Useful entanglement can be extracted from noisy graph states

TL;DR

This work develops a stabilizer-based approach to extract high-fidelity Bell pairs and GHZ states from noisy graph states, aiming to enable robust measurement-based quantum computation with limited connectivity. It models realistic noise—uncorrelated and correlated edge noise, local Z flips, and fusion-gate imperfections—and analyzes how carefully designed measurement patterns and postselection mitigate these effects. The authors introduce fidelity and a fidelity susceptibility α to quantify robustness, and demonstrate that twisted and crazy graph structures offer improved noise resilience, including regimes where first-order noise can be canceled. The framework provides a general, connectivity-aware method for noise reduction in MBQC, with implications for scalable quantum networks and various experimental platforms.

Abstract

Cluster states and graph states in general offer a useful model of the stabilizer formalism and a path toward the development of measurement-based quantum computation. Their defining structure - the stabilizer group - encodes all possible correlations that can be observed during measurement. The measurement outcomes which are consistent with the stabilizer structure make error correction possible. Here, we leverage both properties to design feasible families of states that can be used as robust building blocks of quantum computation. This procedure reduces the effect of experimentally relevant noise models on the extraction of smaller entangled states from the larger noisy graph state. In particular, we study the extraction of Bell pairs from linearly extended graph states - this has the immediate consequence for state teleportation across the graph. We show that robust entanglement can be extracted by proper design of the linear graph with only a minimal overhead of the physical qubits. This scenario is relevant to systems in which the entanglement can be created between neighboring sites. The results shown in this work provide a mathematical framework for noise reduction in measurement-based quantum computation. With proper connectivity structures, the effect of noise can be minimized for a large class of realistic noise processes.

Figures (12)

• Figure 1: Any two vertices of a graph state associated with a connected graph can be turned into a Bell pair by sequential measurements. a) A grid graph describing the underlying structure of a cluster state. The goal is to generate a Bell pair between the qubits associated with the big red vertices. The chosen path is highlighted in red. b) We measure the qubits associated with the vertices on the path between the red vertices in the $X$ basis and the qubits adjacent to the chosen path in the $Z$ basis. c) The post-measurement state has Bell correlations between the qubits associated with the red vertices. This procedure can be formalized to the so-called X-protocol hahn2019quantum.
• Figure 2: a) The square graph defines the associated graph state $\ket{\square}$ which, as any graph state, possesses an associated structure of the stabilizer$S_{\square}$ composed of specific strings of Pauli operators. Expectation value of any Pauli string $O$ is nonzero if and only if one of $\pm O$ belongs to $S_{\square}$. $X_1 X_3$ is a stabilizer operator of $\ket{\square}$ and therefore it's expectation value is 1. b) Graph states corresponding to the state defined in a) being altered by noise -- here, the noise may remove certain edges (dashed lines), as in the exemplary $\ket{\sqcap}$ defined by the graph shown in the top left corner. Noise processes change the stabilizer group (e.g., by graph modification shown here), which allows for verifiable preparation.
• Figure 3: Partial local measurements on a graph state induce correlations in the remaining qubits. a) If the 4-vertex path graph state is measured according to the shown measurement pattern, that is, measuring the inner vertices in the $X$-basis, we get a connected two-vertex graph. b) The graph here is not connected: the edge $\{2,3\}$ is absent, possibly due to the scenario outlined in \ref{['sec:uenoise']}. If the corresponding graph state is measured according to the same measurement pattern, we get a disconnected two-vertex graph. Measurements can never induce entanglement between parties that were separable before.
• Figure 4: Considered families of graphs for the task of Bell pair extraction, parameterized by the length of the internal part $n$. The terminal qubits, designated for the creation of a Bell pair, are denoted with filled nodes. From the top: path graph, twisted pair, crazy graph.
• Figure 5: Crazy graph of length 3 together with an all-$X$ measurement pattern possesses embedded error-checking and correction structures. Middle: each layer is associated with an $X\otimes X$ stabilizer, which can be used to confirm the proper preparation of the state. Bottom: even if any single internal edge (dotted line between layers 1 and 2) is missing, the terminal qubits correlations are not affected: the operators shown are stabilizers, regardless of whether the edge is present or not.

Theorems & Definitions (7)

• Lemma 1
• proof
• Corollary 2
• Lemma 3
• proof
• Lemma 4
• proof