A Perfectly Distributable Quantum-Classical Algorithm for Estimating Triangular Balance in a Signed Edge Stream

Abstract

We develop a perfectly distributable quantum-classical streaming algorithm that processes signed edges to efficiently estimate the counts of triangles of diverse signed configurations in the single pass edge stream. Our approach introduces a quantum sketch register for processing the signed edge stream, together with measurement operators for query-pair calls in the quantum estimator, while a complementary classical estimator accounts for triangles not captured by the quantum procedure. This hybrid design yields a polynomial space advantage over purely classical approaches, extending known results from unsigned edge stream data to the signed setting. We quantify the lack of balance on random signed graph instances, showcasing how the classical and hybrid algorithms estimate balance in practice.

Figures (5)

• Figure 1: Quantum-Classical Estimation of Balance in the Streaming Model. An overview of our hybrid quantum-classical algorithm for estimating balance.
• Figure 2: Triangle Types and their Balance in a Signed Network. Triangle types labeled by their number of positive edges, with triangles with odd number of plus edges categorized as balanced while those with even positive edges are categorized as unbalanced. Under structural balance theory, real-life network dynamics may evolve towards more balanced local substructures. As such, counting triangle types is essential for analyzing networks.
• Figure 3: Circuit Diagram for Quantum Sketching Algorithm. The circuit for the quantum sketchpad is described by 3 types of action to update the quantum state as described in \ref{['subsec:T1_q_estimator']}. First, $\mathtt{create}$ is applied to generate the initial sketchpad, depicted is the case that $\log(2m)$ is a power of 2 such that Hadamard gates can be used. For each signed edge $(vw,\sigma)$ in the stream, $\mathtt{query\_edges}$ is applied for every $u \in |V|$ related to $vw$ and $\sigma$. If one of the PVMs results in a measurement besides $\perp$, the algorithm halts and the value is returned. Lastly, $\mathtt{insert}$ is applied to add $(vw,\sigma)$ and $(wv,\sigma)$ to the sketchpad and we proceed to the next edge.
• Figure 4: A Schematic for Signed Triangle Estimation with Distributed Resources. During the mapping, based on Alg. \ref{['alg:t1_estimator_hybrid']}, we allocate QPUs and CPUs with their respective initial (empty) sketchpads. During the Graph Streaming, each edge is passed to these distributed CPUs and QPUs. At step $\ell$ in the stream, based on the hash functions $g(\ell)$, $h_I(\ell)$ and $h_V$, each CPU or QPU applies a query (see \ref{['fig:q_sketch_circ']}). Each CPU and QPU will terminate asynchronously and provide their estimate. During the reduction, we calculate the median-of-means (MoM in diagram) based on parameters $\delta$ and $\epsilon$.
• Figure 5: Violin Plot Comparing the Performance of Classical and Quantum-Classical Triangular Balance Estimators. Comparison between $\overline{\mathcal{B}}_{\triangle}^H$ and $\overline{\mathcal{B}}_{\triangle}^C$ outputted by \ref{['alg:bal_estimator_hybrid']} versus \ref{['alg:bal_estimator_classical']}, respectively. These algorithms are tested over random Erdős--Rényi instances of sizes $n \in \{30,40,50\}$. The relative error is computed as $|\overline{\mathcal{B}}_{\triangle}^A - \mathcal{B}_{\triangle}|/\mathcal{B}_{\triangle}$ for $A \in \{H,C\}$. Both algorithms are ran with accuracy parameter $\epsilon = 0.1$ and the performance is better than expected. The hybrid algorithm slightly outperforming the purely classical but does experience higher variance. This is to be expected, as there are two sources of error for the hybrid algorithm, on from the quantum and one from the classical estimators.

Theorems & Definitions (27)

• Corollary 1
• Lemma 2: Analysis of \ref{['alg:t1_greater_k_sub']}
• Corollary 3
• Lemma 4: Performance of \ref{['alg:t1_less_k_sub']}
• Corollary 5
• Theorem 6
• proof
• Corollary 7
• Conjecture 8
• Lemma 9