Clifford
class Clifford(data, validate=True)
An N-qubit unitary operator from the Clifford group.
Representation
An N-qubit Clifford operator is stored as a length 2N StabilizerTable using the convention from reference [1].
- Rows 0 to N-1 are the destabilizer group generators
- Rows N-1 to 2N-1 are the stabilizer group generators.
The internal StabilizerTable for the Clifford can be accessed using the table attribute. The destabilizer or stabilizer rows can each be accessed as a length-N Stabilizer table using destabilizer and stabilizer attributes.
A more easily human readible representation of the Clifford operator can be obtained by calling the to_dict() method. This representation is also used if a Clifford object is printed as in the following example
from qiskit import QuantumCircuit
from qiskit.quantum_info import Clifford
# Bell state generation circuit
qc = QuantumCircuit(2)
qc.h(0)
qc.cx(0, 1)
cliff = Clifford(qc)
# Print the Clifford
print(cliff)
# Print the Clifford destabilizer rows
print(cliff.destabilizer)
# Print the Clifford stabilizer rows
print(cliff.stabilizer)Clifford: Stabilizer = ['+XX', '+ZZ'], Destabilizer = ['+IZ', '+XI']
StabilizerTable: ['+IZ', '+XI']
StabilizerTable: ['+XX', '+ZZ']
Circuit Conversion
Clifford operators can be initialized from circuits containing only the following Clifford gates: IGate, XGate, YGate, ZGate, HGate, SGate, SdgGate, CXGate, CZGate, SwapGate. They can be converted back into a QuantumCircuit, or Gate object using the to_circuit() or to_instruction() methods respectively. Note that this decomposition is not necessarily optimal in terms of number of gates.
A minimally generating set of gates for Clifford circuits is the HGate and SGate gate and either the CXGate or CZGate two-qubit gate.
Clifford operators can also be converted to Operator objects using the to_operator() method. This is done via decomposing to a circuit, and then simulating the circuit as a unitary operator.
References
- S. Aaronson, D. Gottesman, Improved Simulation of Stabilizer Circuits, Phys. Rev. A 70, 052328 (2004). arXiv:quant-ph/0406196
Initialize an operator object.
Attributes
atol
The default absolute tolerance parameter for float comparisons.
destabilizer
Return the destabilizer block of the StabilizerTable.
dim
Return tuple (input_shape, output_shape).
num_qubits
Return the number of qubits if a N-qubit operator or None otherwise.
qargs
Return the qargs for the operator.
rtol
The relative tolerance parameter for float comparisons.
stabilizer
Return the stabilizer block of the StabilizerTable.
table
Return StabilizerTable
Methods
__call__
Clifford.__call__(qargs)
Return a clone with qargs set
__getitem__
Clifford.__getitem__(key)
Return a stabilizer Pauli row
__mul__
Clifford.__mul__(other)
add
Clifford.add(other)
Return the linear operator self + other.
DEPRECATED: use operator + other instead.
Parameters
other (BaseOperator) – an operator object.
Returns
the operator self + other.
Return type
BaseOperator
adjoint
Clifford.adjoint()
Return the conjugate transpose of the Clifford
compose
Clifford.compose(other, qargs=None, front=False)
Return the composed operator.
Parameters
- other (Clifford) – an operator object.
- qargs (list or None) – a list of subsystem positions to apply other on. If None apply on all subsystems [default: None].
- front (bool) – If True compose using right operator multiplication, instead of left multiplication [default: False].
Returns
The operator self @ other.
Return type
Raises
QiskitError – if operators have incompatible dimensions for composition.
Additional Information:
Composition (@) is defined as left matrix multiplication for matrix operators. That is that A @ B is equal to B * A. Setting front=True returns right matrix multiplication A * B and is equivalent to the dot() method.
conjugate
Clifford.conjugate()
Return the conjugate of the Clifford.
copy
Clifford.copy()
Make a deep copy of current operator.
dot
Clifford.dot(other, qargs=None)
Return the right multiplied operator self * other.
Parameters
- other (Clifford) – an operator object.
- qargs (list or None) – a list of subsystem positions to apply other on. If None apply on all subsystems [default: None].
Returns
The operator self * other.
Return type
Raises
QiskitError – if operators have incompatible dimensions for composition.
expand
Clifford.expand(other)
Return the tensor product operator other ⊗ self.
Parameters
other (Clifford) – an operator object.
Returns
the tensor product operator other ⊗ self.
Return type
from_circuit
static Clifford.from_circuit(circuit)
Initialize from a QuantumCircuit or Instruction.
Parameters
circuit (QuantumCircuit orInstruction) – instruction to initialize.
Returns
the Clifford object for the instruction.
Return type
Raises
QiskitError – if the input instruction is non-Clifford or contains classical register instruction.
from_dict
static Clifford.from_dict(obj)
Load a Clifford from a dictionary
from_label
static Clifford.from_label(label)
Return a tensor product of single-qubit Clifford gates.
Parameters
label (string) – single-qubit operator string.
Returns
The N-qubit Clifford operator.
Return type
Raises
QiskitError – if the label contains invalid characters.
Additional Information:
The labels correspond to the single-qubit Cliffords are
-
- Label
- Stabilizer
- Destabilizer
-
"I"- +Z
- +X
-
"X"- -Z
- +X
-
"Y"- -Z
- -X
-
"Z"- +Z
- -X
-
"H"- +X
- +Z
-
"S"- +Z
- +Y
input_dims
Clifford.input_dims(qargs=None)
Return tuple of input dimension for specified subsystems.
is_unitary
Clifford.is_unitary()
Return True if the Clifford table is valid.
multiply
Clifford.multiply(other)
Return the linear operator other * self.
DEPRECATED: use other * operator instead.
Parameters
other (complex) – a complex number.
Returns
the linear operator other * self.
Return type
BaseOperator
Raises
NotImplementedError – if subclass does not support multiplication.
output_dims
Clifford.output_dims(qargs=None)
Return tuple of output dimension for specified subsystems.
power
Clifford.power(n)
Return the compose of a operator with itself n times.
Parameters
n (int) – the number of times to compose with self (n>0).
Returns
the n-times composed operator.
Return type
BaseOperator
Raises
QiskitError – if the input and output dimensions of the operator are not equal, or the power is not a positive integer.
reshape
Clifford.reshape(input_dims=None, output_dims=None)
Return a shallow copy with reshaped input and output subsystem dimensions.
Arg:
input_dims (None or tuple): new subsystem input dimensions.
If None the original input dims will be preserved [Default: None].
output_dims (None or tuple): new subsystem output dimensions.
If None the original output dims will be preserved [Default: None].
Returns
returns self with reshaped input and output dimensions.
Return type
BaseOperator
Raises
QiskitError – if combined size of all subsystem input dimension or subsystem output dimensions is not constant.
set_atol
classmethod Clifford.set_atol(value)
Set the class default absolute tolerance parameter for float comparisons.
set_rtol
classmethod Clifford.set_rtol(value)
Set the class default relative tolerance parameter for float comparisons.
subtract
Clifford.subtract(other)
Return the linear operator self - other.
DEPRECATED: use operator - other instead.
Parameters
other (BaseOperator) – an operator object.
Returns
the operator self - other.
Return type
BaseOperator
tensor
Clifford.tensor(other)
Return the tensor product operator self ⊗ other.
Parameters
other (Clifford) – a operator subclass object.
Returns
the tensor product operator self ⊗ other.
Return type
to_circuit
Clifford.to_circuit()
Return a QuantumCircuit implementing the Clifford.
For N <= 3 qubits this is based on optimal CX cost decomposition from reference [1]. For N > 3 qubits this is done using the general non-optimal compilation routine from reference [2].
Returns
a circuit implementation of the Clifford.
Return type
References
- S. Bravyi, D. Maslov, Hadamard-free circuits expose the structure of the Clifford group, arXiv:2003.09412 [quant-ph]
- S. Aaronson, D. Gottesman, Improved Simulation of Stabilizer Circuits, Phys. Rev. A 70, 052328 (2004). arXiv:quant-ph/0406196
to_dict
Clifford.to_dict()
Return dictionary represenation of Clifford object.
to_instruction
Clifford.to_instruction()
Return a Gate instruction implementing the Clifford.
to_matrix
Clifford.to_matrix()
Convert operator to Numpy matrix.
to_operator
Clifford.to_operator()
Convert to an Operator object.
transpose
Clifford.transpose()
Return the transpose of the Clifford.