Figure 1

The figure shows graph states: nodes correspond to qubits, and connections (“edges”) correspond to phase entanglement. (a) The “building block” resources that have been considered by previous authors. (b) Illustration of how one would synthesize the TPC state (depicted in upper left, with open circles versus filled circles denoting the two sublattices) by fusing together snowflake resource objects. One would use a quarter of the structure to bond with each of four adjacent objects; basically the same approach applies for the star or the cross geometries. After the bonding stage, there must be a “pruning” phase where we remove all but the core nodes and thus simplify down to the target lattice. (c) One minor revision in the case of the snowflake is that the final round involves fusing qubits that are one step from the “core” in order to generate a 4-node.Reuse & Permissions

Figure 2

Our principle results: threshold estimates analytically inferred from numerical data in Ref. 22. The lines define the parameter regimes where fault-tolerant QIP is possible. Here, for simplicity, we set all gate errors, including both single-qubit and two-qubit errors, to be equally likely. This probability is denoted $p_G$ and is plotted on the vertical axis. (Memory errors due to gradual decoherence are excluded, and are shown in Fig. 3). Meanwhile, the probability $p_h$ of an entangling operation failing in a heralded fashion is plotted on the horizontal axis. Note that $p_h$ can be very high, exceeding 90% if the snowflake strategy is employed.Reuse & Permissions

Figure 3

(a) The effect of introducing memory errors, i.e., finite gradual decoherence. (b) The size of the resource objects required to achieve the thresholds.Reuse & Permissions