# Source code for weylchamber.cartan_decomposition

```
import numpy as np
import qutip
from qutip import Qobj
from .coordinates import to_magic, from_magic, c1c2c3, canonical_gate
from ._types import Gate, Tuple
__all__ = ['cartan_decomposition']
[docs]def cartan_decomposition(U: Gate) -> Tuple[Qobj, Qobj, Qobj]:
r"""Calculate the Cartan Decomposition of the given U in U(4)
.. math::
\Op{U} = \Op{k}_1 \Op{A} \Op{k}_2
up to a global phase factor $(\det \Op{U})^{\frac{1}{4}}$.
Args:
U: Two-qubit quantum gate. Must be unitary.
Returns:
tuple $(\Op{k}_1, \Op{A}, \Op{k}_2)$ where
$\Op{k}_1$ is the left local operations in SU(2) x SU(2),
$\Op{A}$ is non-local operations, in SU(4), and $\Op{k}_2$
is the right local operations in SU(2) x SU(2).
Notes:
If you are working with a logical subspace, you should unitarize U
before calculating the Cartan decomposition
References:
* D. Reich. Optimising the nonlocal content of a two-qubit gate.
Diploma Thesis. FU Berlin, 2010. Appendix E
* Zhang et al. PRA 67, 042313 (2003)
"""
if isinstance(U, qutip.Qobj):
U = U.full()
U = np.array(U) # in U(4)
Utilde = U / np.linalg.det(U)**0.25 # U in SU(4)
found_branch = False
# The fourth root has four branches; the correct solution could be in
# any one of them
for branch in range(4):
UB = to_magic(Utilde).full() # in Bell basis
m = UB.T @ UB
# The F-matrix can be calculated according to Eq (21) in PRA 67, 042313
# It is a diagonal matrix containing the squares of the eigenvalues of
# m
c1, c2, c3 = c1c2c3(Utilde)
F1 = np.exp(np.pi * 0.5j * (+c1 - c2 + c3))
F2 = np.exp(np.pi * 0.5j * (+c1 + c2 - c3))
F3 = np.exp(np.pi * 0.5j * (-c1 - c2 - c3))
F4 = np.exp(np.pi * 0.5j * (-c1 + c2 + c3))
Fd = np.array([F1, F2, F3, F4])
F = np.array(np.diag(Fd))
# Furthermore, we have Eq (22), giving the eigen-decomposition of the
# matrix m. This gives us the matrix O_2.T of the eigenvectors of m
Fsq, O_2_transposed = np.linalg.eig(m)
ord1 = np.argsort(np.angle(Fd**2)) # sort according to complex phase
ord2 = np.argsort(np.angle(Fsq)) # ... (absolute value is 1)
diff = np.sum(np.abs((Fd**2)[ord1] - Fsq[ord2]))
# Do Fd**2 and Fsq contain the same values (irrespective of order)?
if diff < 1.0e-12:
found_branch = True
break
else:
Utilde *= 1.0j
# double check that we managed to find a branch (just to be 100% safe)
assert(found_branch), \
"Couldn't find correct branch of fourth root in mapping U(4) -> SU(4)"
# Getting the entries of F from Eq (21) instead of by taking the square
# root of Fsq has the benefit that we don't have to worry about whether we
# need to take the positive or negative root.
# However, we do need to make sure that the eigenstates are ordered to
# correspond to F1, F2, F3, F4
# After reordering, we need to transpose to get O_2 itself
reordered = np.array(np.zeros((4, 4)), dtype=np.complex128)
order = []
for i in range(4):
for j in range(4):
if (abs(Fd[i]**2 - Fsq[j]) < 1.0e-12):
if j not in order:
order.append(j)
assert len(order) == 4, "Couldn't order O_2" # should not happen
# Reorder using the order we just figured out, and transpose
for i in range(4):
reordered[:, i] = O_2_transposed[:, order[i]]
O_2 = reordered.transpose()
# Now that we have O_2 and F, completing the Cartan decomposition is
# straightforward, following along Appendix E of Daniel's thesis
k2 = from_magic(O_2)
O_1 = UB @ O_2.transpose() @ F.conjugate().transpose()
k1 = from_magic(O_1)
A = canonical_gate(c1, c2, c3)
# Check our results
assert(np.max(np.abs(O_1 @ O_1.transpose() - np.identity(4))) < 1.0e-12), \
"O_1 not orthogonal"
assert(np.max(np.abs(O_2 @ O_2.transpose() - np.identity(4))) < 1.0e-12), \
"O_2 not orthogonal"
err = (k1 * A * k2 - qutip.Qobj(Utilde, dims=[[2, 2], [2, 2]])).norm()
assert(err < 1e-12), "Cartan Decomposition Failed"
return k1, A, k2
```