I've been reading this paper about the procedure to simulate a many-body quantum system on a quantum device. I got confused by Figure 1. on page 3, and the 3 steps explained below the figure.

It seems to me that given a quantum many-body Hamiltonian, we will first rewrite it in 2nd quantization, and transform the terms into a sum of Pauli matrices using the Jordan-Wigner transform. Then the trotter decomposition follows. This agrees with the 3 steps explained on the paper:

enter image description here

However in fig 1, the JW transform happens after trotterization. So I'm confused about which step goes first? Does JW transform act on the Hamiltonian, $H$, or the system under time-evolution, $U = e^{-iHt}$?

Thanks for the help!

enter image description here


1 Answer 1


Short answer:

When you do the Jordan-Wigner transformation, you essentially insert a linear combination of tensor products of Pauli matrices for each fermionic creation and annihilation operator. As there is a one-to-one correspondence (you're basically just "rewriting" what $a_p^\dagger$, $a_r$ stand for on the quantum computer), it is irrelevant whether you do this before or after trotterization.

Long answer:

In second quantization, in general, each $h_i$ is a linear combination of creation and annihilation operators $$ h_i=\sum_{pq\dots rs} c_{i,pq\dots rs} a_p^\dagger a_q^\dagger \dots a_r a_s $$ where $p,q,\dots,r,s$ each go from 1 to $M$, where $M$ is the number of fermionic modes.

Now, performing the Jordan-Wigner mapping, each operator $a_p^\dagger$, $a_r$ is replaced by a linear combination of tensor products of Pauli spin matrices. We thus get a "quantum computer representation" of the operator $h_i$.

It does not matter at all when we perform the Jordan-Wigner transformation. As $a_p^\dagger$, $a_r$ will be replaced by a linear combination of tensor products of Pauli spin matrices, it is just a question of notation whether you do that before or after trotterization.

On a side note: Operators which are local acting on fermionic systems (acting on a constant number of fermionic modes, e.g. $a_p^\dagger a_r$ acts on two fermionic modes) are not necessarily local when represented on a quantum computer (e.g. $a_p^\dagger a_r$ acts on $O(M)$ qubits, see for example here). As $h_i$ should be local operators, performing the Jordan-Wigner transformation might turn it into a highly non-local operator. There are other valid transformations between second quantized operators and qubit operators that get $O(M)$ down to $O(log(M))$ (most notably, the Bravyi-Kitaev transformation). From that point of view, it does indeed matter when to perform a transformation (and which to perform) - but that has more to do with computational efficiency.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.