去量子化量子奇异值变换：硬度及其在量子化学与量子PCP猜想中的应用
Dequantizing the Quantum Singular Value Transformation: Hardness and Applications to Quantum Chemistry and the Quantum PCP Conjecture
AI总结：
本文研究了经典模拟量子奇异值变换（QSVT）的难度，提出对低次多项式相关的稀疏矩阵QSVT进行常数精度去量子化的技术，证明了量子化学中基态能量估计问题在常数精度下可经典高效求解，而在逆多项式精度下为BQP-complete，揭示了量子优越性源于精度提升。
摘要：
量子奇异值变换（QSVT）是一项近期提出的技术，它提供了一个统一框架来描述迄今为止发现的大多数量子算法，并可能促成新型量子算法的开发。在本文中，我们研究了经典模拟QSVT的难度。Chia, Gilyén, Li, Lin, Tang和Wang（STOC 2020）的一项近期结果表明，对于低秩矩阵，QSVT可以被高效地“去量子化”，并讨论了其对量子机器学习的启示。在本研究中，为了确立量子算法在量子化学中的优越性以及在量子PCP猜想上取得进展，我们重点关注QSVT应用中考虑的另一主要矩阵类别，即稀疏矩阵。我们首先展示了如何以任意小的常数精度，高效地“去量子化”与低次多项式相关的QSVT。我们应用该技术设计了经典算法，以常数精度估计稀疏矩阵的奇异值。我们特别指出，量子化学量子算法所考虑的一个核心计算问题（当给定一个足够接近基态的状态作为额外输入时，估计局部哈密顿量的基态能量）可以在经典计算机上以常数精度高效解决。作为补充结果，我们证明了在逆多项式精度下，同一问题变为BQP-complete。这为量子算法在化学领域的优越性提供了理论证据，并有力地表明这种优越性源于量子设定中可达到的更高精度。我们还讨论了这种去量子化技术如何可能有助于在核心的量子PCP猜想上取得进展。
英文摘要：
The Quantum Singular Value Transformation (QSVT) is a recent technique that gives a unified framework to describe most quantum algorithms discovered so far, and may lead to the development of novel quantum algorithms. In this paper we investigate the hardness of classically simulating the QSVT. A recent result by Chia, Gilyén, Li, Lin, Tang and Wang (STOC 2020) showed that the QSVT can be efficiently "dequantized" for low-rank matrices, and discussed its implication to quantum machine learning. In this work, motivated by establishing the superiority of quantum algorithms for quantum chemistry and making progress on the quantum PCP conjecture, we focus on the other main class of matrices considered in applications of the QSVT, sparse matrices. We first show how to efficiently "dequantize", with arbitrarily small constant precision, the QSVT associated with a low-degree polynomial. We apply this technique to design classical algorithms that estimate, with constant precision, the singular values of a sparse matrix. We show in particular that a central computational problem considered by quantum algorithms for quantum chemistry (estimating the ground state energy of a local Hamiltonian when given, as an additional input, a state sufficiently close to the ground state) can be solved efficiently with constant precision on a classical computer. As a complementary result, we prove that with inverse-polynomial precision, the same problem becomes BQP-complete. This gives theoretical evidence for the superiority of quantum algorithms for chemistry, and strongly suggests that said superiority stems from the improved precision achievable in the quantum setting. We also discuss how this dequantization technique may help make progress on the central quantum PCP conjecture.
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