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I have a problem relevant to the circuit depth of a $\text{CCCX}$ gate, but I can't find any literature on decomposing this gate into basic gates without an ancillary qubit. Is this possible, and are there any sources where I can read about this?

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The seminal paper Barenco et al., Elementary gates for quantum computation is a great resource for decompositions of many quantum gates, including multi-control Toffoli gates, into elementary gates. (Notably, it contains both constructions with and without ancillas.)

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