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 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 |psirangle 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/(varepsilon2τ4)) samples and expleft(O(n/τ4)right) / varepsilon2 time. In the regime of τ constant, this algorithm estimates stabilizer fidelity substantially faster than the na"ive exp(O(n2))-time brute-force algorithm over all stabilizer states. - In the special case of τ > cos2(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.