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How to implement CCH gate in quantum computers available in clouds? If there is not any gate directly available for it, what are the possible ways to represent CCH?

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  • $\begingroup$ Welcome to Quantum Computing SE! When you say 'represent', do you mean a way of graphically drawing such a gate, or are you asking how to implement it using other gates (which are directly available)? $\endgroup$ – Mithrandir24601 Jan 5 at 10:31
  • $\begingroup$ Yes I meant implement using directly available gates. Thank you for the correction. $\endgroup$ – Ankit Raj Jan 6 at 3:44
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Assuming you've got Toffoli and single-qubit rotations, you can implement the following: enter image description here

This basically works because if either of the controls is not $|1\rangle$, the Toffoli does nothing and the two single-qubit unitaries cancel each other. Whereas, if both controls are $|1\rangle$, then the net gate on the target qubit is $$ (\cos\frac{\pi}{8}I+i\sin\frac{\pi}{8}Y)X(\cos\frac{\pi}{8}I-i\sin\frac{\pi}{8}Y)=X(\cos\frac{\pi}{4}I-i\sin\frac{\pi}{4}Y)=\frac{X+Z}{\sqrt2}=H. $$

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A brute force solution :). You can also obtain CCH via qiskit's basic gates with help of get_controlled_circuit method.

from qiskit import *
from qiskit.aqua.utils.controlled_circuit import get_controlled_circuit

q_reg = QuantumRegister(3, 'q')
qc_h = QuantumCircuit(q_reg)
qc_ch = QuantumCircuit(q_reg)
qc_cch = QuantumCircuit(q_reg)

qc_h.h(q_reg[0])

qc_ch += get_controlled_circuit(qc_h, q_reg[1])

qc_cch += get_controlled_circuit(qc_ch, q_reg[2])

print(qc_cch.qasm())

Note that it may be not the optimal gate set for representing the $CCH$ gate, because get_controlled_circuit, how I understand, doesn't optimize the obtained gate set. Also, just info, $H = u2(0, \pi)$, where $u2$ is one of the basis gates of qiskit:

$$ u2(\phi, \lambda) = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & -e^{i \lambda} \\ e^{i\phi} & e^{i(\phi + \lambda)} \end{pmatrix} $$

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