def main(argv):
if len(argv) > 1:
raise app.UsageError('Too many command-line arguments.')
num_experiments = 10
depth = 8
recursion = 4
print('SK algorithm - depth: {}, recursion: {}, experiments: {}'.
format(depth, recursion, num_experiments))
base = [to_su2(ops.Hadamard()), to_su2(ops.Tgate())]
gates = create_unitaries(base, depth)
sum_dist = 0.0
for i in range(num_experiments):
U = (ops.RotationX(2.0 * np.pi * random.random()) @
ops.RotationY(2.0 * np.pi * random.random()) @
ops.RotationZ(2.0 * np.pi * random.random()))
U_approx = sk_algo(U, gates, recursion)
dist = trace_dist(U, U_approx)
sum_dist += dist
phi1 = U(state.zero)
phi2 = U_approx(state.zero)
print('[{:2d}]: Trace Dist: {:.4f} State: {:6.4f}%'.
format(i, dist,
100.0 * (1.0 - np.real(np.dot(phi1, phi2.conj())))))
print('Gates: {}, Mean Trace Dist:: {:.4f}'.
format(len(gates), sum_dist / num_experiments))
def gc_decomp(U):
"""Group Commutator Decomposition."""
def diagonalize(U):
_, V = np.linalg.eig(U)
return ops.Operator(V)
# Because of moderate numerical instability, it can happen
# that the trace is just a tad over 2.000000. If this happens,
# we tolerate it and set the trace to exactly 2.000000.
tr = np.trace(U)
if tr > 2.0:
tr = 2.0
# We know how to compute theta from u_to_bloch().
theta = 2.0 * np.arccos(np.real(tr / 2))
# The angle phi comes from eq 10 in 'The Solovay-Kitaev Algorithm' by
# Dawson, Nielsen.
phi = 2.0 * np.arcsin(np.sqrt(np.sqrt((0.5 - 0.5 * np.cos(theta / 2)))))
axis, _ = u_to_bloch(U)
V = ops.RotationX(phi)
if axis[2] < 0:
W = ops.RotationY(2 * np.pi - phi)
else:
W = ops.RotationY(phi)
V1 = diagonalize(U)
V2 = diagonalize(V @ W @ V.adjoint() @ W.adjoint())
S = V1 @ V2.adjoint()
V_tilde = S @ V @ S.adjoint()
W_tilde = S @ W @ S.adjoint()
return V_tilde, W_tilde
def gc_decomp(U):
"""Group commutator decomposition."""
def diagonalize(U):
_, V = np.linalg.eig(U)
return ops.Operator(V)
# Get axis and theta for the operator.
axis, theta = u_to_bloch(U)
# The angle phi comes from eq 10 in 'The Solovay-Kitaev Algorithm' by
# Dawson, Nielsen. It is fully derived in the book section on the
# theorem and algorithm.
phi = 2.0 * np.arcsin(np.sqrt(
np.sqrt((0.5 - 0.5 * np.cos(theta / 2)))))
V = ops.RotationX(phi)
if axis[2] > 0:
W = ops.RotationY(2 * np.pi - phi)
else:
W = ops.RotationY(phi)
Ud = diagonalize(U)
VWVdWdd = diagonalize(V @ W @ V.adjoint() @ W.adjoint())
S = Ud @ VWVdWdd.adjoint()
V_hat = S @ V @ S.adjoint()
W_hat = S @ W @ S.adjoint()
return V_hat, W_hat
def random_gates(min_length, max_length, num_experiments):
"""Just create random sequences, find the best."""
base = [to_su2(ops.Hadamard()), to_su2(ops.Tgate())]
U = (ops.RotationX(2.0 * np.pi * random.random()) @
ops.RotationY(2.0 * np.pi * random.random()) @
ops.RotationZ(2.0 * np.pi * random.random()))
min_dist = 1000
for _ in range(num_experiments):
seq_length = min_length + random.randint(0, max_length)
U_approx = ops.Identity()
for _ in range(seq_length):
g = random.randint(0, 1)
U_approx = U_approx @ base[g]
dist = trace_dist(U, U_approx)
min_dist = min(dist, min_dist)
phi1 = U(state.zero)
phi2 = U_approx(state.zero)
print('Trace Dist: {:.4f} State: {:6.4f}%'.
format(min_dist,
100.0 * (1.0 - np.real(np.dot(phi1, phi2.conj())))))
def hipster_single():
"""Single-qubit Hipster Technique."""
# This is a nice trick, outlined in this paper on "Hipster":
# https://arxiv.org/pdf/1601.07195.pdf
#
# The observation is that to apply a single-qubit gate to a
# gubit with index i, take the binary representation of inidices and
# apply the transformation matrix to the elements according
# to the power of 2 index. Generally:
# "Performing a single-qubit gate on qubit k of n-qubit quantum
# register applies G to pairs of amplitudes whose indices differ in
# k-th bits of their binary index".
#
# For example, for a 2-qubit system, to apply a gate to qubit 0:
# apply G to
# q11, q12 psi[0], psi[1]
# q21, q22 psi[2], psi[3]
#
# To apply to qubit 1:
# q11, q12 psi[0], psi[2]
# q21, q22 psi[1], psi[3]
#
# 'Outer loop' jumps by 2**(nbits+1)
# 'Inner loop' jumps by 2**k
#
# To maintain the qubit / index ordering of this infrastructure,
# the qubit index in the paper is reversed to the qubit index here.
# (Hence the (nbits - qubit - 1) above)
#
# Make sure that for sample gates and all states the transformations
# are identical.
#
for gate in (ops.PauliX(), ops.PauliZ(), ops.Hadamard(),
ops.RotationX(0.5)):
nbits = 5
for bits in helper.bitprod(nbits):
psi = state.bitstring(*bits)
qubit = random.randint(0, nbits - 1)
# Full matrix (O(n*n).
op = ops.Identity(qubit) * gate * ops.Identity(nbits - qubit - 1)
psi1 = op(psi)
# Single Qubit (O(n))
psi = apply_single_gate(gate, qubit, psi)
if not psi.is_close(psi1):
raise AssertionError('Invalid Single Gate Application.')
def test_global_phase(self):
"""Exercise 4.14 in Nielson, Chuang, HTH == phase*rotX(pi/4)."""
h = ops.Hadamard()
op = h(ops.Tgate()(h))
# If equal up to a global phase, all values should be equal.
phase = op / ops.RotationX(math.pi / 4)
self.assertTrue(
math.isclose(phase[0, 0].real, phase[0, 1].real, abs_tol=1e-6))
self.assertTrue(
math.isclose(phase[0, 0].imag, phase[0, 1].imag, abs_tol=1e-6))
self.assertTrue(
math.isclose(phase[0, 0].real, phase[1, 0].real, abs_tol=1e-6))
self.assertTrue(
math.isclose(phase[0, 0].imag, phase[1, 0].imag, abs_tol=1e-6))
self.assertTrue(
math.isclose(phase[0, 0].real, phase[1, 1].real, abs_tol=1e-6))
self.assertTrue(
math.isclose(phase[0, 0].imag, phase[1, 1].imag, abs_tol=1e-6))
def test_control_equalities(self):
"""Exercise 4.31 Nielson, Chung."""
i, x, y, z = ops.Pauli()
x1 = x * i
x2 = i * x
y1 = y * i
y2 = i * y
z1 = z * i
z2 = i * z
c = ops.Cnot(0, 1)
theta = 25.0 * math.pi / 180.0
rx2 = i * ops.RotationX(theta)
rz1 = ops.RotationZ(theta) * i
self.assertTrue(c(x1(c)).is_close(x1(x2)))
self.assertTrue((c @ x1 @ c).is_close(x1 @ x2))
self.assertTrue((c @ y1 @ c).is_close(y1 @ x2))
self.assertTrue((c @ z1 @ c).is_close(z1))
self.assertTrue((c @ x2 @ c).is_close(x2))
self.assertTrue((c @ y2 @ c).is_close(z1 @ y2))
self.assertTrue((c @ z2 @ c).is_close(z1 @ z2))
self.assertTrue((rz1 @ c).is_close(c @ rz1))
self.assertTrue((rx2 @ c).is_close(c @ rx2))
def rx(self, idx: int, theta: float):
self.apply1(ops.RotationX(theta), idx, 'rx', val=theta)
def test_double_rot(self):
"""Make sure rotations add up."""
rx = ops.RotationX(45.0 / 180.0 * math.pi)
self.assertTrue(
(rx @ rx).is_close(ops.RotationX(90.0 / 180.0 * math.pi)))
def rx(self, idx, theta):
self.apply1(ops.RotationX(theta), idx, 'rx', val=theta)