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This question is two-fold and considers general $n$-qubit operations on a quantum computer.

First, can a general $n$-qubit operation be implemented on a quantum computer without the use of ancilla qubits?

Second, can ancilla qubits help in reducing the depth of such general $n$-qubit operations and if yes, what is the relation between the two?

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    $\begingroup$ This is definitely a good question, I think we need some more background info to contextualize it. For example, is it okay to approximate the n-qubit operation with error? What is our gate set? Do you have specific classes of operations in mind? $\endgroup$ – C. Kang Sep 17 '20 at 20:21
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    $\begingroup$ It would be interesting to know the difference between an approximation with error and an exact decomposition. For the gate-set, assume Pauli, H, CNOT, S, T, i.e., the 'standard' gate set. No specifics on the operations itself is assumed, though partial answers for specific operations are also useful. $\endgroup$ – nippon Sep 18 '20 at 6:38
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    $\begingroup$ Have you looked at the Solovay Kitaev algorithm? I believe it discusses this problem more arxiv.org/abs/quant-ph/0505030 $\endgroup$ – C. Kang Sep 18 '20 at 14:45

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