Implementing a simple Jordan Wigner model

[marekgluza]

• for the nearest neighbor Hubbard we need correct JW formulas like here https://arxiv.org/pdf/2005.09000#section*.18
• we need to figure out how to compile terms of the form $S^+i Z{i+1} S^-_{i+2}$
• @jeongrak-son @wrightjandrew is there any other way to get that string out of simple operators other than $S^+ = (X+iY)/2$? If I expand the sum then I would use by brute force something like TFIM 2 CNOT not 3 CNOT compiling #70 and rotate after I have the right string length but maybe there is a better way than splitting the sum?

[jeongrak-son]

If you expand, the term (with the h.c.) is already quite simple: $\frac{1}{2}(X_{i}\otimes Z_{i+1}\otimes X_{i+2} + Y_{i} \otimes Z_{i+1}\otimes Y_{i+2})$, or equivalently $\frac{1}{2}Z_{i+1}\otimes (X_{i}\otimes X_{i+2} + Y_{i}\otimes Y_{i+2})$. Two terms commute, so when you exponentiate, you can write $e^{-iHt} = e^{-i\frac{t}{2}X\otimes Z\otimes X}e^{-i\frac{t}{2}Y\otimes Z\otimes Y}$.

[marekgluza]

thanks @jeongrak-son the commuting question was what I was wondering about

you got 2 and not 4 terms - will I not additionally get $X\otimes Z \otimes Y$?

[jeongrak-son]

It will be cancelled by the hermitian conjugate.
$(X-iY)Z(X+iY)+(X+iY)Z(X-iY) = (XZX + YZY - iYZX +iXZY) + (XZX + YZY + iYZX -iXZY) = 2(XZX+YZY)$.

[marekgluza]

brilliant, thank you @jeongrak-son this is very helpful and I think this is going to be the right way to find a solution

• as you established there are 2 terms only
• they commute

to me this is simple enough to code now

• @renatomello maybe you know some trick in literature? open question remaining: we have a 3 qubit gate. What CZ cost would you expect for this one? Are there any minimality results for JW gates like that? For a Majorana gate it's I think simple enough because it will be only 1 term like $X\otimes Z \otimes X$ but here we have 2. Is compression possible or we need to treat them individually and that's optimal?