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arXiv:2604.15030

A NISQ-friendly Coined Quantum Walk Algorithm for Chaos-based Cryptographic Applications

Natalie Gibson, Niklas Keckman, Andrea Marchesin, Matti Raasakka, Ilkka Tittonen

quant-phSubmitted Apr 16, 2026• Announced Apr 17, 2026• 0 citations
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Abstract

We present a novel lackadaisical alternating quantum walk (LAQW) algorithm whose circuit depth scales as $\mathcal{O}(n^2+nt)$ for a $n\times n$ lattice over $t$ time steps. We show that this is a significant depth reduction compared to the existing controlled alternating quantum walk (CAQW) model, which has a circuit depth that scales as $\mathcal{O}(n^2t)$ (Li et al., 2017, arXiv:1707.07389). This makes the implementation of the LAQW viable for Noisy Intermediate-scale Quantum (NISQ) devices. We then showcase the applicability of the LAQW algorithm by proposing a chaos-based symmetric-key generation scheme. Our approach uses the LAQW as a quantum entropy source from which reproducible random bitstring sequences are generated using the underlying probability distribution and subsequent post-processing methods. We provide a comprehensive evaluation of the LAQW algorithm and demonstrate the reproducibility of 128-bit keys under simulated quantum noise provided by IBM's FakeTorino backend. A direct comparison with the CAQW model, which has been used in image encryption and hash function schemes (Li et al., 2017, arXiv:1707.07389; Abd EL-Latif et al., 2020, ScienceDirect; Abd El-Latif, Abd El-Atty, and Venegas-Andraca, 2020, ScienceDirect), highlights the potential and usefulness of the LAQW model in cryptographic applications.

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27 pages, 7 figures

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We present a novel lackadaisical alternating quantum walk (LAQW) algorithm whose circuit depth scales as O(n2+nt)\mathcal{O}(n^2+nt)O(n2+nt) for a n×nn\times nn×n lattice over ttt time steps. We show that this is a significant depth reduction compared to the existing controlled alternating quantum walk (CAQW) model, which has a circuit depth that scales as O(n2t)\mathcal{O}(n^2t)O(n2t) (Li et al., 2017, arXiv:1707.07389). This makes the implementation of the LAQW viable for Noisy Intermediate-scale Quantum (NISQ) devices. We then showcase the applicability of the LAQW algorithm by proposing a chaos-based symmetric-key generation scheme. Our approach uses the LAQW as a quantum entropy source from which reproducible random bitstring sequences are generated using the underlying probability distribution and subsequent post-processing methods. We provide a comprehensive evaluation of the LAQW algorithm and demonstrate the reproducibility of 128-bit keys under simulated quantum noise provided by IBM's FakeTorino backend. A direct comparison with the CAQW model, which has been used in image encryption and hash function schemes (Li et al., 2017, arXiv:1707.07389; Abd EL-Latif et al., 2020, ScienceDirect; Abd El-Latif, Abd El-Atty, and Venegas-Andraca, 2020, ScienceDirect), highlights the potential and usefulness of the LAQW model in cryptographic applications.
Compatibility with noisy intermediate-scale quantum (NISQ) devices is crucial for the realistic implementation of quantum cryptographic protocols. We investigate a cryptographic scheme based on discrete-time quantum walks (DTQWs) on cyclic graphs that exploits Parrondo dynamics, wherein periodic evolution emerges from a deterministic sequence of individually chaotic coin operators. We construct an explicit quantum circuit realization tailored to NISQ architectures and analyze its performance through numerical simulations in Qiskit under both ideal and noisy conditions. Protocol performance is quantified using probability distributions, Hellinger fidelity, and total variation distance. To assess security at the circuit level, we model intercept-resend and man-in-the- middle attacks and evaluate the resulting quantum bit error rate. In the absence of adversarial intervention, the protocol enables reliable message recovery, whereas eavesdropping induces characteristic disturbances that disrupt the periodic reconstruction mechanism. We further examine hardware feasibility on contemporary NISQ processors, specifically ibm_torino, incorporating qubit connectivity and state-transfer constraints into the circuit design. Our analysis demonstrates that communication between spatially separated logical modules increases circuit depth via SWAP operations, leading to cumulative noise effects. By exploring hybrid state-transfer strategies, we show that qubit selection and connectivity play a decisive role in determining fidelity and overall protocol performance, highlighting hardware-dependent trade-offs in NISQ implementations.
Advancements in noisy intermediate-scale quantum (NISQ) computing are steadily pushing these systems toward outperforming classical supercomputers on specific well-defined computational tasks. In this work we explore and control quantum chaos in NISQ systems using discrete-time quantum walks (DTQWs) on cyclic graphs. To efficiently implement quantum walks on NISQ hardware, we employ the quantum Fourier transform to diagonalize the conditional shift operator, optimizing circuit depth and fidelity. We experimentally realize the transition from quantum chaos to order via DTQW dynamics on both odd and even cyclic graphs, specifically 3- and 4-cycle graphs, using the counterintuitive Parrondo paradox strategy across three different NISQ devices. While the 4-cycle graphs exhibit high-fidelity quantum evolution, the 3-cycle implementation shows significant fidelity improvement when augmented with dynamical decoupling pulses. Our results demonstrate a practical approach to probing and harnessing controlled chaotic dynamics on real quantum hardware, laying the groundwork for future quantum algorithms and cryptographic protocols based on quantum walks.
Designing efficient and secure cryptosystems has been a preoccupation for many scientists and engineers for a long time wherein they use chaotic systems to design new cryptosystems. While one dimensional (1-D) chaotic maps possess powerful properties compared to higher dimension ones, they are vulnerable to various attacks due to their small key space, chaotic discontinuous ranges, and degradation in chaotic dynamical behaviours. Moreover, when simulated on a computer, every such chaotic system produces a periodic cycle. Meanwhile, quantum random walks exhibit the potential for deployment in efficient cryptosystem design, which makes it an excellent solution for this problem. In this context, we present a new method for constructing substitution boxes (S-boxes) based on cascaded quantum-inspired quantum walks and chaos inducement. The performance of the proposed S-box scheme is investigated via established S-box evaluation criterion and outcomes suggest that the constructed S-box has significant qualities for viable applications information security. Further, we present an efficient scheme for pseudo-random numbers generation (PRNG) whose sustainability over long periods remedies the periodicity problem associated with traditional cryptographic applications. Furthermore, by combining the two mechanisms, an atypical image encryption scheme is introduced. Simulation results and analysis validate that the proposed image encryption algorithm will offer gains in many cryptographic applications.
Through introducing controlled alternate quantum walks, we present controlled alternate quantum walks (CAQW) based quantum hash function. CAQW based quantum hash function have excellent security, outstanding statistical performance and splendid expansibility. Furthermore, due to the structure of alternate quantum walks, implementing CAQW based quantum hash function significantly reduces the resources necessary for its feasible experimental realization than implementing other quantum hash functions.
In this research work, we offer a quantum public key encryption protocol scheme that is constructed via lackadaisical quantum walks (LQWs) on cycle graphs. Our proposed protocol scheme involves the laziness parameter or loop-weight (l) at each vertex and coin phase (θ) as tunable variables, which effectively increases the performance of the cryptographic model. The sender encrypts the message using LQW evolution which is governed by shift operator and 3 dimensional coin operator. The receiver, given final quantum state as public key and correct parameters as secrete key, is able to decrypt the message with high fidelity. Any differences in the parameters shows a drastic drop in fidelity due to the sensitive behavior of quantum walk dynamics dependent on secret key variables. We also study the correctness, security and efficiency of the proposed protocol scheme. The findings show promising efficiency and protection against eavesdropping. The method proposed in the work introduces a novel method for QPKE, leveraging the robust mathematical framework of quantum walks.
Lackadaisical quantum walks (LQW) extend discrete-time quantum walks (DTQW) by introducing weighted self-loops, enabling improved spatial-search performance through controlled localization of the walker. Despite substantial theoretical progress, practical gate-level implementations suitable for quantum hardware remain largely unexplored, limiting evaluation under realistic architectural constraints, noise, and resource requirements. In this work, we present a gate-level implementation framework for lackadaisical quantum walk search. The proposed construction encodes the position and coin spaces into qubit registers, and realizes the walk dynamics through oracle, coin, and flip-flop shift operations. We validate the circuit by reproducing the expected search behavior for single and multiple marked vertices and by analyzing the effect of the self-loop weight on the success probability. We further evaluate the implementation under realistic noisy settings using superconducting hardware's noise models. Logical resource analysis shows that, for grids ranging from 8×88\times88×8 to 64×6464\times6464×64, the transpiled gate count increases from 3.63×1053.63\times10^{5}3.63×105 to 4.38×1064.38\times10^{6}4.38×106 and the circuit depth from 2.13×1052.13\times10^{5}2.13×105 to 2.56×1062.56\times10^{6}2.56×106. Finally, fault-tolerant resource estimates based on a surface-code model using the Microsoft Quantum Resource Estimator demonstrate the substantial space-time trade-off associated with magic-state production. The results establish a practical circuit-level pathway for implementing the LQW search and provide a basis for evaluating its performance.