def _compute_num_cnots(U):
r"""Compute the number of CNOTs required to implement a U in SU(4). This is based on
the trace of
.. math::
\gamma(U) = (E^\dag U E) (E^\dag U E)^T,
and follows the arguments of this paper: https://arxiv.org/abs/quant-ph/0308045.
"""
u = math.dot(Edag, math.dot(U, E))
gammaU = math.dot(u, math.T(u))
trace = math.trace(gammaU)
# Case: 0 CNOTs (tensor product), the trace is +/- 4
# We need a tolerance of around 1e-7 here in order to work with the case where U
# is specified with 8 decimal places.
if math.allclose(trace, 4, atol=1e-7) or math.allclose(
trace, -4, atol=1e-7):
return 0
# To distinguish between 1/2 CNOT cases, we need to look at the eigenvalues
evs = math.linalg.eigvals(gammaU)
sorted_evs = math.sort(math.imag(evs))
# Case: 1 CNOT, the trace is 0, and the eigenvalues of gammaU are [-1j, -1j, 1j, 1j]
# Checking the eigenvalues is needed because of some special 2-CNOT cases that yield
# a trace 0.
if math.allclose(trace, 0j, atol=1e-7) and math.allclose(
sorted_evs, [-1, -1, 1, 1]):
return 1
# Case: 2 CNOTs, the trace has only a real part (or is 0)
if math.allclose(math.imag(trace), 0.0, atol=1e-7):
return 2
# For the case with 3 CNOTs, the trace is a non-zero complex number
# with both real and imaginary parts.
return 3
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
def _decomposition_2_cnots(U, wires):
r"""If 2 CNOTs are required, we can write the circuit as
-╭U- = -A--╭X--RZ(d)--╭X--C-
-╰U- = -B--╰C--RX(p)--╰C--D-
We need to find the angles for the Z and X rotations such that the inner
part has the same spectrum as U, and then we can recover A, B, C, D.
"""
# Compute the rotation angles
u = math.dot(Edag, math.dot(U, E))
gammaU = math.dot(u, math.T(u))
evs, _ = math.linalg.eig(gammaU)
# These choices are based on Proposition III.3 of
# https://arxiv.org/abs/quant-ph/0308045
# There is, however, a special case where the circuit has the form
# -╭U- = -A--╭C--╭X--C-
# -╰U- = -B--╰X--╰C--D-
#
# or some variant of this, where the two CNOTs are adjacent.
#
# What happens here is that the set of evs is -1, -1, 1, 1 and we can write
# -╭U- = -A--╭X--SZ--╭X--C-
# -╰U- = -B--╰C--SX--╰C--D-
# where SZ and SX are square roots of Z and X respectively. (This
# decomposition comes from using Hadamards to flip the direction of the
# first CNOT, and then decomposing them and merging single-qubit gates.) For
# some reason this case is not handled properly with the full algorithm, so
# we treat it separately.
sorted_evs = math.sort(math.real(evs))
if math.allclose(sorted_evs, [-1, -1, 1, 1]):
interior_decomp = [
qml.CNOT(wires=[wires[1], wires[0]]),
qml.S(wires=wires[0]),
qml.SX(wires=wires[1]),
qml.CNOT(wires=[wires[1], wires[0]]),
]
# S \otimes SX
inner_matrix = S_SX
else:
# For the non-special case, the eigenvalues come in conjugate pairs.
# We need to find two non-conjugate eigenvalues to extract the angles.
x = math.angle(evs[0])
y = math.angle(evs[1])
# If it was the conjugate, grab a different eigenvalue.
if math.allclose(x, -y):
y = math.angle(evs[2])
delta = (x + y) / 2
phi = (x - y) / 2
interior_decomp = [
qml.CNOT(wires=[wires[1], wires[0]]),
qml.RZ(delta, wires=wires[0]),
qml.RX(phi, wires=wires[1]),
qml.CNOT(wires=[wires[1], wires[0]]),
]
RZd = qml.RZ(math.cast_like(delta, 1j), wires=0).matrix
RXp = qml.RX(phi, wires=0).matrix
inner_matrix = math.kron(RZd, RXp)
# We need the matrix representation of this interior part, V, in order to
# decompose U = (A \otimes B) V (C \otimes D)
V = math.dot(math.cast_like(CNOT10, U),
math.dot(inner_matrix, math.cast_like(CNOT10, U)))
# Now we find the A, B, C, D in SU(2), and return the decomposition
A, B, C, D = _extract_su2su2_prefactors(U, V)
A_ops = zyz_decomposition(A, wires[0])
B_ops = zyz_decomposition(B, wires[1])
C_ops = zyz_decomposition(C, wires[0])
D_ops = zyz_decomposition(D, wires[1])
return C_ops + D_ops + interior_decomp + A_ops + B_ops