Improved Stabilizer Estimation via Bell Difference Sampling

Abstract: 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 $\Omega(n)$ $T$-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 $|\psi\rangle$ that has fidelity at least $\tau$ with some stabilizer state, we give an algorithm that outputs a succinct description of a stabilizer state that witnesses fidelity at least $\tau - \varepsilon$. The algorithm uses $O(n/(\varepsilon<sup>2\tau<sup>4))$ samples and $$\exp\left(O(n/\tau^4)\right) / \varepsilon<sup>2$$ time. In the regime of $\tau$ constant, this algorithm estimates stabilizer fidelity substantially faster than the na\"ive $\exp(O(n<sup>2))$-time brute-force algorithm over all stabilizer states. - In the special case of $\tau &gt; \cos<sup>2(\pi/8)$, 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.