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- W2952133313 abstract "We prove that, to compute a Boolean function $f$ on $N$ variables with error probability $epsilon$, any quantum black-box algorithm has to query at least $frac{1 - 2sqrt{epsilon}}{2} rho_f N = frac{1 - 2sqrt{epsilon}}{2} bar{S}_f$ times, where $rho_f$ is the average influence of variables in $f$, and $bar{S}_f$ is the average sensitivity. It's interesting to contrast this result with the known lower bound of $Omega (sqrt{S_f})$, where $S_f$ is the sensitivity of $f$. This lower bound is tight for some functions. We also show for any polynomial $tilde{f}$ that approximates $f$ with error probability $epsilon$, $deg(tilde{f}) ge 1/4 (1 - frac{3 epsilon}{1 + epsilon})^2 rho_f N$. This bound can be better than previous known lower bound of $Omega(sqrt{BS_f})$ for some functions. Our technique may be of intest itself: we apply Fourier analysis to functions mapping ${0, 1}^N$ to unit vectors in a Hilbert space. From this viewpoint, the state of the quantum computer at step $t$ can be written as $sum_{sin {0, 1}^N, |s| le t} hat{phi}_s (-1)^ {s cdot x}$, which is handy for lower bound analysis." @default.
- W2952133313 created "2019-06-27" @default.
- W2952133313 creator A5022499594 @default.
- W2952133313 date "1999-04-29" @default.
- W2952133313 modified "2023-09-27" @default.
- W2952133313 title "Lower bounds of quantum black-box complexity and degree of approximation polynomials by influence of Boolean variables" @default.
- W2952133313 hasPublicationYear "1999" @default.
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