How to generate all stabilizer states numerically?

Question

I would like to obtain a list of all stabilizer states in the given dimension (not necessarily qubit systems). What is an efficient way of generating this list numerically?

Answer 1

First of all, keep in mind that there is no efficient way of generating them, simply because the number of stabiliser states increases super-exponentially with the number of qudits $n$ (like $p^{O(n^2)}$).

Second, I can only give you hints how to do it here, since a detailed treatise would take too long. Sorry. As far as I know, there is also no literature with details on implementation where I can point you to. I'm currently writing a tutorial on the stabiliser formalism which will contain more details and possibly even pseudo-code. But the pre-print will not be out before beginning of next year.

We consider stabiliser states in prime dimension $p$ (if you want prime-power, like e.g. for MUBs, that's similar). Define $\omega:=\exp(2\pi i /p)$ to be the standard choice of a primitive $p$-th root of unity.

State vector representation

For $p\neq 2$, the Hilbert space vectors representing stabiliser states can be chosen to be of the following form (Refs. [1,2]) $$ |K, q, b\rangle := \frac{1}{|K|} \sum_{x\in K} \omega^{q(x)+b\cdot x} |x\rangle, $$ where $K\subset\mathbb F_p^n$ is an affine subspace, $q$ is a quadratic form on $K$, and $b\in\mathbb F_p^n$.

For $p=2$, this is $$ |K, q, b \rangle := \frac{1}{|K|} \sum_{x\in K} i^{b\cdot x}\omega^{q(x)} |x\rangle, $$ i.e. the linear part is computed mod 4 instead of mod 2.

Now you can enumerate all stabiliser state by iterating over all possible affine subspaces $K$, and then over quadratic and linear forms on $K$. The later part is easy once you have fixed a basis of $K$.

Density matrix representation

The projectors of stabiliser states are determined by their stabiliser group $S$ as follows $$ P(S) = \frac{1}{p^n} \sum_{g\in S} g. $$ The stabiliser groups modulo signs correspond to so-called Lagrangian subspaces $L\subset \mathbb F_p^{2n}$ (see my answer here). These are $n$-dimensional linear subspaces such that the symplectic form of any two elements vanishes (Ref. [3]): $$ 0 = [v,w] := \sum_{i=1}^{n} v_i w_{n+i} - w_i v_{n+i}, \quad \forall v,w\in L. $$ Any vector $v\in L$ correspond to a Pauli/Weyl operator $W(v)$ and the above equation expresses that those should commute.

Now, for $p\neq 2$, we can rewrite the projector as $$ P(S) = P(L,h) = \frac{1}{p^n} \sum_{v\in L} \omega^{[h,v]} W(v). $$ From this representation, we can see that the coefficients of $P(L,h)$ in the Weyl basis (i.e. its Bloch vector) are given by $\omega^{[h,v]}$ if $v\in L$ and 0 otherwise. Hence, one can again iterate over Lagrangian subspaces $L$, and then over the vectors $h$. The latter part is again simplest in a basis of $L$. In the case $p\neq 2$, one can alternatively use the representation of stabiliser states by their Wigner functions.

For $p= 2$, this is a bit uglier. The coefficients here have the form $$ P(S) = P(L,h) = \frac{1}{2^n} \sum_{v\in L} (-1)^{\alpha(v)} W(v). $$ For a suitable function $\alpha$. To compute $\alpha$, one can fix it arbitrarily on a basis of $L$ (i.e. choose signs for the generators of $S$) and then compute the coefficients of all other elements from that using $$ \alpha(v+w) = \beta(v,w) - \alpha(v) - \alpha(w) $$ where $\beta$ is some horrible function determining whether you get an additional sign when you multiply two Pauli operators. Its explicit form is e.g. given in a paper of mine (Ref. [4, App. A])), but also elsewhere.

Literature

1. Dehaene, Jeroen, and Bart De Moor. “The Clifford Group, Stabilizer States, and Linear and Quadratic Operations over GF(2).” Physical Review A 68, no. 4 (2003). https://doi.org/10.1103/PhysRevA.68.042318.
2. Gross, D., and M. Van den Nest. “The LU-LC Conjecture, Diagonal Local Operations and Quadratic Forms over GF(2),” (2007). https://arxiv.org/abs/0707.4000v2.
3. Gross, D. “Hudson’s Theorem for Finite-Dimensional Quantum Systems.” Journal of Mathematical Physics 47, no. 12 (2006): 122107. https://doi.org/10.1063/1.2393152.
4. Heimendahl, Arne, Markus Heinrich, and David Gross. “The Axiomatic and the Operational Approaches to Resource Theories of Magic Do Not Coincide.” arXiv:2011.11651 (2020). http://arxiv.org/abs/2011.11651.