Improved Stabilizer Estimation via Bell Difference Sampling

• URL: http://arxiv.org/abs/2304.13915v3
• Date: Fri, 29 Mar 2024 19:56:54 GMT
• Title: Improved Stabilizer Estimation via Bell Difference Sampling
• Authors: Sabee Grewal, Vishnu Iyer, William Kretschmer, Daniel Liang,
• Abstract summary: We study the complexity of learning quantum states in various models with respect to the stabilizer formalism. We prove that $Omega(n)$ $T$gates are necessary for any Clifford+$T$ circuit to prepare pseudorandom quantum states. We show that a modification of the above algorithm runs in time.
• Score: 0.43123403062068827
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $O(n/(\varepsilon^2\tau^4))$ samples and $\exp\left(O(n/\tau^4)\right) / \varepsilon^2$ time. In the regime of $\tau$ constant, this algorithm estimates stabilizer fidelity substantially faster than the na\"ive $\exp(O(n^2))$-time brute-force algorithm over all stabilizer states. - In the special case of $\tau > \cos^2(\pi/8)$, 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.

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