def Ising_prog_3(n, t):
'''Definition of a function to obtain a sequence of gates
for an Ising chain with H = -0.5*(1/|i-j|)*sigma_x_i*sigma_x_j
- 0.5*sigma_x_j - 0.5*sigma_z_j where index i j run from 0th
to (n-1)th site and i is always smaller than j. Here 3rd-order
Trotter approximant is applied.
param n: number of qubits (n>=2).
param t: evolution time.'''
from pyquil.paulis import exponentiate
factor = -0.5
prog = Program()
# The coefficients of exponential operators
coef_lis = [7 / 24, 2 / 3, 3 / 4, -2 / 3, -1 / 24, 1]
for i in np.arange(len(comb_xx(n))):
prog_ini = Program()
prog_ini += trotterize(t * factor * coef_lis[0] * comb_xx(n)[i],
t * factor * coef_lis[0] * comb_x(n)[i],
trotter_order=3)
prog_ini += exponentiate(t * factor * coef_lis[1] * comb_z(n)[i])
prog_ini += trotterize(t * factor * coef_lis[2] * comb_xx(n)[i],
t * factor * coef_lis[2] * comb_x(n)[i],
trotter_order=3)
prog_ini += exponentiate(t * factor * coef_lis[3] * comb_z(n)[i])
prog_ini += trotterize(t * factor * coef_lis[4] * comb_xx(n)[i],
t * factor * coef_lis[4] * comb_x(n)[i],
trotter_order=3)
prog_ini += exponentiate(t * factor * coef_lis[5] * comb_z(n)[i])
prog += prog_ini
return prog
def test_trotterize():
term_one = PauliTerm("X", 0, 1.0)
term_two = PauliTerm("Z", 0, 1.0)
with pytest.raises(ValueError):
trotterize(term_one, term_two, trotter_order=0)
with pytest.raises(ValueError):
trotterize(term_one, term_two, trotter_order=5)
prog = trotterize(term_one, term_one)
result_prog = Program().inst([H(0), RZ(2.0, 0), H(0), H(0),
RZ(2.0, 0), H(0)])
assert prog == result_prog
# trotter_order 1 steps 1
prog = trotterize(term_one, term_two, trotter_steps=1)
result_prog = Program().inst([H(0), RZ(2.0, 0), H(0), RZ(2.0, 0)])
assert prog == result_prog
# trotter_order 1 steps 2
prog = trotterize(term_one, term_two, trotter_steps=2)
result_prog = Program().inst([H(0), RZ(1.0, 0), H(0), RZ(1.0, 0),
H(0), RZ(1.0, 0), H(0), RZ(1.0, 0)])
assert prog == result_prog
# trotter_order 2 steps 1
prog = trotterize(term_one, term_two, trotter_order=2)
result_prog = Program().inst([H(0), RZ(1.0, 0), H(0), RZ(2.0, 0),
H(0), RZ(1.0, 0), H(0)])
assert prog == result_prog
# trotter_order 2 steps 2
prog = trotterize(term_one, term_two, trotter_order=2, trotter_steps=2)
result_prog = Program().inst([H(0), RZ(0.5, 0), H(0), RZ(1.0, 0),
H(0), RZ(0.5, 0), H(0),
H(0), RZ(0.5, 0), H(0), RZ(1.0, 0),
H(0), RZ(0.5, 0), H(0)])
assert prog == result_prog
# trotter_order 3 steps 1
prog = trotterize(term_one, term_two, trotter_order=3, trotter_steps=1)
result_prog = Program().inst([H(0), RZ(14.0 / 24, 0), H(0), RZ(4.0 / 3.0, 0),
H(0), RZ(1.5, 0), H(0), RZ(-4.0 / 3.0, 0),
H(0), RZ(-2.0 / 24, 0), H(0), RZ(2.0, 0)])
assert prog == result_prog
def error(order, time_step_length):
a_pauli = time_step_length * sZ(0) * sY(1) * sX(2)
a_program = a_pauli.program
b_pauli = time_step_length * sX(0) * sZ(1) * sY(2)
b_program = b_pauli.program
num_qubits = len(a_program.get_qubits())
assert num_qubits == len(b_program.get_qubits())
a = program_unitary(a_program, num_qubits)
b = program_unitary(b_program, num_qubits)
a_plus_b = a + b
exp_a_plus_b = expmi(time_step_length * a_plus_b)
trotter_program = trotterize(a_pauli, b_pauli, trotter_order=order)
trotter = program_unitary(trotter_program, num_qubits)
return np.linalg.norm(exp_a_plus_b - trotter, np.inf)
def Ising_prog_4(n, t):
'''Definition of a function to obtain a sequence of gates
for an Ising chain with H = -0.5*(1/|i-j|)*sigma_x_i*sigma_x_j
- 0.5*sigma_x_j - 0.5*sigma_z_j where index i j run from 0th
to (n-1)th site and i is always smaller than j. Here 4th-order
Trotter approximant is applied.
param n: number of qubits (n>=2).
param t: evolution time.'''
from pyquil.paulis import exponentiate
factor = -0.5
prog = Program()
# The coefficient k2
coef_k = 1 / (4 - 4**(1 / 3))
for i in np.arange(len(comb_xx(n))):
prog_ini = Program()
prog_ini += trotterize(t * factor * (coef_k / 2) * comb_xx(n)[i],
t * factor * (coef_k / 2) * comb_x(n)[i],
trotter_order=4)
prog_ini += exponentiate(t * factor * coef_k * comb_z(n)[i])
prog_ini += trotterize(t * factor * coef_k * comb_xx(n)[i],
t * factor * coef_k * comb_x(n)[i],
trotter_order=4)
prog_ini += exponentiate(t * factor * coef_k * comb_z(n)[i])
prog_ini += trotterize(
t * factor * ((1 - 3 * coef_k) / 2) * comb_xx(n)[i],
t * factor * ((1 - 3 * coef_k) / 2) * comb_x(n)[i],
trotter_order=4)
prog_ini += exponentiate(t * factor * (1 - 4 * coef_k) * comb_z(n)[i])
prog_ini += trotterize(
t * factor * ((1 - 3 * coef_k) / 2) * comb_xx(n)[i],
t * factor * ((1 - 3 * coef_k) / 2) * comb_x(n)[i],
trotter_order=4)
prog_ini += exponentiate(t * factor * coef_k * comb_z(n)[i])
prog_ini += trotterize(t * factor * coef_k * comb_xx(n)[i],
t * factor * coef_k * comb_x(n)[i],
trotter_order=4)
prog_ini += exponentiate(t * factor * coef_k * comb_z(n)[i])
prog_ini += trotterize(t * factor * (coef_k / 2) * comb_xx(n)[i],
t * factor * (coef_k / 2) * comb_x(n)[i],
trotter_order=4)
prog += prog_ini
return prog
def test_trotterize():
q = QubitPlaceholder.register(8)
term_one = PauliTerm("X", q[0], 1.0)
term_two = PauliTerm("Z", q[0], 1.0)
with pytest.raises(ValueError):
trotterize(term_one, term_two, trotter_order=0)
with pytest.raises(ValueError):
trotterize(term_one, term_two, trotter_order=5)
prog = trotterize(term_one, term_one)
result_prog = Program().inst(
[H(q[0]),
RZ(2.0, q[0]),
H(q[0]),
H(q[0]),
RZ(2.0, q[0]),
H(q[0])])
assert address_qubits(prog) == address_qubits(result_prog)
# trotter_order 1 steps 1
prog = trotterize(term_one, term_two, trotter_steps=1)
result_prog = Program().inst(
[H(q[0]), RZ(2.0, q[0]),
H(q[0]), RZ(2.0, q[0])])
assert address_qubits(prog) == address_qubits(result_prog)
# trotter_order 1 steps 2
prog = trotterize(term_one, term_two, trotter_steps=2)
result_prog = Program().inst([
H(q[0]),
RZ(1.0, q[0]),
H(q[0]),
RZ(1.0, q[0]),
H(q[0]),
RZ(1.0, q[0]),
H(q[0]),
RZ(1.0, q[0])
])
assert address_qubits(prog) == address_qubits(result_prog)
# trotter_order 2 steps 1
prog = trotterize(term_one, term_two, trotter_order=2)
result_prog = Program().inst([
H(q[0]),
RZ(1.0, q[0]),
H(q[0]),
RZ(2.0, q[0]),
H(q[0]),
RZ(1.0, q[0]),
H(q[0])
])
assert address_qubits(prog) == address_qubits(result_prog)
# trotter_order 2 steps 2
prog = trotterize(term_one, term_two, trotter_order=2, trotter_steps=2)
result_prog = Program().inst([
H(q[0]),
RZ(0.5, q[0]),
H(q[0]),
RZ(1.0, q[0]),
H(q[0]),
RZ(0.5, q[0]),
H(q[0]),
H(q[0]),
RZ(0.5, q[0]),
H(q[0]),
RZ(1.0, q[0]),
H(q[0]),
RZ(0.5, q[0]),
H(q[0])
])
assert address_qubits(prog) == address_qubits(result_prog)
# trotter_order 3 steps 1
prog = trotterize(term_one, term_two, trotter_order=3, trotter_steps=1)
result_prog = Program().inst([
H(q[0]),
RZ(14.0 / 24, q[0]),
H(q[0]),
RZ(4.0 / 3.0, q[0]),
H(q[0]),
RZ(1.5, q[0]),
H(q[0]),
RZ(-4.0 / 3.0, q[0]),
H(q[0]),
RZ(-2.0 / 24, q[0]),
H(q[0]),
RZ(2.0, q[0])
])
assert address_qubits(prog) == address_qubits(result_prog)
