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Suppose we start with $|00...0\rangle$. We want to build an equal superposition over $|0\rangle + ... + |n-1\rangle$.

When $n=2^m$ for some $m$, I know I can do this using $H^{\otimes m}$.

What is the general circuit for this (i.e. in case $n$ is not power of 2)?

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You can use a single step of amplitude amplification, with a less-than-N oracle, to get to a uniform distribution.

Example Quirk Circuit

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Source: https://arxiv.org/abs/1805.03662

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