Breaking RSA Using Shor's Algorithm

Most of the logical qubits in our construction spend most of their time sitting still, waiting for other qubits to be processed. It should be possible to store these qubits in a more compact, but less computationally convenient, form. A simple example is that resting qubits do not need the "padding layer" between qubits implicit in Figure 8, because this layer is only there to make surgery easier. Another example is that there may to be ways to pack lattice surgery qubits that use less area while preserving the code distance (e.g. the "bulbs" shown in Figure 9).

Figure 9: A possible dense packing for idle qubits, using $1.5d^2 + O(d)$ physical qubits instead of $2(d + 1)^2$. In the diagram, there is one distance 13 logical qubit per "bulb". The X and Z observables of one of the logical qubits are shown in red and blue respectively. The pattern continues past the cut-off at the left and right sides. 

One could also imagine encoding the resting qubits into a block code. The difficulty is in finding a block code that a) works with small groups of qubits, b) can be encoded and decoded fast enough and compactly enough to fit into the existing computation, and c) is sufficiently better than the surface code that the benefits (reduced code distance within the surface code) outweigh the costs (redundant additional qubits). 

Finally, there are completely different ways of storing information in the surface code. For example, qubits stored using dislocations could be denser than qubits stored using lattice surgery. However, dislocations also create runs of stabilizers over not-quite-adjacent physical qubits. Measuring these stabilizers requires additional physical operations, creating more opportunities for errors, and so errors will propagate more quickly along these runs. 

Do the "qubit houses" in Figure 9 actually work, or is there some unexpected error mechanism? Can dislocation qubits be packed more tightly than lattice surgery qubits while achieving an equivalent logical error rate? Is there a block code that is sufficiently beneficial when layered over surface code qubits? Until careful simulations are done it will be unclear what the answer to these questions is, and so we leave the answers to future work.