Simulating the $d=5$ colour code magic state cultivation results with $|{\text{cat}_n}\rangle$ states stabiliser decomposition

Question

In the magic state cultivation paper, the authors mentioned: "due to a lack of better alternatives, we are forced to rely on Assumption 3.1." - which is to replace all the $T$ gates in the simulation with $S$ gates instead. However, they proceed to talk about the "dangers of using S as a proxy for T" in their figure 22. In general, I think they had issues verifying the simulation results when a $d=5$ colour code is used.

I counted 38 total $T$ or $T^{\dagger}$ gates in their $d=5$ colour code version. In principle they can use the $|\text{cat}_{n=39}\rangle$ stabiliser decomposition from this thesis to simulate the $d=5$ colour code cultivation results at a cost of 39366 times slower simulations. Albeit the higher cost to simulate, it seems like this can be parallelised. Note that $_1\langle T|\text{cat}_{n=39}\rangle = |T\rangle^{\otimes 38}$ (use ZX copy rule).

Has anyone thought of somehow incorporating Stim with PyZX or equivalent for stabiliser decompositions? 39366 classical CPU cores isn't too expensive in the grand scheme of things. Would be nice to actually verify their results.

Thesis on $|\text{cat}_{n=39}\rangle$: (Cutting-Edge Graphical Stabiliser Decompositions for Classical Simulation of Quantum Circuits, J Codsi, A thesis submitted for the degree of MSc in Mathematics and Foundations of Computer Science, Trinity 2022)

Answer 1

39366 classical CPU cores isn't too expensive in the grand scheme of things

Actually, that is too expensive. You're suggesting spending thirty million dollars on this simulation.

If you look at the data available on zenodo you will find that the S gate simulation for the d=5 injection at p=0.001 noise consumed 130188883.9 core seconds. That's roughly four cpu core years. (It's that high because a trillion shots were needed. Estimating a 1-in-a-billion-error-rate requires taking tens of billion shots, and this a heralded error rate so you also divide by the postselection rate which was around 1% for the best error rates.)

At current cloud computing rates, a cpu core year costs roughly 200USD. So those trillion shots cost around 800USD. You're suggesting multiplying this number by 39366, paying tens of millions of dollars for one curve in a paper with many curves.

As much as I would love the simulations to not rely on "in small cases, multiplying S by 2 gives something close to T, so let's simulate S and multiply by 2", I'm not going to spend tens of millions of dollars on fixing it. A much cheaper plan is to research cheaper ways to do the simulation, then do those.

(Caveat: the raw core seconds number includes both decoding [not multiplied by 39366] and simulation [yes multiplied by 39366]. So maybe only millions of dollars? Still.)