Phase Transitions and Noise Robustness of Quantum Graph States

TL;DR

The paper shows that the fidelity between an ideal graph state and its IID Pauli-noised counterpart can be mapped to the partition function of a classical spin system, enabling efficient analysis via mean-field theory, transfer matrices, and Monte Carlo simulations. It reveals phase transitions in fidelity driven by graph degree and spatial dimensionality: in 2D, transitions occur for $d\ge6$, while in 3D they appear for $d\ge5$, with fully connected graphs exhibiting no sharp transition. Lower connectivity and dimensionality yield smoother crossovers and greater robustness, whereas high dimensionality or high connectivity induces fragility. The work provides a scalable framework for robustness assessment of graph states relevant to MBQC and quantum information tasks, and suggests directions for extending the approach to other noise models and stabilizer states.

Abstract

Graph states are entangled states that are essential for quantum information processing, including measurement-based quantum computation. As experimental advances enable the realization of large-scale graph states, efficient fidelity estimation methods are crucial for assessing their robustness against noise. However, calculations of exact fidelity become intractable for large systems due to the exponential growth in the number of stabilizers. In this work, we show that the fidelity between any ideal graph state and its noisy counterpart under IID Pauli noise can be mapped to the partition function of a classical spin system, enabling efficient computation via statistical mechanical techniques, including transfer matrix methods and Monte Carlo simulations. Using this approach, we analyze the fidelity for regular graph states under depolarizing noise and uncover the emergence of phase transitions in fidelity between the pure-state regime and the noise-dominated regime governed by both the connectivity (degree) and spatial dimensionality of the graph state. Specifically, in 2D, phase transitions occur only when the degree satisfies $d\ge 6$, while in 3D they already appear at $d\ge 5$. However, for graph states with excessively high degree, such as fully connected graphs, the phase transition disappears, suggesting that extreme connectivity suppresses critical behavior. These findings reveal that robustness of graph states against noise is determined by their connectivity and spatial dimensionality. Graph states with lower degree and/or dimensionality, which exhibit a smooth crossover rather than a sharp transition, demonstrate greater robustness, while highly connected or higher-dimensional graph states are more fragile. Extreme connectivity, as the fully connected graph state possesses, restores robustness.

Figures (8)

• Figure 1: Illustration of how the local operator $\tau_i\in\{X,Y,Z,I\}$ acting on qubit $i$ in a stabilizer $S_\ell$ is determined by the presence or absence of $g_i$ ($s_i=\pm 1$) and the parity of the number of stabilizer generators (spin $\uparrow$s) connected to qubit $i$. In the presence of $g_i$, (a) $\tau_i=X$ when the number of generators is even, and (b) $\tau_i=Y$ when the number of generators is odd. In the absence of $g_i$, (c) $\tau_i=I$ when the number of generators is even, and (d) $\tau_i=Z$ when the number of generators is odd.
• Figure 2: The $p$-dependence of the inverse temperature $\beta$: $p\to 0$ corresponds to the low-temperature region ($\beta\rightarrow\infty$), while $p\to 3/4$ corresponds to the high-temperature region ($\beta\rightarrow 0$).
• Figure 3: Mean-field results for the fidelity $F^{1/n}$ of 1D (a), 2D (b), and 3D (c) cluster states under depolarizing noise (red curves). The blue curves show the corresponding results obtained using the transfer matrix method for the 1D cluster state with $n=1000$ qubits (a), and Monte Carlo simulations for the 2D cluster state with $n_x=n_y=60$ (b), and the 3D cluster state with $n_x=n_y=n_z=6$ (c). The dashed magenta lines correspond to $1-p$.
• Figure 4: (a) 1D cluster state, where black dots represent qubits and black lines indicate $CZ$ gates. (b) Fidelity, (c) internal energy, and (d) specific heat as functions of the noise parameter $p$ for the 1D cluster state with number of qubits $n=1000$. The solid red curves in (b), (c), and (d) show the results obtained using the transfer matrix method. The dashed magenta line in (b) corresponds to $1-p$. $k_{\rm B}$ denotes the Boltzmann constant.
• Figure 5: 2D $d$-regular graph states for $3\le d\le 8$. Each black dot and line indicate a qubit and $CZ$ gate, respectively.