Operator locality in the quantum simulation of fermionic models

Simulating fermionic lattice models with qubits requires mapping fermionic degrees of freedom to qubits. The simplest method for this task, the Jordan-Wigner transformation, yields strings of Pauli operators acting on an extensive number of qubits. This overhead can be a hindrance to implementation of qubit-based quantum simulators, especially in the analog context. Here we thus review and analyze alternative fermion-to-qubit mappings, including the two approaches by Bravyi and Kitaev and the auxiliary fermion transformation. The Bravyi-Kitaev transform is reformulated in terms of a classical data structure and generalized to achieve a further locality improvement for local fermionic models on a rectangular lattice. We conclude that the most compact encoding of the fermionic operators can be done using ancilla qubits with the auxiliary fermion scheme. Without introducing ancillas, a variant of the Bravyi-Kitaev transform provides the most compact fermion-to-qubit mapping for Hubbard-like models.

Figure 1

Fenwick tree of depth 3 for $N=7$. The structure can be constructed by taking the first node and making it dependent on contents of the node half way (rounded down) in the lattice and proceeding recursively for halves of the site array. The example here is illustrated for $N=7$. Odd $N$ has been chosen in order to show a construction of the mapping for $N$ not being a power of 2, a restriction implicitly imposed in [17]. The content of the white boxes corresponds to the information stored in each node.

Figure 2

Fenwick tree of depth 3 for $N=8$. Fenwick trees for $N=2^d$ can also be described by a partial ordering on tree node indices. Suppose we write the indices in binary as in the tree on the right. Then a bit string with $h>0$ zeros labels a child of another bit string with $h-1$ zeros given by flipping the last 0 of the string to 1. For example, 101, 011, and 110 are all children of 111. This construction manifests a possible connection to algebraic coding, as every path from the root to a leaf gives a Gray code [22, 23]. Other definitions can be found, but working out examples is the fastest way to familiarize oneself with the construction.

Figure 3

An example of $c_9$ Majorana operator mapped with the Bravyi-Kitaev method. The white colored nodes of the Fenwick tree correspond to $P\left(9\right)=\{7,8\}$ and are the qubits to which Pauli $Z$ operators are applied. The black nodes are in $U\left(9\right)=\{11,15\}$ to which $X$ is applied.

Figure 4

Recursion steps of the Fenwick tree algorithm shown for $N=8$. The bottom case corresponds to the JW transformation.

Figure 5

Locality of vertical hopping terms as a function of the segment tree size for $W=64$. The optimal tree size is $W/2$. The hopping term localities of segments of $W=64$ and $W=32$ sizes are given by 14 and 13 in this lattice.

Figure 6

Left: Fermionic site ordering used with the BK scheme. Qubits ($•$) are on edges of the lattice. Right: A stabilizer on the $(5,9,6,10)$ plaquette. There are nine such stabilizers for a $4\times 4$ lattice corresponding to the nine plaquettes. If a state is a simultaneous eigenstate of all stabilizers, it encodes a physical fermionic state. If a qubit operator is to be applied to the edge outside the lattice, ignore it.

Figure 7

Example of LSFS generators $A_{(9,10)}$ and $A_{(6,10)}$.

Figure 8

Left: Snake pattern used for $G_1$ in a $w=l=3$ in the auxiliary fermion method. Right: Optimal JW ordering for geometrically local fermionic models.

Figure 9

Left: Degrees of fermionic sites in the $3\times 3$ 2D Hubbard lattice. Middle: Degrees along the linear path $G_1$ through each sublattice (right). The nonlocal degree of each node determines the number of auxiliary modes to be used.

Figure 10

The worst case for hopping operator locality, if all-to-all couplings are allowed is the hopping from the first to the last node. In the diagram above, this corresponds to $0\leftrightarrow 15$—the nodes with raising and lowering operators are colored white. The set $F\left(15\right)$ of all children of node 15 corresponds to the black nodes. The update set $U\left(0\right)$ of 0 is the set of square nodes. This implies that $XZ=iY$ will be applied at node 7.

Figure 11

Worst-case locality of qubit operators with lattice size in the 2D Hubbard model.