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Explicit lower bounds on strong quantum simulation
IEEE Transactions on Information Theory ( IF 2.5 ) Pub Date : 2020-09-01 , DOI: 10.1109/tit.2020.3004427
Cupjin Huang , Michael Newman , Mario Szegedy

We consider the problem of classical strong (amplitude-wise) simulation of $n$ -qubit quantum circuits, and identify a subclass of simulators we call monotone. This subclass encompasses almost all prominent simulation techniques. We prove an unconditional (i.e. without relying on any complexity-theoretic assumptions) and explicit $(n-2)(2^{n-3}-1)$ lower bound on the running time of simulators within this subclass. Assuming the Strong Exponential Time Hypothesis (SETH), we further remark that a universal simulator computing any amplitude to precision $2^{-n}/2$ must take at least $2^{n - o(n)}$ time. We then compare strong simulators to existing SAT solvers, and identify the time-complexity below which a strong simulator would improve on state-of-the-art general SAT solving. Finally, we investigate Clifford+ $T$ quantum circuits with $t~T$ -gates. Using the sparsification lemma, we identify a time complexity lower bound of $2^{2.2451\times 10^{-8}t}$ below which a strong simulator would improve on state-of-the-art 3-SAT solving. This also yields a conditional exponential lower bound on the growth of the stabilizer rank of magic states.

中文翻译:

强量子模拟的显式下界

我们考虑经典强(幅度)模拟的问题 $n$ -qubit 量子电路,并确定我们称为单调的模拟器子类。这个子类几乎涵盖了所有著名的模拟技术。我们证明一个无条件的 (即不依赖任何复杂性理论假设)和 明确的 $(n-2)(2^{n-3}-1)$ 此子类中模拟器运行时间的下限。假设强指数时间假设(SETH),我们进一步指出,通用模拟器计算任何 幅度精度 $2^{-n}/2$ 必须至少 $2^{n - o(n)}$ 时间。然后,我们将强大的模拟器与现有的 SAT 求解器进行比较,并确定一个强大的模拟器将在最先进的通用 SAT 求解上改进的时间复杂度。最后,我们调查 Clifford+ $T$ 量子电路 $t~T$ - 盖茨。使用稀疏引理,我们确定了一个时间复杂度下界 $2^{2.2451\times 10^{-8}t}$ 在此之下,强大的模拟器将改进最先进的 3-SAT 求解。这也产生了魔法状态稳定器等级增长的条件指数下限。
更新日期:2020-09-01
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