Local spin operators for fermion simulations

Digital quantum simulation of fermionic systems is important in the context of chemistry and physics. Simulating fermionic models on general purpose quantum computers requires imposing a fermionic algebra on qubits. The previously studied Jordan-Wigner and Bravyi-Kitaev transformations are two techniques for accomplishing this task. Here, we reexamine an auxiliary fermion construction which maps fermionic operators to local operators on qubits. The local simulation is performed by relaxing the requirement that the number of qubits should match the number of single-particle states. Instead, auxiliary sites are introduced to enable nonconsecutive fermionic couplings to be simulated with constant low-rank tensor products on qubits. The additional number of auxiliary qubits required per fermionic degree of freedom depends only on the degree of connectivity of the Hamiltonian. We connect the auxiliary fermion construction to topological models and give examples of the construction.

Figure 1

Auxiliary sites are introduced in order to cancel strings of $Z$ operators on nonlocal couplings. In the graph above, we add auxiliary fermionic sites $1^{\prime }$ and $5^{\prime }$ for sites 1 and 5, as the fermionic coupling (dashed line) would map into nonlocal coupling on qubits with Jordan-Wigner transformation. The fermionic auxiliary sites are ordered right after the sites of the original model—in the graph above, the ordering changes as $12345\to {11}^{\prime }{23455}^{\prime }$ with the introduction of the auxiliary sites. We assumed no knowledge about the excitation parity sector of the model.

Figure 2

The $K_4$ graph is the completely connected graph with four sites. The solid lines indicate the linear indexing and the dashed edges indicate nonlocal couplings.

Figure 3

The two-dimensional (2D) Hubbard model provides an illustration of the advantage of the auxiliary fermion simulations over Jordan-Wigner and Bravyi-Kitaev transformed operators. On the left, the $L=3$ model is depicted with linear indexing graph $G_1$ in curved bold lines and with dotted lines indicating the Hubbard coupling graph $G$. On the right, $D=\text{deg}\left(G\right)$ is the degree of the sites in $G$, similarly $D_1=\text{deg}\left(G_1\right)$. Their difference gives the nonlocal degree $D_{\text{nl}}$. This translates into the number of auxiliary fermions needed at each site following $N_{\text{aux}}=\mathrm{ceil}(D_{\text{nl}}/2)$.