Quantum Walks-Based Adaptive Distribution Generation with Efficient CUDA-Q Acceleration

TL;DR

This work presents a Quantum Walks-Based Adaptive Distribution Generator (QWs-based ADG) that fuses variational quantum circuits with split-step and entangled quantum walks to learn target probability distributions. Implemented on the CUDA-Q GPU framework, the approach optimizes coin parameters via a classical loop to shape 1D distributions and extends to 2D pattern generation through entangled coin spaces. The results show accurate modeling of diverse distributions, including a log-normal density for option pricing, and high-fidelity 8×8 digit patterns, with performance benefiting from GPU acceleration. The work advances practical quantum-inspired generative modeling and points toward scalable quantum-assisted finance and image synthesis on near-term hardware.

Abstract

We present a novel Adaptive Distribution Generator that leverages a quantum walks-based approach to generate high precision and efficiency of target probability distributions. Our method integrates variational quantum circuits with discrete-time quantum walks, specifically, split-step quantum walks and their entangled extensions, to dynamically tune coin parameters and drive the evolution of quantum states towards desired distributions. This enables accurate one-dimensional probability modeling for applications such as financial simulation and structured two-dimensional pattern generation exemplified by digit representations(0~9). Implemented within the CUDA-Q framework, our approach exploits GPU acceleration to significantly reduce computational overhead and improve scalability relative to conventional methods. Extensive benchmarks demonstrate that our Quantum Walks-Based Adaptive Distribution Generator achieves high simulation fidelity and bridges the gap between theoretical quantum algorithms and practical high-performance computation.

Figures (5)

• Figure 1: Two representations of the SSQWs concept and framework. In (a), the SSQW is depicted as a branching process over multiple layers, illustrating how coin parameters guide the walker’s position probabilities. In (b), a parameterized SSQW layer is shown, where two coin operations ($C_{\vec{\theta_1}}$ and $C_{\vec{\theta_2}}$) and shift operators ($T^+$ and $T^-$) evolve the quantum state, and a classical optimizer iteratively refines the coin parameters to achieve a desired probability distribution.
• Figure 2: Conceptual diagram illustrating two entangled quantum walkers. Each walker occupies one spatial dimension (e.g., $x$ and $y$), and both walkers share a joint coin Hilbert space of dimension $4$. The entangled coin operator acts on the two-qubit coin state, inducing correlated movements in the 2D position space and enabling more intricate interference patterns.
• Figure 3: Representative results for the six one-dimensional target distributions: NVDA returns, beta, binomial, bimodal, exponential, and poisson. Each set of plots shows (top) a boxplot of final optimization errors, (middle) average computation time, and (bottom) a comparison between the QWs-based ADG-simulated distribution and the target distribution.
• Figure 4: CUDA‑Q–simulated log‑normal distribution from our QWs‑based ADG, used for pricing a European call option with $S=6.0$, $\sigma=0.4$, $r=0.04$, and $T=90/365$.
• Figure 5: Examples of 2D digit generation (0--9) using entangled quantum walkers. Each subfigure shows the final QWs-based ADG-simulated distribution for a specific digit on an 8$\times$8 grid. The coin spaces of two quantum walkers are entangled to capture the spatial structure of each digit.