Boolean Hamiltonian gate (quantumlib#4309)

Following quantumlib#4282, the present PR would like to add a gate that allows computing the Hamiltonian from a Boolean expression. Note that while the decomposition is somewhat efficient, more optimizations are possible and planned in a follow-up PR.

cirq/__init__.py

@@ -175,6 +175,7 @@
     BaseDensePauliString,
     bit_flip,
     BitFlipChannel,
+    BooleanHamiltonian,
     CCX,
     CCXPowGate,
     CCZ,

cirq/json_resolver_cache.py

@@ -52,6 +52,7 @@ def two_qubit_matrix_gate(matrix):
         'AsymmetricDepolarizingChannel': cirq.AsymmetricDepolarizingChannel,
         'BitFlipChannel': cirq.BitFlipChannel,
         'BitstringAccumulator': cirq.work.BitstringAccumulator,
+        'BooleanHamiltonian': cirq.BooleanHamiltonian,
         'ProductState': cirq.ProductState,
         'CCNotPowGate': cirq.CCNotPowGate,
         'CCXPowGate': cirq.CCXPowGate,

cirq/ops/__init__.py

@@ -18,6 +18,10 @@
     ArithmeticOperation,
 )
 
+from cirq.ops.boolean_hamiltonian import (
+    BooleanHamiltonian,
+)
+
 from cirq.ops.clifford_gate import (
     PauliTransform,
     SingleQubitCliffordGate,

cirq/ops/boolean_hamiltonian.py

@@ -0,0 +1,160 @@
+# Copyright 2021 The Cirq Developers
+#
+# Licensed under the Apache License, Version 2.0 (the "License");
+# you may not use this file except in compliance with the License.
+# You may obtain a copy of the License at
+#
+#     https://www.apache.org/licenses/LICENSE-2.0
+#
+# Unless required by applicable law or agreed to in writing, software
+# distributed under the License is distributed on an "AS IS" BASIS,
+# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+# See the License for the specific language governing permissions and
+# limitations under the License.
+"""Represents Boolean functions as a series of CNOT and rotation gates. The Boolean functions are
+passed as Sympy expressions and then turned into an optimized set of gates.
+
+References:
+[1] On the representation of Boolean and real functions as Hamiltonians for quantum computing
+    by Stuart Hadfield, https://arxiv.org/pdf/1804.09130.pdf
+[2] https://www.youtube.com/watch?v=AOKM9BkweVU is a useful intro
+[3] https://github.com/rsln-s/IEEE_QW_2020/blob/master/Slides.pdf
+"""
+
+from typing import cast, Any, Dict, Generator, List, Sequence, Tuple
+
+import sympy.parsing.sympy_parser as sympy_parser
+
+import cirq
+from cirq import value
+from cirq.ops import raw_types
+from cirq.ops.linear_combinations import PauliSum, PauliString
+
+
+@value.value_equality
+class BooleanHamiltonian(raw_types.Operation):
+    """An operation that represents a Hamiltonian from a set of Boolean functions."""
+
+    def __init__(
+        self,
+        qubit_map: Dict[str, 'cirq.Qid'],
+        boolean_strs: Sequence[str],
+        theta: float,
+    ):
+        """Builds a BooleanHamiltonian.
+
+        For each element of a sequence of Boolean expressions, the code first transforms it into a
+        polynomial of Pauli Zs that represent that particular expression. Then, we sum all the
+        polynomials, thus making a function that goes from a series to Boolean inputs to an integer
+        that is the number of Boolean expressions that are true.
+
+        For example, if we were using this gate for the unweighted max-cut problem that is typically
+        used to demonstrate the QAOA algorithm, there would be one Boolean expression per edge. Each
+        Boolean expression would be true iff the vertices on that are in different cuts (i.e. it's)
+        an XOR.
+
+        Then, we compute exp(-j * theta * polynomial), which is unitary because the polynomial is
+        Hermitian.
+
+        Args:
+            boolean_strs: The list of Sympy-parsable Boolean expressions.
+            qubit_map: map of string (boolean variable name) to qubit.
+            theta: The evolution time (angle) for the Hamiltonian
+        """
+        self._qubit_map: Dict[str, 'cirq.Qid'] = qubit_map
+        self._boolean_strs: Sequence[str] = boolean_strs
+        self._theta: float = theta
+
+    def with_qubits(self, *new_qubits: 'cirq.Qid') -> 'BooleanHamiltonian':
+        return BooleanHamiltonian(
+            {cast(cirq.NamedQubit, q).name: q for q in new_qubits},
+            self._boolean_strs,
+            self._theta,
+        )
+
+    @property
+    def qubits(self) -> Tuple[raw_types.Qid, ...]:
+        return tuple(self._qubit_map.values())
+
+    def num_qubits(self) -> int:
+        return len(self._qubit_map)
+
+    def _value_equality_values_(self):
+        return self._qubit_map, self._boolean_strs, self._theta
+
+    def _json_dict_(self) -> Dict[str, Any]:
+        return {
+            'cirq_type': self.__class__.__name__,
+            'qubit_map': self._qubit_map,
+            'boolean_strs': self._boolean_strs,
+            'theta': self._theta,
+        }
+
+    @classmethod
+    def _from_json_dict_(cls, qubit_map, boolean_strs, theta, **kwargs):
+        return cls(qubit_map, boolean_strs, theta)
+
+    def _decompose_(self):
+        boolean_exprs = [sympy_parser.parse_expr(boolean_str) for boolean_str in self._boolean_strs]
+        hamiltonian_polynomial_list = [
+            PauliSum.from_boolean_expression(boolean_expr, self._qubit_map)
+            for boolean_expr in boolean_exprs
+        ]
+
+        return _get_gates_from_hamiltonians(
+            hamiltonian_polynomial_list, self._qubit_map, self._theta
+        )
+
+
+def _get_gates_from_hamiltonians(
+    hamiltonian_polynomial_list: List['cirq.PauliSum'],
+    qubit_map: Dict[str, 'cirq.Qid'],
+    theta: float,
+) -> Generator['cirq.Operation', None, None]:
+    """Builds a circuit according to [1].
+
+    Args:
+        hamiltonian_polynomial_list: the list of Hamiltonians, typically built by calling
+            PauliSum.from_boolean_expression().
+        qubit_map: map of string (boolean variable name) to qubit.
+        theta: A single float scaling the rotations.
+    Yields:
+        Gates that are the decomposition of the Hamiltonian.
+    """
+    combined = sum(hamiltonian_polynomial_list, PauliSum.from_pauli_strings(PauliString({})))
+
+    qubit_names = sorted(qubit_map.keys())
+    qubits = [qubit_map[name] for name in qubit_names]
+    qubit_indices = {qubit: i for i, qubit in enumerate(qubits)}
+
+    hamiltonians = {}
+    for pauli_string in combined:
+        w = pauli_string.coefficient.real
+        qubit_idx = tuple(sorted(qubit_indices[qubit] for qubit in pauli_string.qubits))
+        hamiltonians[qubit_idx] = w
+
+    def _apply_cnots(prevh: Tuple[int, ...], currh: Tuple[int, ...]):
+        cnots: List[Tuple[int, int]] = []
+
+        cnots.extend((prevh[i], prevh[-1]) for i in range(len(prevh) - 1))
+        cnots.extend((currh[i], currh[-1]) for i in range(len(currh) - 1))
+
+        # TODO(tonybruguier): At this point, some CNOT gates can be cancelled out according to:
+        # "Efficient quantum circuits for diagonal unitaries without ancillas" by Jonathan Welch,
+        # Daniel Greenbaum, Sarah Mostame, Alán Aspuru-Guzik
+        # https://arxiv.org/abs/1306.3991
+
+        for gate in (cirq.CNOT(qubits[c], qubits[t]) for c, t in cnots):
+            yield gate
+
+    previous_h: Tuple[int, ...] = ()
+    for h, w in hamiltonians.items():
+        yield _apply_cnots(previous_h, h)
+
+        if len(h) >= 1:
+            yield cirq.Rz(rads=(theta * w)).on(qubits[h[-1]])
+
+        previous_h = h
+
+    # Flush the last CNOTs.
+    yield _apply_cnots(previous_h, ())

cirq/ops/boolean_hamiltonian_test.py

@@ -0,0 +1,85 @@
+# Copyright 2021 The Cirq Developers
+#
+# Licensed under the Apache License, Version 2.0 (the "License");
+# you may not use this file except in compliance with the License.
+# You may obtain a copy of the License at
+#
+#     https://www.apache.org/licenses/LICENSE-2.0
+#
+# Unless required by applicable law or agreed to in writing, software
+# distributed under the License is distributed on an "AS IS" BASIS,
+# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+# See the License for the specific language governing permissions and
+# limitations under the License.
+import itertools
+import math
+
+import numpy as np
+import pytest
+import sympy.parsing.sympy_parser as sympy_parser
+
+import cirq
+
+
+@pytest.mark.parametrize(
+    'boolean_str',
+    [
+        'x0',
+        '~x0',
+        'x0 ^ x1',
+        'x0 & x1',
+        'x0 | x1',
+        'x0 & x1 & x2',
+        'x0 & x1 & ~x2',
+        'x0 & ~x1 & x2',
+        'x0 & ~x1 & ~x2',
+        '~x0 & x1 & x2',
+        '~x0 & x1 & ~x2',
+        '~x0 & ~x1 & x2',
+        '~x0 & ~x1 & ~x2',
+        'x0 ^ x1 ^ x2',
+        'x0 | (x1 & x2)',
+        'x0 & (x1 | x2)',
+        '(x0 ^ x1 ^ x2) | (x2 ^ x3 ^ x4)',
+        '(x0 ^ x2 ^ x4) | (x1 ^ x2 ^ x3)',
+        'x0 & x1 & (x2 | x3)',
+        'x0 & ~x2',
+        '~x0 & x2',
+        'x2 & ~x0',
+        '~x2 & x0',
+        '(x2 | x1) ^ x0',
+    ],
+)
+def test_circuit(boolean_str):
+    boolean_expr = sympy_parser.parse_expr(boolean_str)
+    var_names = cirq.parameter_names(boolean_expr)
+    qubits = [cirq.NamedQubit(name) for name in var_names]
+
+    # We use Sympy to evaluate the expression:
+    n = len(var_names)
+
+    expected = []
+    for binary_inputs in itertools.product([0, 1], repeat=n):
+        subed_expr = boolean_expr
+        for var_name, binary_input in zip(var_names, binary_inputs):
+            subed_expr = subed_expr.subs(var_name, binary_input)
+        expected.append(bool(subed_expr))
+
+    # We build a circuit and look at its output state vector:
+    circuit = cirq.Circuit()
+    circuit.append(cirq.H.on_each(*qubits))
+
+    hamiltonian_gate = cirq.BooleanHamiltonian(
+        {q.name: q for q in qubits}, [boolean_str], 0.1 * math.pi
+    )
+    assert hamiltonian_gate.with_qubits(*qubits) == hamiltonian_gate
+
+    assert hamiltonian_gate.num_qubits() == n
+
+    circuit.append(hamiltonian_gate)
+
+    phi = cirq.Simulator().simulate(circuit, qubit_order=qubits, initial_state=0).state_vector()
+    actual = np.arctan2(phi.real, phi.imag) - math.pi / 2.0 > 0.0
+
+    # Compare the two:
+    np.testing.assert_array_equal(actual, expected)

cirq/protocols/json_test_data/BooleanHamiltonian.json

@@ -0,0 +1,15 @@
+[
+  {
+    "cirq_type": "BooleanHamiltonian",
+    "qubit_map": {
+      "q0": {
+        "cirq_type": "NamedQubit",
+        "name": "q0"
+      }
+    },
+    "boolean_strs": [
+      "q0"
+    ],
+    "theta": 0.20160913
+  }
+]

cirq/protocols/json_test_data/BooleanHamiltonian.repr

@@ -0,0 +1 @@
+[cirq.BooleanHamiltonian({'q0': cirq.NamedQubit('q0')}, ['q0'], 0.20160913)]