The one-way model of quantum computation is an alternative to the circuit model. A one-way computation is driven entirely by successive adaptive measurements of a pre-prepared entangled resource state. For each measurement, only one outcome is desired; hence a fundamental question is whether some intended measurement scheme can be performed in a robustly deterministic way. So-called flow structures witness robust determinism by providing instructions for correcting undesired outcomes. Pauli flow is one of the broadest of these structures and has been studied extensively. 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. We simplify these definitions by providing a new algebraic interpretation of Pauli flow. This involves defining two matrices arising from the adjacency matrix of the underlying graph: the flow-demand matrix $M$ and the order-demand matrix $N$. We show that Pauli flow exists if and only if there is a right inverse $C$ of $M$ such that the product $NC$ forms the adjacency matrix of a directed acyclic graph. From the newly defined algebraic interpretation, we obtain $\mathcal{O}(n^3)$ algorithms for finding Pauli flow, improving on the previous $\mathcal{O}(n^4)$ bound for finding generalised flow, a weaker variant of flow, and $\mathcal{O}(n^5)$ bound for finding Pauli flow. We also introduce a first lower bound for the Pauli flow-finding problem, by linking it to the matrix invertibility and multiplication problems over $\mathbb{F}_2$.

翻译：量子计算的单向量子计算模型是电路模型的一种替代方案。单向量子计算完全通过对预先制备的纠缠资源态进行连续的自适应测量来驱动。对于每次测量，仅期望获得特定结果；因此一个基本问题是，某些预期的测量方案是否能够以鲁棒的确定性方式执行。所谓的流结构通过提供纠正非期望结果的指令来验证鲁棒确定性。泡利流是这类结构中最具普适性的之一，并已得到广泛研究。已知当流结构存在时，可以在多项式时间内找到它们；然而，其冗长复杂的定义常常阻碍了相关研究工作。我们通过为泡利流提供一种新的代数解释来简化这些定义。这涉及从底层图的邻接矩阵导出两个矩阵：流需求矩阵$M$和阶需求矩阵$N$。我们证明泡利流存在的充要条件是：存在$M$的右逆$C$，使得乘积$NC$构成有向无环图的邻接矩阵。基于新定义的代数解释，我们获得了寻找泡利流的$\mathcal{O}(n^3)$算法，改进了先前寻找广义流（流的一种较弱变体）的$\mathcal{O}(n^4)$界限，以及寻找泡利流的$\mathcal{O}(n^5)$界限。我们还通过将该问题与$\mathbb{F}_2$上的矩阵可逆性及乘法问题相关联，首次提出了泡利流寻找问题的下界。