Can anyone explain this in-place modular addition circuit from the paper "Quantum Resource Estimates for Computing Elliptic Curve Discrete Logarithms" by Martin Roetteler et. al (https://arxiv.org/pdf/1706.06752.pdf). What does that '>' sign mean in the last circuit component? Also, what do those black arrowheads imply leaving from the circuit components?
They're trying to say this (working quirk link):
In pseudocode it would be this:
# perform y += x (mod R)
# the $ operator means to concatenate two quint registers' qubits
borrow g = quint(len=1)
let c = quint(len=1, val=0)
y$c += x
y$c$g -= R
if c:
y$g += R
del c = x <= y
unborrow g
It's kind of complicated to explain why it works. The key thing to realize is that you can fold a comparison into a subtraction by appending a bit to the high part of the target register, because when a subtraction underflows it flips that bit.
You don't actually need g for this circuit. Maybe it's filling some underlying purpose (like making the additions be the same size), but if not then the code can be simplified to:
let c = quint(len=1, val=0)
y$c += x - R
if c:
y += R
del c = x <= y
