Random Compiler for Fast Hamiltonian Simulation

The dynamics of a quantum system can be simulated using a quantum computer by breaking down the unitary into a quantum circuit of one and two qubit gates. The most established methods are the Trotter-Suzuki decompositions, for which rigorous bounds on the circuit size depend on the number of terms L in the system Hamiltonian and the size of the largest term in the Hamiltonian Λ. Consequently, the Trotter-Suzuki method is only practical for sparse Hamiltonians. Trotter-Suzuki is a deterministic compiler but it was recently shown that randomized compiling offers lower overheads. Here we present and analyze a randomized compiler for Hamiltonian simulation where gate probabilities are proportional to the strength of a corresponding term in the Hamiltonian. This approach requires a circuit size independent of L and Λ, but instead depending on λ the absolute sum of Hamiltonian strengths (the ℓ_{1} norm). Therefore, it is especially suited to electronic structure Hamiltonians relevant to quantum chemistry. Considering propane, carbon dioxide, and ethane, we observe speed-ups compared to standard Trotter-Suzuki of between 306× and 1591× for physically significant simulation times at precision 10^{-3}. Performing phase estimation at chemical accuracy, we report that the savings are similar.