some visuals to complement 3 recent papers on logical gate schemes for arbitrary quantum LDPC codes. :)

Thiago put together a nice plot summarizing how this result fits into other results about bounds of time/space overhead vs locality. In terms of this plot, the paper's goal was to put a circle in the O(1) column, but not in the "all" row.

there must be some trickle down effect onto efficacy of two stages?

ah, P_{1,17} is ~e^14 so it's still cubic suppression, but the qutrit code offers much better constants.

Trading hundreds of qubits for just 20 qutrits isn't a big ask in terms of temporarily switching dimension of the system either.

where is this from?
if one throws out everything but the leading order, error^16 suppression sounds pretty good...

Curious how did you get the initial depth as 4 specifically? guessing here that a minimum of 3 time steps arise from pairwise gates between 3 pairs of qubits, and an extra gate somewhere?
Will check out the relevant section for technical details

I see, it's depth 1 exactly, not just depth O(1). That would be a pretty specialized code indeed.
Pieceably transversal CCZs (which are applicable to arbitrary CSS codes) are targeted, but they would need a depth of O(d).
Interesting to see how much better tailored targeted codes can do!

quick edit in my previous comment: the adapters were designed for LDPC regime, but simplifications for higher-weight codes would only make it easier to teleport info. But this is all side-tracking from the main story, for which I'll check out your paper.

ah so the motivation is towards code construction tailored towards these gates rather than a generic gate scheme. Pretty neat! One can still hook these codes up to arbitrary LDPC codes using our adapters (arxiv.org/abs/2410.03628) for using these more generically in other codes.

Congrats on these results on a very interesting problem -- Looking forward to reading this work! Are the check weights bounded during the CCZ circuit? (If not by constants, then what is the extent to which they increase?)

Congrats! Curious, how did you verify the quantum computer's output is correct? (due to inability to reproduce this classically in feasible time)

ahhh it's LDPC specific. thanks for clarifying!

thanks! ok i see why good soundness isn't contained in the good confinement condition (as small syndromes can be caused by large errors).
Reg the other way around (good confinement ⊆ good soundness), it's also somehow implicit that small errors trigger a small syndrome?

my current guess it's the same inequality (f(|σ(e)|) ≥ |e|_red) , but enforced on a smaller set of errors to get a weaker condition.