Hamiltonian representation of binary functions

examples/hamiltonian_representation.py

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+import math
+from typing import Dict, Tuple
+
+from sympy.logic.boolalg import And, Not, Or, Xor
+from sympy.core.symbol import Symbol
+
+import cirq
+
+# References:
+# [1] On the representation of Boolean and real functions as Hamil tonians for quantum computing
+#     by StuartHadfield
+# [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
+
+# TODO(tonybruguier): Hook up this code with the QAOA example so that we can solve generic problems
+# instead of just the max-cut example.
+
+
+class HamiltonianList:
+    """A container class of Boolean function as equation (2) or [1]"""
+
+    def __init__(self, hamiltonians: Dict[Tuple[int, ...], float]):
+        self.hamiltonians = {h: w for h, w in hamiltonians.items() if math.fabs(w) > 1e-12}
+
+    def __str__(self):
+        # For run-to-run identicalness, we sort the keys lexicographically.
+        return "; ".join(
+            "%.2f.%s" % (self.hamiltonians[h], ".".join("Z_%d" % (d) for d in h) if h else 'I')
+            for h in sorted(self.hamiltonians)
+        )
+
+    def __add__(self, other):
+        return self._signed_add(other, 1.0)
+
+    def __sub__(self, other):
+        return self._signed_add(other, -1.0)
+
+    def _signed_add(self, other, sign: float):
+        hamiltonians = self.hamiltonians.copy()
+        for h, w in other.hamiltonians.items():
+            if h not in hamiltonians:
+                hamiltonians[h] = sign * w
+            else:
+                hamiltonians[h] += sign * w
+        return HamiltonianList(hamiltonians)
+
+    def __rmul__(self, other: float):
+        return HamiltonianList({k: other * w for k, w in self.hamiltonians.items()})
+
+    def __mul__(self, other):
+        hamiltonians = {}
+        for h1, w1 in self.hamiltonians.items():
+            for h2, w2 in other.hamiltonians.items():
+                h = tuple(set(h1).symmetric_difference(h2))
+                w = w1 * w2
+                if h not in hamiltonians:
+                    hamiltonians[h] = w
+                else:
+                    hamiltonians[h] += w
+        return HamiltonianList(hamiltonians)
+
+    @staticmethod
+    def O():
+        return HamiltonianList({})
+
+    @staticmethod
+    def I():
+        return HamiltonianList({(): 1.0})
+
+    @staticmethod
+    def Z(i: int):
+        return HamiltonianList({(i,): 1.0})
+
+
+def build_hamiltonian_from_boolean(boolean_expr, name_to_id) -> HamiltonianList:
+    """Builds the Hamiltonian representation of Boolean expression as per [1]:
+
+    Args:
+        boolean_expr: A Sympy expression containing symbols and Boolean operations
+        name_to_id: A dictionary from symbol name to an integer, typically built by calling
+            get_name_to_id().
+
+    Return:
+        The HamiltonianList that represents the Boolean expression.
+    """
+
+    if isinstance(boolean_expr, (And, Not, Or, Xor)):
+        sub_hamiltonians = [
+            build_hamiltonian_from_boolean(sub_boolean_expr, name_to_id)
+            for sub_boolean_expr in boolean_expr.args
+        ]
+        # We apply the equalities of theorem 1 of [1].
+        if isinstance(boolean_expr, And):
+            hamiltonian = HamiltonianList.I()
+            for sub_hamiltonian in sub_hamiltonians:
+                hamiltonian = hamiltonian * sub_hamiltonian
+        elif isinstance(boolean_expr, Not):
+            assert len(sub_hamiltonians) == 1
+            hamiltonian = HamiltonianList.I() - sub_hamiltonians[0]
+        elif isinstance(boolean_expr, Or):
+            hamiltonian = HamiltonianList.O()
+            for sub_hamiltonian in sub_hamiltonians:
+                hamiltonian = hamiltonian + sub_hamiltonian - hamiltonian * sub_hamiltonian
+        elif isinstance(boolean_expr, Xor):
+            hamiltonian = HamiltonianList.O()
+            for sub_hamiltonian in sub_hamiltonians:
+                hamiltonian = hamiltonian + sub_hamiltonian - 2.0 * hamiltonian * sub_hamiltonian
+        return hamiltonian
+    elif isinstance(boolean_expr, Symbol):
+        # Table 1 of [1], entry for 'x' is '1/2.I - 1/2.Z'
+        i = name_to_id[boolean_expr.name]
+        return 0.5 * HamiltonianList.I() - 0.5 * HamiltonianList.Z(i)
+    else:
+        raise ValueError(f'Unsupported type: {type(boolean_expr)}')
+
+
+def get_name_to_id(boolean_expr):
+    # For run-to-run identicalness, we sort the symbol name lexicographically.
+    symbol_names = sorted(symbol.name for symbol in boolean_expr.free_symbols)
+    return {symbol_name: i for i, symbol_name in enumerate(symbol_names)}
+
+
+def build_circuit_from_hamiltonian(hamiltonian, name_to_id, theta):
+    qubits = [cirq.NamedQubit(name) for name in name_to_id.keys()]
+    circuit = cirq.Circuit()
+
+    circuit.append(cirq.H.on_each(*qubits))
+
+    for h, w in hamiltonian.hamiltonians.items():
+        for i in range(1, len(h)):
+            circuit.append(cirq.CNOT(qubits[h[i]], qubits[h[0]]))
+
+        if len(h) >= 1:
+            circuit.append(cirq.Rz(rads=(theta * w)).on(qubits[h[0]]))
+
+        for i in range(1, len(h)):
+            circuit.append(cirq.CNOT(qubits[h[i]], qubits[h[0]]))
+
+    return circuit, qubits

examples/hamiltonian_representation_test.py

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+import math
+
+import numpy as np
+import pytest
+from sympy.parsing.sympy_parser import parse_expr
+
+import cirq
+import examples.hamiltonian_representation as hr
+
+# These are some of the entries of table 1.
+@pytest.mark.parametrize(
+    'boolean_expr,hamiltonian',
+    [
+        ('x', '0.50.I; -0.50.Z_0'),
+        ('~x', '0.50.I; 0.50.Z_0'),
+        ('x0 ^ x1', '0.50.I; -0.50.Z_0.Z_1'),
+        ('x0 & x1', '0.25.I; -0.25.Z_0; 0.25.Z_0.Z_1; -0.25.Z_1'),
+        ('x0 | x1', '0.75.I; -0.25.Z_0; -0.25.Z_0.Z_1; -0.25.Z_1'),
+        ('x0 ^ x1 ^ x2', '0.50.I; -0.50.Z_0.Z_1.Z_2'),
+    ],
+)
+def test_build_hamiltonian_from_boolean(boolean_expr, hamiltonian):
+    boolean = parse_expr(boolean_expr)
+    name_to_id = hr.get_name_to_id(boolean)
+    actual = hr.build_hamiltonian_from_boolean(boolean, name_to_id)
+    assert hamiltonian == str(actual)
+
+
+def test_unsupported_op():
+    not_a_boolean = parse_expr('x * x')
+    name_to_id = hr.get_name_to_id(not_a_boolean)
+    with pytest.raises(ValueError, match='Unsupported type'):
+        hr.build_hamiltonian_from_boolean(not_a_boolean, name_to_id)
+
+
+@pytest.mark.parametrize(
+    'boolean_expr, expected',
+    [
+        ('x', [False, True]),
+        ('~x', [True, False]),
+        ('x0 ^ x1', [False, True, True, False]),
+        ('x0 & x1', [False, False, False, True]),
+        ('x0 | x1', [False, True, True, True]),
+        ('x0 & ~x1 & x2', [False, False, False, False, False, True, False, False]),
+    ],
+)
+def test_circuit(boolean_expr, expected):
+    boolean = parse_expr(boolean_expr)
+    name_to_id = hr.get_name_to_id(boolean)
+    hamiltonian = hr.build_hamiltonian_from_boolean(boolean, name_to_id)
+
+    theta = 0.1 * math.pi
+    circuit, qubits = hr.build_circuit_from_hamiltonian(hamiltonian, name_to_id, theta)
+
+    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
+
+    np.testing.assert_array_equal(actual, expected)