def test_singular_values_from_theta_values_for_two_by_two_identity_matrix(
self):
"""Tests the correct theta values (in the range [0, 1] are measured for the identity matrix.
These theta values are 0.25 and 0.75, or 0.01 and 0.11 as binary decimals, respectively.
The identity matrix A = [[1, 0], [0, 1]] has singular value 1 and Froebenius norm sqrt(2). It follows that
sigma / ||A||_F = 1 / sqrt(2)
Since cos(pi * theta) = sigma / ||A||_F,
we must have cos(pi * theta) = 1 / sqrt(2),
which means that theta = - 0.25 or theta = 0.25. After mapping from the interval [-1/2, 1/2] to the interval
[0, 1] via
theta ----> theta (if 0 <= theta <= 1 / 2)
theta ----> theta + 1 (if -1 / 2 <= theta < 0)
(which is what we measure in QSVE), the possible outcomes are thus 0.25 and 0.75. These correspond to binary
decimals 0.01 and 0.10, respectively.
This test does QSVE on the identity matrix using 2, 3, 4, 5, and 6 precision qubits for QPE.
"""
# Define the identity matrix
matrix = np.identity(2)
# Create the QSVE instance
qsve = QSVE(matrix)
for nprecision_bits in [2, 3, 4, 5, 6]:
# Get the circuit to perform QSVE with terminal measurements on the QPE register
circuit = qsve.create_circuit(nprecision_bits=nprecision_bits,
terminal_measurements=True)
# Run the quantum circuit for QSVE
sim = BasicAer.get_backend("qasm_simulator")
job = execute(circuit, sim, shots=10000)
# Get the output bit strings from QSVE
res = job.result()
counts = res.get_counts()
thetas_binary = np.array(list(counts.keys()))
# Make sure there are only two measured values
self.assertEqual(len(thetas_binary), 2)
# Convert the measured angles to floating point values
thetas = [
qsve.binary_decimal_to_float(binary_decimal, big_endian=False)
for binary_decimal in thetas_binary
]
thetas = [qsve.convert_measured(theta) for theta in thetas]
# Make sure the theta values are correct
self.assertEqual(len(thetas), 2)
self.assertIn(0.25, thetas)
self.assertIn(-0.25, thetas)
def test_singular_values_two_by_two_three_pi_over_eight_singular_vector2(
self):
"""Tests computing the singular values of the matrix
A = [[cos(3 * pi / 8), 0],
[0, sin(3 * pi / 8)]]
with an input singular vector. Checks that only one singular value is present in the measurement outcome.
"""
# Define the matrix
matrix = np.array([[np.cos(3 * np.pi / 8), 0],
[0, np.sin(3 * np.pi / 8)]])
# Do the classical SVD. (Note: We could just access the singular values from the diagonal matrix elements.)
_, sigmas, _ = np.linalg.svd(matrix)
# Get the quantum circuit for QSVE
for nprecision_bits in range(3, 7):
qsve = QSVE(matrix)
circuit = qsve.create_circuit(nprecision_bits=nprecision_bits,
init_state_row_and_col=[0, 1, 0, 0],
terminal_measurements=True)
# Run the quantum circuit for QSVE
sim = BasicAer.get_backend("qasm_simulator")
job = execute(circuit, sim, shots=10000)
# Get the output bit strings from QSVE
res = job.result()
counts = res.get_counts()
thetas_binary = np.array(list(counts.keys()))
# Convert from the binary strings to theta values
computed = [
qsve.convert_measured(qsve.binary_decimal_to_float(bits))
for bits in thetas_binary
]
# Convert from theta values to singular values
qsigmas = [
np.cos(np.pi * theta) for theta in computed if theta > 0
]
# Sort the sigma values for comparison
sigmas = list(sorted(sigmas))
qsigmas = list(sorted(qsigmas))
# Make sure the quantum solution is close to the classical solution
self.assertTrue(np.allclose(sigmas[1], qsigmas))