SQUWALS: A Szegedy QUantum WALks Simulator

TL;DR

The paper presents SQUWALS, a memory-efficient classical simulator for Szegedy's quantum walk that scales as $\mathcal{O}(N^2)$ in both time and memory, enabling accurate simulation on arbitrary graphs with dense transition matrices. It introduces a matrix-state representation and building-block operators (reflections, swaps, and oracles) to implement Szegedy-like unitaries efficiently, including extensions with complex phases and semiclassical/mixed-state dynamics. The authors provide a detailed methodology for simulating semiclassical Szegedy walks and mixed states, and they demonstrate the library's capabilities with high-level applications such as quantum PageRank. The work offers practical impact by enabling error-free classical testing of Szegedy-based algorithms and facilitating scalable exploration of quantum-walk-inspired optimization and learning tasks, with future plans for GPU acceleration and expanded algorithms.

Abstract

Szegedy's quantum walk is an algorithm for quantizing a general Markov chain. It has plenty of applications such as many variants of optimizations. In order to check its properties in an error-free environment, it is important to have a classical simulator. However, the current simulation algorithms require a great deal of memory due to the particular formulation of this quantum walk. In this paper we propose a memory-saving algorithm that scales as $\mathcal{O}(N^2)$ with the size $N$ of the graph. We provide additional procedures for simulating Szegedy's quantum walk over mixed states and also the Semiclassical Szegedy walk. With these techniques we have built a classical simulator in Python called SQUWALS. We show that our simulator scales as $\mathcal{O}(N^2)$ in both time and memory resources. This package provides some high-level applications for algorithms based on Szegedy's quantum walk, as for example the quantum PageRank.

Figures (9)

• Figure 1: How to represent a vector state by a matrix state for a network with $N=3$ nodes. The blocks of the vector, indexed by the computational basis of the first register, correspond to the columns of the matrix. Beware that the indices in the coefficients $a_{ij}$ are swapped with respect to the ones of the usual matrix notation $\Phi_{ij}$.
• Figure 2: How to condense all the information of the $\left|\psi_i\right>$ states in a matrix for a network with $N=3$ nodes. Each column $i$ of the matrix $\Psi$ corresponds to the non-null block of each $\left|\psi_i\right>$ state.
• Figure 3: How to obtain the vector $C$ in \ref{['Ci_2']} for a network with $N=3$ nodes. The element-wise multiplication between $\Phi$ and $\Psi$ results in a matrix whose $i$-th column has the product of the non-null elements of $\left|\psi_i\right>$ and the elements of $\left|\phi\right>$. Thus, summing the elements of each column, the coefficients $C_i$ are obtained.
• Figure 4: How to obtain the parallel component of the state $\left|\phi\right>$ for a network with $N=3$ nodes. The element-wise product of $C$ and $\Psi$ results in a matrix where each column represents the non-null elements of $C_i\left|\psi_i\right>$, which ends up representing the vector $\left|\phi_\parallel\right>$.
• Figure 5: How to apply the oracle operators for a network with $N=3$ nodes. In this case nodes $0$ and $2$ are being marked. In order to mark them in the first (second) register, the corresponding columns (rows) are multiplied by $-1$.