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arXiv:2506.11225 · Physical review. E

Controlling quantum chaos via Parrondo strategies on noisy intermediate-scale quantum hardware

Alison M. Rath, Dinesh Kumar Panda, Colin Benjamin

Published Nov 18, 2025• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/m89r-2dy5

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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.
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.
It has been shown classically that combining two chaotic random walks can yield an ordered (periodic) walk. Our aim in this paper is to find a quantum analog for this rather counterintuitive result. We study the chaotic and periodic nature of cyclic quantum walks and focus on a unique situation wherein a periodic quantum walk on 3-cycle graph is generated via a deterministic combination of two chaotic quantum walks on the same graph. We extend our results to even numbered cyclic graphs, specifically a 4-cycle graph too. Our results will be relevant in quantum cryptography and quantum chaos control.
We present a generalized and efficient quantum circuit framework for implementing discrete-time quantum walks (DTQWs) on Cayley graphs of arbitrary dimension. Building on the Boundary QFT scheme of Razzoli et al., we introduce a systematic multi-stage decomposition of the shift operator for 1D Cayley graphs across three classes of generating sets: inverse-closed without involutions, inverse-closed with an involution, and non-inverse-closed. The decomposition hierarchically factorizes the QFT-diagonalized shift operator into structured block components, progressively reducing the control degree of the required rotation gates and replacing high-degree multi-qubit controlled operations with collections of lower-degree equivalents. We extend this construction to ddd-dimensional torus graphs and provide explicit circuit implementations for an 8-Cayley graph and a Z16×Z8\mathbb{Z}_{16} \times \mathbb{Z}_8Z16​×Z8​ torus graph as concrete illustrations. Gate complexity analysis using the linear CNOT scaling of Rosa et al. demonstrates that the decomposed implementation achieves a substantial reduction in upper-bound CNOT cost relative to the naive implementation within the regime k≤64k \leq 64k≤64 for inverse-closed graphs and k≤16k \leq 16k≤16 for non-inverse-closed graphs, where kkk denotes the degree of the generating set. Benchmarking further reveals that this efficiency gain is largely insensitive to the system size NNN, identifying kkk as the dominant resource parameter for the shift operator. These results provide a scalable and hardware-conscious pathway toward practical DTQW implementations on near-term quantum devices.
Quantum walks have proven to be a universal model for quantum computation and to provide speed-up in certain quantum algorithms. The discrete-time quantum walk (DTQW) model, among others, is one of the most suitable candidates for circuit implementation due to its discrete nature. Current implementations, however, are usually characterized by quantum circuits of large size and depth, which leads to a higher computational cost and severely limits the number of time steps that can be reliably implemented on current quantum computers. In this work, we propose an efficient and scalable quantum circuit implementing the DTQW on the 2n-cycle based on the diagonalization of the conditional shift operator. For t time steps of the DTQW, the proposed circuit requires only O(n2+nt) two-qubit gates compared to the O(n2t) of the current most efficient implementation based on quantum Fourier transforms. We test the proposed circuit on an IBM quantum device for a Hadamard DTQW on the 4-cycle and 8-cycle characterized by periodic dynamics and by recurrent generation of maximally entangled single-particle states. Experimental results are meaningful well beyond the regime of few time steps, paving the way for reliable implementation and use on quantum computers.
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.
Quantum walks provide a versatile framework for probing the structural and dynamical properties of complex systems ranging from biological networks to synthetic materials. However, their realization on current noisy pre-fault-tolerant quantum computers is fundamentally limited by decoherence. Conventional dense encodings of graph structures require prohibitively deep circuits, making them incompatible with existing hardware. Here we introduce an algorithm that leverages symmetry-sector encoding and trades circuit depth for the number of qubits, while integrating symmetry-respecting postselection as an effective noise-mitigation strategy. This combination enables us to execute practical quantum-walk circuits for biological network motifs on actual quantum hardware. We benchmark the proposed methodology against known state-of-the-art circuit architectures, highlighting significant reduction of circuit depth in our approach at the cost of moderate qubit overhead. Utilizing 40 qubits, we implement quantum walks on complex graphs containing up to 17 nodes and 20 edges -- the largest experiment on superconducting hardware to date, with the Hellinger fidelity exceeding 87% throughout 7 steps. We present a proof-of-principle case study that illustrates how experimentally obtained quantum-walk dynamics on a protein-protein-interaction network can be applied to prioritizing disease-associated genes. We discuss the framework scalability in the pre-fault-tolerant era and its potential for studying larger biological networks.