Stabilizer states are fundamental families of quantum states with crucial applications such as error correction, quantum computation, and simulation of quantum circuits. In this talk, we study the problem of testing how close or far a quantum state is to a stabilizer state. We present two contributions: First, we improve the state-of-the-art parameters for the tolerant testing of stabilizer states. In particular, we show that there is an efficient quantum primitive to distinguish if the maximum fidelity of a quantum state with a stabilizer state is ≥ ϵ1 or ≤ ϵ2, given that one of them is the case, provided that ϵ_2 ≤ ϵ_1^O(1). This result improves the parameters in a previous work of Arunachalam and Dutt [AD24], which assumed ϵ_12 ≤ e^(−1/ϵ_1^O(1)). Our proof technique extends the toolsets developed in [AD24] by applying a random Clifford map, which balances the characteristic function of a quantum state, enabling the use of standard proof techniques from higher-order Fourier analysis for Boolean functions, where improved testing bounds are available. Second, we study the problem of testing low stabilizer rank states. We show that if for an infinite family of quantum states stabilizer rank is lower than a constant independent of system size, then stabilizer fidelity is lower bounded by an absolute constant. Using a result of [GIKL22], one of the implications of this result is that low approximate stabilizer rank states are not pseudo-random.
Based on joint work with Mehrdad Tahmasbi; for preprint visit https://arxiv.org/pdf/2410.24202. To be presented at STOC 2025.Â
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