Toward Minimum Graphic Parity Networks

Quantum circuits composed of CNOT and $R_z$ are fundamental building blocks of many quantum algorithms, so optimizing the synthesis of such quantum circuits is crucial. We address this problem from a theoretical perspective by studying the graphic parity network synthesis problem. A graphic parity network for a graph $G$ is a quantum circuit composed solely of CNOT gates where each edge of $G$ is represented in the circuit, and the final state of the wires matches the original input. We aim to synthesize graphic parity networks with the minimum number of gates, specifically for quantum algorithms addressing combinatorial optimization problems with Ising formulations. We demonstrate that a graphic parity network for a connected graph with $n$ vertices and $m$ edges requires at least $m+n-1$ gates. This lower bound can be improved to $m+\Omega(m) = m+\Omega(n^{1.5})$ when the shortest cycle in the graph has a length of at least five. We complement this result with a simple randomized algorithm that synthesizes a graphic parity network with expected $m + O(n^{1.5}\sqrt{\log n})$ gates. Additionally, we begin exploring connected graphs that allow for graphic parity networks with exactly $m+n-1$ gates. We conjecture that all such graphs belong to a newly defined graph class. Furthermore, we present a linear-time algorithm for synthesizing minimum graphic parity networks for graphs within this class. However, this graph class is not closed under taking induced subgraphs, and we show that recognizing it is $\textsf{NP}$-complete, which is complemented with a fixed-parameter tractable algorithm parameterized by the treewidth.