Quantum Analog of Shannon's Lower Bound Theorem

TL;DR

This paper establishes a quantum version of Shannon's lower bound theorem, demonstrating that most Boolean functions require large quantum circuits, and introduces a novel approach using real algebraic geometry to bound circuit realizations.

Contribution

It presents the first quantum analog of Shannon's classical lower bound, extending the understanding of quantum circuit complexity with new mathematical tools.

Findings

Most Boolean functions require quantum circuits of size (2^n/n)

Uncountably infinite quantum circuits of fixed size exist

Real algebraic geometry bounds the number of realizable quantum circuit configurations

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.