def _recursive_builder(self, operation, gate_name, control_qubits, target_qubit):
"""Helper function used to define the controlled gate recursively. It uses the algorithm in
the reference above. Namely it recursively constructs a controlled gate by applying a
controlled square root of the gate, followed by a toffoli gate with len(control_qubits) - 1
controls, applying the controlled adjoint of the square root, another toffoli with
len(control_qubits) - 1 controls, and finally another controlled copy of the gate.
:param numpy.ndarray operation: The matrix for the unitary to be controlled.
:param String gate_name: The name for the gate being controlled.
:param Sequence control_qubits: The qubits that are the controls.
:param Qubit or Int target_qubit: The qubit that the gate should be applied to.
:return: The intermediate Program being built.
:rtype: Program
"""
control_true = np.kron(ONE_PROJECTION, operation)
control_false = np.kron(ZERO_PROJECTION, np.eye(2, 2))
control_root_true = np.kron(ONE_PROJECTION, sqrtm(operation, disp=True))
controlled_gate = control_true + control_false
controlled_root_gate = control_root_true + control_false
sqrt_name = self.format_gate_name(SQRT_PREFIX, gate_name)
controlled_subprogram = pq.Program()
control_gate = pq.Program()
# For the base case, we check to see if there is just one control qubit.
if len(control_qubits) == 1:
control_name = self.format_gate_name(CONTROL_PREFIX, gate_name)
control_gate = self._defgate(control_gate, control_name, controlled_gate)
control_gate.inst((control_name, control_qubits[0], target_qubit))
return control_gate
else:
control_sqrt_name = self.format_gate_name(CONTROL_PREFIX, sqrt_name)
control_gate = self._defgate(control_gate, control_sqrt_name, controlled_root_gate)
control_gate.inst((control_sqrt_name, control_qubits[-1], target_qubit))
# Here we recurse to build a toffoli gate on n - 1 of the qubits.
n_minus_one_toffoli = self._recursive_builder(NOT_GATE,
NOT_GATE_LABEL,
control_qubits[:-1],
control_qubits[-1])
# We recurse to build a controlled sqrt of the target_gate, excluding the last control.
n_minus_one_controlled_sqrt = self._recursive_builder(sqrtm(operation, disp=True),
sqrt_name,
control_qubits[:-1],
target_qubit)
controlled_subprogram += control_gate
controlled_subprogram += n_minus_one_toffoli
controlled_subprogram += control_gate.dagger()
controlled_subprogram += n_minus_one_toffoli
controlled_subprogram += n_minus_one_controlled_sqrt
return controlled_subprogram
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