Rings or algebras with many nilpotent elements and efficient computation

Crossposted from quantum.SEwhere comment appears to suggest that solving modulo 2 mightbe possible.

Searching the web for '"quantum computer" nilpotent'returns many results, so maybe the question is ontopicfor this site.

Can a quantum computer solve the following mathematicalproblem:

This is related to an open problem, so likely the answer is negative.

The problem is Cycle Enumeration using Nilpotent AdjacencyMatrices with Algorithm Runtime Comparisons pp 2-3

Is there commutative ring or commutative algebra $R$ with the following properties:

1. There are $n$ nilpotent elements $a_i$ satisfying $a_i^2=0$
2. $a_1 a_2 \cdots a_n \ne 0$.
3. Computation in $R$ is efficient: for an $n$ by $n$ matrix $M$with entries zero and $a_i$, for natural $m$ we can compute$M^m$ in time polynomial in $nm$.

If we omit the efficiency constraint, the answer is easy:

Take $R=K[a_1,a_2,...a_n]/(a_1^2,a_2^2,...a_n^2)$ for any ring $K$.

If we omit commutativity, there are solutions with matrices.