用于汉明权重投影的对数深度量子电路
Logarithmic-Depth Quantum Circuits for Hamming Weight Projections
AI总结：
本研究针对n量子比特系统的汉明权重投影测量问题，提出若干相干投影量子算法，分析了电路深度与宽度的权衡，在具备比特重置能力下实现了O(log n)深度、O(n)控制比特的最优构造，且仅用单双量子比特门。
摘要：
固定汉明权重的纯态是计算基态的叠加，其中叠加态里的每个比特串都包含相同数量的1。给定形式为$\u210b = (\u2102_2)^{\u2297 n}$的希尔伯特空间（即n量子比特系统），恒等算子可分解为多个向固定汉明权重子空间投影的投影算子之和。本研究提出了若干量子算法，可对输入纯态实现相干汉明权重投影测量，即算法的测量后态为输入态向对应固定汉明权重子空间的投影。我们分析了对应量子电路的深度-宽度权衡关系，可通过增加控制量子比特的数量来降低电路深度。对于n量子比特输入，假设具备量子比特重置能力，深度最优算法使用$O(n)$个控制量子比特，对应电路深度为$O(\u27e8(n))$。此外，所提出的算法构造仅使用单量子比特门和双量子比特门。
英文摘要：
A pure state of fixed Hamming weight is a superposition of computational basis states such that each bitstring in the superposition has the same number of ones. Given a Hilbert space of the form $\mathcal{H} = (\mathbb{C}_2)^{\otimes n}$, or an $n$-qubit system, the identity operator can be decomposed as a sum of projectors onto subspaces of fixed Hamming weight. In this work, we propose several quantum algorithms that realize a coherent Hamming weight projective measurement on an input pure state, meaning that the post-measurement state of the algorithm is the projection of the input state onto the corresponding subspace of fixed Hamming weight. We analyze a depth-width trade-off for the corresponding quantum circuits, allowing for a depth reduction of the circuits at the cost of more control qubits. For an $n$-qubit input, the depth-optimal algorithm uses $O(n)$ control qubits and the corresponding circuit has depth $O(\log (n))$, assuming that we have the ability to perform qubit resets. Furthermore, the proposed algorithm construction uses only one- and two-qubit gates.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
