Near-deterministic creation of universal cluster states with probabilistic Bell measurements and three-qubit resource states

We develop a scheme for generating a universal qubit cluster state using probabilistic Bell measurements without the need for feed-forward. Borrowing ideas from percolation theory, we numerically show that using unambiguous Bell measurements that succeed with 75% success probability one could build a cluster state with an underlying pyrochlore geometry such that the probability of having a spanning cluster approaches unity in the limit of infinite lattice size. The initial resources required for the generation of a universal state in our protocol are three-qubit cluster states that are within experimental reach and are a minimal resource for a Bell-measurement-based percolation proposal. Since single and multiphoton losses can be detected in Bell measurements, our protocol raises the prospect of a fully error-robust scheme.

Figure 1

The effect of a successful Bell measurement (measuring qubits 1 and $1^{\prime }$). A square node represents a polarization qubit with a Hadamard applied to it, while a dashed oblong around two qubits denotes a Bell measurement. The italicized labels $\mathcal{A}$, $\mathcal{B}$, $\mathcal{C}$, and $\mathcal{D}$ denote graphs for which the products of Pauli $Z$ operators are represented as $A$, $B$, $C$, and $D$, respectively, in the text. The arguments presented in the text apply to all neighbors of the measured qubits, and hence, other neighbors inherit entanglement just as is shown for qubits $i$ and $j$. Hence, $\bigcup$ represents the union of all graphs over all possible combinations of neighbors of qubit 1 ($i$) and neighbors of qubit $1^{\prime }$ ($j$).

Figure 2

Fusing two triangular clusters with Bell measurements results in a four-qubit tetrahedron upon success (denoted by “s”). Failure of the measurement (denoted by “f”, and implying an ambiguous measurement outcome) results in two Bell pairs (up to local operations), which can still be used in subsequent fusions shown later.

Figure 3

The three-qubit chain is used as an intermediary to fuse two tetrahedra together to form a “bow tie.” Since there are two Bell measurements involved, the theoretical success probability of this step is $56.25\%$.

Figure 4

Three-qubit chains are used as intermediaries to fuse five tetrahedra together. The failure event is an example of what happens when the pair of fusions at the top and the top left fail.

Figure 5

Simulation of spanning-cluster generation using three-qubit cluster states. Figure (d) shows a representative outcome on a unit cell, while figures (a) and (b) show a fully connected pyrochlore cell, which is what the protocol would produce if Bell measurements were deterministic.

Figure 6

Percolation probability versus the success probability of Bell measurement for various pyrochlore lattice sizes. Each lattice size was sampled 200 times.

Figure 7

Percolation probability versus the success probability of Bell measurement. Large number of trials (1000), large lattice sizes, and a small step size in the fusion probability (0.001) allow us to estimate the percolation threshold to be between 69.5% and 70%.