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Chemical Research in Chinese Universities

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

Lingyun Wan, Jie Liu, Jinlong Yang

Published Sep 20, 2025• 1 citation
Publisher
DOI
10.1007/s40242-025-5149-y

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We show the applicability of the Cartan decomposition of Lie algebras to quantum circuits. This approach can be used to synthesize circuits that can efficiently implement any desired unitary operation. Our method finds explicit quantum circuit representations of the algebraic generators of the relevant Lie algebras allowing the direct implementation of a Cartan decomposition on a quantum computer. The construction is recursive and allows us to expand any circuit down to generators and rotation matrices on individual qubits, where through our recursive algorithm we find that the generators themselves can be expressed with controlled-not (cnot) and swap gates explicitly. Our approach is independent of the standard cnot implementation and can be easily adapted to other cross-qubit circuit elements. In addition to its versatility, we also achieve near-optimal counts when working with cnot gates, achieving an asymptotic cnot cost of 21164n\frac{21}{16}{4}^{n}1621​4n for nnn qubits.