Adds cirq.num_cnots_required and cirq.to_special (#2892)

Adds cirq.num_two_qubit_gates_required and cirq.to_special. 

- `cirq.num_cnots_required`: Based on simple linear algebra users can calculate the minimum required two-qubit gates (CZ, CNOT) to implement a two-qubit unitary. 
- `cirq.to_special`: converts a unitary to a special unitary

Context: this is the first PR breaking up #2873.

For an overview of the high level design decisions of the whole project see: https://drive.google.com/open?id=1SDEtttIAaTwfV9AUs7XAxeW3AUd0qLoY.

Author: balopat

cirq/__init__.py

@@ -144,6 +144,7 @@
     match_global_phase,
     matrix_commutes,
     matrix_from_basis_coefficients,
+    num_cnots_required,
     partial_trace,
     partial_trace_of_state_vector_as_mixture,
     PAULI_BASIS,
@@ -156,6 +157,7 @@
     sub_state_vector,
     targeted_conjugate_about,
     targeted_left_multiply,
+    to_special,
     unitary_eig,
     wavefunction_partial_trace_as_mixture,
 )

cirq/linalg/__init__.py

@@ -35,6 +35,7 @@
     KakDecomposition,
     kron_factor_4x4_to_2x2s,
     map_eigenvalues,
+    num_cnots_required,
     unitary_eig,
     scatter_plot_normalized_kak_interaction_coefficients,
     so4_to_magic_su2s,
@@ -89,5 +90,6 @@
     sub_state_vector,
     targeted_conjugate_about,
     targeted_left_multiply,
+    to_special,
     wavefunction_partial_trace_as_mixture,
 )

cirq/linalg/decompositions.py

@@ -26,7 +26,7 @@
 
 from cirq import value, protocols
 from cirq._compat import proper_repr
-from cirq.linalg import combinators, diagonalize, predicates
+from cirq.linalg import combinators, diagonalize, predicates, transformations
 
 if TYPE_CHECKING:
     import cirq
@@ -36,8 +36,16 @@
                   [0, 1j, 1, 0],
                   [0, 1j, -1, 0],
                   [1, 0, 0, -1j]]) * np.sqrt(0.5)
+
 MAGIC_CONJ_T = np.conj(MAGIC.T)
 
+# yapf: disable
+YY = np.array([[0, 0, 0, -1],
+               [0, 0, 1, 0],
+               [0, 1, 0, 0],
+               [-1, 0, 0, 0]])
+# yapf: enable
+
 
 def _phase_matrix(angle: float) -> np.ndarray:
     return np.diag([1, np.exp(1j * angle)])
@@ -992,3 +1000,48 @@ def _canonicalize_kak_vector(k_vec: np.ndarray, atol: float) -> np.ndarray:
     k_vec[need_diff, 2] *= -1
 
     return k_vec
+
+
+def num_cnots_required(u: np.ndarray, atol: float = 1e-8) -> int:
+    """Returns the min number of CNOT/CZ gates required by a two-qubit unitary.
+
+    See Proposition III.1, III.2, III.3 in Shende et al. “Recognizing Small-
+    Circuit Structure in Two-Qubit Operators and Timing Hamiltonians to Compute
+    Controlled-Not Gates”.  https://arxiv.org/abs/quant-ph/0308045
+
+    Args:
+        u: a two-qubit unitary
+    Returns:
+        the number of CNOT or CZ gates required to implement the unitary
+    """
+    if u.shape != (4, 4):
+        raise ValueError(f"Expected unitary of shape (4,4), instead "
+                         f"got {u.shape}")
+    g = _gamma(transformations.to_special(u))
+    # see Fadeev-LeVerrier formula
+    a3 = -np.trace(g)
+    # no need to check a2 = 6, as a3 = +-4 only happens if the eigenvalues are
+    # either all +1 or -1, which unambiguously implies that a2 = 6
+    if np.abs(a3 - 4) < atol or np.abs(a3 + 4) < atol:
+        return 0
+    # see Fadeev-LeVerrier formula
+    a2 = (a3 * a3 - np.trace(g @ g)) / 2
+    if np.abs(a3) < atol and np.abs(a2 - 2) < atol:
+        return 1
+    if np.abs(a3.imag) < atol:
+        return 2
+    return 3
+
+
+def _gamma(u: np.ndarray) -> np.ndarray:
+    """Gamma function to convert u to the magic basis.
+
+    See Definition IV.1 in Shende et al. "Minimal Universal Two-Qubit CNOT-based
+    Circuits." https://arxiv.org/abs/quant-ph/0308033
+
+    Args:
+        u: a member of SU(4)
+    Returns:
+        u @ yy @ u.T @ yy, where yy = Y ⊗ Y
+    """
+    return u @ YY @ u.T @ YY

cirq/linalg/decompositions_test.py

@@ -741,3 +741,38 @@ def test_kak_decompose(unitary: np.ndarray):
     np.testing.assert_allclose(cirq.unitary(circuit), unitary, atol=1e-8)
     assert len(circuit) == 5
     assert len(list(circuit.all_operations())) == 8
+
+
+def test_num_two_qubit_gates_required():
+    for i in range(4):
+        assert cirq.num_cnots_required(
+            _two_qubit_circuit_with_cnots(i).unitary()) == i
+
+    assert cirq.num_cnots_required(np.eye(4)) == 0
+
+
+def test_num_two_qubit_gates_required_invalid():
+    with pytest.raises(ValueError, match="(4,4)"):
+        cirq.num_cnots_required(np.array([[1]]))
+
+
+def _two_qubit_circuit_with_cnots(num_cnots=3, a=None, b=None):
+    random.seed(32123)
+    if a is None or b is None:
+        a, b = cirq.LineQubit.range(2)
+
+    def random_one_qubit_gate():
+        return cirq.PhasedXPowGate(phase_exponent=random.random(),
+                                   exponent=random.random())
+
+    def one_cz():
+        return [
+            cirq.CZ.on(a, b),
+            random_one_qubit_gate().on(a),
+            random_one_qubit_gate().on(b),
+        ]
+
+    return cirq.Circuit([
+        random_one_qubit_gate().on(a),
+        random_one_qubit_gate().on(b), [one_cz() for _ in range(num_cnots)]
+    ])

cirq/linalg/transformations.py

@@ -504,3 +504,20 @@ def sub_state_vector(state_vector: np.ndarray,
 @deprecated(deadline='v0.10.0', fix='Use `cirq.sub_state_vector` instead.')
 def subwavefunction(*args, **kwargs):
     return sub_state_vector(*args, **kwargs)
+
+
+def to_special(u: np.ndarray) -> np.ndarray:
+    """Converts a unitary matrix to a special unitary matrix.
+
+    All unitary matrices u have |det(u)| = 1.
+    Also for all d dimensional unitary matrix u, and scalar s:
+        det(u * s) = det(u) * s^(d)
+    To find a special unitary matrix from u:
+        u * det(u)^{-1/d}
+
+    Args:
+        u: the unitary matrix
+    Returns:
+        the special unitary matrix
+    """
+    return u * (np.linalg.det(u)**(-1 / len(u)))

cirq/linalg/transformations_test.py

@@ -671,3 +671,10 @@ def test_deprecated():
         # pylint: disable=unexpected-keyword-arg,no-value-for-parameter
         _ = cirq.partial_trace_of_state_vector_as_mixture(wavefunction=a,
                                                           keep_indices=[0])
+
+
+def test_to_special():
+    u = cirq.testing.random_unitary(4)
+    su = cirq.to_special(u)
+    assert not cirq.is_special_unitary(u)
+    assert cirq.is_special_unitary(su)

rtd_docs/api.rst

@@ -671,6 +671,7 @@ Algebra and Representation
     cirq.match_global_phase
     cirq.matrix_commutes
     cirq.matrix_from_basis_coefficients
+    cirq.num_cnots_required
     cirq.partial_trace
     cirq.partial_trace_of_state_vector_as_mixture
     cirq.reflection_matrix_pow
@@ -679,6 +680,7 @@ Algebra and Representation
     cirq.sub_state_vector
     cirq.targeted_conjugate_about
     cirq.targeted_left_multiply
+    cirq.to_special
     cirq.unitary_eig
     cirq.AxisAngleDecomposition
     cirq.Duration