# Fixed Depth Hamiltonian Simulation via Cartan Decomposition

Physical Review Letters（2021）

Abstract

Simulating spin systems on classical computers is challenging for large systems due to the significant memory requirements. This makes Hamiltonian simulation by quantum computers a promising option due to the direct representation of quantum states in terms of its qubits. Standard algorithms for time evolution on quantum computers require circuits whose depth grows with time. We present a new algorithm, based on Cartan decomposition of the Lie algebra generated by the Hamiltonian, that generates a circuit with time complexity O(1) for ordered and disordered models of n spins. We highlight our algorithm for special classes of models where an O(n^2)-gate circuit emerges. Compared to product formulas with significantly larger gate counts, our algorithm drastically improves simulation precision. Our algebraic technique sheds light on quantum algorithms and will reduce gate requirements for near term simulation.

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