On the Exponential Sample Complexity of the Quantum State Sign Estimation Problem

TL;DR

This paper proves that estimating the relative sign of an n-qubit quantum state requires exponential sample complexity, as an efficient solution would imply a polynomial-time solution to NP-complete problems like 3-SAT.

Contribution

It establishes a fundamental exponential lower bound on the sample complexity for quantum sign estimation, linking it to computational hardness.

Findings

Quantum sign estimation has exponential sample complexity.

Efficient sign estimation would solve NP-complete problems in polynomial time.

The result connects quantum measurement complexity with computational complexity theory.

Abstract

We demonstrate that the ability to estimate the relative sign of an arbitrary $n$-qubit quantum state (with real amplitudes), given only $k$ copies of that state, would yield a $kn$-query algorithm for unstructured search. Thus the quantum sample complexity of sign estimation must be exponential: $\Omega(2^{n/2}/n)$. In particular, we show that an efficient procedure for solving the sign estimation problem would allow for a polynomial time solution to the NP-complete problem 3-SAT.