qiskit.quantum_info.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 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 readable 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

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

Initialize an operator object.

__init__(data, validate=True)[source]¶: Initialize an operator object.

Methods

`__init__`(data[, validate])	Initialize an operator object.
`adjoint`()	Return the adjoint of the Operator.
`compose`(other[, qargs, front])	Return the operator composition with another Clifford.
`conjugate`()	Return the conjugate of the Clifford.
`copy`()	Make a deep copy of current operator.
`dot`(other[, qargs])	Return the right multiplied operator self * other.
`expand`(other)	Return the reverse-order tensor product with another Clifford.
`from_circuit`(circuit)	Initialize from a QuantumCircuit or Instruction.
`from_dict`(obj)	Load a Clifford from a dictionary
`from_label`(label)	Return a tensor product of single-qubit Clifford gates.
`input_dims`([qargs])	Return tuple of input dimension for specified subsystems.
`is_unitary`()	Return True if the Clifford table is valid.
`output_dims`([qargs])	Return tuple of output dimension for specified subsystems.
`power`(n)	Return the compose of a operator with itself n times.
`reshape`([input_dims, output_dims, num_qubits])	Return a shallow copy with reshaped input and output subsystem dimensions.
`tensor`(other)	Return the tensor product with another Clifford.
`to_circuit`()	Return a QuantumCircuit implementing the Clifford.
`to_dict`()	Return dictionary representation of Clifford object.
`to_instruction`()	Return a Gate instruction implementing the Clifford.
`to_matrix`()	Convert operator to Numpy matrix.
`to_operator`()	Convert to an Operator object.
`transpose`()	Return the transpose of the Clifford.

Attributes

`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.
`settings`	Return operator settings.
`stabilizer`	Return the stabilizer block of the StabilizerTable.
`table`	Return StabilizerTable

adjoint()[source]¶: Return the adjoint of the Operator.

compose(other, qargs=None, front=False)[source]¶

Return the operator composition with another Clifford.

Parameters

other (Clifford) – a Clifford object.
qargs (list or None) – Optional, 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 composed Clifford.

Return type

Clifford

Raises

QiskitError – if other cannot be converted to an operator, or has incompatible dimensions for specified subsystems.

Note

Composition (&) by default is defined as left matrix multiplication for matrix operators, while dot() is defined as right matrix multiplication. That is that A & B == A.compose(B) is equivalent to B.dot(A) when A and B are of the same type.

Setting the front=True kwarg changes this to right matrix multiplication and is equivalent to the dot() method A.dot(B) == A.compose(B, front=True).

conjugate()[source]¶: Return the conjugate of the Clifford.

copy()¶: Make a deep copy of current operator.

property destabilizer¶: Return the destabilizer block of the StabilizerTable.

property dim¶: Return tuple (input_shape, output_shape).

dot(other, qargs=None)¶

Return the right multiplied operator self * other.

Parameters

other (Operator) – an operator object.
qargs (list or None) – Optional, a list of subsystem positions to apply other on. If None apply on all subsystems (default: None).

Returns

The right matrix multiplied Operator.

Return type

Operator

expand(other)[source]¶

Return the reverse-order tensor product with another Clifford.

Parameters

other (Clifford) – a Clifford object.

Returns

the tensor product \(b \otimes a\), where \(a\): is the current Clifford, and \(b\) is the other Clifford.

Return type

Clifford

static from_circuit(circuit)[source]¶

Initialize from a QuantumCircuit or Instruction.

Parameters: circuit (QuantumCircuit or Instruction) – instruction to initialize.
Returns: the Clifford object for the instruction.
Return type: Clifford
Raises: QiskitError – if the input instruction is non-Clifford or contains classical register instruction.

static from_dict(obj)[source]¶: Load a Clifford from a dictionary

static from_label(label)[source]¶

Return a tensor product of single-qubit Clifford gates.

Parameters: label (string) – single-qubit operator string.
Returns: The N-qubit Clifford operator.
Return type: Clifford
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(qargs=None)¶: Return tuple of input dimension for specified subsystems.

is_unitary()[source]¶: Return True if the Clifford table is valid.

property num_qubits¶: Return the number of qubits if a N-qubit operator or None otherwise.

output_dims(qargs=None)¶: Return tuple of output dimension for specified subsystems.

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: Pauli
Raises: QiskitError – if the input and output dimensions of the operator are not equal, or the power is not a positive integer.

property qargs¶: Return the qargs for the operator.

reshape(input_dims=None, output_dims=None, num_qubits=None)¶

Return a shallow copy with reshaped input and output subsystem dimensions.

Parameters

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].
num_qubits (None or int) – reshape to an N-qubit operator [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.

property settings¶: Return operator settings.

property stabilizer¶: Return the stabilizer block of the StabilizerTable.

property table¶: Return StabilizerTable

tensor(other)[source]¶

Return the tensor product with another Clifford.

Parameters

other (Clifford) – a Clifford object.

Returns

the tensor product \(a \otimes b\), where \(a\): is the current Clifford, and \(b\) is the other Clifford.

Return type

Clifford

Note

The tensor product can be obtained using the ^ binary operator. Hence a.tensor(b) is equivalent to a ^ b.

to_circuit()[source]¶

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: QuantumCircuit

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()[source]¶: Return dictionary representation of Clifford object.

to_instruction()[source]¶: Return a Gate instruction implementing the Clifford.

to_matrix()[source]¶: Convert operator to Numpy matrix.

to_operator()[source]¶: Convert to an Operator object.

transpose()[source]¶: Return the transpose of the Clifford.