def _convert_to_su4(U):
r"""Check unitarity of a 4x4 matrix and convert it to :math:`SU(4)` if the determinant is not 1.
Args:
U (array[complex]): A matrix, presumed to be :math:`4 \times 4` and unitary.
Returns:
array[complex]: A :math:`4 \times 4` matrix in :math:`SU(4)` that is
equivalent to U up to a global phase.
"""
# Check unitarity
if not math.allclose(
math.dot(U, math.T(math.conj(U))), math.eye(4), atol=1e-7):
raise ValueError("Operator must be unitary.")
# Compute the determinant
det = math.linalg.det(U)
# Convert to SU(4) if it's not close to 1
if not math.allclose(det, 1.0):
exp_angle = -1j * math.cast_like(math.angle(det), 1j) / 4
U = math.cast_like(U, det) * math.exp(exp_angle)
return U
def _convert_to_su2(U):
r"""Check unitarity of a matrix and convert it to :math:`SU(2)` if possible.
Args:
U (array[complex]): A matrix, presumed to be :math:`2 \times 2` and unitary.
Returns:
array[complex]: A :math:`2 \times 2` matrix in :math:`SU(2)` that is
equivalent to U up to a global phase.
"""
# Check unitarity
if not math.allclose(
math.dot(U, math.T(math.conj(U))), math.eye(2), atol=1e-7):
raise ValueError("Operator must be unitary.")
# Compute the determinant
det = U[0, 0] * U[1, 1] - U[0, 1] * U[1, 0]
# Convert to SU(2) if it's not close to 1
if not math.allclose(det, [1.0]):
exp_angle = -1j * math.cast_like(math.angle(det), 1j) / 2
U = math.cast_like(U, exp_angle) * math.exp(exp_angle)
return U
def _decomposition_3_cnots(U, wires):
r"""The most general form of this decomposition is U = (A \otimes B) V (C \otimes D),
where V is as depicted in the circuit below:
-╭U- = -C--╭X--RZ(d)--╭C---------╭X--A-
-╰U- = -D--╰C--RY(b)--╰X--RY(a)--╰C--B-
"""
# First we add a SWAP as per v1 of arXiv:0308033, which helps with some
# rearranging of gates in the decomposition (it will cancel out the fact
# that we need to add a SWAP to fix the determinant in another part later).
swap_U = np.exp(1j * np.pi / 4) * math.dot(math.cast_like(SWAP, U), U)
# Choose the rotation angles of RZ, RY in the two-qubit decomposition.
# They are chosen as per Proposition V.1 in quant-ph/0308033 and are based
# on the phases of the eigenvalues of :math:`E^\dagger \gamma(U) E`, where
# \gamma(U) = (E^\dag U E) (E^\dag U E)^T.
# The rotation angles can be computed as follows (any three eigenvalues can be used)
u = math.dot(Edag, math.dot(swap_U, E))
gammaU = math.dot(u, math.T(u))
evs, _ = math.linalg.eig(gammaU)
# We will sort the angles so that results are consistent across interfaces.
angles = math.sort([math.angle(ev) for ev in evs])
x, y, z = angles[0], angles[1], angles[2]
# Compute functions of the eigenvalues; there are different options in v1
# vs. v3 of the paper, I'm not entirely sure why. This is the version from v3.
alpha = (x + y) / 2
beta = (x + z) / 2
delta = (z + y) / 2
# This is the interior portion of the decomposition circuit
interior_decomp = [
qml.CNOT(wires=[wires[1], wires[0]]),
qml.RZ(delta, wires=wires[0]),
qml.RY(beta, wires=wires[1]),
qml.CNOT(wires=wires),
qml.RY(alpha, wires=wires[1]),
qml.CNOT(wires=[wires[1], wires[0]]),
]
# We need the matrix representation of this interior part, V, in order to
# decompose U = (A \otimes B) V (C \otimes D)
#
# Looking at the decomposition above, V has determinant -1 (because there
# are 3 CNOTs, each with determinant -1). The relationship between U and V
# requires that both are in SU(4), so we add a SWAP after to V. We will see
# how this gets fixed later.
#
# -╭V- = -╭X--RZ(d)--╭C---------╭X--╭SWAP-
# -╰V- = -╰C--RY(b)--╰X--RY(a)--╰C--╰SWAP-
RZd = qml.RZ(math.cast_like(delta, 1j), wires=wires[0]).matrix
RYb = qml.RY(beta, wires=wires[0]).matrix
RYa = qml.RY(alpha, wires=wires[0]).matrix
V_mats = [
CNOT10,
math.kron(RZd, RYb), CNOT01,
math.kron(math.eye(2), RYa), CNOT10, SWAP
]
V = math.convert_like(math.eye(4), U)
for mat in V_mats:
V = math.dot(math.cast_like(mat, U), V)
# Now we need to find the four SU(2) operations A, B, C, D
A, B, C, D = _extract_su2su2_prefactors(swap_U, V)
# At this point, we have the following:
# -╭U-╭SWAP- = --C--╭X-RZ(d)-╭C-------╭X-╭SWAP--A
# -╰U-╰SWAP- = --D--╰C-RZ(b)-╰X-RY(a)-╰C-╰SWAP--B
#
# Using the relationship that SWAP(A \otimes B) SWAP = B \otimes A,
# -╭U-╭SWAP- = --C--╭X-RZ(d)-╭C-------╭X--B--╭SWAP-
# -╰U-╰SWAP- = --D--╰C-RZ(b)-╰X-RY(a)-╰C--A--╰SWAP-
#
# Now the SWAPs cancel, giving us the desired decomposition
# (up to a global phase).
# -╭U- = --C--╭X-RZ(d)-╭C-------╭X--B--
# -╰U- = --D--╰C-RZ(b)-╰X-RY(a)-╰C--A--
A_ops = zyz_decomposition(A, wires[1])
B_ops = zyz_decomposition(B, wires[0])
C_ops = zyz_decomposition(C, wires[0])
D_ops = zyz_decomposition(D, wires[1])
# Return the full decomposition
return C_ops + D_ops + interior_decomp + A_ops + B_ops