def proj(self, u, vec):
"""Returns projection of vectors on a tangen space
of the manifold.
Args:
u: complex valued tensor of shape (..., n, r),
a set of points from the manifold.
vec: complex valued tensor of shape (..., n, r),
a set of vectors to be projected.
Returns:
complex valued tensor of shape (..., n, r),
a set of projected vectors.
Note:
The complexity is O(nr^2)."""
# projection onto the tangent space of ||u||_F = 1
vec_proj = vec - u * tf.reduce_sum(
tf.math.conj(u) * vec, axis=(-2, -1))[..., tf.newaxis, tf.newaxis]
# projection onto the horizontal space
uu = adj(u) @ u
Omega = lyap_symmetric(uu, adj(u) @ vec_proj - adj(vec_proj) @ u)
return vec_proj - u @ Omega
def proj(self, u, vec):
"""Returns projection of vectors on a tangen space
of the manifold.
Args:
u: complex valued tensor of shape (..., n ** 2, k),
a set of points from the manifold.
vec: complex valued tensor of shape (..., n ** 2, k),
a set of vectors to be projected.
Returns:
complex valued tensor of shape (..., n ** 2, k),
a set of projected vectors."""
k = u.shape[-1]
n = int(math.sqrt(u.shape[-2]))
shape = u.shape[:-2]
# projection onto the tangent space of the Stiefel manifold
vec_mod = tf.reshape(vec, shape + (n, k * n))
vec_mod = tf.linalg.matrix_transpose(vec_mod)
u_mod = tf.reshape(u, shape + (n, k * n))
u_mod = tf.linalg.matrix_transpose(u_mod)
vec_mod = vec_mod - 0.5 * u_mod @ (adj(u_mod) @ vec_mod +\
adj(vec_mod) @ u_mod)
vec_mod = tf.linalg.matrix_transpose(vec_mod)
vec_mod = tf.reshape(vec_mod, shape + (n ** 2, k))
# projection onto the horizontal space
uu = adj(u) @ u
Omega = lyap_symmetric(uu, adj(u) @ vec_mod - adj(vec_mod) @ u)
return vec_mod - u @ Omega
def proj(self, u, vec):
"""Returns projection of vectors on a tangen space
of the manifold.
Args:
u: complex valued tensor of shape (..., m, n, n),
a set of points from the manifold.
vec: complex valued tensor of shape (..., m, n, n),
a set of vectors to be projected.
Returns:
complex valued tensor of shape (..., m, n, n),
a set of projected vectors.
Note:
The complexity is O(mn^3)."""
n = tf.shape(u)[-1]
m = tf.shape(u)[-3]
shape = tf.shape(u)[:-3]
size = tf.size(shape)
idx = tf.range(size)
# projection onto the tangent space of the Stiefel manifold
vec_mod = tf.transpose(
vec,
shape_conc(idx, (size + 2)[tf.newaxis], size[tf.newaxis],
(size + 1)[tf.newaxis]))
vec_mod = tf.reshape(
vec_mod, shape_conc(shape, (n * m)[tf.newaxis], n[tf.newaxis]))
u_mod = tf.transpose(
u,
shape_conc(idx, (size + 2)[tf.newaxis], size[tf.newaxis],
(size + 1)[tf.newaxis]))
u_mod = tf.reshape(
u_mod, shape_conc(shape, (n * m)[tf.newaxis], n[tf.newaxis]))
vec_mod = vec_mod - 0.5 * u_mod @ (adj(u_mod) @ vec_mod +\
adj(vec_mod) @ u_mod)
vec_mod = tf.reshape(
vec_mod,
shape_conc(shape, n[tf.newaxis], m[tf.newaxis], n[tf.newaxis]))
vec_mod = tf.transpose(
vec_mod,
shape_conc(idx, (size + 1)[tf.newaxis], (size + 2)[tf.newaxis],
size[tf.newaxis]))
# projection onto the horizontal space (POVM element-wise)
uu = adj(u) @ u
Omega = lyap_symmetric(uu, adj(u) @ vec_mod - adj(vec_mod) @ u)
return vec_mod - u @ Omega
def proj(self, u, vec):
"""Returns projection of vectors on a tangen space
of the manifold.
Args:
u: complex valued tensor of shape (..., n ** 2, k),
a set of points from the manifold.
vec: complex valued tensor of shape (..., n ** 2, k),
a set of vectors to be projected.
Returns:
complex valued tensor of shape (..., n ** 2, k),
a set of projected vectors.
Note:
The complexity O(kn^3+k^2n^2)"""
k = tf.shape(u)[-1]
n = tf.cast(tf.math.sqrt(tf.cast(tf.shape(u)[-2], dtype=tf.float32)),
dtype=tf.int32)
shape = tf.shape(u)[:-2]
# projection onto the tangent space of the Stiefel manifold
vec_mod = tf.reshape(
vec, shape_conc(shape, n[tf.newaxis], (k * n)[tf.newaxis]))
vec_mod = tf.linalg.matrix_transpose(vec_mod)
u_mod = tf.reshape(
u, shape_conc(shape, n[tf.newaxis], (k * n)[tf.newaxis]))
u_mod = tf.linalg.matrix_transpose(u_mod)
vec_mod = vec_mod - 0.5 * u_mod @ (adj(u_mod) @ vec_mod +\
adj(vec_mod) @ u_mod)
vec_mod = tf.linalg.matrix_transpose(vec_mod)
vec_mod = tf.reshape(
vec_mod, shape_conc(shape, (n**2)[tf.newaxis], k[tf.newaxis]))
# projection onto the horizontal space
uu = adj(u) @ u
Omega = lyap_symmetric(uu, adj(u) @ vec_mod - adj(vec_mod) @ u)
return vec_mod - u @ Omega
def proj(self, u, vec):
"""Returns projection of vectors on a tangen space
of the manifold.
Args:
u: complex valued tensor of shape (..., n, n),
a set of points from the manifold.
vec: complex valued tensor of shape (..., n, n),
a set of vectors to be projected.
Returns:
complex valued tensor of shape (..., n, n),
a set of projected vectors"""
# projection onto the tangent space of ||u||_F = 1
vec_proj = vec - u * tf.linalg.trace(adj(u) @ vec)[..., tf.newaxis,
tf.newaxis]
# projection onto the horizontal space
uu = adj(u) @ u
Omega = lyap_symmetric(uu, adj(u) @ vec_proj - adj(vec_proj) @ u)
return vec_proj - u @ Omega
def proj(self, u, vec):
"""Returns projection of vectors on a tangen space
of the manifold.
Args:
u: complex valued tensor of shape (..., m, n, n),
a set of points from the manifold.
vec: complex valued tensor of shape (..., m, n, n),
a set of vectors to be projected.
Returns:
complex valued tensor of shape (..., m, n, n),
a set of projected vectors."""
n = u.shape[-1]
m = u.shape[-3]
shape = u.shape[:-3]
size = len(shape)
idx = tuple(range(size))
# projection onto the tangent space of the Stiefel manifold
vec_mod = tf.transpose(vec, idx + (size + 2, size, size + 1))
vec_mod = tf.reshape(vec_mod, shape + (n * m, n))
u_mod = tf.transpose(u, idx + (size + 2, size, size + 1))
u_mod = tf.reshape(u_mod, shape + (n * m, n))
vec_mod = vec_mod - 0.5 * u_mod @ (adj(u_mod) @ vec_mod +\
adj(vec_mod) @ u_mod)
vec_mod = tf.reshape(vec_mod, shape + (n, m, n))
vec_mod = tf.transpose(vec_mod, idx + (size + 1, size + 2, size))
# projection onto the horizontal space (POVM element-wise)
uu = adj(u) @ u
Omega = lyap_symmetric(uu, adj(u) @ vec_mod - adj(vec_mod) @ u)
return vec_mod - u @ Omega