Randomized SearchRank: A Semiclassical Approach to a Quantum Search Engine

TL;DR

A modification of the quantum SearchRank algorithm is proposed, replacing the underlying Szegedy quantum walk with a semiclassical walk, and this new algorithm is called randomized SearchRank, since it corresponds to a quantum walk over a randomized mixed state.

Abstract

The quantum SearchRank algorithm is a promising tool for a future quantum search engine based on PageRank quantization. However, this algorithm loses its functionality when the $N/M$ ratio between the network size $N$ and the number of marked nodes $M$ is sufficiently large. We propose a modification of the algorithm, replacing the underlying Szegedy quantum walk with a semiclassical walk. To maintain the same time complexity as the quantum SearchRank algorithm we propose a simplification of the algorithm. This new algorithm is called Randomized SearchRank, since it corresponds to a quantum walk over a randomized mixed state. The performance of the SearchRank algorithms is first analyzed on an example network, and then statistically on a set of different networks of increasing size and different number of marked nodes. On the one hand, to test the search ability of the algorithms, it is computed how the probability of measuring the marked nodes decreases with $N/M$ for the quantum SearchRank, but remarkably it remains at a high value around $0.9$ for our semiclassical algorithms, solving the quantum SearchRank problem. The time complexity of the algorithms is also analyzed, obtaining a quadratic speedup with respect to the classical ones. On the other hand, the ranking functionality of the algorithms has been investigated, obtaining a good agreement with the classical PageRank distribution. Finally, the dependence of these algorithms on the intrinsic PageRank damping parameter has been clarified. Our results suggest that this parameter should be below a threshold so that the execution time does not increase drastically.

Figures (14)

• Figure 1: Semiclassical Szegedy's walk of class II. Let us denote $x_{t_c}$ as the position of the walker at classical time step $t_c$. Thus, we start at node $x_0$. We prepare the proxy state $\left|\psi_{x_0}\right>$ in \ref{['psi_i']} and perform the quantum evolution, parameterized by the quantum time $t_q$ being the number of applications of the quantum walk evolution operator. After measuring the second register the system collapses to a particular node $x_1$ in the second register. The remaining information in the first register, represented by a question mark, plays no role in the algorithm, so we do not worry about it. Before proceeding to the next step, the system must be reset. To this end, the first register is erased, so that it is forced non-unitarily to be in $\left|0\right>_1$. After that, we use the measured information about the node $x_1$ to prepare with a suitable unitary evolution the new proxy state, $\left|\psi_{x_1}\right>$ in \ref{['psi_i']}, completing the reset of the system. This constitutes a classical step of the semiclassical walk, and the process is repeated the number of classical steps $t_c$ as desired.
• Figure 2: Quantum circuit diagrams of SearchRank algorithms. In the Semiclassical SearchRank (upper panel in green), the first step consists on the initialization of the mixed state $\rho$, the quantum evolution of $t_q$ times the unitary operator $W_Q$, and a measurement in the second register. After that, each classical step consists on a reset of the system depending on the previous measurement, the quantum evolution, and a measurement. In total, $t_c^*$ classical steps are carried out until convergence. The blue dashed box represents the Randomized SearchRank, which is a simplified semiclassical algorithm with only one classical step. In the quantum SearchRank (bottom left panel in red), the initial state $\left|\Psi^{(0)}\right>$ is prepared, the quantum evolution is performed and the system is measured. The right dashed box is a legend explaining the meaning of the different elements (quantum circuits) from which the SearchRank algorithms are constructed. In particular, notice that the reset operation is a combination of a measurement and evolution operation.
• Figure 3: Scale-free network with 32 nodes. The inner (green) nodes correspond to the main nodes. The middle (orange) nodes correspond to secondary nodes. The outer (blue) nodes correspond to residual nodes without links pointing to them.
• Figure 4: PageRank and SearchRank distributions of the network with 32 nodes of Figure \ref{['F:graph']}. The marked nodes (2, 7, 13 and 21) have a highlighted color. In the three SearchRank algorithms the marked nodes have an amplified importance.
• Figure 5: Probability of measuring one of the marked nodes versus the quantum time for the three SearchRank algorithms applied in the scale-free graph with 32 nodes of Figure \ref{['F:graph']}. The first maximum of each curve is marked with a dot, and they are surrounded by a circle. The vertical dashed line represents the reference time $\sqrt{N/M}$. The horizontal dashed lines represent the probability of the marked nodes in the classical (black) and quantum (blue) PageRank distributions.