Abstract

Time evolution of quantum systems is a challenging task for classical computers due to the significant memory requirements. This makes quantum computers a great alternative, but due to the noise generated by the current NISQ hardware, one needs to optimize the depth of the circuit generated. Trotterization is widely used for time evolution, leading to deep quantum circuits, hence not NISQ friendly. We present two methods that use Lie algebraic relations of circuit elements of the given Trotter decomposition (or Hamiltonian elements), leading to efficient quantum circuits with depth that is independent of the simulation time. The first method is based on Cartan decomposition of the Hamiltonian algebra, the algebra generated by elements in the given Hamiltonian [1]. It provides fixed depth time evolution circuits for any time independent Hamiltonian and generated useful results for DMFT on the Hubbard model [2]. The second method allows us to compress any given circuit if the circuit elements follow 3 local algebraic rules we name as fusion, turnover, and commutation [3, 4]. While this allows us to compress Trotter circuits for models such as time dependent 1-D XY, transverse field XY models and 1-D transverse field Ising model; the method is not limited to time evolution only. It can be used to compress any circuit that consists of elements that follow the listed 3 rules and can be incorporated into a circuit compiler software in the near future.

[1] E. Kökcü et al., Phys. Rev. Lett. 129, 070501 (2022) [2] T. Steckmann et al., arXiv:2112.05688 [3] E. Kökcü et al., Phys. Rev. A 105, 032420 (2022) [4] D. Camps et al., SIAM J. Matrix Analysis and App. 43, 1084 (2022)