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Quantum Analog of Shannon's Lower Bound Theorem

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arxiv 2308.13091 v1 pith:A3CAKJKV submitted 2023-08-24 quant-ph cs.CC

classification quant-phcs.CC
keywords classicalquantumanalognumberrealresultshannonsize
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abstract

Shannon proved that almost all Boolean functions require a circuit of size $\Theta(2^n/n)$. 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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