def _decompose_(self):
if not self._has_unitary_():
return None
return [
*([] if self.coefficient == 1 else
[global_phase_op.GlobalPhaseOperation(self.coefficient)]),
*[self[q].on(q) for q in self.qubits],
]
def _decompose_(self, qubits):
if not self._has_unitary_():
return NotImplemented
result = [
PAULI_GATES[p].on(q) for p, q in zip(self.pauli_mask, qubits) if p
]
if self.coefficient != 1:
result.append(
global_phase_op.GlobalPhaseOperation(self.coefficient))
return result
def _decompose_(self, qubits: Sequence['cirq.Qid']) -> 'cirq.OP_TREE':
"""Decompose the n-qubit diagonal gates into CNOT and Rz gates.
A 3 qubits decomposition looks like
0: ───────────────────────────────────X───Rz(6)───X───Rz(7)───X───Rz(5)───X───Rz(4)───
│ │ │ │
1: ───────────X───Rz(3)───X───Rz(2)───@───────────┼───────────@───────────┼───────────
│ │ │ │
2: ───Rz(1)───@───────────@───────────────────────@───────────────────────@───────────
where the angles in Rz gates are corresponding to the fast-walsh-Hadamard transfrom
of diagonal_angles in the Gray Code order.
For n qubits decomposition looks similar but with 2^n-1 Rz gates and 2^n-2 CNOT gates.
The algorithm is implemented according to the paper:
Welch, Jonathan, et al. "Efficient quantum circuits for diagonal unitaries without
ancillas." New Journal of Physics 16.3 (2014): 033040.
https://iopscience.iop.org/article/10.1088/1367-2630/16/3/033040/meta
"""
if protocols.is_parameterized(self):
return NotImplemented
n = self._num_qubits_()
hat_angles = _fast_walsh_hadamard_transform(
self._diag_angles_radians) / (2**n)
# There is one global phase shift between unitary matrix of the diagonal gate and the
# decomposed gates. On its own it is not physically observable. However, if using this
# diagonal gate for sub-system like controlled gate, it is no longer equivalent. Hence,
# we add global phase.
decomposed_circ: List[Any] = [
global_phase_op.GlobalPhaseOperation(np.exp(1j * hat_angles[0]))
]
for i, bit_flip in _gen_gray_code(n):
decomposed_circ.extend(
self._decompose_for_basis(i, bit_flip, -hat_angles[i], qubits))
return decomposed_circ
