  We present a paradigm for constructing approximate quantum circuits from
reversible classical circuits that operate on many possible encodings of an
input and send almost all encodings of that input to an encoding of the correct
output. We introduce oblivious carry runways, which use piecewise addition
circuits to perform approximate encoded additions and reduce the asymptotic
depth of addition to $O(\lg \lg n)$. We show that the coset representation of
modular integers (Zalka 2006) is an approximate encoded modular addition, and
that it can be used in combination with oblivious carry runways. We prove error
bounds on these approximate representations, and use them to construct
2s-complement adders and modular adders with lower costs than in previous work
at register sizes relevant in practice.
