Toward end-to-end quantum simulation of rapidly distorted turbulence

TL;DR

This work develops an end-to-end quantum framework for simulating rapidly distorted turbulence by marrying rapid distortion theory with a linear combination of Hamiltonian simulations. It constructs a unitary quantum evolution for the linearized RDT dynamics, enables efficient state preparation of turbulence initial conditions, and provides measurement schemes for Reynolds stresses and energy spectra. The approach yields polynomial-time quantum resource scaling and potential speedups over classical turbulence simulations for large grids, while grounding the method in physically meaningful linear turbulence dynamics. It also presents a concrete numerical validation on a 3D rapidly distorted shear flow, showing qualitative and quantitative agreement with ground truth and establishing a baseline quantum-resource benchmark (roughly 26–28 qubits and 2100 TS steps) for such turbulent problems. The study acknowledges limitations due to deep circuits and the use of a linearized model, outlining future directions toward stochastic extensions and more realistic, inhomogeneous turbulence scenarios on fault-tolerant quantum hardware.

Abstract

We propose an end-to-end quantum algorithm to simulate rapidly distorted turbulence via linear combination of Hamiltonian (LCHS). The algorithm comprises three primary stages: the efficient preparation of an initial turbulent state with a prescribed energy spectrum, its subsequent time evolution via LCHS, and the direct measurement of key turbulence statistics. Our analysis indicates that the algorithm can offer a practical quantum speedup over the classical simulation methods for a sufficiently large computational grid. We evaluate the quantum resource requirements for simulating a minimal instance of non-trivial turbulence with classical validation. The numerical results show excellent agreement with ground-truth solutions, capturing both the qualitative evolution of turbulent fields and the quantitative behavior of statistics, including the Reynolds stresses and the fluctuating velocity spectrum. Despite its linearity, rapidly distorted turbulence captures essential turbulence mechanisms and may inform the development of quantum algorithms for the Navier-Stokes equations. Our work establishes a foundation for addressing more complex turbulent phenomena on future fault-tolerant quantum computers.

Figures (13)

• Figure 1: Schematic of the quantum simulation of turbulence using RDT. (a) An initial field of homogeneous turbulence is subjected to (b) an instantaneous, strong shear flow characterized by a mean velocity $U_x$ and mean vorticity $\varOmega_z$. This scenario corresponds to the rapid distortion limit, where the strong mean shear, $\varOmega_z\gg |\boldsymbol{\omega}|$, acts on a timescale much shorter than the intrinsic nonlinear timescale of the turbulence. (c) Consequently, an initially straight mean-vortex tube is distorted by the turbulent velocity fluctuations, leading to the formation of a kink. (d) An end-to-end quantum algorithm for simulating the non-unitary RDT dynamics via a linear combination of Hamiltonian simulations. (e) The kink constitutes a new, small-scale, locally rotating structure, corresponding to the generation of new fluctuation vorticity. Through this mechanism, kinetic energy is extracted from the mean flow and transferred to the turbulent fluctuations, as a primary mechanism for turbulence production.
• Figure 2: Quantum circuits for state preparation. (a) Decomposition of the operator $U_A$ for amplitude loading, as defined in Eq. \ref{['eq:Amplitude_loading_state']}. (b) Circuit implementation of the unitary $U_j$ from Eq. \ref{['eq:U_j_amplitude']}, corresponding to the sequence in Eq. \ref{['eq:amplitude_sequence']}. (c) Circuit implementation of the unitary $U_{k_\perp}$ for wavevector loading, corresponding to the sequence in Eq. \ref{['eq:wavevector_loading_state']}. (d) Decomposition of each constituent operator $U_{k_\perp, m}$. (e) Circuit implementation of the unitary $U_{\phi}$ for phase loading, corresponding to the sequence in Eq. \ref{['eq:phase_sequence']}. The grey boxes denote control qubits or registers without specifying the control state ($| 0 \rangle$ or $| 1 \rangle$).
• Figure 3: Quantum circuit for time evolution. Following the state preparation in Eq. \ref{['eq:state_preparation_target']}, the total evolution time $t$ is discretized into $N_t$ small time steps, implemented by the operator in Eq. \ref{['eq:U_select']}. Each constituent operator $V_{j,l}$ from Eq. \ref{['eq:opt_each']} is constructed using an oracle $O_{h,l}$ (e.g., $O_{23}$ in Eq. \ref{['eq:O_23']}) that computes the required rotation angles as arithmetic functions of the state index. The desired final state $| \tilde{\mathcal{U}}(t) \rangle$ is then obtained in the main register by applying the inverse operation $U_{\mathrm{coef}}^\dagger$ to the ancillary register and projecting it onto its ground state, with success probability $P \approx \| \tilde{\boldsymbol{\mathcal{U}}}(t) \|_2^2$. The grey boxes denote control qubits or registers without specifying the control state ($| 0 \rangle$ or $| 1 \rangle$).
• Figure 4: As a function of $\beta$, the integral $I_\beta$ in Eq. \ref{['eq:P_success_est_1']} attains a minimum value of $1.09$ at $\beta\approx 0.44$.
• Figure 5: Quantum circuits for implementing the off-diagonal components of $U_{ij}$ in Eq. \ref{['eq:U_ij']}.