Remove Deprecated Quil classes (quantumlib#5710)

- Removes QuilFormatter, QuilOutput, and QuilXQubitGate
    
Changes all typing of QuilFormatter to string.Formatter

Author: dstrain115

cirq/__init__.py

@@ -74,7 +74,6 @@
     PointOptimizationSummary,
     PointOptimizer,
     QasmOutput,
-    QuilOutput,
     TextDiagramDrawer,
     Unique,
 )
@@ -606,8 +605,6 @@
     qasm,
     QasmArgs,
     qid_shape,
-    quil,
-    QuilFormatter,
     read_json_gzip,
     read_json,
     resolve_parameters,

cirq/circuits/__init__.py

@@ -18,8 +18,6 @@
 
 from cirq.circuits.qasm_output import QasmOutput
 
-from cirq.circuits.quil_output import QuilOutput
-
 from cirq.circuits.circuit import AbstractCircuit, Alignment, Circuit
 from cirq.circuits.circuit_dag import CircuitDag, Unique
 from cirq.circuits.circuit_operation import CircuitOperation

cirq/circuits/circuit.py

@@ -57,7 +57,6 @@
 from cirq.circuits.circuit_operation import CircuitOperation
 from cirq.circuits.insert_strategy import InsertStrategy
 from cirq.circuits.qasm_output import QasmOutput
-from cirq.circuits.quil_output import QuilOutput
 from cirq.circuits.text_diagram_drawer import TextDiagramDrawer
 from cirq.circuits.moment import Moment
 from cirq.protocols import circuit_diagram_info_protocol
@@ -1311,12 +1310,6 @@ def _to_qasm_output(
             version='2.0',
         )
 
-    def _to_quil_output(
-        self, qubit_order: 'cirq.QubitOrderOrList' = ops.QubitOrder.DEFAULT
-    ) -> 'cirq.QuilOutput':
-        qubits = ops.QubitOrder.as_qubit_order(qubit_order).order_for(self.all_qubits())
-        return QuilOutput(operations=self.all_operations(), qubits=qubits)
-
     def to_qasm(
         self,
         header: Optional[str] = None,
@@ -1335,9 +1328,6 @@ def to_qasm(
 
         return str(self._to_qasm_output(header, precision, qubit_order))
 
-    def to_quil(self, qubit_order: 'cirq.QubitOrderOrList' = ops.QubitOrder.DEFAULT) -> str:
-        return str(self._to_quil_output(qubit_order))
-
     def save_qasm(
         self,
         file_path: Union[str, bytes, int],