As mentioned in the comment above there is a relevant section of Nielsen and Chuang that discusses circuits which perform this sort of action (p191-193, 10th aniversary edition). The dersired transformation can be done using a classical reversible circuit with the {X, CnX} gateset.
I'm going to show how to implement such permutation circuits in pytket using the ToffoliBox construction. This implementation uses multiplexor operations to reduce the number of multi-qubit gates required.
Here is an example from the user manual.
Let's perform the following permutation of basis states
\begin{gather}
|001\rangle \longmapsto |111\rangle \\
|111\rangle \longmapsto |001\rangle \\
|100\rangle \longmapsto |000\rangle \\
|000\rangle \longmapsto |100\rangle
\end{gather}
We can construct this permutation as a dictionary of key-value pairs.
from pytket.circuit import ToffoliBox
# Specify the desired permutation of the basis states
mapping = {
(0, 0, 1): (1, 1, 1),
(1, 1, 1): (0, 0, 1),
(1, 0, 0): (0, 0, 0),
(0, 0, 0): (1, 0, 0),
}
# Define box to perform the permutation
perm_box = ToffoliBox(permutation=mapping)
For correctness if a basis state appears as a key in the dictionary then it must also appear as a value.
Let's now test this out by preparing the three qubit Werner state $|W\rangle$ and see if our ToffoliBox
shuffles our basis states as expected.
\begin{equation}
|W\rangle = \frac{1}{\sqrt{3}} \big(|001\rangle + |010\rangle + |100\rangle \big)
\end{equation}
from pytket.circuit import Circuit, StatePreparationBox
import numpy as np
werner_state = 1 / np.sqrt(3) * np.array([0, 1, 1, 0, 1, 0, 0, 0])
werner_state_box = StatePreparationBox(werner_state)
state_circ = Circuit(3)
state_circ.add_gate(werner_state_box, [0, 1, 2])
This is not the most efficent way to prepare $|W\rangle$ but will serve for illustration purposes.
# Verify state preperation
np.round(state_circ.get_statevector().real, 3) # 1/sqrt(3) approx 0.577
As expected we get
array([-0. , 0.577, 0.577, 0. , 0.577, 0. , 0. , 0. ])
Now lets add our ToffoliBox
which we defined in the first code block to our state preparation circuit and check our statevector again

state_circ.add_gate(perm_box, [0, 1, 2])
np.round(state_circ.get_statevector().real, 3)
After the permutation box is applied we now get...
array([0.577, 0. , 0.577, 0. , 0. , 0. , 0. , 0.577])
corresponding to the modified state $|W'\rangle$
\begin{equation}
|W'\rangle = \frac{1}{\sqrt{3}} \big(|000\rangle + |010\rangle + |111\rangle \big)
\end{equation}
We see that our ToffoliBox
has performed the desired permutation.