Polynomial-time tolerant testing stabilizer states

TL;DR

It is shown that for every ε1>0 and ε2≤ ε1C, there is a poly(1/ε1)-sample and n· poly(1/ε1)-time algorithm that decides which is the case (where C>1 is a universal constant).

Abstract

We consider the following task: suppose an algorithm is given copies of an unknown $n$-qubit quantum state $|ψ\rangle$ promised $(i)$ $|ψ\rangle$ is $\varepsilon_1$-close to a stabilizer state in fidelity or $(ii)$ $|ψ\rangle$ 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.

Figures (3)

• Figure 1: Illustration of sets, their properties and theorem statements obtained en route to show an inverse theorem of the Gowers-$3$ norm of quantum states. (a) Sets obtained starting from $S_1=\{x \in \mathbb{F}_2^{2n}:2^n p_\Psi(x) \geq \gamma \}$ as a consequence of high Gowers-$3$ norm of an $n$-qubit quantum state $\ket{\psi}$ i.e., $\textsc{Gowers}\left({\ket{\psi}},{3}\right)^8 \geq \gamma$, are depicted. Each of the sets $S_1$, $S$, $S'$, and group $V$ shown are large with sizes being greater than $\mathsf{poly}(\gamma) \cdot 2^n$. Proof strategy of the inverse theorem is indicated by the order of theorems and their implications. (b) Illustration of a stabilizer covering of the group $V$ obtained in (a) is shown along with the theorem commenting on the size of the stabilizer covering.
• Figure 2: Quantum circuit for estimating $\textsc{Gowers}\left({\ket{\psi}},{3}\right)^{8}$ given access to copies of $\ket{\psi}$ and $\ket{\psi^\star}$.
• Figure 3: Quantum circuit for estimating $\mathop{\mathbb{E}}_{x \sim q_\Psi}\left[|\langle \psi | W_x | \psi \rangle|^2 \right]$ given access to copies of $\ket{\psi}$.

Theorems & Definitions (69)

• theorem 1.1
• conjecture 1.2
• proof
• lemma 2.2: Chernoff bound
• lemma 2.3: Fact 3.2, grewal2022low
• proposition 2.4: Proposition 3.3, grewal2022low
• proposition 2.5: Theorem 3.2, gross2021schur
• lemma 2.6
• proof
• corollary 2.7