Randomly Compiled Quantum Simulation with Exponentially Reduced Circuit Depths

The quantum stochastic drift protocol, also known as qDRIFT, has become a popular algorithm for implementing time-evolution of quantum systems using randomised compiling. In this work we develop qFLO, a higher order randomised algorithm for time-evolution. To estimate an observable expectation value at time $T$ to precision $\epsilon$, we show it is sufficient to use circuit depths of $O(T^2\log (1\mathrm{/}\epsilon ))$ – an exponential improvement over standard qDRIFT requirements with respect to $\epsilon$. The protocol achieves this using $O(1\mathrm{/}{\epsilon }^2)$ repeated runs of the standard qDRIFT protocol combined with classical post-processing in the form of Richardson extrapolation. Notably, it requires no ancillary qubits or additional control gates making it especially promising for near-term quantum devices. Furthermore, it is well-conditioned and inherits many desirable properties of randomly compiled simulation methods, including circuit depths that do not explicitly depend on the number of terms in the Hamiltonian.