Multiple Qubits Clifford Gate (quantumlib#4791)

Add initial Clifford Gate with multiple qubits. Compared with SingleQubitCliffordGate, it has fewer functionalities since we cannot enumerate all of them with PauliGates and several special single qubit properties like Bloch rotation no longer exist. Anyway, it provides several basic interactions:
1. It uses Clifford tableau as underlying data representation (different from the state representation).
2. It can be constructed from a tableau or list of operations (`_has_stabilizer_effect_` only). All Clifford gates can be built through \{S, H, CNOT\}, so we can construct any Clifford Gate from the list of operations. We just cannot pre-define it.
3. Decomposing into several basic operations.
4. Get unitary matrix through decomposing (we cannot do this in a reverse way from unitary to Clifford gate :( ).
5. Know how to interact with ActOnCliffordTableauArgs, i.e. it should be able to use with CliffordTableau simulator (Looks like we don't have that in cirq yet?  @daxfohl will add that? see quantumlib#4639 and quantumlib#4748.).

This PR is part of efforts for quantumlib#3639. Context: this PR doesn't introduce any new algorithms but the key methods are already implemented in quantumlib#4183 and quantumlib#4096.

cirq-core/cirq/__init__.py

@@ -204,6 +204,7 @@
     CCNOT,
     CCNotPowGate,
     ClassicallyControlledOperation,
+    CliffordGate,
     CNOT,
     CNotPowGate,
     ControlledGate,
@@ -707,7 +708,7 @@
     contrib,
 )
 
-# deprecate cirq.ops.moment and related attributes
+# deprecate cirq.ops and related attributes
 
 from cirq import _compat
 

cirq-core/cirq/json_resolver_cache.py

@@ -66,6 +66,7 @@ def _parallel_gate_op(gate, qubits):
         'CircuitOperation': cirq.CircuitOperation,
         'ClassicallyControlledOperation': cirq.ClassicallyControlledOperation,
         'ClassicalDataDictionaryStore': cirq.ClassicalDataDictionaryStore,
+        'CliffordGate': cirq.CliffordGate,
         'CliffordState': cirq.CliffordState,
         'CliffordTableau': cirq.CliffordTableau,
         'CNotPowGate': cirq.CNotPowGate,

cirq-core/cirq/ops/__init__.py

@@ -19,6 +19,7 @@
 )
 
 from cirq.ops.clifford_gate import (
+    CliffordGate,
     PauliTransform,
     SingleQubitCliffordGate,
 )

cirq-core/cirq/ops/clifford_gate.py

@@ -12,19 +12,44 @@
 # See the License for the specific language governing permissions and
 # limitations under the License.
 
-from typing import Any, cast, Dict, NamedTuple, Optional, Sequence, Tuple, TYPE_CHECKING, Union
+from typing import (
+    Any,
+    cast,
+    Dict,
+    List,
+    NamedTuple,
+    Optional,
+    Sequence,
+    Tuple,
+    TYPE_CHECKING,
+    Union,
+)
 
 import numpy as np
 
 from cirq import protocols, value, linalg, qis
 from cirq._doc import document
-from cirq.ops import common_gates, gate_features, named_qubit, pauli_gates, phased_x_z_gate
+from cirq._import import LazyLoader
+from cirq.ops import (
+    common_gates,
+    gate_features,
+    identity,
+    named_qubit,
+    raw_types,
+    pauli_gates,
+    phased_x_z_gate,
+)
 from cirq.ops.pauli_gates import Pauli
 from cirq.type_workarounds import NotImplementedType
 
 if TYPE_CHECKING:
     import cirq
 
+# Lazy imports to break circular dependencies.
+devices = LazyLoader("devices", globals(), "cirq.devices")
+sim = LazyLoader("sim", globals(), "cirq.sim")
+transformers = LazyLoader("transformers", globals(), "cirq.transformers")
+
 PauliTransform = NamedTuple('PauliTransform', [('to', Pauli), ('flip', bool)])
 document(PauliTransform, """+X, -X, +Y, -Y, +Z, or -Z.""")
 
@@ -571,3 +596,285 @@ def _circuit_diagram_info_(
         SingleQubitCliffordGate.Z: SingleQubitCliffordGate.Z_nsqrt,
     },
 }
+
+
+class CommonCliffordGateMetaClass(value.ABCMetaImplementAnyOneOf):
+    """A metaclass used to lazy initialize several common Clifford Gate as class attributes."""
+
+    @property
+    def I(cls):
+        if getattr(cls, '_I', None) is None:
+            cls._I = cls._generate_clifford_from_known_gate(1, identity.I)
+        return cls._I
+
+    @property
+    def X(cls):
+        if getattr(cls, '_X', None) is None:
+            cls._Z = cls._generate_clifford_from_known_gate(1, pauli_gates.X)
+        return cls._Z
+
+    @property
+    def Y(cls):
+        if getattr(cls, '_X', None) is None:
+            cls._Z = cls._generate_clifford_from_known_gate(1, pauli_gates.Y)
+        return cls._Z
+
+    @property
+    def Z(cls):
+        if getattr(cls, '_X', None) is None:
+            cls._Z = cls._generate_clifford_from_known_gate(1, pauli_gates.Z)
+        return cls._Z
+
+    @property
+    def H(cls):
+        if getattr(cls, '_H', None) is None:
+            cls._H = cls._generate_clifford_from_known_gate(1, common_gates.H)
+        return cls._H
+
+    @property
+    def S(cls):
+        if getattr(cls, '_S', None) is None:
+            cls._S = cls._generate_clifford_from_known_gate(1, common_gates.S)
+        return cls._S
+
+    @property
+    def CNOT(cls):
+        if getattr(cls, '_CNOT', None) is None:
+            cls._CNOT = cls._generate_clifford_from_known_gate(2, common_gates.CNOT)
+        return cls._CNOT
+
+    @property
+    def CZ(cls):
+        if getattr(cls, '_CZ', None) is None:
+            cls._CZ = cls._generate_clifford_from_known_gate(2, common_gates.CZ)
+        return cls._CZ
+
+    @property
+    def SWAP(cls):
+        if getattr(cls, '_SWAP', None) is None:
+            cls._SWAP = cls._generate_clifford_from_known_gate(2, common_gates.SWAP)
+        return cls._SWAP
+
+
+class CommonCliffordGates(metaclass=CommonCliffordGateMetaClass):
+
+    # We need to use the lazy initialization of these common gates since they need to use
+    # cirq.sim, which can not be imported when
+    @classmethod
+    def _generate_clifford_from_known_gate(
+        cls, num_qubits: int, gate: raw_types.Gate
+    ) -> 'CliffordGate':
+        qubits = devices.LineQubit.range(num_qubits)
+        t = qis.CliffordTableau(num_qubits=num_qubits)
+        args = sim.ActOnCliffordTableauArgs(
+            tableau=t, qubits=qubits, prng=np.random.RandomState(), log_of_measurement_results={}
+        )
+
+        protocols.act_on(gate, args, qubits, allow_decompose=False)
+        return CliffordGate.from_clifford_tableau(args.tableau)
+
+    @classmethod
+    def from_clifford_tableau(cls, tableau: qis.CliffordTableau) -> 'CliffordGate':
+        """Create the CliffordGate instance from Clifford Tableau.
+
+        Args:
+            tableau: A CliffordTableau to define the effect of Clifford Gate applying on
+            the stabilizer state or Pauli group. The meaning of tableau here is
+                    To  X   Z    sign
+            from  X  [ X_x Z_x | r_x ]
+            from  Z  [ X_z Z_z | r_z ]
+            Each row in the Clifford tableau indicates how the transformation of original
+            Pauli gates to the new gates after applying this Clifford Gate.
+
+        Returns:
+            A CliffordGate instance, which has the transformation defined by
+            the input tableau.
+
+        Raises:
+            ValueError: When input tableau is wrong type or the tableau does not
+            satisfy the symplectic property.
+        """
+        if not isinstance(tableau, qis.CliffordTableau):
+            raise ValueError('Input argument has to be a CliffordTableau instance.')
+        if not tableau._validate():
+            raise ValueError('It is not a valid Clifford tableau.')
+        return CliffordGate(_clifford_tableau=tableau)
+
+    @classmethod
+    def from_op_list(
+        cls, operations: Sequence[raw_types.Operation], qubit_order: Sequence[raw_types.Qid]
+    ) -> 'CliffordGate':
+        """Construct a new Clifford gates from several known operations.
+
+        Args:
+            operations: A list of cirq operations to construct the Clifford gate.
+                The combination order is the first element in the list applies the transformation
+                on the stabilizer state first.
+            qubit_order: Determines how qubits are ordered when decomposite the operations.
+
+        Returns:
+            A CliffordGate instance, which has the transformation on the stabilizer
+            state equivalent to the composition of operations.
+
+        Raises:
+            ValueError: When one or more operations do not have stabilizer effect.
+        """
+        for op in operations:
+            if op.gate and op.gate._has_stabilizer_effect_():
+                continue
+            raise ValueError(
+                "Clifford Gate can only be constructed from the "
+                "operations that has stabilizer effect."
+            )
+
+        base_tableau = qis.CliffordTableau(len(qubit_order))
+        args = sim.clifford.ActOnCliffordTableauArgs(
+            tableau=base_tableau,
+            qubits=qubit_order,
+            prng=np.random.RandomState(0),  # unused
+            log_of_measurement_results={},  # unused
+        )
+        for op in operations:
+            protocols.act_on(op, args, allow_decompose=True)
+
+        return CliffordGate.from_clifford_tableau(args.tableau)
+
+    @classmethod
+    def _from_json_dict_(cls, n, rs, xs, zs, **kwargs):
+        _clifford_tableau = qis.CliffordTableau._from_json_dict_(
+            n,
+            rs,
+            xs,
+            zs,
+        )
+        return cls(_clifford_tableau=_clifford_tableau)
+
+
+def _pad_tableau(
+    clifford_tableau: qis.CliffordTableau, num_qubits_after_padding: int, axes: List[int]
+) -> qis.CliffordTableau:
+    """Roughly, this function copies self.tabluea into the "identity" matrix."""
+    # Sanity check
+    if len(set(axes)) != clifford_tableau.n:
+        raise ValueError(
+            "Input axes of padding should match with the number of qubits in the input tableau."
+        )
+    if clifford_tableau.n > num_qubits_after_padding:
+        raise ValueError(
+            "The number of qubits in the input tableau should not be larger than "
+            "num_qubits_after_padding."
+        )
+
+    padded_tableau = qis.CliffordTableau(num_qubits_after_padding)
+    v_index = np.concatenate((np.asarray(axes), num_qubits_after_padding + np.asarray(axes)))
+
+    padded_tableau.xs[np.ix_(v_index, axes)] = clifford_tableau.xs
+    padded_tableau.zs[np.ix_(v_index, axes)] = clifford_tableau.zs
+    padded_tableau.rs[v_index] = clifford_tableau.rs
+    return padded_tableau
+
+
+@value.value_equality
+class CliffordGate(raw_types.Gate, CommonCliffordGates):
+    """Clifford rotation for N-qubit."""
+
+    def __init__(
+        self,
+        *,
+        _clifford_tableau: qis.CliffordTableau,
+    ) -> None:
+        # We use the Clifford tableau to represent a Clifford gate.
+        # It is crucial to note that the meaning of tableau here is different
+        # from the one used to represent a Clifford state (Of course, they are related).
+        # A) We have to use the full 2n * (2n + 1) matrix
+        # B) The meaning of tableau here is
+        #                 X   Z    sign
+        #     from  X  [ X_x Z_x | r_x ]
+        #     from  Z  [ X_z Z_z | r_z ]
+        # Each row in the Clifford tableau means the transformation of original Pauli gates.
+        # For example, take a 2 * (2+1) tableau as example:
+        #         X       Z     r
+        #  XI  [ 1  0 | 1  0  | 0 ]
+        #  IX  [ 0  0 | 1  1  | 0 ]
+        #  ZI  [ 0  0 | 1  0  | 1 ]
+        #  IZ  [ 1  0 | 1  1  | 0 ]
+        # Take the third row as example: this means the ZI gate after the this gate,
+        # more precisely the conjugate transformation of ZI by this gate, becomes -ZI.
+        # (Note the real clifford tableau has to satify the Symplectic property.
+        # here is just for illustration)
+        self._clifford_tableau = _clifford_tableau.copy()
+
+    @property
+    def clifford_tableau(self):
+        return self._clifford_tableau
+
+    def _json_dict_(self) -> Dict[str, Any]:
+        json_dict = self._clifford_tableau._json_dict_()
+        return json_dict
+
+    def _value_equality_values_(self):
+        return self.clifford_tableau
+
+    def _num_qubits_(self):
+        return self.clifford_tableau.n
+
+    def _has_stabilizer_effect_(self) -> Optional[bool]:
+        # By definition, Clifford Gate should always return True.
+        return True
+
+    def __pow__(self, exponent) -> 'CliffordGate':
+        if exponent == -1:
+            return CliffordGate.from_clifford_tableau(self.clifford_tableau.inverse())
+        if exponent > 0 and int(exponent) == exponent:
+            base_tableau = self.clifford_tableau.copy()
+            for _ in range(int(exponent) - 1):
+                base_tableau = base_tableau.then(self.clifford_tableau)
+            return CliffordGate.from_clifford_tableau(base_tableau)
+        if exponent < 0 and int(exponent) == exponent:
+            base_tableau = self.clifford_tableau.copy()
+            for _ in range(int(-exponent) - 1):
+                base_tableau = base_tableau.then(self.clifford_tableau)
+            return CliffordGate.from_clifford_tableau(base_tableau.inverse())
+
+        return NotImplemented
+
+    def __repr__(self) -> str:
+        return f"Clifford Gate with Tableau:\n {self.clifford_tableau._str_full_()}"
+
+    def _commutes_(self, other: Any, atol: float) -> Union[bool, NotImplementedType, None]:
+        # Note even if we assume two gates define the tabluea based on the same qubit order,
+        # the following approach cannot judge it:
+        # self.clifford_tableau.then(other.clifford_tableau) == other.clifford_tableau.then(
+        #     self.clifford_tableau
+        # )
+        # For example: X.then(Z) and Z.then(X) both return same tableau
+        # it is because Clifford tableau ignores the global phase information.
+        return NotImplemented
+
+    def _decompose_(self, qubits: Sequence['cirq.Qid']) -> List[raw_types.Operation]:
+        return transformers.analytical_decompositions.decompose_clifford_tableau_to_operations(
+            list(qubits), self.clifford_tableau
+        )
+
+    def _act_on_(self, args: 'cirq.ActOnArgs', qubits: Sequence['cirq.Qid']) -> bool:
+
+        # Note the computation complexity difference between _decompose_ and _act_on_.
+        # Suppose this Gate has `m` qubits, args has `n` qubits, and the decomposition of
+        # this operation into `k` operations:
+        #   1. Direct act_on is O(n^3) -- two matrices multiplication
+        #   2. Decomposition is O(m^3)+O(k*n^2) -- Decomposition complexity + k * One/two-qubits Ops
+        # So when m << n, the decomposition is more efficient.
+        if isinstance(args, sim.clifford.ActOnCliffordTableauArgs):
+            axes = args.get_axes(qubits)
+            # This padding is important and cannot be omitted.
+            padded_tableau = _pad_tableau(self._clifford_tableau, len(args.qubits), axes)
+            args._state = args.tableau.then(padded_tableau)
+            return True
+
+        if isinstance(args, sim.clifford.ActOnStabilizerCHFormArgs):
+            # Do we know how to apply CliffordTableau on ActOnStabilizerCHFormArgs?
+            # It should be unlike because CliffordTableau ignores the global phase but CHForm
+            # is aimed to fix that.
+            return NotImplemented
+
+        return NotImplemented