How can we code a ccz (control control z, equivalent of Toffoli on Z and not on X) gate in Qiskit language?

Question

How can we code in Qiskit, a ccz gate ?

In python (in https://qiskit.org/textbook/ch-algorithms/grover.html), we see three instructions : qc.h(nqubits-1) qc.mct(list(range(nqubits-1)), nqubits-1) # multi-controlled-toffoli qc.h(nqubits-1)

In qiskit , a Hadamard gate can be coded as : circuit.h(qreg[i]). But what is it this "mct" gate ? i've not seen it in Qiskit documentation.

Answer 1

You can use the MCMT (multi-controlled multi-target gate) class to do something like this. To execute CCZ gate, I can do it as follow:

from qiskit.circuit.library import MCMT
qr= QuantumRegister(3)
circ=QuantumCircuit(qr)
circ = MCMT('cz',2,1)
print(circ)
q_0: ─■─
      │ 
q_1: ─■─
      │ 
q_2: ─■─
        

And if you want to see how the circuit would look using more elementary gates, you can decompose it:

print(circ.decompose().decompose())

                                               ┌───┐    ┌─────────────┐     »
q_0: ──────────────────────────────────────────┤ X ├────┤ U(0,0,-π/4) ├──■──»
     ┌────────────┐                            └─┬─┘    └─────────────┘  │  »
q_1: ┤ U(0,0,π/4) ├──■───────────────────■───────■───────────────────────┼──»
     ├────────────┤┌─┴─┐┌─────────────┐┌─┴─┐┌──────────┐ ┌────────────┐┌─┴─┐»
q_2: ┤ U(0,0,π/4) ├┤ X ├┤ U(0,0,-π/4) ├┤ X ├┤ U(0,0,0) ├─┤ U(0,0,π/4) ├┤ X ├»
     └────────────┘└───┘└─────────────┘└───┘└──────────┘ └────────────┘└───┘»
«                             ┌───┐     ┌────────────┐                         »
«q_0: ────────────────■───────┤ X ├─────┤ U(0,0,π/4) ├──■───────────────────■──»
«                     │       └─┬─┘     └────────────┘  │                   │  »
«q_1: ────────────────┼─────────■───────────────────────┼───────────────────┼──»
«     ┌────────────┐┌─┴─┐┌─────────────┐┌────────────┐┌─┴─┐┌─────────────┐┌─┴─┐»
«q_2: ┤ U(0,0,π/4) ├┤ X ├┤ U(0,-π/2,0) ├┤ U(0,0,π/4) ├┤ X ├┤ U(0,0,-π/4) ├┤ X ├»
«     └────────────┘└───┘└─────────────┘└────────────┘└───┘└─────────────┘└───┘»
«                 
«q_0: ────────────
«                 
«q_1: ────────────
«     ┌──────────┐
«q_2: ┤ U(0,0,0) ├
«     └──────────┘