Calculation of the Green’s Function on Near-term Quantum Computers via Cartan Decomposition

Abstract

Accurate computation of the Green’s function is crucial for connecting experimental observations to the underlying quantum states. A major challenge in evaluating the Green’s function in the time domain is the efficient simulation of quantum state evolution under a given Hamiltonian, a task that becomes exponentially complex for strongly correlated systems on classical computers. Quantum computing provides a promising pathway to overcome this challenge by enabling efficient simulation of the time evolution operator. However, for near-term quantum devices with limited coherence times and fidelity, the deep quantum circuits required to implement time-evolution operators present a significant challenge for practical applications. In this work, we introduce an efficient algorithm for computing Green’s functions via Cartan decomposition, which requires only fixed-depth quantum circuits for arbitrarily long time simulations. Additionally, analytical gradients are formulated to accelerate the Cartan decomposition by leveraging a unitary transformation in the factorized form. The new algorithm is applied to simulating longtime Green’s functions for the Fermi-Hubbard and transverse-field Ising models, extracting the spectral functions through Fourier transformation.

Keywords

Quantum computing / Green’s function / Cartan decomposition / Time evolution operator

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Jilin University, The Editorial Department of Chemical Research in Chinese Universities and Springer-Verlag GmbH