Quantum Information

qiskit.quantum_info.average_gate_fidelity(channel, target=None, require_cp=True, require_tp=False)

Return the average gate fidelity of a noisy quantum channel.

The average gate fidelity $F_{\text{ave}}$ is given by

where $F_{\text{pro}}(\mathcal{E}, U)$ is the process_fidelity() of the input quantum channel $\mathcal{E}$ with a target unitary $U$, and $d$ is the dimension of the channel.

Parameters

• channel (QuantumChannel or Operator) – noisy quantum channel.
• target (Operator or None) – target unitary operator. If None target is the identity operator [Default: None].
• require_cp (bool) – check if input and target channels are completely-positive and if non-CP log warning containing negative eigenvalues of Choi-matrix [Default: True].
• require_tp (bool) – check if input and target channels are trace-preserving and if non-TP log warning containing negative eigenvalues of partial Choi-matrix $Tr_{\text{out}}[\mathcal{E}] - I$ [Default: True].

Returns

The average gate fidelity $F_{\text{ave}}$.

Return type

float

Raises

QiskitError – if the channel and target do not have the same dimensions, or have different input and output dimensions.

qiskit.quantum_info.process_fidelity(channel, target=None, require_cp=True, require_tp=True)

Return the process fidelity of a noisy quantum channel.

The process fidelity $F_{\text{pro}}(\mathcal{E}, \mathcal{F})$ between two quantum channels $\mathcal{E}, \mathcal{F}$ is given by

where $F$ is the state_fidelity(), $\rho_{\mathcal{E}} = \Lambda_{\mathcal{E}} / d$ is the normalized Choi matrix for the channel $\mathcal{E}$, and $d$ is the input dimension of $\mathcal{E}$.

When the target channel is unitary this is equivalent to

where $S_{\mathcal{E}}, S_{U}$ are the SuperOp matrices for the input quantum channel $\mathcal{E}$ and target unitary $U$ respectively, and $d$ is the input dimension of the channel.

Parameters

• channel (Operator or QuantumChannel) – input quantum channel.
• target (Operator or QuantumChannel or None) – target quantum channel. If None target is the identity operator [Default: None].
• require_cp (bool) – check if input and target channels are completely-positive and if non-CP log warning containing negative eigenvalues of Choi-matrix [Default: True].
• require_tp (bool) – check if input and target channels are trace-preserving and if non-TP log warning containing negative eigenvalues of partial Choi-matrix $Tr_{\text{out}}[\mathcal{E}] - I$ [Default: True].

Returns

The process fidelity $F_{\text{pro}}$.

Return type

float

Raises

QiskitError – if the channel and target do not have the same dimensions.

qiskit.quantum_info.gate_error(channel, target=None, require_cp=True, require_tp=False)

Return the gate error of a noisy quantum channel.

The gate error $E$ is given by the average gate infidelity

where $F_{\text{ave}}(\mathcal{E}, U)$ is the average_gate_fidelity() of the input quantum channel $\mathcal{E}$ with a target unitary $U$.

Parameters

• channel (QuantumChannel) – noisy quantum channel.
• target (Operator or None) – target unitary operator. If None target is the identity operator [Default: None].
• require_cp (bool) – check if input and target channels are completely-positive and if non-CP log warning containing negative eigenvalues of Choi-matrix [Default: True].
• require_tp (bool) – check if input and target channels are trace-preserving and if non-TP log warning containing negative eigenvalues of partial Choi-matrix $Tr_{\text{out}}[\mathcal{E}] - I$ [Default: True].

Returns

The average gate error $E$.

Return type

float

Raises

QiskitError – if the channel and target do not have the same dimensions, or have different input and output dimensions.

qiskit.quantum_info.diamond_norm(choi, solver='SCS', **kwargs)

Return the diamond norm of the input quantum channel object.

This function computes the completely-bounded trace-norm (often referred to as the diamond-norm) of the input quantum channel object using the semidefinite-program from reference [1].

Parameters

• choi (Choi or QuantumChannel) – a quantum channel object or Choi-matrix array.
• solver (str) – The solver to use.
• kwargs – optional arguments to pass to CVXPY solver.

Returns

The completely-bounded trace norm $\|\mathcal{E}\|_{\diamond}$.

Return type

float

Raises

QiskitError – if CVXPY package cannot be found.

Additional Information:

The input to this function is typically not a CPTP quantum channel, but rather the difference between two quantum channels $\|\Delta\mathcal{E}\|_\diamond$ where $\Delta\mathcal{E} = \mathcal{E}_1 - \mathcal{E}_2$.

Reference:

J. Watrous. “Simpler semidefinite programs for completely bounded norms”, arXiv:1207.5726 [quant-ph] (2012).

qiskit.quantum_info.state_fidelity(state1, state2, validate=True)

Return the state fidelity between two quantum states.

The state fidelity $F$ for density matrix input states $\rho_1, \rho_2$ is given by

If one of the states is a pure state this simplifies to $F(\rho_1, \rho_2) = \langle\psi_1|\rho_2|\psi_1\rangle$, where $\rho_1 = |\psi_1\rangle\!\langle\psi_1|$.

Parameters

• state1 (Statevector orDensityMatrix) – the first quantum state.
• state2 (Statevector orDensityMatrix) – the second quantum state.
• validate (bool) – check if the inputs are valid quantum states [Default: True]

Returns

The state fidelity $F(\rho_1, \rho_2)$.

Return type

float

Raises

QiskitError – if validate=True and the inputs are invalid quantum states.

qiskit.quantum_info.purity(state, validate=True)

Calculate the purity of a quantum state.

The purity of a density matrix $\rho$ is

Parameters

• state (Statevector orDensityMatrix) – a quantum state.
• validate (bool) – check if input state is valid [Default: True]

Returns

the purity $Tr[\rho^2]$.

Return type

float

Raises

QiskitError – if the input isn’t a valid quantum state.

qiskit.quantum_info.concurrence(state)

Calculate the concurrence of a quantum state.

The concurrence of a bipartite Statevector $|\psi\rangle$ is given by

where $\rho_0 = Tr_1[|\psi\rangle\!\langle\psi|]$ is the reduced state from by taking the partial_trace() of the input state.

For density matrices the concurrence is only defined for 2-qubit states, it is given by:

where $\lambda _1 \ge \lambda _2 \ge \lambda _3 \ge \lambda _4$ are the ordered eigenvalues of the matrix $R=\sqrt{\sqrt{\rho }(Y\otimes Y)\overline{\rho}(Y\otimes Y)\sqrt{\rho}}$.

Parameters

state (Statevector orDensityMatrix) – a 2-qubit quantum state.

Returns

The concurrence.

Return type

float

Raises

• QiskitError – if the input state is not a valid QuantumState.
• QiskitError – if input is not a bipartite QuantumState.
• QiskitError – if density matrix input is not a 2-qubit state.

qiskit.quantum_info.entropy(state, base=2)

Calculate the von-Neumann entropy of a quantum state.

The entropy $S$ is given by

Parameters

• state (Statevector orDensityMatrix) – a quantum state.
• base (int) – the base of the logarithm [Default: 2].

Returns

The von-Neumann entropy S(rho).

Return type

float

Raises

QiskitError – if the input state is not a valid QuantumState.

qiskit.quantum_info.entanglement_of_formation(state)

Calculate the entanglement of formation of quantum state.

The input quantum state must be either a bipartite state vector, or a 2-qubit density matrix.

Parameters

state (Statevector orDensityMatrix) – a 2-qubit quantum state.

Returns

The entanglement of formation.

Return type

float

Raises

• QiskitError – if the input state is not a valid QuantumState.
• QiskitError – if input is not a bipartite QuantumState.
• QiskitError – if density matrix input is not a 2-qubit state.

qiskit.quantum_info.mutual_information(state, base=2)

Calculate the mutual information of a bipartite state.

The mutual information $I$ is given by:

where $\rho_A=Tr_B[\rho_{AB}], \rho_B=Tr_A[\rho_{AB}]$, are the reduced density matrices of the bipartite state $\rho_{AB}$.

Parameters

• state (Statevector orDensityMatrix) – a bipartite state.
• base (int) – the base of the logarithm [Default: 2].

Returns

The mutual information $I(\rho_{AB})$.

Return type

float

Raises

• QiskitError – if the input state is not a valid QuantumState.
• QiskitError – if input is not a bipartite QuantumState.

qiskit.quantum_info.partial_trace(state, qargs)

Return reduced density matrix by tracing out part of quantum state.

If all subsystems are traced over this returns the trace() of the input state.

Parameters

• state (Statevector orDensityMatrix) – the input state.
• qargs (list) – The subsystems to trace over.

Returns

The reduced density matrix.

Return type

DensityMatrix

Raises

QiskitError – if input state is invalid.

qiskit.quantum_info.schmidt_decomposition(state, qargs)

Return the Schmidt Decomposition of a pure quantum state.

For an arbitrary bipartite state:

its Schmidt Decomposition is given by the single-index sum over k:

where $|u_k\rangle_A$ and $|v_k\rangle_B$ are an orthonormal set of vectors in their respective spaces $A$ and $B$, and the Schmidt coefficients $\lambda_k$ are positive real values.

Parameters

• state (Statevector orDensityMatrix) – the input state.
• qargs (list) – the list of Input state positions corresponding to subsystem $B$.

Returns

list of tuples (s, u, v), where s (float) are the Schmidt coefficients $\lambda_k$, and u (Statevector), v (Statevector) are the Schmidt vectors $|u_k\rangle_A$, $|u_k\rangle_B$, respectively.

Return type

list

Raises

• QiskitError – if Input qargs is not a list of positions of the Input state.
• QiskitError – if Input qargs is not a proper subset of Input state.

qiskit.quantum_info.shannon_entropy(pvec, base=2)

Compute the Shannon entropy of a probability vector.

The shannon entropy of a probability vector $\vec{p} = [p_0, ..., p_{n-1}]$ is defined as

where $b$ is the log base and (default 2), and $0 \log_b(0) \equiv 0$.

Parameters

• pvec (array_like) – a probability vector.
• base (int) – the base of the logarithm [Default: 2].

Returns

The Shannon entropy H(pvec).

Return type

float

qiskit.quantum_info.commutator(a, b)

Compute commutator of a and b.

Parameters

• a (OperatorTypeT) – Operator a.
• b (OperatorTypeT) – Operator b.

Returns

The commutator

Return type

OperatorTypeT

qiskit.quantum_info.anti_commutator(a, b)

Compute anti-commutator of a and b.

Parameters

• a (OperatorTypeT) – Operator a.
• b (OperatorTypeT) – Operator b.

Returns

The anti-commutator

Return type

OperatorTypeT

qiskit.quantum_info.double_commutator(a, b, c, *, commutator=True)

Compute symmetric double commutator of a, b and c.

See also Equation (13.6.18) in [1].

If commutator is True, it returns

If commutator is False, it returns

Parameters

• a (OperatorTypeT) – Operator a.
• b (OperatorTypeT) – Operator b.
• c (OperatorTypeT) – Operator c.
• commutator (bool) – If True compute the double commutator, if False the double anti-commutator.

Returns

The double commutator

Return type

OperatorTypeT

References

[1]: R. McWeeny.

Methods of Molecular Quantum Mechanics. 2nd Edition, Academic Press, 1992. ISBN 0-12-486552-6.

qiskit.quantum_info.random_statevector(dims, seed=None)

Generator a random Statevector.

The statevector is sampled from the uniform distribution. This is the measure induced by the Haar measure on unitary matrices.

Parameters

• dims (int ortuple) – the dimensions of the state.
• seed (int or np.random.Generator) – Optional. Set a fixed seed or generator for RNG.

Returns

the random statevector.

Return type

Statevector

Reference:

K. Zyczkowski and H. Sommers (2001), “Induced measures in the space of mixed quantum states”, J. Phys. A: Math. Gen. 34 7111.

qiskit.quantum_info.random_density_matrix(dims, rank=None, method='Hilbert-Schmidt', seed=None)

Generator a random DensityMatrix.

Parameters

• dims (int ortuple) – the dimensions of the DensityMatrix.
• rank (int or None) – Optional, the rank of the density matrix. The default value is full-rank.
• method (string) – Optional. The method to use. ‘Hilbert-Schmidt’: (Default) sample from the Hilbert-Schmidt metric. ‘Bures’: sample from the Bures metric.
• seed (int or np.random.Generator) – Optional. Set a fixed seed or generator for RNG.

Returns

the random density matrix.

Return type

DensityMatrix

Raises

QiskitError – if the method is not valid.

qiskit.quantum_info.random_unitary(dims, seed=None)

Return a random unitary Operator.

The operator is sampled from the unitary Haar measure.

Parameters

• dims (int ortuple) – the input dimensions of the Operator.
• seed (int or np.random.Generator) – Optional. Set a fixed seed or generator for RNG.

Returns

a unitary operator.

Return type

Operator

qiskit.quantum_info.random_hermitian(dims, traceless=False, seed=None)

Return a random hermitian Operator.

The operator is sampled from Gaussian Unitary Ensemble.

Parameters

• dims (int ortuple) – the input dimension of the Operator.
• traceless (bool) – Optional. If True subtract diagonal entries to return a traceless hermitian operator (Default: False).
• seed (int or np.random.Generator) – Optional. Set a fixed seed or generator for RNG.

Returns

a Hermitian operator.

Return type

Operator

qiskit.quantum_info.random_pauli(num_qubits, group_phase=False, seed=None)

Return a random Pauli.

Parameters

• num_qubits (int) – the number of qubits.
• group_phase (bool) – Optional. If True generate random phase. Otherwise the phase will be set so that the Pauli coefficient is +1 (default: False).
• seed (int or np.random.Generator) – Optional. Set a fixed seed or generator for RNG.

Returns

a random Pauli

Return type

Pauli

qiskit.quantum_info.random_clifford(num_qubits, seed=None)

Return a random Clifford operator.

The Clifford is sampled using the method of Reference [1].

Parameters

• num_qubits (int) – the number of qubits for the Clifford
• seed (int or np.random.Generator) – Optional. Set a fixed seed or generator for RNG.

Returns

a random Clifford operator.

Return type

Clifford

Reference:

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

qiskit.quantum_info.random_quantum_channel(input_dims=None, output_dims=None, rank=None, seed=None)

Return a random CPTP quantum channel.

This constructs the Stinespring operator for the quantum channel by sampling a random isometry from the unitary Haar measure.

Parameters

• input_dims (int ortuple) – the input dimension of the channel.
• output_dims (int ortuple) – the input dimension of the channel.
• rank (int) – Optional. The rank of the quantum channel Choi-matrix.
• seed (int or np.random.Generator) – Optional. Set a fixed seed or generator for RNG.

Returns

a quantum channel operator.

Return type

Stinespring

Raises

QiskitError – if rank or dimensions are invalid.

qiskit.quantum_info.random_cnotdihedral(num_qubits, seed=None)

Return a random CNOTDihedral element.

Parameters

• num_qubits (int) – the number of qubits for the CNOTDihedral object.
• seed (int or RandomState) – Optional. Set a fixed seed or generator for RNG.

Returns

a random CNOTDihedral element.

Return type

CNOTDihedral

qiskit.quantum_info.random_pauli_list(num_qubits, size=1, seed=None, phase=True)

Return a random PauliList.

Parameters

• num_qubits (int) – the number of qubits.
• size (int) – Optional. The length of the Pauli list (Default: 1).
• seed (int or np.random.Generator) – Optional. Set a fixed seed or generator for RNG.
• phase (bool) – If True the Pauli phases are randomized, otherwise the phases are fixed to 0. [Default: True]

Returns

a random PauliList.

Return type

PauliList

qiskit.quantum_info.hellinger_distance(dist_p, dist_q)

Computes the Hellinger distance between two counts distributions.

Parameters

• dist_p (dict) – First dict of counts.
• dist_q (dict) – Second dict of counts.

Returns

Distance

Return type

float

References

Hellinger Distance @ wikipedia

qiskit.quantum_info.hellinger_fidelity(dist_p, dist_q)

Computes the Hellinger fidelity between two counts distributions.

The fidelity is defined as $\left(1-H^{2}\right)^{2}$ where H is the Hellinger distance. This value is bounded in the range [0, 1].

This is equivalent to the standard classical fidelity $F(Q,P)=\left(\sum_{i}\sqrt{p_{i}q_{i}}\right)^{2}$ that in turn is equal to the quantum state fidelity for diagonal density matrices.

Parameters

• dist_p (dict) – First dict of counts.
• dist_q (dict) – Second dict of counts.

Returns

Fidelity

Return type

float

Example

from qiskit import QuantumCircuit, execute, BasicAer
from qiskit.quantum_info.analysis import hellinger_fidelity
 
qc = QuantumCircuit(5, 5)
qc.h(2)
qc.cx(2, 1)
qc.cx(2, 3)
qc.cx(3, 4)
qc.cx(1, 0)
qc.measure(range(5), range(5))
 
sim = BasicAer.get_backend('qasm_simulator')
res1 = execute(qc, sim).result()
res2 = execute(qc, sim).result()
 
hellinger_fidelity(res1.get_counts(), res2.get_counts())

References

Quantum Fidelity @ wikipedia Hellinger Distance @ wikipedia

qiskit.quantum_info.two_qubit_cnot_decompose(*args, **kwargs)

Return type

QuantumCircuit

qiskit.quantum_info.decompose_clifford(clifford, method=None)

DEPRECATED: Decompose a Clifford operator into a QuantumCircuit.

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 greedy compilation routine from reference [3], which typically yields better CX cost compared to the AG method in [2].

Parameters

• clifford (Clifford) – a clifford operator.
• method (str) – Optional, a synthesis method (‘AG’ or ‘greedy’). If set this overrides optimal decomposition for N <=3 qubits.

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
3. Sergey Bravyi, Shaohan Hu, Dmitri Maslov, Ruslan Shaydulin, Clifford Circuit Optimization with Templates and Symbolic Pauli Gates, arXiv:2105.02291 [quant-ph]