Quantum circuits and low-degree polynomials over F_2

TL;DR

This paper establishes a correspondence between quantum circuits with specific gates and low-degree polynomials over F_2, enabling classical simulation and hardness results based on polynomial properties.

Contribution

It introduces a novel link between quantum circuit amplitudes and polynomial zero counting, facilitating new classical simulation techniques and complexity insights.

Findings

Quantum circuits correspond to degree-3 polynomials over F_2.

Classical hardness results are derived from quantum circuit concepts.

Efficient classical algorithms are developed for certain quantum circuit classes.

Abstract

In this work we explore a correspondence between quantum circuits and low-degree polynomials over the finite field F_2. Any quantum circuit made up of Hadamard, Z, controlled-Z and controlled-controlled-Z gates gives rise to a degree-3 polynomial over F_2 such that calculating quantum circuit amplitudes is equivalent to counting zeroes of the corresponding polynomial. We exploit this connection, which is especially clean and simple for this particular gate set, in two directions. First, we give proofs of classical hardness results based on quantum circuit concepts. Second, we find efficient classical simulation algorithms for certain classes of quantum circuits based on efficient algorithms for classes of polynomials.