Adaptive Framework for Failure-Aware Protocols in Fusion-Based Graph-State Generation

TL;DR

This work considers the generation of photonic graph states in a linear optics setting where sequential non-deterministic fusion measurements are used to build large graph states out of small linear clusters and develops a framework to optimize the building process using graph theoretic characterizations of fusion networks.

Abstract

We consider the generation of photonic graph states in a linear optics setting where sequential non-deterministic fusion measurements are used to build large graph states out of small linear clusters and develop a framework to optimize the building process using graph theoretic characterizations of fusion networks. We present graph state generation protocols for linear cluster resource states and Type-I/Type-II fusions which are adaptive to fusion failure, that is, they reuse leftover graph states in the remaining building process. To estimate hardware costs, we interpret our protocols as finite Markov processes. This viewpoint allows to cast the expected number of fusion measurements until success as a first passage problem. We then deploy a pipeline of polynomial algorithms to optimize arbitrary graph states, extract fusion networks and find beneficial orderings of fusions with the goal of lowering the corresponding mean first passage times. We evaluate our pipeline for different initial resource states and fusion mechanisms with varying success probabilities. Results show that our strategies can reduce the fusion overhead by several orders of magnitude when compared to simple repeat until success protocols, especially for realistic fusion success probabilities between 50-75 %.

Figures (12)

• Figure 1: A schematic overview of the graph state generation process. We consider resource generating modules (RSG), consisting of small entangled units, like 2 or 3-qubit linear cluster states serving as input for a sequential buildup process using fusions (F) on adjacent qubits.
• Figure 2: Examples of graph-theoretic rewrites on two simple graphs. Left: a local complementation operation is applied at vertex $u$ which toggles the edges between neighbors of $u$. Right: a pivoting operation on edge $\{u,v\}$ toggling edges between exclusive neighbors of $u$, $v$ and the shared neighbors of $u$ and $v$. Further, $u$ and $v$ switch their neighbors.
• Figure 3: Successful Type-I (left) and Type-II fusion (with observables $\{XZ,ZX\}$) on the middle qubits of two linear clusters with three qubits. The fusion is represented as a dashed blue edge with its endpoints indicating which qubits are inputs of the fusion gate. Upon success, we contract the blue edge in the Type-I case, and apply a Pivot + vertex deletions in the Type-II case.
• Figure 4: Different (non-isomorphic) fusion networks $H,H'$ for the same target graph $G$ with either Bell-pairs and Type-I fusion, or 3-qubit linear clusters and Type-II ($\{XZ,ZX\}$) fusion.
• Figure 5: Adding an edge $\{a,b\}\in G$ to a fusion network for both Type-I and Type-II networks. Characters in brackets denote the labeling before adding the edge, characters without brackets the updated labeling.

Theorems & Definitions (5)

• Definition 1
• Definition 2
• Theorem 1
• Definition 3
• Definition 4