The degree of a polynomial representing (or approximating) a function f is a lower bound for the quantum query complexity of f. This observation has been a source of many lower bounds on quantum algorithms. It has been an open problem whether this lower bound is tight. This is why Boolean functions are needed with a high number of essential variables and a low polynomial degree. Unfortunately, it is a well-known problem to construct such functions. The best separation between these two complexity measures of a Boolean function was exhibited by Ambainis [5]. He constructed functions with polynomial degree M and number of variables Ω(M{sup}2). We improve such a separation to become exponential. On the other hand, we use a computerized exhaustive search to prove tightness of this bound.
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