def test_representation_coeff_of_nonexistant_operation():
qbit = cirq.LineQubit(0)
ideal = cirq.Circuit(cirq.X(qbit))
noisy_xop = NoisyOperation.from_cirq(circuit=cirq.X,
channel_matrix=np.zeros(shape=(4, 4)))
decomp = OperationRepresentation(ideal=ideal,
basis_expansion=dict([(noisy_xop, 0.5)]))
noisy_zop = NoisyOperation.from_cirq(circuit=cirq.Z,
channel_matrix=np.zeros(shape=(4, 4)))
with pytest.raises(ValueError, match="does not appear in the basis"):
decomp.coeff_of(noisy_zop)
def test_print_operation_representation_two_qubits_neg():
qreg = cirq.LineQubit.range(2)
ideal = cirq.Circuit(cirq.CNOT(*qreg))
noisy_a = NoisyOperation.from_cirq(
circuit=cirq.Circuit(cirq.H.on_each(qreg[0]), cirq.CNOT(
*qreg), cirq.H.on_each(qreg[1])))
noisy_b = NoisyOperation.from_cirq(
circuit=cirq.Circuit(cirq.Z.on_each(qreg[1])))
decomp = OperationRepresentation(
ideal=ideal,
basis_expansion={
noisy_a: -0.5,
noisy_b: 1.5,
},
)
print(str(decomp))
expected = f"""
0: ───@───
│
1: ───X─── ={" "}
-0.500
0: ───H───@───────
│
1: ───────X───H───
+1.500
1: ───Z───"""
# Remove initial newline
expected = expected[1:]
assert str(decomp) == expected
def _represent_operation_with_amplitude_damping_noise(
ideal_operation: Circuit,
noise_level: float,
) -> OperationRepresentation:
r"""Returns the quasi-probability representation of the input
single-qubit ``ideal_operation`` with respect to a basis of noisy
operations.
Any ideal single-qubit unitary followed by local amplitude-damping noise
of equal ``noise_level`` is assumed to be in the basis of implementable
operations.
The representation is based on the analytical result presented
in :cite:`Takagi2020`.
Args:
ideal_operation: The ideal operation (as a QPROGRAM) to represent.
noise_level: The noise level of each amplitude damping channel.
Returns:
The quasi-probability representation of the ``ideal_operation``.
.. note::
The input ``ideal_operation`` is typically a QPROGRAM with a single
gate but could also correspond to a sequence of more gates.
This is possible as long as the unitary associated to the input
QPROGRAM, followed by a single final amplitude damping channel, is
physically implementable.
.. note::
The input ``ideal_operation`` must be a ``cirq.Circuit``.
"""
if not isinstance(ideal_operation, Circuit):
raise NotImplementedError(
"The input ideal_operation must be a cirq.Circuit.", )
qubits = ideal_operation.all_qubits()
if len(qubits) == 1:
q = tuple(qubits)[0]
eta_0 = (1 + np.sqrt(1 - noise_level)) / (2 * (1 - noise_level))
eta_1 = (1 - np.sqrt(1 - noise_level)) / (2 * (1 - noise_level))
eta_2 = -noise_level / (1 - noise_level)
etas = [eta_0, eta_1, eta_2]
post_ops = [[], Z(q), reset(q)]
else:
raise ValueError( # pragma: no cover
"Only single-qubit operations are supported." # pragma: no cover
) # pragma: no cover
# Basis of implementable operations as circuits
imp_op_circuits = [ideal_operation + Circuit(op) for op in post_ops]
# Build basis_expantion
expansion = {NoisyOperation(c): a for c, a in zip(imp_op_circuits, etas)}
return OperationRepresentation(ideal_operation, expansion)
def test_representation_bad_type_for_basis_expansion():
ideal = cirq.Circuit(cirq.H(cirq.LineQubit(0)))
noisy_xop = NoisyOperation.from_cirq(circuit=cirq.X,
channel_matrix=np.zeros(shape=(4, 4)))
with pytest.raises(TypeError, match="All keys of `basis_expansion` must"):
OperationRepresentation(ideal=ideal,
basis_expansion=dict([(1.0, noisy_xop)]))
def find_optimal_representation(
ideal_operation: QPROGRAM,
noisy_basis: NoisyBasis,
tol: float = 1.0e-8,
initial_guess: Optional[np.ndarray] = None,
) -> OperationRepresentation:
r"""Returns the ``OperationRepresentaiton`` of the input ideal operation
which minimizes the one-norm of the associated quasi-probability
distribution.
More precicely, it solve the following optimization problem:
.. math::
\min_{{\eta_\alpha}} = \sum_\alpha |\eta_\alpha|,
\text{ such that }
\mathcal G = \sum_\alpha \eta_\alpha \mathcal O_\alpha,
where :math:`\{\mathcal O_j\}` is the input basis of noisy operations.
Args:
ideal_operation: The ideal operation to represent.
noisy_basis: The ``NoisyBasis`` in which the ``ideal_operation``
should be represented. It must contain ``NoisyOperation`` objects
which are initialized with a numerical superoperator matrix.
tol: The error tolerance for each matrix element
of the represented operation.
initial_guess: Optional initial guess for the coefficients
:math:`\{ \eta_\alpha \}``.
Returns: The optimal OperationRepresentation.
"""
ideal_cirq_circuit, _ = convert_to_mitiq(ideal_operation)
ideal_matrix = kraus_to_super(
cast(List[np.ndarray], kraus(ideal_cirq_circuit)))
basis_set = noisy_basis.elements
try:
basis_matrices = [noisy_op.channel_matrix for noisy_op in basis_set]
except ValueError as err:
if str(err) == "The channel matrix is unknown.":
raise ValueError(
"The input noisy_basis should contain NoisyOperation objects"
" which are initialized with a numerical superoperator matrix."
)
else:
raise err # pragma no cover
# Run numerical optimization problem
quasi_prob_dist = minimize_one_norm(
ideal_matrix,
basis_matrices,
tol=tol,
initial_guess=initial_guess,
)
basis_expansion = {op: eta for op, eta in zip(basis_set, quasi_prob_dist)}
return OperationRepresentation(ideal_operation, basis_expansion)
def test_representation_sample_zero_coefficient():
ideal = cirq.Circuit(cirq.H(cirq.LineQubit(0)))
noisy_xop = NoisyOperation.from_cirq(circuit=cirq.X,
channel_matrix=np.zeros(shape=(4, 4)))
noisy_zop = NoisyOperation.from_cirq(circuit=cirq.Z,
channel_matrix=np.zeros(shape=(4, 4)))
decomp = OperationRepresentation(
ideal=ideal,
basis_expansion={
noisy_xop: 0.5,
noisy_zop: 0.0, # This should never be sampled.
},
)
random_state = np.random.RandomState(seed=1)
for _ in range(500):
noisy_op, sign, coeff = decomp.sample(random_state=random_state)
assert sign == 1
assert coeff == 0.5
assert np.allclose(noisy_op.ideal_unitary, cirq.unitary(cirq.X))
def test_equal_method_of_representations():
q = cirq.LineQubit(0)
ideal = cirq.Circuit(cirq.H(q))
noisy_xop_a = NoisyOperation(
ideal=cirq.Circuit(cirq.X(q)), real=np.zeros(shape=(4, 4)),
)
noisy_zop_a = NoisyOperation(
ideal=cirq.Circuit(cirq.Z(q)), real=np.zeros(shape=(4, 4)),
)
rep_a = OperationRepresentation(
ideal=ideal, basis_expansion={noisy_xop_a: 0.5, noisy_zop_a: 0.5},
)
noisy_xop_b = NoisyOperation(
ideal=cirq.Circuit(cirq.X(q)), real=np.ones(shape=(4, 4)),
)
noisy_zop_b = NoisyOperation(
ideal=cirq.Circuit(cirq.Z(q)), real=np.ones(shape=(4, 4)),
)
rep_b = OperationRepresentation(
ideal=ideal, basis_expansion={noisy_xop_b: 0.5, noisy_zop_b: 0.5},
)
# Equal representation up to real superoperators
assert rep_a == rep_b
# Different ideal
ideal_b = cirq.Circuit(cirq.X(q))
rep_b = OperationRepresentation(
ideal=ideal_b, basis_expansion={noisy_xop_b: 0.5, noisy_zop_b: 0.5},
)
assert rep_a != rep_b
# Different type
q_b = qiskit.QuantumRegister(1)
ideal_b = qiskit.QuantumCircuit(q_b)
ideal_b.x(q_b)
rep_b = OperationRepresentation(
ideal=ideal_b, basis_expansion={noisy_xop_b: 0.5, noisy_zop_b: 0.5},
)
assert rep_a != rep_b
# Different length
rep_b = OperationRepresentation(
ideal=ideal, basis_expansion={noisy_xop_b: 0.5},
)
assert rep_a != rep_b
# Different operations
noisy_diff = NoisyOperation(ideal=cirq.Circuit(cirq.H(q)))
rep_b = OperationRepresentation(
ideal=ideal, basis_expansion={noisy_xop_b: 0.5, noisy_diff: 0.5},
)
assert rep_a != rep_b
# Different coefficients
rep_b = OperationRepresentation(
ideal=ideal, basis_expansion={noisy_xop_b: 0.7, noisy_zop_b: 0.5},
)
assert rep_a != rep_b
def get_test_representation():
ideal = cirq.Circuit(cirq.H(cirq.LineQubit(0)))
noisy_xop = NoisyOperation.from_cirq(
ideal=cirq.X, real=np.zeros(shape=(4, 4))
)
noisy_zop = NoisyOperation.from_cirq(
ideal=cirq.Z, real=np.zeros(shape=(4, 4))
)
decomp = OperationRepresentation(
ideal=ideal, basis_expansion={noisy_xop: 0.5, noisy_zop: -0.5}
)
return ideal, noisy_xop, noisy_zop, decomp
def test_print_cirq_operation_representation():
ideal = cirq.Circuit(cirq.H(cirq.LineQubit(0)))
noisy_xop = NoisyOperation.from_cirq(circuit=cirq.X,
channel_matrix=np.zeros(shape=(4, 4)))
noisy_zop = NoisyOperation.from_cirq(circuit=cirq.Z,
channel_matrix=np.zeros(shape=(4, 4)))
# Positive first coefficient
decomp = OperationRepresentation(
ideal=ideal,
basis_expansion={
noisy_xop: 0.5,
noisy_zop: 0.5,
},
)
expected = r"0: ───H─── = 0.500*(0: ───X───)+0.500*(0: ───Z───)"
assert str(decomp) == expected
# Negative first coefficient
decomp = OperationRepresentation(
ideal=ideal,
basis_expansion={
noisy_xop: -0.5,
noisy_zop: 1.5,
},
)
expected = r"0: ───H─── = -0.500*(0: ───X───)+1.500*(0: ───Z───)"
assert str(decomp) == expected
# Empty representation
decomp = OperationRepresentation(ideal=ideal, basis_expansion={})
expected = r"0: ───H─── = 0.000"
assert str(decomp) == expected
# Small coefficient approximation
decomp = OperationRepresentation(
ideal=ideal,
basis_expansion={
noisy_xop: 1.00001,
noisy_zop: 0.00001
},
)
expected = r"0: ───H─── = 1.000*(0: ───X───)"
assert str(decomp) == expected
# Small coefficient approximation different position
decomp = OperationRepresentation(
ideal=ideal,
basis_expansion={
noisy_xop: 0.00001,
noisy_zop: 1.00001
},
)
expected = r"0: ───H─── = 1.000*(0: ───Z───)"
# Small coefficient approximation different position
decomp = OperationRepresentation(
ideal=ideal,
basis_expansion={noisy_xop: 0.00001},
)
expected = r"0: ───H─── = 0.000"
assert str(decomp) == expected
def test_print_cirq_operation_representation():
ideal = cirq.Circuit(cirq.H(cirq.LineQubit(0)))
noisy_xop = NoisyOperation.from_cirq(
ideal=cirq.X, real=np.zeros(shape=(4, 4))
)
noisy_zop = NoisyOperation.from_cirq(
ideal=cirq.Z, real=np.zeros(shape=(4, 4))
)
decomp = OperationRepresentation(
ideal=ideal, basis_expansion={noisy_xop: 0.5, noisy_zop: 0.5, },
)
expected = r"0: ───H─── = 0.500*0: ───X───+0.500*0: ───Z───"
assert str(decomp) == expected
def represent_operation_with_global_depolarizing_noise(
ideal_operation: QPROGRAM,
noise_level: float) -> OperationRepresentation:
r"""As described in [Temme2017]_, this function maps an
``ideal_operation`` :math:`\mathcal{U}` into its quasi-probability
representation, which is a linear combination of noisy implementable
operations :math:`\sum_\alpha \eta_{\alpha} \mathcal{O}_{\alpha}`.
This function assumes a depolarizing noise model and, more precicely,
that the following noisy operations are implementable
:math:`\mathcal{O}_{\alpha} = \mathcal{D} \circ \mathcal P_\alpha
\circ \mathcal{U}`, where :math:`\mathcal{U}` is the unitary associated
to the input ``ideal_operation`` acting on :math:`k` qubits,
:math:`\mathcal{P}_\alpha` is a Pauli operation and
:math:`\mathcal{D}(\rho) = (1 - \epsilon) \rho + \epsilon I/2^k` is a
depolarizing channel (:math:`\epsilon` is a simple function of
``noise_level``).
For a single-qubit ``ideal_operation``, the representation is as
follows:
.. math::
\mathcal{U}_{\beta} = \eta_1 \mathcal{O}_1 + \eta_2 \mathcal{O}_2 +
\eta_3 \mathcal{O}_3 + \eta_4 \mathcal{O}_4
.. math::
\eta_1 =1 + \frac{3}{4} \frac{\epsilon}{1- \epsilon},
\qquad \mathcal{O}_1 = \mathcal{D} \circ \mathcal{I} \circ \mathcal{U}
\eta_2 =- \frac{1}{4}\frac{\epsilon}{1- \epsilon} , \qquad
\mathcal{O}_2 = \mathcal{D} \circ \mathcal{X} \circ \mathcal{U}
\eta_3 =- \frac{1}{4}\frac{\epsilon}{1- \epsilon} , \qquad
\mathcal{O}_3 = \mathcal{D} \circ \mathcal{Y} \circ \mathcal{U}
\eta_4 =- \frac{1}{4}\frac{\epsilon}{1- \epsilon} , \qquad
\mathcal{O}_4 = \mathcal{D} \circ \mathcal{Z} \circ \mathcal{U}
It was proven in [Takagi2020]_ that, under suitable assumptions,
this representation is optimal (minimum 1-norm).
Args:
ideal_operation: The ideal operation (as a QPROGRAM) to represent.
noise_level: The noise level (as a float) of the depolarizing channel.
Returns:
The quasi-probability representation of the ``ideal_operation``.
.. note::
This representation is based on the ideal assumption that one
can append Pauli gates to a noisy operation without introducing
additional noise. For a beckend which violates this assumption,
it remains a good approximation for small values of ``noise_level``.
.. note::
The input ``ideal_operation`` is typically a QPROGRAM with a single
gate but could also correspond to a sequence of more gates.
This is possible as long as the unitary associated to the input
QPROGRAM, followed by a single final depolarizing channel, is
physically implementable.
"""
circ, in_type = convert_to_mitiq(ideal_operation)
post_ops: List[List[Operation]]
qubits = circ.all_qubits()
# The single-qubit case: linear combination of 1Q Paulis
if len(qubits) == 1:
q = tuple(qubits)[0]
epsilon = 4 / 3 * noise_level
alpha_pos = 1 + ((3 / 4) * epsilon / (1 - epsilon))
alpha_neg = -(1 / 4) * epsilon / (1 - epsilon)
alphas = [alpha_pos] + 3 * [alpha_neg]
post_ops = [[]] # for alpha_pos, we do nothing, rather than I
post_ops += [[P(q)] for P in [X, Y, Z]] # 1Q Paulis
# The two-qubit case: linear combination of 2Q Paulis
elif len(qubits) == 2:
q0, q1 = qubits
epsilon = 16 / 15 * noise_level
alpha_pos = 1 + ((15 / 16) * epsilon / (1 - epsilon))
alpha_neg = -(1 / 16) * epsilon / (1 - epsilon)
alphas = [alpha_pos] + 15 * [alpha_neg]
post_ops = [[]] # for alpha_pos, we do nothing, rather than I x I
post_ops += [[P(q0)] for P in [X, Y, Z]] # 1Q Paulis for q0
post_ops += [[P(q1)] for P in [X, Y, Z]] # 1Q Paulis for q1
post_ops += [[Pi(q0), Pj(q1)] for Pi in [X, Y, Z]
for Pj in [X, Y, Z]] # 2Q Paulis
else:
raise ValueError("Can only represent single- and two-qubit gates."
"Consider pre-compiling your circuit.")
# Basis of implementable operations as circuits
imp_op_circuits = [circ + Circuit(op) for op in post_ops]
# Convert back to input type
imp_op_circuits = [convert_from_mitiq(c, in_type) for c in imp_op_circuits]
# Build basis_expantion
expansion = {NoisyOperation(c): a for c, a in zip(imp_op_circuits, alphas)}
return OperationRepresentation(ideal_operation, expansion)
def represent_operation_with_local_depolarizing_noise(
ideal_operation: QPROGRAM,
noise_level: float) -> OperationRepresentation:
r"""As described in [Temme2017]_, this function maps an
``ideal_operation`` :math:`\mathcal{U}` into its quasi-probability
representation, which is a linear combination of noisy implementable
operations :math:`\sum_\alpha \eta_{\alpha} \mathcal{O}_{\alpha}`.
This function assumes a (local) single-qubit depolarizing noise model even
for multi-qubit operations. More precicely, it assumes that the following
noisy operations are implementable :math:`\mathcal{O}_{\alpha} =
\mathcal{D}^{\otimes k} \circ \mathcal P_\alpha \circ \mathcal{U}`,
where :math:`\mathcal{U}` is the unitary associated
to the input ``ideal_operation`` acting on :math:`k` qubits,
:math:`\mathcal{P}_\alpha` is a Pauli operation and
:math:`\mathcal{D}(\rho) = (1 - \epsilon) \rho + \epsilon I/2` is a
single-qubit depolarizing channel (:math:`\epsilon` is a simple function
of ``noise_level``).
More information about the quasi-probability representation for a
depolarizing noise channel can be found in:
:func:`represent_operation_with_global_depolarizing_noise`.
Args:
ideal_operation: The ideal operation (as a QPROGRAM) to represent.
noise_level: The noise level of each depolarizing channel.
Returns:
The quasi-probability representation of the ``ideal_operation``.
.. note::
The input ``ideal_operation`` is typically a QPROGRAM with a single
gate but could also correspond to a sequence of more gates.
This is possible as long as the unitary associated to the input
QPROGRAM, followed by a single final depolarizing channel, is
physically implementable.
.. [Temme2017] : Kristan Temme, Sergey Bravyi, Jay M. Gambetta,
"Error mitigation for short-depth quantum circuits,"
*Phys. Rev. Lett.* **119**, 180509 (2017),
(https://arxiv.org/abs/1612.02058).
"""
circ, in_type = convert_to_mitiq(ideal_operation)
qubits = circ.all_qubits()
if len(qubits) == 1:
return represent_operation_with_global_depolarizing_noise(
ideal_operation,
noise_level,
)
# The two-qubit case: tensor product of two depolarizing channels.
elif len(qubits) == 2:
q0, q1 = qubits
# Single-qubit representation coefficients.
epsilon = noise_level * 4 / 3
c_neg = -(1 / 4) * epsilon / (1 - epsilon)
c_pos = 1 - 3 * c_neg
imp_op_circuits = []
alphas = []
# The zero-pauli term in the linear combination
imp_op_circuits.append(circ)
alphas.append(c_pos * c_pos)
# The single-pauli terms in the linear combination
for qubit in qubits:
for pauli in [X, Y, Z]:
imp_op_circuits.append(circ + Circuit(pauli(qubit)))
alphas.append(c_neg * c_pos)
# The two-pauli terms in the linear combination
for pauli_0, pauli_1 in product([X, Y, Z], repeat=2):
imp_op_circuits.append(circ + Circuit(pauli_0(q0), pauli_1(q1)))
alphas.append(c_neg * c_neg)
else:
raise ValueError("Can only represent single- and two-qubit gates."
"Consider pre-compiling your circuit.")
# Convert back to input type
imp_op_circuits = [convert_from_mitiq(c, in_type) for c in imp_op_circuits]
# Build basis_expantion
expansion = {NoisyOperation(c): a for c, a in zip(imp_op_circuits, alphas)}
return OperationRepresentation(ideal_operation, expansion)