Diagrammatic reasoning via languages such as ZX Calculus is a powerful tool in theoretical quantum computing. However, physical quantum computers do not understand diagrams and require explicit quantum circuits as inputs. Finding a circuit equivalent to the given diagram is known as circuit extraction. Such problems in graphical calculi form a limitation preventing the wider adoption of diagrammatic reasoning. Computational complexity theory formalises the difficulty of problems. For ZX diagrams, the circuit extraction problem is #P-hard and unlikely to have an efficient algorithm. Therefore, partial solutions are critical. Currently, efficient circuit extraction is only possible for diagrams containing a structure of so-called Pauli flow. This structure is necessary and sufficient for robust determinism in measurement-based quantum computation (MBQC) – a model of quantum computation, an alternative to circuits, where adaptive single-qubit measurements drive computation. Thus, Pauli flow links MBQC, ZX Calculus, and the circuit model. It is known how to find flow structures in polynomial time when they exist; nevertheless, their lengthy and complex definitions often hinder working with them. In this thesis, I present contributions to two connected topics: the hardness of problems in graphical calculi and Pauli flow as a method for overcoming circuit extraction. First, I show that circuit extraction can be #P-hard for phase-free ZH diagrams, extending the previous result for ZX diagrams. I also establish a relation between phase-free ZH and
NP⧣P class. This involves weakening the oracle, crafting a complete problem, and encoding the problem into phase-free ZH. My next contribution concerns Pauli flow. I simplify the Pauli flow definition by providing a new algebraic interpretation. This involves defining two matrices arising from the adjacency matrix of the underlying graph. Then, Pauli flow corresponds to the existence of a right-inverse of the first matrix, which, when multiplied by the second matrix, results in a matrix of a directed acyclic graph. Based on the newly defined algebraic interpretation, I obtain O(n
3) algorithms for finding Pauli flow, improving on the previous O(n
4) bounds. I also introduce a lower bound for Pauli flow-finding by linking it to linear algebra problems over F
2. Finally, when given an unlabelled open graph, I show how to find measurement labelling resulting in Pauli flow.
The increasing complexity of 5G and emerging 6G wireless networks has intensified the need for scalable optimization techniques capable of addressing computationally challenging combinatorial problems. While quantum optimization offers a promising alternative, current near-term quantum hardware remains limited by qubit availability, noise, and scalability constraints. Pauli Correlation Encoding (PCE) has emerged as a qubit-efficient optimization framework that addresses these challenges through polynomial variable compression. However, the impact of compression on representability and solution recovery remains poorly understood. In this work, we investigate the hypothesis that increasing compression ratios introduce representability constraints arising from algebraic closure relationships among encoded Pauli observables. Using Maximum Independent Set benchmark problems, we evaluate multiple PCE ansatz designs and compression ratios. We find that ansatz selection is critical to both solution quality and feasibility, with two ansatz families consistently outperforming alternatives and achieving approximation ratios approaching 0.9. Furthermore, increasing circuit repetitions provides limited benefit and can degrade performance. Most notably, reducing compression by increasing the number of available qubits significantly improves raw solution quality, while post-processed performance remains comparatively stable. These findings suggest that representability effects play an important role in compressed PCE encodings and provide new insight into the relationship among compression, ansatz design, decoding, and solution quality in scalable quantum optimization.