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This paper simulates duality mode to solve sudoku puzzles. The last diagram in the paper (shown below) illustrates multi-qubit controlled gates with two Z/X gates. I wonder if Qiskit can directly implement these gates.

enter image description here

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    $\begingroup$ The two Z gates in a single circuit is just a shorthand for two separate Z gates, both with the same set of controls. Likewise for the X Gate. $\endgroup$
    – Frank Yellin
    Commented Nov 29, 2022 at 5:53

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As @FrankYellin mentioned in the comments, this is theoretically equivalent to two MCXGate applied successively (up to some HGate for the $Z$ version).

However, you shouldn't worry about this, a framework's goal is to let it deal with low-level implementation, while you should care about what operation you want to perform. Fundamentally, what you're looking for is a multi-controlled $XX$ gate.

Qiskit has an RXXGate, but this will include an unwanted relative phase. You can create the $XX$ gate like this:

from qiskit import QuantumCircuit

circ = QuantumCircuit(2)
circ.x([0, 1])
xx_gate = circ.to_gate(label="XXGate")

Now that we have our $XX$ gate, we can control it out of other qubits as desired, like so:

from qiskit import QuantumRegister

# Be careful about Qiskit's little-endian convention
controlled_xx_gate = xx_gate.control(7, "MCXXGate", "0000100")

alpha = QuantumRegister(4, "alpha")
beta = QuantumRegister(4, "beta")
gamma = QuantumRegister(4, "gamma")

circ = QuantumCircuit(alpha, beta, gamma)
circ.h([alpha[-1], beta[-1], gamma[-1]])

# Using HXH = Z
circ.h([beta[0], gamma[0]])
# Be careful about Qiskit's little-endian convention
circ.append(controlled_xx_gate, [*alpha[:-1], beta[1], beta[-1], gamma[1], gamma[-1], beta[0], gamma[0]])
circ.h([beta[0], gamma[0]])

# Be careful about Qiskit's little-endian convention
circ.append(controlled_xx_gate, [*alpha[:-1], beta[1], beta[-1], gamma[1], gamma[-1], beta[0], gamma[0]])

The resulting circuit looks like this:

enter image description here

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  • $\begingroup$ What does "*" do in the last line of code? Which code creates anti-controlled gate? $\endgroup$
    – Yili
    Commented Nov 29, 2022 at 15:51
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    $\begingroup$ @Yili *alpha[:-1] is a shorthand for alpha[0], alpha[1], alpha[2]. More generally, *my_list "unwraps" the list. The controls of the gate are defined via the string "0000100", which means that the gate will be applied if the first 4 qubits are 0, the fifth one 1 and the last ones 0. Note that because of Qiskit's convention to use little-endian, this has to be read from bottom to top wire-wise. $\endgroup$
    – Tristan Nemoz ♦
    Commented Nov 29, 2022 at 18:15
  • $\begingroup$ Which part of codes do you specify which wire(s) to append xx_gate within the gate "MCXXGate"? $\endgroup$
    – Yili
    Commented Nov 30, 2022 at 16:12
  • $\begingroup$ @Yili This one: controlled_xx_gate = xx_gate.control(7, "MCXXGate", "0000100"). This code transforms the xx_gate to a controlled version of it, which I labelled MCXXGate, controlled by the string "0000100". The convention in Qiskit when creating such a gate is that the first inputs are the controls, and the last ones the inputs to the original gate, which is why, when applying it, we pass beta[0] and gamma[0] as the last parameters: these are the registers on which we wish to apply the $XX$ gate $\endgroup$
    – Tristan Nemoz ♦
    Commented Nov 30, 2022 at 16:17

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