Continuous-time quantum walk on a random graph using quantum circuits

TL;DR

This work presents a scalable quantum circuit formalism to simulate CTQW on random graph structures, especially focusing on Erd\H{o}s-R\'enyi random graphs, and efficiently implements the time evolution of the graph Laplacian, using the Trotterization scheme.

Abstract

Quantum walks, particularly continuous-time quantum walks (CTQW), have emerged as powerful tools for modeling quantum transport, simulating complex dynamics, and developing quantum algorithms with potential speedups over classical counterparts. In this work, we present a scalable quantum circuit formalism to simulate CTQW on random graph structures, especially focusing on Erdős-Rényi random graphs. Our quantum circuit construction efficiently implements the time evolution of the graph Laplacian, using the Trotterization scheme. We investigate key dynamical properties, \emph{i.e.,} the localization behavior of the CTQW. Our quantum circuit implementation over random graph ensures that the circuit design can work on any graph structure, thereby laying the foundation for realizing CTQW-based quantum simulations efficiently.

Figures (9)

• Figure 1: (a) Random graph $G(N, p)$ with $N = 5$ and each edge is present independently with probability $p = 0.4$. (b–f) Decomposition of the original graph into subgraphs, each corresponding to a distinct $1$-sparse Hamiltonian representation.
• Figure 2: Fidelity plot of the 6-qubit quantum circuit simulating continuous-time quantum walk on Erdős-Rényi graphs for four different edge probabilities $p = 0.1, 0.4, 0.7, 1.0$. The simulation is performed using two Trotter step sizes $\delta t = 10^{-2}$ (dashed lines) and $\delta t = 10^{-3}$ (solid lines). Fidelity is computed against the exact unitary evolution operator $\exp(-iHt)$ using Eq. \ref{['eq:fidelity']}. The results demonstrate that smaller Trotter step sizes yield higher circuit fidelity over longer evolution times, with fidelity degrading more rapidly for higher connectivity (larger $p$).
• Figure 3: Cutoff Time ($\tau_c$) at which the quantum circuit fidelity $\sim 0.95$ plotted against the number of qubits $n$, for different edge probabilities $p$ in the underlying Erdős-Rényi graph. (a) Results for Trotter time step $\delta t = 10^{-2}$. (b) Same for $\delta t = 10^{-3}$. The fidelity decays more rapidly with increasing number of qubits $n$, and the decay is further for graphs with higher connectivity $p$ and larger Trotter step size $\delta t$.
• Figure 4: Combined analysis of the cutoff time ($\tau_c$) for fidelity decay (falls below $95\%$) plotted against the number of qubits $n$, including data from both Fig. \ref{['fig:fidelity_02']} and Fig. \ref{['fig:fidelity_03']}. Each curve corresponds to a different Erdős-Rényi graph connectivity $p$ and Trotter step size $\delta t$. The straight lines represent exponential fits of the form $T(n) \sim e^{m n + c}$, with fitted slope ($m$) mentioned in the legend.
• Figure 5: Scaling analysis of the Trotterization error ($\varepsilon_{\delta t}$) at a single Trotter step $\delta t$ as a function of qubit number $n$. Theoretical upper bound of Trotter error ($\varepsilon_{\delta t}$), given by $\delta t^2 \cdot \epsilon \cdot 2^{2n - 1}$, is also fitted with straight lines, showing a slope of $\sim 1.39$ for both $\delta t$ values.

Theorems & Definitions (5)

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