Add an option to build a PauliSum from a Sympy Boolean expression (qu…

…antumlib#4282)

As part of the the review quantumlib#3989 we could use the ability to build a PauliSum from a Boolean expression. This PR makes available that function (smaller change).

cirq-core/cirq/ops/linear_combinations.py

@@ -14,6 +14,7 @@
 from collections import defaultdict
 from typing import (
     AbstractSet,
+    Dict,
     Iterable,
     Mapping,
     Optional,
@@ -27,6 +28,9 @@
 import numbers
 
 import numpy as np
+from sympy.logic.boolalg import And, Not, Or, Xor
+from sympy.core.expr import Expr
+from sympy.core.symbol import Symbol
 
 from cirq import linalg, protocols, qis, value
 from cirq._doc import document
@@ -398,6 +402,56 @@ def from_pauli_strings(cls, terms: Union[PauliString, List[PauliString]]) -> 'Pa
             termdict[key] += pstring.coefficient
         return cls(linear_dict=value.LinearDict(termdict))
 
+    @classmethod
+    def from_boolean_expression(
+        cls, boolean_expr: Expr, qubit_map: Dict[str, 'cirq.Qid']
+    ) -> 'PauliSum':
+        """Builds the Hamiltonian representation of a Boolean expression.
+
+        This is based on "On the representation of Boolean and real functions as Hamiltonians for
+        quantum computing" by Stuart Hadfield, https://arxiv.org/abs/1804.09130
+
+        Args:
+            boolean_expr: A Sympy expression containing symbols and Boolean operations
+            qubit_map: map of string (boolean variable name) to qubit.
+
+        Return:
+            The PauliString that represents the Boolean expression.
+        """
+        if isinstance(boolean_expr, Symbol):
+            # In table 1, the entry for 'x' is '1/2.I - 1/2.Z'
+            return cls.from_pauli_strings(
+                [
+                    PauliString({}, 0.5),
+                    PauliString({qubit_map[boolean_expr.name]: pauli_gates.Z}, -0.5),
+                ]
+            )
+
+        if isinstance(boolean_expr, (And, Not, Or, Xor)):
+            sub_pauli_sums = [
+                cls.from_boolean_expression(sub_boolean_expr, qubit_map)
+                for sub_boolean_expr in boolean_expr.args
+            ]
+            # We apply the equalities of theorem 1.
+            if isinstance(boolean_expr, And):
+                pauli_sum = cls.from_pauli_strings(PauliString({}, 1.0))
+                for sub_pauli_sum in sub_pauli_sums:
+                    pauli_sum = pauli_sum * sub_pauli_sum
+            elif isinstance(boolean_expr, Not):
+                assert len(sub_pauli_sums) == 1
+                pauli_sum = cls.from_pauli_strings(PauliString({}, 1.0)) - sub_pauli_sums[0]
+            elif isinstance(boolean_expr, Or):
+                pauli_sum = cls.from_pauli_strings(PauliString({}, 0.0))
+                for sub_pauli_sum in sub_pauli_sums:
+                    pauli_sum = pauli_sum + sub_pauli_sum - pauli_sum * sub_pauli_sum
+            elif isinstance(boolean_expr, Xor):
+                pauli_sum = cls.from_pauli_strings(PauliString({}, 0.0))
+                for sub_pauli_sum in sub_pauli_sums:
+                    pauli_sum = pauli_sum + sub_pauli_sum - 2.0 * pauli_sum * sub_pauli_sum
+            return pauli_sum
+
+        raise ValueError(f'Unsupported type: {type(boolean_expr)}')
+
     @property
     def qubits(self) -> Tuple[raw_types.Qid, ...]:
         qs = {q for k in self._linear_dict.keys() for q, _ in k}

cirq-core/cirq/ops/linear_combinations_test.py

@@ -18,6 +18,7 @@
 import numpy as np
 import pytest
 import sympy
+import sympy.parsing.sympy_parser as sympy_parser
 
 import cirq
 import cirq.testing
@@ -1352,6 +1353,42 @@ def test_pauli_sum_pow():
         assert cirq.approx_eq(psum ** 0, identity)
 
 
+# Using the entries of table 1 of https://arxiv.org/abs/1804.09130 as golden values.
+@pytest.mark.parametrize(
+    'boolean_expr,expected_pauli_sum',
+    [
+        ('x', ['(-0.5+0j)*Z(x)', '(0.5+0j)*I']),
+        ('~x', ['(0.5+0j)*I', '(0.5+0j)*Z(x)']),
+        ('x0 ^ x1', ['(-0.5+0j)*Z(x0)*Z(x1)', '(0.5+0j)*I']),
+        (
+            'x0 & x1',
+            ['(-0.25+0j)*Z(x0)', '(-0.25+0j)*Z(x1)', '(0.25+0j)*I', '(0.25+0j)*Z(x0)*Z(x1)'],
+        ),
+        (
+            'x0 | x1',
+            ['(-0.25+0j)*Z(x0)', '(-0.25+0j)*Z(x0)*Z(x1)', '(-0.25+0j)*Z(x1)', '(0.75+0j)*I'],
+        ),
+        ('x0 ^ x1 ^ x2', ['(-0.5+0j)*Z(x0)*Z(x1)*Z(x2)', '(0.5+0j)*I']),
+    ],
+)
+def test_from_boolean_expression(boolean_expr, expected_pauli_sum):
+    boolean = sympy_parser.parse_expr(boolean_expr)
+    qubit_map = {name: cirq.NamedQubit(name) for name in sorted(cirq.parameter_names(boolean))}
+    actual = cirq.PauliSum.from_boolean_expression(boolean, qubit_map)
+    # Instead of calling str() directly, first make sure that the items are sorted. This is to make
+    # the unit test more robut in case Sympy would result in a different parsing order. By sorting
+    # the individual items, we would have a canonical representation.
+    actual_items = list(sorted(str(pauli_string) for pauli_string in actual))
+    assert expected_pauli_sum == actual_items
+
+
+def test_unsupported_op():
+    not_a_boolean = sympy_parser.parse_expr('x * x')
+    qubit_map = {name: cirq.NamedQubit(name) for name in cirq.parameter_names(not_a_boolean)}
+    with pytest.raises(ValueError, match='Unsupported type'):
+        cirq.PauliSum.from_boolean_expression(not_a_boolean, qubit_map)
+
+
 def test_imul_aliasing():
     q0, q1, q2 = cirq.LineQubit.range(3)
     psum1 = cirq.X(q0) + cirq.Y(q1)

docs/operators_and_observables.ipynb

@@ -78,7 +78,8 @@
     "    print(\"installed cirq.\")\n",
     "    import cirq\n",
     "    \n",
-    "import numpy as np"
+    "import numpy as np\n",
+    "import sympy.parsing.sympy_parser as sympy_parser"
    ]
   },
   {
@@ -740,6 +741,28 @@
     "psum.matrix()"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "ba00a12b9fbd"
+   },
+   "source": [
+    "A Pauli sum can also be constructed from a `Sympy` Boolean expression. This is based on \"On the representation of Boolean and real functions as Hamiltonians for quantum computing\" by Stuart Hadfield (https://arxiv.org/abs/1804.09130)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "id": "8da2814b60f3"
+   },
+   "outputs": [],
+   "source": [
+    "psum = cirq.PauliSum.from_boolean_expression(\n",
+    "    sympy_parser.parse_expr('x0 ^ x1'),\n",
+    "    {'x0': cirq.NamedQubit('q0'), 'x1': cirq.NamedQubit('q1')})"
+   ]
+  },
   {
    "cell_type": "markdown",
    "metadata": {