translation of BMPs into Binary Decision Diagrams (BDDs), and derive an elementary set of operations used to manipulate and combine BMPs that are analogous to those introduced by Bryant for BDDs. Both BDDs and BMPs are practical tools when the complexity of these representations, as measured by the maximum bond dimension of a BMP (or the accumulated bond dimension across the BMP matrix train) and the number of nodes of a BDD, remains polynomial in the number of bits, $n$. In both cases, controlling the complexity hinges on optimizing the order of the Boolean variables. BMPs offer the advantage that their construction and manipulation rely on simple linear algebra -- a compelling feature that can facilitate the development of open-source libraries that are both more flexible and easier to use than those currently available for BDDs. An initial implementation of a BMP library is available on GitHub, with the expectation that the close conceptual connection to TT and MPS techniques will motivate further development of BMP methods by researchers in these fields, potentially enabling novel applications to classical and quantum computing.

A Matrix Product State Representation of Boolean Functions

The research paper titled "A Matrix Product State Representation of Boolean Functions" investigates a novel representation of Boolean functions using the Binary Matrix Product (BMP) formalism. The authors draw an analogy between BMPs and the tensor-train (TT) and matrix product state (MPS) methods widely used in applied mathematics and quantum many-body physics. The core idea of the paper is to employ BMPs as a means to enhance computational efficiency in representing and manipulating Boolean functions.

The motivation for introducing BMPs is to leverage straightforward linear algebra tools to develop flexible and easily modifiable computational packages, contrasting with the existing, more complex Binary Decision Diagrams (BDDs). The authors provide a systematic approach for transforming BMP representations into BDDs, and vice versa, establishing the equivalence between these data structures when normal forms are considered.

Core Concepts and Methodology

1. Binary Matrix Product Representation (BMP): The paper introduces BMPs as an analogous concept to matrix product states, focusing on representing Boolean functions through a train of binary matrices. This approach is designed to handle classical computational problems with enhanced efficiency, both in terms of representation and manipulation.

2. Normal Form and Operations: BMPs, when fully compressed, function as canonical normal forms. This attribute ensures that given a specific Boolean variable order, the representation is unique (up to permutations within matrices). The paper details fundamental operations on BMPs such as CLEAN, APPLY, RESTRICT, and JOIN, which are pivotal for synthesizing and manipulating Boolean circuits primarily through simple linear algebra operations.

3. Comparison with Binary Decision Diagrams (BDDs): The authors establish a direct translation mechanism between BMPs and BDDs. While BDDs are a common compressed representation of Boolean functions, the BMP approach provides an alternative with its linear algebra-based toolkit, potentially lowering the barrier for the development of efficient open-source libraries.

4. Variable Order Optimization: The complexity of BMPs is closely linked to the chosen ordering of Boolean variables. The authors address this optimization problem with both exact and heuristic methods. The exact approach is acknowledged as computationally intensive, while heuristics — such as sifting — offer practical solutions by dynamically ordering variables to minimize the size of the BMP.

Practical Implications and Future Directions

The introduction of BMPs for Boolean function computation emphasizes the potential for creating more accessible and versatile computational tools. By simplifying the operations through linear algebra, BMPs can attract broader adoption outside the traditional domain of BDDs — particularly in fields like random classical circuits and quantum circuit simulation.

The paper also highlights the potential for BMPs and their related operations to be expanded upon by the applied mathematics and quantum physics communities. This linkage may prompt exploratory research into classical and quantum computing applications that benefit from classical approaches inspired by quantum methodologies.

In conclusion, while this representation and related operations do not immediately surpass existing BDD methods, they proffer a compelling computational framework that simplifies implementation and could lead to developments in representing and handling Boolean functions in varied computational contexts. Researchers in both computational science and quantum physics may find value in exploring BMPs further, particularly as they seek efficient methods to manage complex Boolean function representations.