Improved Stabilizer Estimation via Bell Difference Sampling
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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⟩ 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 - ε. The algorithm uses O(n/(ε^2τ^4)) samples and exp(O(n/τ^4)) / ε^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. - We improve the soundness analysis of the stabilizer state property testing algorithm due to Gross, Nezami, and Walter [Comms. Math. Phys. 385 (2021)]. As an application, 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.
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