Coined Quantum Walks on Complex Networks for Quantum Computers

TL;DR

The paper tackles implementing coined discrete-time quantum walks on irregular complex networks where varying node degrees complicate circuit design.It introduces a dual-register encoding that enables a SWAP-based shift, reducing resource overhead relative to edge-encoding approaches, and implements the design in Qmod.Through simulations on ER, WS, and BA networks, the circuit depth scales approximately as $D \approx 40 N^{1.9}$, independent of topology, and time evolution scales as $t^{0.86}$.Hardware experiments on IBM Torino show that topology-aware synthesis helps for larger Watts–Strogatz graphs but can incur overhead for small graphs, indicating topology-aware design is crucial for practical graph-based quantum algorithms as devices scale toward fault tolerance.

Abstract

We propose a quantum circuit design for implementing coined quantum walks on complex networks. In complex networks, the coin and shift operators depend on the varying degrees of the nodes, which makes circuit construction more challenging than for regular graphs. To address this issue, we use a dual-register encoding. This approach enables a simplified shift operator and reduces the resource overhead compared to previous methods. We implement the circuit using Qmod, a high-level quantum programming language, and evaluated the performance through numerical simulations on Erdős-Rényi, Watts-Strogatz, and Barabási-Albert models. The results show that the circuit depth scales as approximately $N^{1.9}$ regardless of the network topology. Furthermore, we execute the proposed circuits on the ibm\_torino superconducting quantum processor for Watts-Strogatz models with $N=4$ and $N=8$. The experiments show that hardware-aware optimization slightly improved the $L_1$ distance for the larger graph, whereas connectivity constraints imposed overhead for the smaller one. These results indicate that while current NISQ devices are limited to small-scale validations, the polynomial scaling of our framework makes it suitable for larger-scale implementations in the early fault-tolerant quantum computing era.

Figures (6)

• Figure 1: The definition of coined quantum walk on complex networks. $j$ is the neighbor of node $i$.
• Figure 2: The quantum circuit for implementing the coined quantum walk on complex networks for the walker step as $t=1$. The first dashed box from the left side represents the initial state of the quantum walker $\ket{\psi(0)}$. The half-filled circle denotes a control qubit based on its $0/1$ binary state. The second dashed-box represents coin operator $\hat{C} = \sum_i\ket{i}\bra{i}\otimes \hat{C}_i$. The third dashed box represents the swap operator as the shift operator $\hat{S}$.
• Figure 3: The probability histogram of quantum walk on Erdös–Rényi random graph with $n=10$ nodes and edge probability $p=0.3$. The exact and circuit in the legend represent the exact value and simulation value by the state vector simulator for the quantum walk algorithm, respectively. (a) $t=1$ (b) $t=2$ (c) $t=3$ and (d)$t=4$.
• Figure 4: (a) The node-dependent depths of coined quantum walk circuits for the ER model, WS model, and BA model. The circuits are evaluated at time step $t=1$. (b) The time-dependent depths of coined quantum walk circuits for the ER model, WS model, and BA model. The circuits are evaluated for $N=2^5$.
• Figure 5: Time-evolution of the probability distribution for a quantum walk on two different WS models. The histograms compare the exact theoretical simulation (red bars) with experimental results obtained from the ibm_torino backend (10000 shots). The experimental data contrast hardware (HW) agnostic (dark blue outline) and HW-aware (cyan filled) synthesis strategies. (a) A WS model with $N=4$, $k=2$, and $\beta=0.2$. (b)–(e) Probability distributions at time steps $t=1$ to $t=4$ for the graph in (a). (f) A WS model with $N=8$, $k=2$, and $\beta=0.2$. (g)–(j) Corresponding probability distributions for time steps $t=1$ to $t=4$.