arXiv:quant-ph/0101004 · Physical Review Letters

Stable Quantum Computation of Unstable Classical Chaos

Bertrand Georgeot, Dima L. Shepelyansky

Published Jun 4, 2001• 43 citations

Abstract

We show on the example of the Arnold cat map that classical chaotic systems can be simulated with exponential efficiency on a quantum computer. Although classical computer errors grow exponentially with time, the quantum algorithm with moderate imperfections is able to simulate accurately the unstable chaotic classical nonlinear dynamics for long times. The algorithm can be easily implemented on systems of a few qubits.

DOI: 10.1103/physrevlett.86.5393

Related papers

Comment on "Stable Quantum Computation of Unstable Classical Chaos"

Christof Zalka

This is a 1-page comment on a wrong paper that recently appeared in PRL (Phys. Rev. Lett. 86 (23), 5393 (2001), also quant-ph/0101004). The authors claim to have shown that using a quantum computer gives an "exponential advantage" for simulating and studying classical chaotic systems.

arXiv:quant-ph/0110019Oct 2, 20011 citation

Comment on “Stable Quantum Computation of Unstable Classical Chaos”

Lajos Diósi

I think the title and content of the recent Letter by Georgeot and Shepelyanski [PRL 86, 5393 (2001), also quant-ph/0101004)] are not correct. As long as the classical Arnold map is considered, the classical computational algorithm can be made exactly equivalent with the quantum one. The claimed advantage of the Letter's quantum algorithm disappears if we correctly restrict the statistical analysis for the classical Arnold system.

arXiv:quant-ph/0110026Physical Review LettersMay 9, 20022 citations

Efficient quantum computing insensitive to phase errors

Bertrand Georgeot, Dima L. Shepelyansky

We show that certain computational algorithms can be simulated on a quantum computer with exponential efficiency and be insensitive to phase errors. Our explicit algorithm simulates accurately the classical chaotic dynamics for exponentially many orbits even when the quantum fidelity drops to zero. Such phase-insensitive algorithms open new possibilities for computation on realistic quantum computers.

arXiv:quant-ph/0102082Feb 16, 20011 citation

Chaos System Simulation based on Quantum Computing

Jincheng Zhang

This paper explores the potential of quantum computing for simulating chaotic systems. Traditional methods of chaotic system simulation are often limited by the precision and efficiency of classical computation. This research proposes leveraging the unique capabilities of quantum computing, specifically quantum chaos algorithms, to overcome these limitations. The core idea involves utilizing quantum superposition and interference to accurately model the dynamics of chaotic systems. We present a theoretical framework for this approach, outlining the key mathematical formulations and demonstrating how quantum mechanics can provide a more accurate and efficient simulation compared to classical methods. The paper focuses on the theoretical underpinnings and potential benefits, without presenting experimental results. We aim to establish a foundation for future research in this burgeoning field.

Sep 3, 20260 citations

Complexity of chaos and quantum computation

Bertrand Georgeot

This paper reviews recent work related to the interplay between quantum information and computation on the one hand and classical and quantum chaos on the other. First, we present several models of quantum chaos that can be simulated efficiently on a quantum computer. Then a discussion of information extraction shows that such models can give rise to complete algorithms including measurements that can achieve an increase in speed compared with classical computation. It is also shown that models of classical chaos can be simulated efficiently on a quantum computer, and again information can be extracted efficiently from the final wave function. The total gain can be exponential or polynomial, depending on the model chosen and the observable measured. The simulation of such systems is also economical in the number of qubits, allowing implementation on present-day quantum computers, some of these algorithms having been already experimentally implemented. The second topic considered concerns the analysis of errors on quantum computers. It is shown that quantum chaos algorithms can be used to explore the effect of errors on quantum algorithms, such as random unitary errors or dissipative errors. Furthermore, the tools of quantum chaos allows a direct analysis of the effects of static errors on quantum computers. Finally, we consider the different resources used by quantum information, and show that quantum chaos has some precise consequences on entanglement generation, which becomes close to maximal. For another resource, interference, a proposal is presented for quantifying it, enabling a discussion on entanglement and interference generation in quantum algorithms.

Mathematical Structures in Computer ScienceNov 23, 20074 citations

Limitations for Quantum Algorithms to Solve Turbulent and Chaotic Systems

Dylan Lewis, Stephan Eidenbenz, Balasubramanya Nadiga +1 more

We investigate the limitations of quantum computers for solving nonlinear dynamical systems. In particular, we tighten the worst-case bounds of the quantum Carleman linearisation (QCL) algorithm [Liu et al., PNAS 118, 2021] answering one of their open questions. We provide a further significant limitation for any quantum algorithm that aims to output a quantum state that approximates the normalized solution vector. Given a natural choice of coordinates for a dynamical system with one or more positive Lyapunov exponents and solutions that grow sub-exponentially, we prove that any such algorithm has complexity scaling at least exponentially in the integration time. As such, an efficient quantum algorithm for simulating chaotic systems or regimes is likely not possible.

arXiv:2307.09593QuantumOct 24, 202413 citations