有限适应性的并行量子优势需要结构
Parallel Quantum Advantage with Limited Adaptivity Requires Structure
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中文总结 AI 辅助
本研究证明大规模并行量子查询算法满足Aaronson与Ambainis的猜想，建立相关模拟定理，还扩展至混合算法及常数适应性轮次量子算法的模拟定理，推进了量子优势与结构相关性的研究。
中文摘要 AI 辅助
Aaronson与Ambainis（《计算理论》，2014）提出猜想：量子查询算法可实现高效的几乎处处经典模拟，即对任意T次查询的量子算法，其接受概率可在(1-δ)比例的输入上以ε加性误差近似，所需经典查询次数为poly(T,1/ε,1/δ)。该猜想表明指数级量子加速仅在足够结构化的输入上成立。本研究在该猜想上取得进展，证明了进行大规模并行量子查询的量子算法满足该猜想。相比之下，Yamakawa与Zhandry（《ACM期刊》，2024）证明仅受限于并行查询的量子算法在采样问题上仍可实现超越经典的指数级加速。本研究通过证明更强结论建立模拟定理：并行查询量子算法无法区分预言机的均匀分布与所谓“稠密分布”抽取的预言机。核心技术贡献为耦合定理，关联预言机均匀分布与稠密分布抽取的预言机。进一步将该方法扩展至纯并行场景之外，获得两类模拟定理：一类是算法包含有限量子查询前缀后接大规模并行量子查询阶段，另一类是混合算法在大规模并行量子查询阶段前进行任意多项式次适应性经典查询。最后，以并行查询模拟定理为基础情形，获得具有常数适应性轮次的量子算法的模拟定理。
英文摘要
Aaronson and Ambainis (Theory of Computing, 2014) conjectured that quantum query algorithms admit efficient almost-everywhere classical simulation: for any $T$-query quantum algorithm, its acceptance probability can be approximated on a $(1-δ)$ fraction of inputs, up to $ε$ additive error, using $\mathrm{poly}(T, 1/ε, 1/δ)$ classical queries. At a high level, the conjecture suggests that exponential quantum speedups are possible only on sufficiently structured inputs. In this work, we make progress on this conjecture by proving it for quantum algorithms that make massively parallel quantum queries. In contrast, Yamakawa and Zhandry (Journal of the ACM, 2024) showed that quantum algorithms restricted to parallel queries can still achieve exponential speedups over classical algorithms for sampling problems. We establish our simulation theorem by proving the stronger statement that parallel-query quantum algorithms cannot distinguish the uniform distribution over oracles from oracles drawn from so-called "dense distributions". Our main technical contribution is a coupling theorem that relates the uniform distribution over oracles to oracles drawn from dense distributions. We further extend this approach beyond the purely parallel setting, obtaining simulation theorems both for algorithms with a bounded quantum-query prefix followed by a massively parallel quantum-query stage, and for hybrid algorithms that make an arbitrary polynomial number of adaptive classical queries before the massively parallel quantum-query stage. Finally, using the parallel-query simulation theorem as a base case, we obtain simulation theorems for quantum algorithms with constant rounds of adaptivity.
