Compressing quantum time evolution circuits of impurity models

Simulation of impurity models, i.e. many body systems with sparse interactions, allows us to study the physics of various materials. While some systems such as the Kondo model are directly represented as an impurity model, other more general systems (e.g. the Hubbard model) can make use of impurity models in embedding methods as DMFT and cluster DMFT. When the number of impurities is small, these models can be solved using conventional computational methods. However, the cost of these methods increases exponentially with the number of impurities, and the need for quantum computation emerges. By using the Trotter-Suzuki formula, quantum circuits for time evolution of these models can be generated with circuit depth scaling as O(n*t) where n is the system size and t is the number of time steps. This results in deep circuits, which are not amenable to near-term and early fault tolerant hardware. To reduce the circuit depth, here we use a recently introduced algebraic compression method, as well as its extension to long range fermion hopping. The method is based on local circuit operations of fusion, commutation and turnover, that are satisfied by the free fermion terms. By using these operations, we show that the time evolution circuits can be compressed down to O(d*t) depth where d is the number of impurities, and t is the number of time steps. We demonstrate our method by simulating the RKKY model to obtain an effective Hamiltonian for two spins in a metal.