Implemented 8n T complexity decomposition of LessThanEqual gate (#6156)

This is the Comparison Oracle from https://static-content.springer.com/esm/art%3A10.1038%2Fs41534-018-0071-5/MediaObjects/41534_2018_71_MOESM1_ESM.pdf

Author: NoureldinYosri

cirq-ft/cirq_ft/algos/__init__.py

@@ -20,6 +20,8 @@
     ContiguousRegisterGate,
     LessThanEqualGate,
     LessThanGate,
+    SingleQubitCompare,
+    BiQubitsMixer,
 )
 from cirq_ft.algos.generic_select import GenericSelect
 from cirq_ft.algos.hubbard_model import PrepareHubbard, SelectHubbard

cirq-ft/cirq_ft/algos/arithmetic_gates.py

@@ -12,8 +12,9 @@
 # See the License for the specific language governing permissions and
 # limitations under the License.
 
-from typing import Iterable, Optional, Sequence, Tuple, Union
+from typing import Iterable, Optional, Sequence, Tuple, Union, List, Iterator
 
+from cirq._compat import cached_property
 import attr
 import cirq
 from cirq_ft import infra
@@ -78,7 +79,7 @@ def _decompose_with_context_(
             return
         adjoint = []
 
-        [are_equal] = context.qubit_manager.qalloc(1)
+        (are_equal,) = context.qubit_manager.qalloc(1)
 
         # Initially our belief is that the numbers are equal.
         yield cirq.X(are_equal)
@@ -130,6 +131,147 @@ def _t_complexity_(self) -> infra.TComplexity:
         )
 
 
+@attr.frozen
+class BiQubitsMixer(infra.GateWithRegisters):
+    """Implements the COMPARE2 (Fig. 1) https://static-content.springer.com/esm/art%3A10.1038%2Fs41534-018-0071-5/MediaObjects/41534_2018_71_MOESM1_ESM.pdf
+
+    This gates mixes the values in a way that preserves the result of comparison.
+    The registers being compared are 2-qubit registers where
+        x = 2*x_msb + x_lsb
+        y = 2*y_msb + y_lsb
+    The Gate mixes the 4 qubits so that sign(x - y) = sign(x_lsb' - y_lsb') where x_lsb' and y_lsb'
+    are the final values of x_lsb' and y_lsb'.
+    """  # pylint: disable=line-too-long
+
+    adjoint: bool = False
+
+    @cached_property
+    def registers(self) -> infra.Registers:
+        return infra.Registers.build(x=2, y=2, ancilla=3)
+
+    def __repr__(self) -> str:
+        return f'cirq_ft.algos.BiQubitsMixer({self.adjoint})'
+
+    def decompose_from_registers(
+        self, *, context: cirq.DecompositionContext, **quregs: Sequence[cirq.Qid]
+    ) -> cirq.OP_TREE:
+        x, y, ancilla = quregs['x'], quregs['y'], quregs['ancilla']
+        x_msb, x_lsb = x
+        y_msb, y_lsb = y
+
+        def _cswap(control: cirq.Qid, a: cirq.Qid, b: cirq.Qid, aux: cirq.Qid) -> cirq.OP_TREE:
+            """A CSWAP with 4T complexity and whose adjoint has 0T complexity.
+
+                A controlled SWAP that swaps `a` and `b` based on `control`.
+            It uses an extra qubit `aux` so that its adjoint would have
+            a T complexity of zero.
+            """
+            yield cirq.CNOT(a, b)
+            yield and_gate.And(adjoint=self.adjoint).on(control, b, aux)
+            yield cirq.CNOT(aux, a)
+            yield cirq.CNOT(a, b)
+
+        def _decomposition():
+            # computes the difference of x - y where
+            #   x = 2*x_msb + x_lsb
+            #   y = 2*y_msb + y_lsb
+            # And stores the result in x_lsb and y_lsb such that
+            #   sign(x - y) = sign(x_lsb - y_lsb)
+            # This decomposition uses 3 ancilla qubits in order to have a
+            # T complexity of 8.
+            yield cirq.X(ancilla[0])
+            yield cirq.CNOT(y_msb, x_msb)
+            yield cirq.CNOT(y_lsb, x_lsb)
+            yield from _cswap(x_msb, x_lsb, ancilla[0], ancilla[1])
+            yield from _cswap(x_msb, y_msb, y_lsb, ancilla[2])
+            yield cirq.CNOT(y_lsb, x_lsb)
+
+        if self.adjoint:
+            yield from reversed(tuple(cirq.flatten_to_ops(_decomposition())))
+        else:
+            yield from _decomposition()
+
+    def __pow__(self, power: int) -> cirq.Gate:
+        if power == 1:
+            return self
+        if power == -1:
+            return BiQubitsMixer(adjoint=not self.adjoint)
+        return NotImplemented  # coverage: ignore
+
+    def _t_complexity_(self) -> infra.TComplexity:
+        if self.adjoint:
+            return infra.TComplexity(clifford=18)
+        return infra.TComplexity(t=8, clifford=28)
+
+    def _has_unitary_(self):
+        return not self.adjoint
+
+
+@attr.frozen
+class SingleQubitCompare(infra.GateWithRegisters):
+    """Applies U|a>|b>|0>|0> = |a> |a=b> |(a<b)> |(a>b)>
+
+    Source: (FIG. 3) in https://static-content.springer.com/esm/art%3A10.1038%2Fs41534-018-0071-5/MediaObjects/41534_2018_71_MOESM1_ESM.pdf
+    """  # pylint: disable=line-too-long
+
+    adjoint: bool = False
+
+    @cached_property
+    def registers(self) -> infra.Registers:
+        return infra.Registers.build(a=1, b=1, less_than=1, greater_than=1)
+
+    def __repr__(self) -> str:
+        return f'cirq_ft.algos.SingleQubitCompare({self.adjoint})'
+
+    def decompose_from_registers(
+        self, *, context: cirq.DecompositionContext, **quregs: Sequence[cirq.Qid]
+    ) -> cirq.OP_TREE:
+        a = quregs['a']
+        b = quregs['b']
+        less_than = quregs['less_than']
+        greater_than = quregs['greater_than']
+
+        def _decomposition() -> Iterator[cirq.Operation]:
+            yield and_gate.And((0, 1), adjoint=self.adjoint).on(*a, *b, *less_than)
+            yield cirq.CNOT(*less_than, *greater_than)
+            yield cirq.CNOT(*b, *greater_than)
+            yield cirq.CNOT(*a, *b)
+            yield cirq.CNOT(*a, *greater_than)
+            yield cirq.X(*b)
+
+        if self.adjoint:
+            yield from reversed(tuple(_decomposition()))
+        else:
+            yield from _decomposition()
+
+    def __pow__(self, power: int) -> cirq.Gate:
+        if not isinstance(power, int):
+            raise ValueError('SingleQubitCompare is only defined for integer powers.')
+        if power % 2 == 0:
+            return cirq.IdentityGate(4)
+        if power < 0:
+            return SingleQubitCompare(adjoint=not self.adjoint)
+        return self
+
+    def _t_complexity_(self) -> infra.TComplexity:
+        if self.adjoint:
+            return infra.TComplexity(clifford=11)
+        return infra.TComplexity(t=4, clifford=16)
+
+
+def _equality_with_zero(
+    context: cirq.DecompositionContext, qubits: Sequence[cirq.Qid], z: cirq.Qid
+) -> cirq.OP_TREE:
+    if len(qubits) == 1:
+        (q,) = qubits
+        yield cirq.X(q)
+        yield cirq.CNOT(q, z)
+        return
+
+    ancilla = context.qubit_manager.qalloc(len(qubits) - 2)
+    yield and_gate.And(cv=[0] * len(qubits)).on(*qubits, *ancilla, z)
+
+
 @attr.frozen
 class LessThanEqualGate(cirq.ArithmeticGate):
     """Applies U|x>|y>|z> = |x>|y> |z ^ (x <= y)>"""
@@ -161,9 +303,130 @@ def __pow__(self, power: int):
     def __repr__(self) -> str:
         return f'cirq_ft.LessThanEqualGate({self.x_bitsize}, {self.y_bitsize})'
 
+    def _decompose_via_tree(
+        self, context: cirq.DecompositionContext, X: Sequence[cirq.Qid], Y: Sequence[cirq.Qid]
+    ) -> cirq.OP_TREE:
+        """Returns comparison oracle from https://static-content.springer.com/esm/art%3A10.1038%2Fs41534-018-0071-5/MediaObjects/41534_2018_71_MOESM1_ESM.pdf
+
+        This decomposition follows the tree structure of (FIG. 2)
+        """  # pylint: disable=line-too-long
+        if len(X) == 1:
+            return
+        if len(X) == 2:
+            yield BiQubitsMixer().on_registers(x=X, y=Y, ancilla=context.qubit_manager.qalloc(3))
+            return
+
+        m = len(X) // 2
+        yield self._decompose_via_tree(context, X[:m], Y[:m])
+        yield self._decompose_via_tree(context, X[m:], Y[m:])
+        yield BiQubitsMixer().on_registers(
+            x=(X[m - 1], X[-1]), y=(Y[m - 1], Y[-1]), ancilla=context.qubit_manager.qalloc(3)
+        )
+
+    def _decompose_with_context_(
+        self, qubits: Sequence[cirq.Qid], context: Optional[cirq.DecompositionContext] = None
+    ) -> cirq.OP_TREE:
+        """Decomposes the gate in a T-complexity optimal way.
+
+        The construction can be broken in 4 parts:
+            1. In case of differing bitsizes then a multicontrol And Gate
+                - Section III.A. https://arxiv.org/abs/1805.03662) is used to check whether
+                the extra prefix is equal to zero:
+                    - result stored in: `prefix_equality` qubit.
+            2. The tree structure (FIG. 2) https://static-content.springer.com/esm/art%3A10.1038%2Fs41534-018-0071-5/MediaObjects/41534_2018_71_MOESM1_ESM.pdf
+                followed by a SingleQubitCompare to compute the result of comparison of
+                the suffixes of equal length:
+                    - result stored in: `less_than` and `greater_than` with equality in qubits[-2]
+            3. The results from the previous two steps are combined to update the target qubit.
+            4. The adjoint of the previous operations is added to restore the input qubits
+                to their original state and clean the ancilla qubits.
+        """  # pylint: disable=line-too-long
+
+        if context is None:
+            context = cirq.DecompositionContext(cirq.ops.SimpleQubitManager())
+
+        lhs, rhs, target = qubits[: self.x_bitsize], qubits[self.x_bitsize : -1], qubits[-1]
+
+        n = min(len(lhs), len(rhs))
+
+        prefix_equality = None
+        adjoint: List[cirq.Operation] = []
+
+        # if one of the registers is longer than the other store equality with |0--0>
+        # into `prefix_equality` using d = |len(P) - len(Q)| And operations => 4d T.
+        if len(lhs) != len(rhs):
+            (prefix_equality,) = context.qubit_manager.qalloc(1)
+            if len(lhs) > len(rhs):
+                for op in cirq.flatten_to_ops(
+                    _equality_with_zero(context, lhs[:-n], prefix_equality)
+                ):
+                    yield op
+                    adjoint.append(cirq.inverse(op))
+            else:
+                for op in cirq.flatten_to_ops(
+                    _equality_with_zero(context, rhs[:-n], prefix_equality)
+                ):
+                    yield op
+                    adjoint.append(cirq.inverse(op))
+
+                yield cirq.X(target), cirq.CNOT(prefix_equality, target)
+
+        # compare the remaing suffix of P and Q
+        lhs = lhs[-n:]
+        rhs = rhs[-n:]
+        for op in cirq.flatten_to_ops(self._decompose_via_tree(context, lhs, rhs)):
+            yield op
+            adjoint.append(cirq.inverse(op))
+
+        less_than, greater_than = context.qubit_manager.qalloc(2)
+        yield SingleQubitCompare().on_registers(
+            a=lhs[-1], b=rhs[-1], less_than=less_than, greater_than=greater_than
+        )
+        adjoint.append(
+            SingleQubitCompare(adjoint=True).on_registers(
+                a=lhs[-1], b=rhs[-1], less_than=less_than, greater_than=greater_than
+            )
+        )
+
+        if prefix_equality is None:
+            yield cirq.X(target)
+            yield cirq.CNOT(greater_than, target)
+        else:
+            (less_than_or_equal,) = context.qubit_manager.qalloc(1)
+            yield and_gate.And([1, 0]).on(prefix_equality, greater_than, less_than_or_equal)
+            adjoint.append(
+                and_gate.And([1, 0], adjoint=True).on(
+                    prefix_equality, greater_than, less_than_or_equal
+                )
+            )
+
+            yield cirq.CNOT(less_than_or_equal, target)
+
+        yield from reversed(adjoint)
+
     def _t_complexity_(self) -> infra.TComplexity:
-        # TODO(#112): This is rough cost that ignores cliffords.
-        return infra.TComplexity(t=4 * (self.x_bitsize + self.y_bitsize))
+        n = min(self.x_bitsize, self.y_bitsize)
+        d = max(self.x_bitsize, self.y_bitsize) - n
+        is_second_longer = self.y_bitsize > self.x_bitsize
+        if d == 0:
+            # When both registers are of the same size the T complexity is
+            # 8n - 4 same as in https://static-content.springer.com/esm/art%3A10.1038%2Fs41534-018-0071-5/MediaObjects/41534_2018_71_MOESM1_ESM.pdf.  pylint: disable=line-too-long
+            return infra.TComplexity(t=8 * n - 4, clifford=46 * n - 17)
+        else:
+            # When the registers differ in size and `n` is the size of the smaller one and
+            # `d` is the difference in size. The T complexity is the sum of the tree
+            # decomposition as before giving 8n + O(1) and the T complexity of an `And` gate
+            # over `d` registers giving 4d + O(1) totaling 8n + 4d + O(1).
+            # From the decomposition we get that the constant is -4 as well as the clifford counts.
+            if d == 1:
+                return infra.TComplexity(t=8 * n, clifford=46 * n + 3 + 2 * is_second_longer)
+            else:
+                return infra.TComplexity(
+                    t=8 * n + 4 * d - 4, clifford=46 * n + 17 * d - 14 + 2 * is_second_longer
+                )
+
+    def _has_unitary_(self):
+        return True
 
 
 @attr.frozen