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

Aditi Rath, Dinesh Kumar Panda, Colin Benjamin

TL;DR

This work addresses how to observe and control quantum chaos on NISQ hardware by implementing discrete-time quantum walks on cyclic graphs and applying Parrondo's paradox to induce order from chaos. The authors develop a QFT-based diagonalization of the shift operator to realize efficient circuits for DTQW on both even (4-cycle) and odd (3-cycle) cycles and verify Parrondo sequences (AABB...) that transition chaotic walks into periodic ones. On three IBM devices, 4-cycle DTQW achieves high fidelity (>$98\%$) up to 25 steps, confirming the chaos-to-order transition in practice, while 3-cycle walks suffer from depth-related noise; dynamical decoupling improves 3-cycle fidelity to the $75$–$95\%$ range but cannot fully overcome gate-depth limitations. This study provides a practical route to harness controlled chaotic dynamics on real quantum hardware, with implications for quantum algorithms and cryptographic protocols based on quantum walks.

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 (DTQW) on cyclic graphs. To efficiently implement quantum walks on NISQ hardware, we employ the quantum Fourier transform (QFT) 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's 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.

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

TL;DR

This work addresses how to observe and control quantum chaos on NISQ hardware by implementing discrete-time quantum walks on cyclic graphs and applying Parrondo's paradox to induce order from chaos. The authors develop a QFT-based diagonalization of the shift operator to realize efficient circuits for DTQW on both even (4-cycle) and odd (3-cycle) cycles and verify Parrondo sequences (AABB...) that transition chaotic walks into periodic ones. On three IBM devices, 4-cycle DTQW achieves high fidelity (>) up to 25 steps, confirming the chaos-to-order transition in practice, while 3-cycle walks suffer from depth-related noise; dynamical decoupling improves 3-cycle fidelity to the – range but cannot fully overcome gate-depth limitations. This study provides a practical route to harness controlled chaotic dynamics on real quantum hardware, with implications for quantum algorithms and cryptographic protocols based on quantum walks.

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 (DTQW) on cyclic graphs. To efficiently implement quantum walks on NISQ hardware, we employ the quantum Fourier transform (QFT) 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's 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.

Paper Structure

This paper contains 21 sections, 32 equations, 24 figures, 6 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Theory
  3. DTQW on cyclic graphs
  4. Parrondo's paradox via quantum walks
  5. Quantum Circuit implementing DTQW on Cycles
  6. Even Cycle
  7. NISQ implementation of Parrondo's strategy on 4-cycle graph
  8. Odd cycle
  9. NISQ implementation of Parrondo's strategy in 3-cycle
  10. Depth and dynamical decoupling in quantum circuits
  11. Analysis
  12. Conclusion
  13. Algorithm for the transition of chaos to order via cyclic DTQWs
  14. Transpiled quantum circuits in NISQ hardware
  15. Calculation of Depth of Proposed 3-cycle quantum circuit
  16. ...and 6 more sections

Figures (24)

  • Figure 1: DTQW on an N-cycle graph.
  • Figure 2: Quantum circuit representation of the $4$-cycle DTQW dynamics. $P$ gates refer to the phase gates (rotation) that are involved in the shift operation, and $C$ is the coin operator.
  • Figure 3: Quantum circuits implementing the sequences (a)$AAAA..$(or $BBBB..$) and (b)$AABB..$ on $4$-cycle for $4$ time steps designed within Qiskit framework.
  • Figure 4: Qubit connectivity of ibm_sherbrooke(Image Source: IBM Quantum ibm Accessed on December 2024).
  • Figure 5: Chaotic QW on $4$-cycle graphs for sequence of chaotic coins (a) $AAA$...and the (b) Hellinger fidelity. The circuits are implemented on ibm_sherbrooke at optimization level 3 for $10^5$ shots with position state being encoded in qubits 46,47, and coin state in qubit 48.
  • ...and 19 more figures