发表机构
Université Paris Cité, CNRS, IRIF; Alfréd Rényi Institute of Mathematics; Université de Bordeaux, CNRS, LaBRI; Université libre de Bruxelles, QuIC（巴黎西岱大学; 阿尔弗雷德·雷尼数学研究所; 波尔多大学; 布鲁塞尔自由大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出用三对角辅助量子比特链实现任意局域哈密顿量的多项式变换，通过不同长度链的加权和逐项逼近，实现特征值滤波和绝热优化，代价为多项式辅助比特和能量尺度。
AI 中文摘要
模拟设备实现一个局域哈密顿量H，但多项式P(H)通常不是局域的，因此设备无法直接实现它。我们将P(H)以任意指定精度实现为单个含时无关局域哈密顿量在一个明确描述的不变子空间上的作用。短链的辅助量子比特附加到H上。一条2m个位点的链具有一个唯一的孤立特征值，它是输入的解析函数，且消失阶数恰好为2m，因为输入必须穿过链并返回，才能移动远端的能量。因此，不同长度的链构成一个三角族，它们的加权和逐项重现指定的多项式。偶数长度的链给出偶部，奇数长度的链给出奇部，因此任何多项式都能实现。我们证明了唯一性、解析性以及对于范数低于任意r<1的输入H，系数界对链长均匀成立。在电路模型中，度2l需要2l次顺序预言调用，而这里的结果是一个局域性比H高2的哈密顿量，其代价是O(l^2 + log2^(1/eps))个辅助量子比特和能量尺度。作为应用，我们过滤一个局域哈密顿量的标记特征态。沿切比雪夫倍增恒等式合成一个平方是可行的，但其能量尺度随度拟多项式增长。相反，将两位点链的精确特征值分支迭代k次，在k+1个辅助量子比特上得到一个滤波器，其误差随k几何衰减，能量尺度与逆通带宽度成多项式关系且与k无关，局域性每阶段增加一。在绝热优化中，使用秩一驱动器时，该构造用具有O(log n)个辅助量子比特和O(log n)体项的哈密顿量替换非局域驱动器，能量尺度与n成多项式关系。包含所有辅助量子比特的模拟重现了理想算法的谱。
英文摘要
An analog device implements a local Hamiltonian H, but a polynomial P (H) is in general not local, so the device cannot implement it. We carry P (H), to any prescribed accuracy, as the action of a single time-independent local Hamiltonian on an explicitly described invariant subspace. Short chains of ancilla qubits are attached to H. A chain of 2m sites has a unique isolated eigenvalue that is an analytic function of the input vanishing to order exactly 2m, because the input must cross the chain and come back before it can shift the energy at the far end. Chains of different lengths therefore form a triangular family, and a weighted sum of them reproduces a prescribed polynomial term by term. Even chains give the even part and odd chains the odd part, so any polynomial is reached. We prove uniqueness, analyticity and coefficient bounds uniform in the chain length for inputs H of norm below any r < 1. Where the circuit model pays degree 2l in 2l sequential oracle calls, the result here is one Hamiltonian of locality two more than that of H, which pays the degree in O(l^2 + log2^(1/eps)) ancillas and in energy scale. As an application we filter a marked eigenstate of a local Hamiltonian. Composing a synthesised square along the Chebyshev doubling identity works, but its energy scale grows quasi-polynomially with the degree. Iterating instead the exact eigenvalue branch of the two-site chain k times gives a filter on k + 1 ancilla qubits whose error decays geometrically in k, at an energy scale polynomial in the inverse passband width and independent of k, the locality growing by one per stage. In adiabatic optimisation with a rank-one driver, the construction replaces that non-local driver, by a Hamiltonian with O(log n) ancillas and O(log n)- body terms, at an energy scale polynomial in n. Simulations including every ancilla reproduce the spectrum of the ideal algorithm.
