量子预计算：级联电路的并行化与Moore-Nilsson猜想不成立
Quantum precomputation: parallelizing cascade circuits and the Moore-Nilsson conjecture is false
AI总结：
本研究提出量子分块预计算技术，证否Moore-Nilsson猜想，将受控酉阶梯电路深度压缩至最优的O(log n)，还给出二维连通电路的最优压缩结果，并推广至更通用的级联电路。
摘要：
由于不可克隆定理等物理约束，并行化是量子算法领域的一大核心挑战。Moore与Nilsson在其关于量子电路复杂度的开创性工作[MN01，1998年公布]中提出的猜想便直观体现了这一点：形式看似简单的酉算子——即受控酉“阶梯”结构——所需的电路最小深度为Ω(n)。若该猜想成立，这一下界将标志着量子并行与经典并行的重大分野，并证明著名的NC≠P猜想的量子对应版本。\n 在本研究中，我们证否了Moore-Nilsson猜想：将该类所有电路的深度压缩至O(log n)，这是最优结果。该并行化是精确的、无需辅助量子比特，且可在poly(n)时间内完成计算。我们还研究了受限于二维连通性的电路，推导出最优深度为O(√n)的压缩方案。\n 更广泛而言，我们提出了量子分块预计算技术，推动了量子并行化研究的进展，该技术与经典动态规划中Arlazarov、Dinič、Kronrod和Faradžev[Arl+70]提出的方法（常被称为“四俄罗斯人法”）有一定相似性。我们还将该技术应用于更通用的“级联”电路，例如对受控log(n)量子比特酉算子的阶梯电路实现了多项式级的深度降低。
英文摘要：
Parallelization is a major challenge in quantum algorithms due to physical constraints like no-cloning. This is vividly illustrated by the conjecture of Moore and Nilsson from their seminal work on quantum circuit complexity [MN01, announced 1998]: unitaries of a deceptively simple form--controlled-unitary "staircases"--require circuits of minimum depth $Ω(n)$. If true, this lower bound would represent a major break from classical parallelism and prove a quantum-native analogue of the famous NC $\neq$ P conjecture. In this work we settle the Moore-Nilsson conjecture in the negative by compressing all circuits in the class to depth $O(\log n)$, which is the best possible. The parallelizations are exact, ancilla-free, and can be computed in poly($n$) time. We also consider circuits restricted to 2D connectivity, for which we derive compressions of optimal depth $O(\sqrt{n})$. More generally, we make progress on the project of quantum parallelization by introducing a quantum blockwise precomputation technique somewhat analogous to the method of Arlazarov, Dinič, Kronrod, and Faradžev [Arl+70] in classical dynamic programming, often called the "Four-Russians method." We apply this technique to more-general "cascade" circuits as well, obtaining for example polynomial depth reductions for staircases of controlled $\log(n)$-qubit unitaries.
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