Polynomial-time tolerant testing stabilizer states

TL;DR

This paper introduces a polynomial-time algorithm for tolerant testing of stabilizer states, utilizing a new quantum Gowers norm and additive combinatorics to distinguish states close to or far from stabilizer states.

Contribution

It develops the first polynomial-time tolerant testing algorithm for stabilizer states, incorporating a novel quantum Gowers norm and inverse theorems.

Findings

Algorithm works with polynomial sample complexity in 1/ε₁

Decides closeness or farness to stabilizer states with high accuracy

Introduces new bounds on stabilizer covering using additive combinatorics

Abstract

We consider the following task: suppose an algorithm is given copies of an unknown $n$-qubit quantum state $|\psi\rangle$ promised $(i)$ $|\psi\rangle$ is $\varepsilon_1$-close to a stabilizer state in fidelity or $(ii)$ $|\psi\rangle$ is $\varepsilon_2$-far from all stabilizer states, decide which is the case. We show that for every $\varepsilon_1>0$ and $\varepsilon_2\leq \varepsilon_1^C$, there is a $\textsf{poly}(1/\varepsilon_1)$-sample and $n\cdot \textsf{poly}(1/\varepsilon_1)$-time algorithm that decides which is the case (where $C>1$ is a universal constant). Our proof includes a new definition of Gowers norm for quantum states, an inverse theorem for the Gowers-$3$ norm of quantum states and new bounds on stabilizer covering for structured subsets of Paulis using results in additive combinatorics.