Suppose I have N register qubits, so they can represent the range [0, 2^N-1]. They are initialized in the all-zeroes state. I want my final state to approximate $$|\phi \rangle = \frac{1}{2^{2^N-1}} \sum_{m=0}^{2^N-1} {2^N-1 \choose m} |m \rangle \, .$$ Can this be done with poly(N) 2-qubit + 1-qubit gates? How good an approximation can I get with poly(N) gates?
2026-03-27 01:45:03.1774575903
Can you make a Gaussian-weighted superposition state with polynomially many gates?
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