Improved Accuracy for Trotter Simulations Using Polynomial Extrapolation

Quantum metrology allow for measuring properties of a quantum system at an optimal scaling known as the Heisenberg limit. However, when the quantum states of interest are prepared using approximate time evolution on a digital quantum computer, the accrued errors will typically deviate from this fundamental limit. In this work, we show how algorithmic errors due to Trotterized time evolution can be mitigated through the use of standard polynomial interpolation techniques. This can be seen as an extrapolation to zero step size, akin to the zero-noise polynomial extrapolation techniques recently developed for mitigating hardware errors. We perform a rigorous error analysis of the interpolation approach for estimating eigenvalues and time-evolved expectation values and show that the Heisenberg limit is achieved up to polylogarithmic factors. Unfortunately, these accuracy gains come at the price of a quadratically worse scaling in simulation time in the case of expectation values, which may be improved with better analysis. Our work suggests that accuracies approaching those of state-of-the-art simulation algorithms may be achieved using Trotter and classical resources alone, for a number of relevant algorithmic tasks.