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Clifford

class Clifford(data, validate=True, copy=True)

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Bases: qiskit.quantum_info.operators.base_operator.BaseOperator, qiskit.quantum_info.operators.mixins.adjoint.AdjointMixin, qiskit.circuit.operation.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

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

Initialize an operator object.

Methods

adjoint

Clifford.adjoint()

Return the adjoint of the Operator.

compose

Clifford.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 @ (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

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 (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

Clifford.expand(other)

Return the reverse-order tensor product with another Clifford.

Parameters

other (Clifford) – a Clifford object.

Returns

the tensor product b⊗ab \otimes a, where aa

is the current Clifford, and bb is the other Clifford.

Return type

Clifford

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

Clifford

Raises

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

from_dict

classmethod 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

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

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.

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

Pauli

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, 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

Clifford.tensor(other)

Return the tensor product with another Clifford.

Parameters

other (Clifford) – a Clifford object.

Returns

the tensor product a⊗ba \otimes b, where aa

is the current Clifford, and bb 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

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

QuantumCircuit

References

  1. S. Bravyi, D. Maslov, Hadamard-free circuits expose the structure of the Clifford group, arXiv:2003.09412 [quant-ph]
  2. 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 representation of Clifford object.

to_instruction

Clifford.to_instruction()

Return a Gate instruction implementing the Clifford.

to_labels

Clifford.to_labels(array=False, mode='B')

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.

LabelPhaseSymplecticMatrixPauli

| "+I" | 0 | [0,0][0, 0] | [1001]\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} | II | | "-I" | 1 | [0,0][0, 0] | [−100−1]\begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} | −I-I | | "X" | 0 | [1,0][1, 0] | [0110]\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} | XX | | "-X" | 1 | [1,0][1, 0] | [0−1−10]\begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix} | −X-X | | "Y" | 0 | [1,1][1, 1] | [01−10]\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} | iYiY | | "-Y" | 1 | [1,1][1, 1] | [0−110]\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} | −iY-iY | | "Z" | 0 | [0,1][0, 1] | [100−1]\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} | ZZ | | "-Z" | 1 | [0,1][0, 1] | [−1001]\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix} | −Z-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 destablizer if “B”, return only stabilizer if “S” and return only destablizer 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

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.

Attributes

destab

The destabilizer array for the symplectic representation.

destab_phase

Return phase of destaibilizer 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.

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 stablizer 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.

symplectic_matrix

Return boolean symplectic matrix.

table

Return StabilizerTable

x

The x array for the symplectic representation.

z

The z array for the symplectic representation.

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