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 ℓ1 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.