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# Near-optimal Circuit construction via Cartan decomposition

Dec 2022

We show the applicability of the Cartan decomposition of Lie algebras toQuantum Circuits. This approach can be used to synthesize circuits that canefficiently reach any desired unitary operation. Our method finds explicitquantum circuit representations of the algebraic generators of the relevant Liealgebras allowing the direct implementation of a Cartan decomposition on aquantum computer. The construction is recursive and allows us to expand anycircuit down to generators and rotation matrices on individual qubits, wherethrough our recursive algorithm we find that the generators themselves can beexpressed with CNOT and SWAP gates explicitly. Our approach is independent ofthe standard CNOT implementation and can be easily adapted to other cross-qubitcircuit elements. In addition to its versatility, we also achieve near-optimalcounts when working with CNOT gates, achieving an asymptotic CNOT cost of$\frac{23}{24}4n$ for $n$ qubits.

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