Clifford#

class qiskit.quantum_info.Clifford(data, validate=True, copy=True)[source]#

Bases: BaseOperator, AdjointMixin, Operation

An N-qubit unitary operator from the Clifford group.

Representation

An N-qubit Clifford operator is stored as a length 2N × (2N+1) boolean tableau 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 boolean tableau for the Clifford can be accessed using the tableau attribute. The destabilizer or stabilizer rows can each be accessed as a length-N Stabilizer table using destab and stab 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.to_labels(mode="D"))

# Print the Clifford stabilizer rows
print(cliff.to_labels(mode="S"))

Clifford: Stabilizer = ['+XX', '+ZZ'], Destabilizer = ['+IZ', '+XI']
['+IZ', '+XI']
['+XX', '+ZZ']

Circuit Conversion

Clifford operators can be initialized from circuits containing only the following Clifford gates: IGate, XGate, YGate, ZGate, HGate, SGate, SdgGate, SXGate, SXdgGate, CXGate, CZGate, CYGate, DXGate, SwapGate, iSwapGate, ECRGate. 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.

Attributes

destab#: The destabilizer array for the symplectic representation.

destab_phase#: Return phase of destabilizer with boolean representation.

destab_x#: The destabilizer x array for the symplectic representation.

destab_z#: The destabilizer z array for the symplectic representation.

destabilizer#: Return the destabilizer block of the StabilizerTable.

Deprecated since version 0.24.0: The property qiskit.quantum_info.operators.symplectic.clifford.Clifford.destabilizer is deprecated as of qiskit-terra 0.24.0. It will be removed no earlier than 3 months after the release date. Use Clifford.destab properties instead.

dim#: Return tuple (input_shape, output_shape).

name#: Unique string identifier for operation type.

num_clbits#: Number of classical bits.

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

phase#: Return phase with boolean representation.

qargs#: Return the qargs for the operator.

stab#: The stabilizer array for the symplectic representation.

stab_phase#: Return phase of stabilizer with boolean representation.

stab_x#: The stabilizer x array for the symplectic representation.

stab_z#: The stabilizer array for the symplectic representation.

stabilizer#: Return the stabilizer block of the StabilizerTable.

Deprecated since version 0.24.0: The property qiskit.quantum_info.operators.symplectic.clifford.Clifford.stabilizer is deprecated as of qiskit-terra 0.24.0. It will be removed no earlier than 3 months after the release date. Use Clifford.stab properties instead.

symplectic_matrix#: Return boolean symplectic matrix.

table#: Return StabilizerTable

Deprecated since version 0.24.0: The property qiskit.quantum_info.operators.symplectic.clifford.Clifford.table is deprecated as of qiskit-terra 0.24.0. It will be removed no earlier than 3 months after the release date. Use Clifford.stab and Clifford.destab properties instead.

x#: The x array for the symplectic representation.

z#: The z array for the symplectic representation.

Methods

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 @ (equivalent to dot()) is defined as right matrix multiplication. That is that A & B == A.compose(B) is equivalent to B @ A == 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()[source]#: Make a deep copy of current operator.

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

Note

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

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.

classmethod 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

classmethod from_matrix(matrix)[source]#

Create a Clifford from a unitary matrix.

Note that this function takes exponentially long time w.r.t. the number of qubits.

Parameters:: matrix (np.array) -- A unitary matrix representing a Clifford to be converted.
Returns:: the Clifford object for the unitary matrix.
Return type:: Clifford
Raises:: QiskitError -- if the input is not a Clifford matrix.

classmethod from_operator(operator)[source]#

Create a Clifford from a operator.

Note that this function takes exponentially long time w.r.t. the number of qubits.

Parameters:: operator (Operator) -- An operator representing a Clifford to be converted.
Returns:: the Clifford object for the operator.
Return type:: Clifford
Raises:: QiskitError -- if the input is not a Clifford operator.

input_dims(qargs=None)#: Return tuple of input dimension for specified subsystems.

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

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.

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.

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_labels(array=False, mode='B')[source]#

Convert a Clifford to a list Pauli (de)stabilizer string labels.

For large Clifford converting using the array=True kwarg will be more efficient since it allocates memory for the full Numpy array of labels in advance.

Table 2 Stabilizer Representations#
Label	Phase	Symplectic	Matrix	Pauli
`"+I"`	0	\([0, 0]\)	\(\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\)	\(I\)
`"-I"`	1	\([0, 0]\)	\(\begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}\)	\(-I\)
`"X"`	0	\([1, 0]\)	\(\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\)	\(X\)
`"-X"`	1	\([1, 0]\)	\(\begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix}\)	\(-X\)
`"Y"`	0	\([1, 1]\)	\(\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}\)	\(iY\)
`"-Y"`	1	\([1, 1]\)	\(\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}\)	\(-iY\)
`"Z"`	0	\([0, 1]\)	\(\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\)	\(Z\)
`"-Z"`	1	\([0, 1]\)	\(\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}\)	\(-Z\)

Parameters:

array (bool) -- return a Numpy array if True, otherwise return a list (Default: False).
mode (Literal["S", "D", "B"]) -- return both stabilizer and destabilizer if "B", return only stabilizer if "S" and return only destabilizer if "D".

Returns:

The rows of the StabilizerTable in label form.

Return type:

list or array

Raises:

QiskitError -- if stabilizer and destabilizer are both False.

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

to_operator()[source]#

Convert to an Operator object.

Return type:: Operator

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