指数压缩且无垃圾的别名采样用于多项式态制备
Exponentially Compressed and Garbage-Free Alias Sampling for Polynomial State Preparation
- Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）
- University of California, Los Angeles（加州大学洛杉矶分校）
- Microsoft Quantum（微软量子）
- Duke University（杜克大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出一种指数压缩且无垃圾的别名采样方法，用于多项式态制备，将成本从指数降至多项式，并开发了辅助寄存器反计算技术，开辟了采样制备初态的新方向。
AI中文摘要：
相干别名采样实现了基于量子查找表的块编码和量子化模拟的重要子程序。相关构造利用数据查找和矩阵结构来实现用于量子奇异值变换的块编码。对于$n$量子比特上的态，构建和存储别名表需要指数于$n$的经典和量子资源。我们证明，当振幅是次数至多为$d$的多项式的样本时，该表可以被相干地表示和评估，其成本关于$n$和$d$为多项式，从$\Theta(2^n)$的经典和量子成本指数级降低。我们使用可逆整数算术构建了一个用于相干计算多项式态别名表的量子电路。通过采用快速算术，我们构建了一个确定性的电路，用于在Clifford+$T$门集上制备多项式态，总门数为$\tilde{O}(ndB+B^2)$，其中算术宽度$B=\tilde{O}(dn+d\log(d+1))$。在固定$n$和$d$的态保真度$\epsilon$的高精度极限下，$B=\Theta(\log(1/\epsilon))$就足够了，总门数变为$O(\log^3(1/\epsilon)\log\log(1/\epsilon))$。此外，别名表的有效表示也可用于经典采样算法。进一步，我们开发了一种新技术来反计算由相干别名采样生成的辅助寄存器。该技术开辟了基于采样的初始态制备的研究方向。
英文摘要：
Coherent alias sampling implement an important subroutine for block encoding and qubitization based quantum simulation using quantum lookup table. Related constructions exploit data lookup and matrix structure to implement block encodings for use in quantum singular value transformation. For a state on $n$ qubits, constructing and storing the alias table requires both classical and quantum resources exponential in $n$. We show that when the amplitudes are samples of a polynomial of degree at most $d$, the table can be represented and evaluated coherently with cost polynomial in $n$, $d$, exponentially reduced from $Θ(2^n)$ classical and quantum cost. We construct a quantum circuit for coherently computing the alias table for polynomial states using reversible integer arithmetic. By employing fast arithmetic, we construct a deterministic circuit for preparing polynomial states over Clifford+$T$ with a complete gate count of $\tilde{O}(ndB+B^2)$, where arithmetic width $B=\tilde{O}(dn+d\log(d+1))$. In the high precision limit for state fidelity $ε$ with fixed $n$ and $d$, it is sufficient have $B=Θ(\log(1/ε))$ and the complete gate count becomes $O(\log^3(1/ε)\log\log(1/ε))$. Additionally, the effective representation of the alias table can also be used for classical sampling algorithms. Furthermore, we develop a novel technique for uncomputing the auxilary register generated by coherent alias sampling. This technique opens a research direction in sampling based initial state preparation.
