Quantum Fast Forwarding of Non-Markovian Quantum Dynamics

In this work, we present a quantum algorithm for simulating non-Markovian quantum dynamics based on the Feynman-Vernon influence functional formalism, which provides an exact description of the dynamics of a quantum system linearly coupled to a harmonic bath. In condensed-phase environments, the bath correlation functions typically decay over time, allowing them to be truncated beyond a finite time scale and enabling an iterative propagation scheme. We implement the finite-memory propagator using a fast-forwarding algorithm that achieves constant circuit depth with respect to the number of propagation steps, while the nonunitary matrices are realized on quantum circuits using unitary dilation. We further exploit the structure of the unitary dilation to show that the additional circuit complexity introduced by the dilation scales polynomially with the number of dilation ancilla qubits, while the dominant computational cost arises from the implementation of the underlying 1-dilation. The algorithm is compared with classical benchmark calculations for the spin-boson model and shows close agreement. Together, these developments establish a quantum-circuit framework for fast forwarding non-Markovian dynamics and identify the principal computational bottlenecks that must be overcome to achieve quantum advantage.