quairkit.database.set¶

quairkit.database.set.pauli_basis(num_qubits)¶

Generate a Pauli basis.

Parameters:¶

num_qubits : int¶

the number of qubits \(n\).

Returns:¶

The Pauli basis of \(\mathbb{C}^{2^n \times 2^n}\), where each tensor is accessible along the first dimension.

Return type:¶

Tensor

num_qubits = 1
basis = pauli_basis(num_qubits)
print(f'The Pauli basis is:\n{basis}')

The Pauli basis is:
tensor([[[ 0.7071+0.0000j,  0.0000+0.0000j],
        [ 0.0000+0.0000j,  0.7071+0.0000j]],

        [[ 0.0000+0.0000j,  0.7071+0.0000j],
        [ 0.7071+0.0000j,  0.0000+0.0000j]],

        [[ 0.0000+0.0000j,  0.0000-0.7071j],
        [ 0.0000+0.7071j,  0.0000+0.0000j]],

        [[ 0.7071+0.0000j,  0.0000+0.0000j],
        [ 0.0000+0.0000j, -0.7071+0.0000j]]])

quairkit.database.set.pauli_group(num_qubits)¶

Generate a Pauli group i.e., an unnormalized Pauli basis.

Parameters:¶

num_qubits : int¶

the number of qubits \(n\).

Returns:¶

The Pauli group of \(\mathbb{C}^{2^n \times 2^n}\), where each tensor is accessible along the first dimension.

Return type:¶

Tensor

num_qubits = 1
group = pauli_group(num_qubits)
print(f'The Pauli group is:\n{group}')

The Pauli group is:
tensor([[[ 1.0000+0.0000j,  0.0000+0.0000j],
        [ 0.0000+0.0000j,  1.0000+0.0000j]],

        [[ 0.0000+0.0000j,  1.0000+0.0000j],
        [ 1.0000+0.0000j,  0.0000+0.0000j]],

        [[ 0.0000+0.0000j,  0.0000-1.0000j],
        [ 0.0000+1.0000j,  0.0000+0.0000j]],

        [[ 1.0000+0.0000j,  0.0000+0.0000j],
        [ 0.0000+0.0000j, -1.0000+0.0000j]]])

quairkit.database.set.pauli_str_basis(pauli_str)¶

Get the state basis with respect to the Pauli string

Parameters:¶

pauli_str : str | List[str]¶

the string composed of ‘i’, ‘x’, ‘y’ and ‘z’ only.

Returns:¶

The state basis of the observable given by the Pauli string

Return type:¶

State

pauli_str = ['x','z']
state_basis = pauli_str_basis(pauli_str)
print(f'The state basis of the observable is:\n{state_basis}')

The state basis of the observable is:

---------------------------------------------------
Backend: state_vector
System dimension: [2]
System sequence: [0]
Batch size: [2, 2]

# 0:
[0.71+0.j 0.71+0.j]
# 1:
[ 0.71+0.j -0.71+0.j]
# 2:
[1.+0.j 0.+0.j]
# 3:
[0.+0.j 1.+0.j]
---------------------------------------------------

quairkit.database.set.pauli_str_povm(pauli_str)¶

Get the povm with respect to the Pauli string

Parameters:¶

pauli_str : str | List[str]¶

the string composed of ‘i’, ‘x’, ‘y’ and ‘z’ only.

Returns:¶

The POVM of the observable given by the Pauli string

Return type:¶

Tensor

pauli_str = ['x','y']
POVM = pauli_str_povm(pauli_str)
print(f'The POVM of the observable is:\n{POVM}')

The POVM of the observable is:
tensor([[[[ 0.5000+0.0000j,  0.5000+0.0000j],
        [ 0.5000+0.0000j,  0.5000+0.0000j]],

        [[ 0.5000+0.0000j, -0.5000+0.0000j],
        [-0.5000+0.0000j,  0.5000+0.0000j]]],


        [[[ 0.5000+0.0000j,  0.0000-0.5000j],
        [ 0.0000+0.5000j,  0.5000+0.0000j]],

        [[ 0.5000+0.0000j,  0.0000+0.5000j],
        [ 0.0000-0.5000j,  0.5000+0.0000j]]]])

quairkit.database.set.qft_basis(num_qubits)¶

Compute the eigenvectors (eigenbasis) of the Quantum Fourier Transform (QFT) matrix.

Parameters:¶

num_qubits : int¶

Number of qubits \(n\) such that \(N = 2^n\).

Returns:¶

A tensor where the first index gives the eigenvector of the QFT matrix.

Return type:¶

State

num_qubits = 2
qft_state = qft_basis(num_qubits)
print(f'The eigenvectors of the QFT matrix is:\n{qft_state}')

The eigenvectors of the QFT matrix is:

---------------------------------------------------
Backend: state_vector
System dimension: [2]
System sequence: [0]
Batch size: [2]

# 0:
[0.92+0.j 0.38+0.j]
# 1:
[-0.38-0.j  0.92+0.j]
---------------------------------------------------

quairkit.database.set.std_basis(num_systems, system_dim=2)¶

Generate all standard basis states for a given number of qubits.

Parameters:¶

num_systems : int¶

number of systems in this state. Alias of num_qubits.

system_dim : List[int] | int¶

dimension of systems. Can be a list of system dimensions or an int representing the dimension of all systems. Defaults to be qubit case.

Returns:¶

A tensor where the first index gives the computational vector

Return type:¶

State

num_systems = 2
system_dim=[1,2]
basis = std_basis(num_systems,system_dim)
print(f'The standard basis states are:\n{basis}')

The standard basis states are:

---------------------------------------------------
Backend: state_vector
System dimension: [1, 2]
System sequence: [0, 1]
Batch size: [2]

# 0:
[1.+0.j 0.+0.j]
# 1:
[0.+0.j 1.+0.j]
---------------------------------------------------

quairkit.database.set.bell_basis()¶

Generate the Bell basis for a 2-qubit system, with each basis state accessible along the first dimension of a tensor.

Returns:¶

A tensor of shape (4, 4, 1), representing the four Bell basis states.

basis=bell_basis()
print(f'The Bell basis for a 2-qubit system are:\n{basis}')

Return type:¶

State

The Bell basis for a 2-qubit system are:

---------------------------------------------------
Backend: state_vector
System dimension: [2, 2]
System sequence: [0, 1]
Batch size: [4]

# 0:
[0.71+0.j 0.  +0.j 0.  +0.j 0.71+0.j]
# 1:
[ 0.71+0.j  0.  +0.j  0.  +0.j -0.71+0.j]
# 2:
[0.  +0.j 0.71+0.j 0.71+0.j 0.  +0.j]
# 3:
[ 0.  +0.j  0.71+0.j -0.71+0.j  0.  +0.j]
---------------------------------------------------

quairkit.database.set.heisenberg_weyl(dim)¶

Generate Heisenberg-Weyl operator for qudit.

The Heisenberg-Weyl operators are defined as \(T(a,b) = e^{-(d+1) \pi i a b/ d}Z^a X^b\).

Parameters:¶

dim : int¶

dimension of qudit

Returns:¶

Heisenberg-Weyl operator for qudit

Return type:¶

Tensor

dim=2
operator=heisenberg_weyl(dim)
print(f'The Heisenberg-Weyl operator for qudit is:\n{operator}')

The Heisenberg-Weyl operator for qudit is:
tensor([[[ 1.0000e+00+0.0000e+00j,  0.0000e+00+0.0000e+00j],
        [ 0.0000e+00+0.0000e+00j,  1.0000e+00+0.0000e+00j]],

        [[ 1.0000e+00+0.0000e+00j,  0.0000e+00+0.0000e+00j],
        [ 0.0000e+00+0.0000e+00j, -1.0000e+00+1.2246e-16j]],

        [[ 0.0000e+00+0.0000e+00j,  1.0000e+00+0.0000e+00j],
        [ 1.0000e+00+0.0000e+00j,  0.0000e+00+0.0000e+00j]],

        [[-0.0000e+00+0.0000e+00j, -1.8370e-16+1.0000e+00j],
        [ 6.1232e-17-1.0000e+00j, -0.0000e+00+0.0000e+00j]]],
    dtype=torch.complex128)

quairkit.database.set.phase_space_point(dim)¶

Generate phase space point operator for qudit

Parameters:¶

dim : int¶

dimension of qudit

Returns:¶

Phase space point operator for qudit

Return type:¶

Tensor

dim=2
operator=phase_space_point(dim)
print(f'The phase space point operator for qudit is:\n{operator}')

The phase space point operator for qudit is:
tensor([[[ 1.0000+0.0000e+00j,  0.5000+5.0000e-01j],
        [ 0.5000-5.0000e-01j,  0.0000+6.1232e-17j]],

        [[ 1.0000+0.0000e+00j, -0.5000-5.0000e-01j],
        [-0.5000+5.0000e-01j,  0.0000+6.1232e-17j]],

        [[ 0.0000+6.1232e-17j,  0.5000-5.0000e-01j],
        [ 0.5000+5.0000e-01j,  1.0000+0.0000e+00j]],

        [[ 0.0000+6.1232e-17j, -0.5000+5.0000e-01j],
        [-0.5000-5.0000e-01j,  1.0000+0.0000e+00j]]], dtype=torch.complex128)

quairkit.database.set.gell_mann(dim)¶

Generate a set of Gell-Mann matrices for a given dimension. These matrices span the entire space dim-by-dim matrices, and they generalize the Pauli operators when dim = 2 and the Gell-Mann operators when dim = 3.

Parameters:¶

dim : int¶

a positive integer indicating the dimension.

Returns:¶

A set of Gell-Mann matrices.

Return type:¶

Tensor

dim=2
matrices=gell_mann(dim)
print(f'The Gell-Mann matrices are:\n{matrices}')

The Gell-Mann matrices are:
tensor([[[ 0.+0.j,  1.+0.j],
        [ 1.+0.j,  0.+0.j]],

        [[ 0.+0.j, -0.-1.j],
        [ 0.+1.j,  0.+0.j]],

        [[ 1.+0.j,  0.+0.j],
        [ 0.+0.j, -1.+0.j]]])