计数族的量子时空下界
A Quantum Time-Space Lower Bound for the Counting Hierarchy
摘要：
我们获得了解决可满足性相关问题的量子算法的第一个非平凡时空下界。我们的界限适用于MajSAT和MajMajSAT，这是完整的问题的第一和第二层次的计数，分别。我们证明了对于每个实数d和每个正实数ε，存在一个实数c>1使得：MajMajSAT不存在在时间n^c内运行的具有有界双侧误差的量子算法，或者MajSAT不存在在时间n^d和空间n^{1-ε}内运行的具有有界双侧误差的量子算法。特别地，MajMajSAT不能通过在时间n^{1+o（1）}和空间n^{1-ε}中运行的具有有界双侧误差的量子算法来求解，对于任何ε>0。关键的技术新颖性是通过具有无限误差的概率机器对具有中间测量的量子计算进行时间和空间有效的模拟。我们还开发了一个模型，特别适合于研究一般的量子计算与同时的时间和空间的界限。然而，我们的论点适用于任何合理的量子计算统一模型。
英文摘要：
We obtain the first nontrivial time-space lower bound for quantum algorithms solving problems related to satisfiability. Our bound applies to MajSAT and MajMajSAT, which are complete problems for the first and second levels of the counting hierarchy, respectively. We prove that for every real d and every positive real epsilon there exists a real c>1 such that either: MajMajSAT does not have a quantum algorithm with bounded two-sided error that runs in time n^c, or MajSAT does not have a quantum algorithm with bounded two-sided error that runs in time n^d and space n^{1-ε}. In particular, MajMajSAT cannot be solved by a quantum algorithm with bounded two-sided error running in time n^{1+o(1)} and space n^{1-ε} for any epsilon>0. The key technical novelty is a time- and space-efficient simulation of quantum computations with intermediate measurements by probabilistic machines with unbounded error. We also develop a model that is particularly suitable for the study of general quantum computations with simultaneous time and space bounds. However, our arguments hold for any reasonable uniform model of quantum computation.
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