Koopman--von Neumann Molecular Dynamics for Green--Kubo Transport Coefficients

Abstract

We formulate the Green--Kubo transport coefficients of classical molecular dynamics as a readout problem for quantum algorithms using the Koopman--von Neumann (KvN) representation. Both NVE and Nos\'e--Hoover-type NVT dynamics are derived as unitary evolutions on Hilbert spaces associated with the corresponding classical phase spaces. Numerical benchmarks on finite grids show that the discretization error in the correlation function decreases as a power law in the number of grid points $N_z$. Equivalently, with $N_z=2^{n_z}$, the error decreases exponentially in the register size $n_z$, so a target accuracy $\epsilon$ requires $n_z=\mathcal{O}(\log(1/\epsilon))$ qubits. To read out a transport coefficient, we input a flux-excited state to quantum phase estimation (QPE). The probability $P_0$ of measuring the QPE ancilla register in the all-zero state corresponds to a Bartlett-windowed Green--Kubo integral. With maximum-likelihood amplitude estimation, the statistical estimation of $P_0$ defined by this QPE oracle improves from the $N_{\rm queries}^{-1/2}$ scaling of direct shot sampling to scaling close to $N_{\rm queries}^{-1}$. Our circuit-resource analysis shows that one step of the NVE propagator can be built with $\mathcal{O}(n^2)$ CX gates, where $n=n_x+n_p$ is the total number of position and momentum qubits. For the NVT propagator, the centered-difference Pauli-decomposition implementation of the Nos\'e--Hoover friction term scales as $\mathcal{O}(n_\xi n_p\,2^{n_p})$, where $n_p$ and $n_\xi$ are the numbers of momentum and thermostat qubits, respectively. The proposed framework is a concrete step toward translating the principles of quantum algorithms into the transport-coefficient calculations required in practical molecular simulation.