Toward Minimum Graphic Parity Networks

2025-09-12

AI Summary

This work addresses the efficient synthesis of quantum circuits for the Ising model: given a connected graph, construct a CNOT+Rz circuit implementing its parity network while minimizing the number of CNOT gates. Method: We introduce the novel notion of “tight graphs”, fully characterizing those graphs achieving the theoretical lower bound $m+n-1$ on CNOT count; develop the first linear-time optimal synthesis algorithm; employ graph-theoretic analysis, parameterized complexity, tree-width–based dynamic programming, and quantum equivalence transformations. Contribution/Results: We prove tight graphs are induced-subgraph-closed and their recognition is NP-complete. For graphs with girth ≥5, we improve the lower bound to $m+Omega(n^{1.5})$, and propose a randomized algorithm achieving expected gate count $m+O(n^{1.5}sqrt{log n})$. Our framework yields optimal synthesis for multiple graph classes, unifying combinatorial and quantum-circuit optimization techniques.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Machine Learning: Probabilistic Circuits and Graphical Models
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

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 $m+Ω(m) = m+Ω(n^{1.5})$ 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 $ extsf{NP}$-complete, which is complemented with a fixed-parameter tractable algorithm parameterized by the treewidth.

Problem

• Optimizing quantum circuit synthesis for CNOT and Rz gates
• Finding minimum gate count for graphic parity networks
• Developing algorithms for efficient quantum combinatorial optimization

Innovation

• Minimum gate quantum circuits synthesis
• Randomized algorithm for network optimization
• Linear-time synthesis for specific graph class