I mean n qubits that are interpreted as a little endian 2s complement integer. Here's a coined walk circuit: https://algassert.com/quirk#circuit=%7B%22cols%22%

I mean n qubits that are interpreted as a little endian 2s complement integer. Here's a coined walk circuit: https://algassert.com/quirk#circuit=%7B%22cols%22%3A%5B%5B%22H%22%5D%2C%5B%22%E2%80%A2%22%2C%22inc9%22%5D%2C%5B%22~s0ae%22%5D%2C%5B1%2C%22Chance9%22%5D%2C%5B%22Chance10%22%5D%5D%2C%22gates%22%3A%5B%7B%22id%22%3A%22~m9ah%22%2C%22name%22%3A%22x2%22%2C%22circuit%22%3A%7B%22cols%22%3A%5B%5B%22H%22%5D%2C%5B%22%E2%80%A2%22%2C%22inc9%22%5D%2C%5B%22H%22%5D%2C%5B%22%E2%80%A2%22%2C%22inc9%22%5D%5D%7D%7D%2C%7B%22id%22%3A%22~34ha%22%2C%22name%22%3A%22x8%22%2C%22circuit%22%3A%7B%22cols%22%3A%5B%5B%22~m9ah%22%5D%2C%5B%22~m9ah%22%5D%2C%5B%22~m9ah%22%5D%2C%5B%22~m9ah%22%5D%5D%7D%7D%2C%7B%22id%22%3A%22~gii%22%2C%22name%22%3A%22x32%22%2C%22circuit%22%3A%7B%22cols%22%3A%5B%5B%22~34ha%22%5D%2C%5B%22~34ha%22%5D%2C%5B%22~34ha%22%5D%2C%5B%22~34ha%22%5D%5D%7D%7D%2C%7B%22id%22%3A%22~p9lg%22%2C%22name%22%3A%22x128%22%2C%22circuit%22%3A%7B%22cols%22%3A%5B%5B%22~gii%22%5D%2C%5B%22~gii%22%5D%2C%5B%22~gii%22%5D%2C%5B%22~gii%22%5D%5D%7D%7D%2C%7B%22id%22%3A%22~hblb%22%2C%22name%22%3A%22x256%22%2C%22circuit%22%3A%7B%22cols%22%3A%5B%5B%22~p9lg%22%5D%2C%5B%22~p9lg%22%5D%5D%7D%7D%2C%7B%22id%22%3A%22~7m3k%22%2C%22name%22%3A%22x64%22%2C%22circuit%22%3A%7B%22cols%22%3A%5B%5B%22~gii%22%5D%2C%5B%22~gii%22%5D%5D%7D%7D%2C%7B%22id%22%3A%22~s0ae%22%2C%22name%22%3A%22x499%22%2C%22circuit%22%3A%7B%22cols%22%3A%5B%5B%22H%22%5D%2C%5B%22%E2%80%A2%22%2C%22inc9%22%5D%2C%5B%22~m9ah%22%5D%2C%5B%22~34ha%22%5D%2C%5B%22~34ha%22%5D%2C%5B%22~gii%22%5D%2C%5B%22~7m3k%22%5D%2C%5B%22~p9lg%22%5D%2C%5B%22~hblb%22%5D%5D%7D%7D%5D%7D https://algassert.com/quirk#circuit=%7B%22cols%22%3A%5B%5B%2... Here's an uncoined walk circuit: https://algassert.com/quirk#circuit=%7B%22cols%22%3A%5B%5B%22H%22%5D%2C%5B%22inc10%22%5D%2C%5B%22H%22%5D%2C%5B%22inc10%22%5D%2C%5B%22~jddj%22%5D%2C%5B%22~jddj%22%5D%2C%5B%22~jddj%22%5D%2C%5B%22~jddj%22%5D%2C%5B%22~jddj%22%5D%2C%5B%22~jddj%22%5D%2C%5B%22~jddj%22%5D%2C%5B%22~jddj%22%5D%2C%5B1%2C%22Chance9%22%5D%2C%5B%22Chance10%22%5D%5D%2C%22gates%22%3A%5B%7B%22id%22%3A%22~inrj%22%2C%22name%22%3A%22x4%22%2C%22circuit%22%3A%7B%22cols%22%3A%5B%5B%22H%22%5D%2C%5B%22inc10%22%5D%2C%5B%22H%22%5D%2C%5B%22inc10%22%5D%2C%5B%22H%22%5D%2C%5B%22inc10%22%5D%2C%5B%22H%22%5D%2C%5B%22inc10%22%5D%5D%7D%7D%2C%7B%22id%22%3A%22~6bnk%22%2C%22name%22%3A%22x16%22%2C%22circuit%22%3A%7B%22cols%22%3A%5B%5B%22~inrj%22%5D%2C%5B%22~inrj%22%5D%2C%5B%22~inrj%22%5D%2C%5B%22~inrj%22%5D%5D%7D%7D%2C%7B%22id%22%3A%22~jddj%22%2C%22name%22%3A%22x64%22%2C%22circuit%22%3A%7B%22cols%22%3A%5B%5B%22~6bnk%22%5D%2C%5B%22~6bnk%22%5D%2C%5B%22~6bnk%22%5D%2C%5B%22~6bnk%22%5D%5D%7D%7D%5D%7D https://algassert.com/quirk#circuit=%7B%22cols%22%3A%5B%5B%2... Because an increment gate decomposes into a controlled increment with the least significant qubit as a control and the rest of the qubits as the target, these circuits are actually almost exactly identical in function. Basically take a coined walk, replace H with H*Z, arbitrarily consider the coin qubit to be part of the register, and that's equivalent to an uncoined walk. At least for this particularly simple system.