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.
atol: The absolute tolerance used to make this judgement.
Returns:
The number of CNOT or CZ gates required to implement the unitary.
Raises:
ValueError: If the shape of `u` is not 4 by 4.
"""
if u.shape != (4, 4):
raise ValueError(
f"Expected unitary of shape (4,4), instead 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 extract_right_diag(u: np.ndarray) -> np.ndarray:
"""Extract a diagonal unitary from a 3-CNOT two-qubit unitary.
Returns a 2-CNOT unitary D that is diagonal, so that U @ D needs only
two CNOT gates in case the original unitary is a 3-CNOT unitary.
See Proposition V.2 in Minimal Universal Two-Qubit CNOT-based Circuits.
https://arxiv.org/abs/quant-ph/0308033
Args:
u: three-CNOT two-qubit unitary
Returns:
diagonal extracted from U
"""
t = _gamma(transformations.to_special(u).T).diagonal()
k = np.real(t[0] + t[3] - t[1] - t[2])
psi = np.arctan2(np.imag(np.sum(t)), k)
f = np.exp(1j * psi)
return np.diag([1, f, f, 1])