Adiabatic Quantum Algorithm for Search Engine Ranking

We propose an adiabatic quantum algorithm for generating a quantum pure state encoding of the PageRank vector, the most widely used tool in ranking the relative importance of internet pages. We present extensive numerical simulations which provide evidence that this algorithm can prepare the quantum PageRank state in a time which, on average, scales polylogarithmically in the number of web pages. We argue that the main topological feature of the underlying web graph allowing for such a scaling is the out-degree distribution. The top-ranked $\log (n)$ entries of the quantum PageRank state can then be estimated with a polynomial quantum speed-up. Moreover, the quantum PageRank state can be used in “$q$-sampling” protocols for testing properties of distributions, which require exponentially fewer measurements than all classical schemes designed for the same task. This can be used to decide whether to run a classical update of the PageRank.

Figure 1

Top panel: The typical adiabatic error $[\epsilon {]}_{\mathrm{ave}}$ scales approximately as $T^{-0.48}$. Results are for a system of size $n=16$ (we checked different sizes obtaining similar results), averaged over 100 random web-graph realizations. Bottom panel: $[\lambda {]}_{\mathrm{ave}}$ scales as $\log \log (n)$, with a prefactor which is approximately 3. Results were averaged over 1000 random web-graph realizations. See text for more details.

Figure 2

Scaling of the inverse of the average minimum gap, $1/[\delta {]}_{\mathrm{ave}}$, for the preferential attachment model (top panel), and copying model (bottom panel). We checked numerically that the same logarithmic scaling holds for the averaged inverse gap $[1/\delta {]}_{\mathrm{ave}}$, with the latter slightly larger than $1/[\delta {]}_{\mathrm{ave}}$ for all graph sizes $n$ [20]. Note that for the copying model the parameter $p$, relative to the in-degree distribution, affects only the prefactor of the scaling.