# Clifford¶

class Clifford(data, validate=True)[source]

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.

Note

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

1. S. Aaronson, D. Gottesman, Improved Simulation of Stabilizer Circuits, Phys. Rev. A 70, 052328 (2004). arXiv:quant-ph/0406196

Initialize an operator object.

Attributes

 Clifford.atol The default absolute tolerance parameter for float comparisons. Clifford.destabilizer Return the destabilizer block of the StabilizerTable. Clifford.dim Return tuple (input_shape, output_shape). Clifford.num_qubits Return the number of qubits if a N-qubit operator or None otherwise. Clifford.qargs Return the qargs for the operator. Clifford.rtol The relative tolerance parameter for float comparisons. Clifford.stabilizer Return the stabilizer block of the StabilizerTable. Clifford.table Return StabilizerTable

Methods

 Clifford.__call__(qargs) Return a clone with qargs set Return a stabilizer Pauli row Clifford.__mul__(other) Clifford.add(other) Return the linear operator self + other. Return the conjugate transpose of the Clifford Clifford.compose(other[, qargs, front]) Return the composed operator. Return the conjugate of the Clifford. Make a deep copy of current operator. Clifford.dot(other[, qargs]) Return the right multiplied operator self * other. Clifford.expand(other) Return the tensor product operator other ⊗ self. Clifford.from_circuit(circuit) Initialize from a QuantumCircuit or Instruction. Load a Clifford from a dictionary Return a tensor product of single-qubit Clifford gates. Clifford.input_dims([qargs]) Return tuple of input dimension for specified subsystems. Return True if the Clifford table is valid. Clifford.multiply(other) Return the linear operator other * self. Clifford.output_dims([qargs]) Return tuple of output dimension for specified subsystems. Return the compose of a operator with itself n times. Clifford.reshape([input_dims, output_dims]) Return a shallow copy with reshaped input and output subsystem dimensions. Clifford.set_atol(value) Set the class default absolute tolerance parameter for float comparisons. Clifford.set_rtol(value) Set the class default relative tolerance parameter for float comparisons. Clifford.subtract(other) Return the linear operator self - other. Clifford.tensor(other) Return the tensor product operator self ⊗ other. Return a QuantumCircuit implementing the Clifford. Return dictionary represenation of Clifford object. Return a Gate instruction implementing the Clifford. Convert operator to Numpy matrix. Convert to an Operator object. Return the transpose of the Clifford. Clifford.__call__(qargs) Return a clone with qargs set Clifford.__mul__(other)