Our growing step is easy to decode and is more efficient than ‘grafting’. We ascribe our efficiency improvement to this.
Well, the above is pretty much the whole story. One then simply needs to get the details right. (See our paper for them.)
For 1. we construct the RP^2 code (a variant of the planar surface code) with fold-duality. We can use this symmetry to easily transform it to a self-dual code (for 2.) and back. (similar to how a color code can be obtained by folding a surface code)
So basically, our protocols work in the following way:
We use the small RP^2 code to hold information and transform it to SRP code (a self-dual code) to double-check the logical state and then back. After preparing a high fidelity T state on the RP^2 code, we grow it to a large rotated surface code.
Zihan Chen
Zihan Chen
Thanks!
Also, it is worth while to think about the total resource cost of running algorithms especially for platforms with non-local connectivity. Trade-offs between Clifford and T on such platforms could be different from platforms with planar connectivity since the former can host transversal Cliffords.
Motivation behind our protocol:
1. To easily grow to a large regular surface code, the small code should be structurally similar to the surface code.
Our MSC protocol uses non-local connectivity available in neutral atom array and ion trap platforms. There is a lot of freedom to do MSC and our protocol still has room for improvements. Moreover, it is important to estimate the resource cost for preparing T states with even lower error rate.
2. To efficiently prepare a high fidelity T state on the small code, we want to use a self-dual CSS code with one logical qubit. Such code also has a transversal H_XY operator so that the correctness of the logical T state can be double-checked efficiently similar to the original MSC protocol.
Zihan Chen
3. Finding a code that simultaneously satisfy 1. and 2. is difficult. So we chose to find two different codes that satisfy 1. and 2. respectively and also a nice way to morph between the two codes.
