Quantum Speedups Require Structure or Depth

2026-08-19

AI Summary

研究解决了量子算法在无结构问题上需要特定深度才能实现超多项式加速的问题，证明了每轮t查询d轮的量子算法大多可以用t^(O(d^2))次经典查询模拟。

Abstract

One of the most basic conjectures in quantum complexity theory states that every $t$-query quantum algorithm can be simulated on most inputs by a $\mathrm{poly}(t)$-query classical algorithm. If true, this would provide broad justification for the need for structure in quantum speedups. We settle this conjecture for parallel quantum algorithms, showing that every $t$-query $d$-round quantum algorithm can be simulated on most inputs with $t^{O(d^2)}$ classical queries. This suggests that for unstructured problems, superpolynomial speedups would require quantum circuits of superconstant depth, and exponential speedups would further require polynomial depth. In contrast, most known speedups for structured problems are achieved by highly parallel, low-depth algorithms. Our techniques also carry new implications for the status of $\mathsf{BPP}$ vs. $\mathsf{BQP}$ relative to a random oracle, a similarly longstanding problem.

Problem

• quantum algorithm
• classical simulation
• query complexity
• parallel quantum algorithms

Innovation

• quantum speedups
• parallel quantum algorithms
• classical simulation
• query complexity
• circuit depth