qiskit.quantum_info.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 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

__init__(data, validate=True)

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

adjoint()

Return the adjoint of the Operator.

compose

compose(other, qargs=None, front=False)

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

conjugate()

Return the conjugate of the Clifford.

copy

copy()

Make a deep copy of current operator.

destabilizer

Return the destabilizer block of the StabilizerTable.

dim

Return tuple (input_shape, output_shape).

dot

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

expand(other)

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

from_circuit

static 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

Clifford

Raises

QiskitError – if the input instruction is non-Clifford or contains classical register instruction.

from_dict

static from_dict(obj)

Load a Clifford from a dictionary

from_label

static 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

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

input_dims(qargs=None)

Return tuple of input dimension for specified subsystems.

is_unitary

is_unitary()

Return True if the Clifford table is valid.

num_qubits

Return the number of qubits if a N-qubit operator or None otherwise.

output_dims

output_dims(qargs=None)

Return tuple of output dimension for specified subsystems.

power

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.

qargs

Return the qargs for the operator.

reshape

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.

settings

Return operator settings.

stabilizer

Return the stabilizer block of the StabilizerTable.

table

Return StabilizerTable

tensor

tensor(other)

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

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

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

to_dict()

Return dictionary representation of Clifford object.

to_instruction

to_instruction()

Return a Gate instruction implementing the Clifford.

to_matrix

to_matrix()

Convert operator to Numpy matrix.

to_operator

to_operator()

Convert to an Operator object.

transpose

transpose()

Return the transpose of the Clifford.

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