Improved Stabilizer Estimation via Bell Difference Sampling

We study the complexity of learning quantum states in various models with respect to the stabilizer formalism and obtain the following results: - We prove that $Ω(n)$ $T$-gates are necessary for any Clifford+$T$ circuit to prepare computationally pseudorandom quantum states, an exponential improvement over the previously known bound. This bound is asymptotically tight if linear-time quantum-secure pseudorandom functions exist. - Given an $n$-qubit pure quantum state $|ψ\rangle$ that has fidelity at least $τ$ with some stabilizer state, we give an algorithm that outputs a succinct description of a stabilizer state that witnesses fidelity at least $τ- \varepsilon$. The algorithm uses $O(n/(\varepsilon^2τ^4))$ samples and $\exp\left(O(n/τ^4)\right) / \varepsilon^2$ time. In the regime of $τ$ constant, this algorithm estimates stabilizer fidelity substantially faster than the naïve $\exp(O(n^2))$-time brute-force algorithm over all stabilizer states. - In the special case of $τ> \cos^2(π/8)$, we show that a modification of the above algorithm runs in polynomial time. - We exhibit a tolerant property testing algorithm for stabilizer states. The underlying algorithmic primitive in all of our results is Bell difference sampling. To prove our results, we establish and/or strengthen connections between Bell difference sampling, symplectic Fourier analysis, and graph theory.