# Why does Parity mapping allow 2 qubit reduction?

when mapping from fermionic to qubit hamiltonian, using Parity mapper allow us to reduce 2 qubit in the final hamiltonian. I would like to understand which kind of symmetries are being used and how it is done. Thank you in advance!

There is a relatively intuitive way to understand 2-qubit reduction using the parity mapping. It uses that the number of $$\alpha$$ (spin up) and $$\beta$$ (spin down) electrons are conserved (because the Hamiltonian commutes with the particle number operator).

In the Jordan-Wigner mapping, we represent a Slater determinant as $$|k_1^\alpha,k_2^\alpha,\dots,k_{M}^\alpha,k_1^\beta,k_2^\beta,\dots,k_{M}^\beta\rangle$$ where $$\sum_{p=1}^M k_p^\alpha=N_\alpha$$ and $$\sum_{p=1}^M k_p^\beta=N_\beta$$.

Going from the Jordan-Wigner mapping to the parity mapping, the following transformation is applied $$|{k_1^\alpha,k_2^\alpha,\dots,k_i^\alpha,\dots, k_M^\alpha,k_1^\beta,k_2^\beta,\dots,k_i^\beta,\dots, k_M^\beta}\rangle \rightarrow |{k_1^\alpha,k_1^\alpha+k_2^\alpha,\dots,\sum_{p=1}^i k_p^\alpha,\sum_{p=1}^M k_p^\alpha+k_1^\beta,\dots,\sum_{p=1}^M k_p^\alpha+\sum_{p=1}^M k_p^\beta} \rangle$$ where all sums are taken mod 2.

Now, as the number of alpha-electrons and beta-electrons is conserved, we know that $$\sum_{p=1}^M k_p^\alpha=N_\alpha$$ and $$\sum_{p=1}^M k_p^\beta=N_\beta$$ . This is true for ANY Slater determinant that preserves the number of alpha-electrons, and we know that the exact ground state is a linear combination of only those Slater determinants. But then, in the parity mapping, the M-th qubit takes the value $$N_\alpha \text{ mod } 2$$ and the 2M-th qubit takes the value $$(N_\alpha+N_\beta) \text{ mod } 2$$. But that means that we already know the correct value of 2 (out of 2M) qubits, and do not have to simulate them, e.g. they can be tapered off.

• Now I understand the physical meaning of the 2-qubit reduction. Thank you very much for the very clear explanation!!! Jun 14, 2022 at 19:53

The Parity mapper 2-qubit reduction uses the Z2Symmetry logic in Qiskit. For any qubit operator the Z2Symmetry logic allows a search to find any symmetries that may be present. In the case of a Fermionic Operator, where we have the block spin mapping and know the number of particles in the system, we know from the mapping itself that it will produce symmetries.

Since the symmetries are known the Z2Symmetry logic can be used to taper the operator without any need to search for them. There may be other symmetries present that the Z2Symmetry logic might find though. If you do Parity mapper, without 2-qubit reduction, and then use Z2Symmetry to find the symmetries it will do so to - the 2-qubit reduction avoids the search since we know they will exist. You can see the code here for the 2-qubit reduction to find out more detail if you want.