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A crucial subroutine in quantum computing is to load the classical data of $N$ complex numbers into the amplitude of a superposed $n=\lceil \log_2N\rceil$-qubit state.
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In pages 198-200, Section 4.5.4 of Ref Nielsen and Chuang 2000 , it is proven that with constant-weight quantum gates and no ancillary qubit, there are states of n n qubits which take Ω ( N / log n ) \Omega(N/\log n) constant-weight operations to approximate to within a fixed distance (Eq.4.85)
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