Fermionic measurement-based quantum computation

Fermions, as a major class of quantum particles, provide platforms for quantum information processing beyond the possibilities of spins or bosons, which have been studied more extensively. One particularly interesting model to study, in view of recent progress in manipulating ultracold fermion gases, is the fermionic version of measurement-based quantum computation (MBQC), which implements full quantum computation with only single-site measurements on a proper fermionic many-body resource state. However, it is not known which fermionic states can be used as the resource states for MBQC and how to find them. In this paper, we generalize the framework of spin MBQC to fermions. In particular, we provide a general formalism to construct many-body entangled fermion resource states for MBQC based on the fermionic projected entangled pair state representation. We give a specific fermionic state which enables universal MBQC and demonstrate that the nonlocality inherent in fermion systems can be properly taken care of with suitable measurement schemes. Such a framework opens up possibilities of finding MBQC resource states which can be more readily realized in the laboratory.

Figure 1

Teleportation of one-qubit unitary gates. The qubits are denoted by dots and the entangled pair is depicted by a dashed line. The big circle represents the input end and qubits inside it are measured together. After measuring qubits 1 and 2 together, information in qubit 1 flows to qubit 3.

Figure 2

Teleportation of the two-qubit controlled operation $U_{ph}$. As shown in the figure, qubit 1 is the control qubit and qubit 5 is the target qubit. After proper three-qubit measurements are implemented on qubits 1, 2, and 3 and 4, 5, and 6, respectively, $U_{ph}$ gets teleported from qubits 1 and 5 to qubits 7 and 8.

Figure 3

Illustration of a 1D PEPS. The virtual space consists of left and right boundaries $|L\rangle$, $|R\rangle$, and virtual spins entangled in ${\sum }_{i=0}^{D-1}|ii\rangle$. The big circle represents an on-site projection where virtual spins inside are projected together to the physical space.

Figure 4

Representation of a 2D PEPS with input modes on the left boundaries.

Figure 5

Teleportation of two-mode fermionic unitary gates. Fermionic modes are depicted as white dots, and entangled mode pairs are represented by a dashed line. Input modes 1 and 3 are in state $|{\psi }_{13}\rangle =\left(m_0\right|\Omega \rangle +m_1{\alpha }_1^{†}{\alpha }_3^{†}|\Omega \rangle )$, where mode 1 carries the information and mode 3 preserves the parity of mode 1.

Figure 6

Teleportation of the four-mode controlled operation. Input modes 1 and 9 are the fermionic analogs of the control and target, and modes 3 and 11 are the parity modes corresponding to 1 and 9, respectively. After measurements on modes $1–6$ and $7–12$ are implemented, the gate is teleported to modes $13–16$.

Figure 7

Representation of the 2D fPEPS simple example resource state. The lattice consists of boundary modes on the left and entangled mode pairs connecting every site.

Figure 8

Labeling of modes on a site. Modes on the left are labeled $\alpha {\alpha }^{\prime }$, modes on the right are $\beta {\beta }^{\prime }$, modes at the top are $\delta {\delta }^{\prime }$, and modes at the bottom are $\gamma {\gamma }^{\prime }$.

Figure 9

Representation of a 2D fPEPS with $k=1$. The lattice has only one bond per direction.

Figure 10

Illustration of boundary changes after measurement on site $(1,1)$. The boundary originally on site $(1,1)$ moves to site $(1,2)$ and site $(2,1)$. The new boundaries are $B_{i,1}(i\ne 1)$ and $R_{11}$.