Mosca and Mukhopadhyay give a Clifford+T circuit for a three-qubit Fredkin (controlled-SWAP) gate:

Clifford+T for Fredkin

This uses four $T$ and three $T^\dagger$ gates at a T-depth of four.

What would the T-count and depth be for a four-qubit CCSWAP gate, e.g., a controlled-controlled-SWAP Gate?


1 Answer 1


You can build a CCSWAP out of 6 T gates:

enter image description here

The reaction depth of this construction is 2. All the T gates apply to observables that commute, and so they can be performed in parallel using ancilla qubits, which is the first reaction. The second reaction is the feedback CZ and CX, which can't be done until the measurement affected by the T gates resolves. This circuit is derived from the CCCZ in arXiv:2106.11513.

The construction you gave for the CSWAP is unnecessarily expensive. The actual T cost should be 4 not 7 and the actual reaction depth should be 2 not 3. Conjugate the CCX from arXiv:1212.5069 or arXiv:1709.06648 by CNOTs and you get a CSWAP.


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