Observability of fidelity decay at the Lyapunov rate in few-qubit quantum simulations

Max D. Porter and Ilon Joseph

Fusion Energy Sciences Program, Lawrence Livermore National Laboratory

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In certain regimes, the fidelity of quantum states will decay at a rate set by the classical Lyapunov exponent. This serves both as one of the most important examples of the quantum-classical correspondence principle and as an accurate test for the presence of chaos. While detecting this phenomenon is one of the first useful calculations that noisy quantum computers without error correction can perform [G. Benenti et al., Phys. Rev. E 65, 066205 (2001)], a thorough study of the quantum sawtooth map reveals that observing the Lyapunov regime is just beyond the reach of present-day devices. We prove that there are three bounds on the ability of any device to observe the Lyapunov regime and give the first quantitatively accurate description of these bounds: (1) the Fermi golden rule decay rate must be larger than the Lyapunov rate, (2) the quantum dynamics must be diffusive rather than localized, and (3) the initial decay rate must be slow enough for Lyapunov decay to be observable. This last bound, which has not been recognized previously, places a limit on the maximum amount of noise that can be tolerated. The theory implies that an absolute minimum of 6 qubits is required. Recent experiments on IBM-Q and IonQ imply that some combination of a noise reduction by up to 100$\times$ per gate and large increases in connectivity and gate parallelization are also necessary. Finally, scaling arguments are given that quantify the ability of future devices to observe the Lyapunov regime based on trade-offs between hardware architecture and performance.

An important milestone for quantum computation is demonstrating the ability to simulate semiclassical dynamics. Even though, for present-day noisy computers, the fidelity of the simulation will decay over time, measuring the decay rate yields valuable information about the underlying dynamics with a quantum speedup over classical algorithms. For example, when the dynamics is chaotic, the fidelity decays at the Lyapunov rate, which controls the butterfly effect, the rate at which classical trajectories exponentially separate in time. In this work, we carefully examine the conditions needed to observe fidelity decay at the Lyapunov rate using noisy quantum computers for the sawtooth map, one of the easiest chaotic systems to simulate. We find that there are three important bounds that together require the computer to have a sufficiently large memory register (at least six qubits) and sufficiently low noise amplitude. After studying the error rates for two state-of-the-art quantum hardware platforms, we conclude that the error is too large by a factor of 10-100 for these platforms. Future hardware with better qubit connectivity and gate parallelization would require less error reduction.

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Cited by

[1] Max D. Porter and Ilon Joseph, "Impact of dynamics, entanglement, and Markovian noise on the fidelity of few-qubit digital quantum simulation", arXiv:2206.04829, (2022).

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