Quantum Analog of Shannon's Lower Bound Theorem

Abstract

Shannon proved that almost all Boolean functions require a circuit of size $\hat{I}˜(2^n/n)$. We prove a quantum analog of this classical result. Unlike in the classical case the number of quantum circuits of any fixed size that we allow is uncountably infinite. Our main tool is a classical result in real algebraic geometry bounding the number of realizable sign conditions of any finite set of real polynomials in many variables.

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