Resolving degeneracies in Google search via quantum stochastic walks

TL;DR

This work addresses degeneracies in classical PageRank and shows that quantum stochastic walks (QSW) can yield a quantum PageRank (QPR) with two practical decoherence schemes: only incoherence (OI) and dephasing with incoherence (DI). By systematically applying QPR to Erdős–Rényi, Watts–Strogatz, scale-free, and spatial networks, the authors demonstrate substantial reductions in rank degeneracies while maintaining convergence times that are often comparable to, or even better than, classical CPR, especially for WS networks. Pure dephasing (PD) is found unsuitable for ranking, while OI and DI provide robust degeneracy resolution across networks, including small eight-vertex graphs used for detailed analysis. The results highlight the potential of quantum-inspired ranking methods to enhance search quality on classical hardware and lay groundwork for future quantum implementations, with practical implications for internet-scale graph problems and network analysis.

Abstract

The Internet is one of the most valuable technologies invented to date. Among them, Google is the most widely used search engine. The PageRank algorithm is the backbone of Google search, ranking web pages according to relevance and recency. We employ quantum stochastic walks (QSWs) to improve the classical PageRank (CPR) algorithm based on classical continuous time random walks. We implement QSW via two schemes: only incoherence and dephasing with incoherence. PageRank using QSW with only incoherence or QSW with dephasing and incoherence best resolves degeneracies that are unresolvable via CPR and with a convergence time comparable to that for CPR, which is generally the minimum. For some networks, the two QSW schemes obtain a convergence time lower than CPR and an almost degeneracy-free ranking compared to CPR.

Figures (23)

• Figure 1: (a) The ER network Zachary Karate club. This network is inbuilt in Mathematica, see Ref. zacharynetwork. Ranks for each vertex for QSW with (b) only incoherence (QPR-OI) and (c) dephasing with incoherence (QPR-DI) for ER network Zachary Karate club. CPR values are also given for comparison.
• Figure 2: (a) A randomly generated ER network Bernoulli graph distribution of 100 vertices. Bernoulli graph distribution is available in Mathematica (see Ref. bernoulligraph). BernoulliGraphDistribution[n,p] in Mathematica shows a graph of $n$ vertices and the probability of an edge existing $p$. We have used BernoulliGraphDistribution[100,0.6]. Ranks for each vertex for QSW with (b) only incoherence (QPR-OI) and (c) dephasing with incoherence (QPR-DI) for a randomly generated Bernoulli graph distribution. CPR values are also given for comparison.
• Figure 3: Convergence time $\tau_{QPR}/\tau_{CPR}$ versus $\omega$ for QSW with (a) only incoherence ($\tau_{QPR-OI}/\tau_{CPR}$) and (b) dephasing with incoherence ($\tau_{QPR-DI}/\tau_{CPR}$) for the ER network Zachary Karate club. Convergence times for CPR are better than QPR.
• Figure 4: Convergence time $\tau_{QPR}/\tau_{CPR}$ versus $\omega$ for QSW with (a) only incoherence ($\tau_{QPR-OI}/\tau_{CPR}$) and (b) dephasing with incoherence ($\tau_{QPR-DI}/\tau_{CPR}$) for the ER network Bernoulli graph distribution. The plots have been averaged over 5 randomly generated Bernoulli graph distribution with $n=100$, $p=0.6$, same as in Fig. \ref{['ER_bernoulli']}. Convergence time for QPR is marginally better than CPR at $\omega=0.9$.
• Figure 5: (a) A randomly generated WS network of 100 vertices. WattsStrogatzGraphDistribution[n,p] in Mathematica shows a graph of $n$ vertices and the probability of rewiring $p$. We have used WattsStrogatzGraphDistribution[100,0.2]. Ranks for each vertex for QSW with (b) only incoherence (QPR-OI) and (c) dephasing with incoherence (QPR-DI) for a randomly generated WS network. CPR values are also given for comparison.