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 $\mathrm{\Lambda }$. 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 $\mathrm{\Lambda }$, but instead depending on $\lambda$ the absolute sum of Hamiltonian strengths (the ${\ell }_1$ 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 $306\times$ and $1591\times$ for physically significant simulation times at precision ${10}^{-3}$. Performing phase estimation at chemical accuracy, we report that the savings are similar.

Figure 1

Pseudocode for the QDRIFT protocol.

Figure 2

The number of gates used to implement $U=\exp (iHt)$ for various $t$ and $\epsilon ={10}^{-3}$ and three different Hamiltonians (energies in Hartree) corresponding to the electronic structure Hamiltonians of propane (in STO-3G basis), carbon dioxide (in 6-31g basis), and ethane (in 6-31g basis). Since the Hamiltonian contains some very small terms, one can argue that conventional Trotter-Suzuki methods would fare better if they truncate the Hamiltonian by eliminating negligible terms. For this reason, whenever simulating to precision $\epsilon$ we also remove from the Hamiltonian the smallest terms with weight summing to $\epsilon$. This makes a fairer comparison, though in practice we found it made no significant difference to performance. For the Suzuki decompositions we choose the best from the first four orders, which is sufficient to find the optimal.