Hamiltonian Simulation via Quantum Assisted Cartan Decomposition and Iterative Block Diagonalization

Generating an efficient time evolution circuit for a given many body Hamiltonian is crucial for simulation of quantum systems via quantum computation. Standard algorithms based on Trotter-Suzuki product formulas result in circuits that grow with increasing simulation time, which is incompatible with the noisy nature of current NISQ devices. Circuits whose depth is independent of simulation time can be generated by Cartan decomposition [1], but this involves finding a local minimum of a cost function, which requires efficient calculation of the cost function, and an efficient search through parameter space. In this work, we advance this method in two aspects. First, we propose a method to calculate the cost function on a quantum computer, which quadratically speeds up its calculation. Secondly, we present a method to partition the parameter space by exploiting the Lie algebraic properties of the cost function, and thus divide the optimization problem into a sequence of smaller optimization problems. With our developments, we are moving towards efficient synthesis of unitaries, in a way that leverages the power of quantum computers, rather than purely offloading the difficult work to classical computer.

[1] E. Kökcü et al. (2022), Phys. Rev. Lett. 129, 070501