def test_random_unitaries_first_moment():
# the first moment should be proportional to P_21/D
N_avg = 50000
D2 = 2
D3 = 3
perm_swap = [1, 0]
# Permutation operators
# SWAP aka P_21 which maps H_1 \otimes H_2 to H_2 \otimes H_1
D2_SWAP = rand_ops.permute_tensor_factors(D2, perm_swap)
D3_SWAP = rand_ops.permute_tensor_factors(D3, perm_swap)
X = rand_ops.haar_rand_unitary(D2)
Id2 = np.matmul(X, np.conjugate(X.T))
Y = rand_ops.haar_rand_unitary(D3)
Id3 = np.matmul(Y, np.conjugate(Y.T))
D2_avg = np.zeros_like(np.kron(Id2, Id2))
D3_avg = np.zeros_like(np.kron(Id3, Id3))
for n in range(0, N_avg):
U2 = rand_ops.haar_rand_unitary(D2)
U2d = np.conjugate(U2.T)
D2_avg += np.kron(U2, U2d) / N_avg
U3 = rand_ops.haar_rand_unitary(D3)
U3d = np.conjugate(U3.T)
D3_avg += np.kron(U3, U3d) / N_avg
# Compute the Frobenius norm of the different between the estimated operator and the answer
assert np.real(la.norm((D2_avg - D2_SWAP / D2), 'fro')) <= 0.01
assert np.real(la.norm((D2_avg - D2_SWAP / D2), 'fro')) >= 0.00
assert np.real(la.norm((D3_avg - D3_SWAP / D3), 'fro')) <= 0.02
assert np.real(la.norm((D3_avg - D3_SWAP / D3), 'fro')) >= 0.00
def generate_abstract_qv_circuit(
depth: int) -> Tuple[List[np.ndarray], np.ndarray]:
"""
Produces an abstract description of the square model circuit of given depth=width used in a
quantum volume measurement.
The description remains abstract as it does not directly reference qubits in a circuit; rather,
the circuit is specified as a list of depth many permutations and depth many layers of two
qubit gates specified as a depth by depth//2 numpy array whose entries are each a haar random
four by four matrix (a single 2 qubit gate). Each permutation is simply a list of the numbers
0 through depth-1, where the number x at index i indicates the qubit in position i should be
moved to position x. The 4 by 4 matrix at gates[i, j] is the gate acting on the qubits at
positions 2j, 2j+1 after the i^th permutation has occurred.
:param depth: the depth, and also width, of the model circuit
:return: the random depth-many permutations and depth by depth//2 many 2q-gates which comprise
the model quantum circuit of [QVol] for a given depth.
"""
# generate a simple list representation for each permutation of the depth many qubits
permutations = [np.random.permutation(range(depth)) for _ in range(depth)]
# generate a matrix representation of each 2q gate in the circuit
num_gates_per_layer = depth // 2 # if odd number of qubits, don't do anything to last qubit
gates = np.asarray(
[[haar_rand_unitary(4) for _ in range(num_gates_per_layer)]
for _ in range(depth)])
return permutations, gates
def single_q_process(request):
if request.param == 'X':
return Program(X(0)), mat.X
elif request.param == 'haar':
u_rand = haar_rand_unitary(2 ** 1, rs=np.random.RandomState(52))
process = Program().defgate("RandUnitary", u_rand)
process += ("RandUnitary", 0)
return process, u_rand
def test_HS_obeys_cauchy_schwarz():
D = 2
Ur = rand_ops.haar_rand_unitary(D)
U = Ur + np.conj(Ur.T)
A = np.eye(D)
B = A / 3
assert dm.hilbert_schmidt_ip(U, U) * dm.hilbert_schmidt_ip(A, A) >= abs(dm.hilbert_schmidt_ip(A, U))**2
assert dm.hilbert_schmidt_ip(U, U) * dm.hilbert_schmidt_ip(B, B) >= abs(dm.hilbert_schmidt_ip(B, U))**2
def two_q_process(request):
if request.param == 'CNOT':
return Program(CNOT(0, 1)), mat.CNOT
elif request.param == 'haar':
u_rand = haar_rand_unitary(2 ** 2, rs=np.random.RandomState(52))
process = Program().defgate("RandUnitary", u_rand)
process += ("RandUnitary", 0, 1)
return process, u_rand
else:
raise ValueError()
def test_random_unitaries_are_unitary():
N_avg = 30
D2 = 2
D3 = 3
avg_trace2 = 0
avg_det2 = 0
avg_trace3 = 0
avg_det3 = 0
for idx in range(0, N_avg):
U2 = rand_ops.haar_rand_unitary(D2)
U3 = rand_ops.haar_rand_unitary(D3)
avg_trace2 += np.trace(np.matmul(U2, np.conjugate(U2.T))) / N_avg
avg_det2 += np.abs(la.det(U2)) / N_avg
avg_trace3 += np.trace(np.matmul(U3, np.conjugate(U3.T))) / N_avg
avg_det3 += np.abs(la.det(U3)) / N_avg
assert np.allclose(avg_trace2, 2.0)
assert np.allclose(avg_det2, 1.0)
assert np.allclose(avg_trace3, 3.0)
assert np.allclose(avg_det3, 1.0)
def test_HS_obeys_linearity():
D = 2
Ur = rand_ops.haar_rand_unitary(D)
U = Ur + np.conj(Ur.T)
A = np.eye(D)
B = A / 3
alpha = 0.17
beta = 0.6713
LHS = alpha * dm.hilbert_schmidt_ip(U, A) + beta * dm.hilbert_schmidt_ip(U, B)
RHS = dm.hilbert_schmidt_ip(U, alpha * A + beta * B)
assert np.allclose(LHS, RHS)
def single_q_tomo_fixture():
qubits = [0]
qc = get_test_qc(n_qubits=len(qubits))
# Generate random unitary
u_rand = haar_rand_unitary(2**1, rs=qc.qam.rs)
state_prep = Program().defgate("RandUnitary", u_rand)
state_prep.inst([("RandUnitary", qubits[0])])
# True state
wfn = NumpyWavefunctionSimulator(n_qubits=1)
psi = wfn.do_gate_matrix(u_rand, qubits=[0]).wf.reshape(-1)
rho_true = np.outer(psi, psi.T.conj())
# Get data from QVM
tomo_expt = generate_state_tomography_experiment(state_prep, qubits)
results = list(
measure_observables(qc=qc, tomo_experiment=tomo_expt, n_shots=100_000))
return results, rho_true
def test_random_unitaries_second_moment():
# the second moment should be proportional to
#
# P_13_24 + P_14_23 P_1423 + P_1324
# ------------------ - -------------------
# ( d^2 -1) d ( d^2 -1)
#
N_avg = 80000
D2 = 2
X = rand_ops.haar_rand_unitary(D2)
Id2 = np.matmul(X, np.conjugate(X.T))
# lists representing the permutations
p_3412 = Permutation([2, 3, 0, 1]).list()
p_4321 = Permutation([3, 2, 1, 0]).list()
p_4312 = Permutation([3, 2, 0, 1]).list()
p_3421 = Permutation([2, 3, 1, 0]).list()
# Permutation operators
P_3412 = rand_ops.permute_tensor_factors(D2, p_3412)
P_4321 = rand_ops.permute_tensor_factors(D2, p_4321)
P_4312 = rand_ops.permute_tensor_factors(D2, p_4312)
P_3421 = rand_ops.permute_tensor_factors(D2, p_3421)
# ^^ convention in https://arxiv.org/pdf/0711.1017.pdf
## For archaeological reasons I will leave this code for those who come next..
## lists representing the permutations
# p_14_23 = Permutation(0, 3)(1, 2).list()
# p_13_24 = Permutation(0, 2)(1, 3).list()
# p_1423 = Permutation([0, 3, 1, 2]).list()
# p_1324 = Permutation([0, 2, 1, 3]).list()
## Permutation operators
# P_14_23 = rand_ops.permute_tensor_factors(D2, p_14_23)
# P_13_24 = rand_ops.permute_tensor_factors(D2, p_13_24)
# P_1423 = rand_ops.permute_tensor_factors(D2, p_1423)
# P_1324 = rand_ops.permute_tensor_factors(D2, p_1324)
## ^^ convention in https://arxiv.org/pdf/0809.3813.pdf
# initalize array
D2_var = np.zeros_like(np.kron(np.kron(np.kron(Id2, Id2), Id2), Id2))
for n in range(0, N_avg):
U2 = rand_ops.haar_rand_unitary(D2)
U2d = np.conjugate(U2.T)
D2_var += np.kron(np.kron(np.kron(U2, U2), U2d), U2d) / N_avg
alpha = 1 / (D2**2 - 1) # 0.3333
beta = 1 / (D2 * (D2**2 - 1)) # 0.1666
theanswer1 = alpha * (P_3412 + P_4321) - beta * (P_4312 + P_3421)
# ^^ Equation 5.17 in https://arxiv.org/pdf/0711.1017.pdf
## For archaeological reasons I will leave this code for those who come next..
# theanswer2 = alpha * (P_13_24 + P_14_23) - beta * (P_1423 + P_1324)
## ^^ Equation at the bottom of page 2 in https://arxiv.org/pdf/0809.3813.pdf
# round and remove tiny imaginary parts
truth = np.around(theanswer1, 2)
estimate = np.around(np.real(D2_var), 2)
# are the estimated operator and the answer close?
print(truth)
print(estimate)
assert np.allclose(truth, estimate)