AI 中文总结
本文提出一种针对克利福德幺正算子的不可知层析成像协议，将哈密顿量的可学习性扩展至泡利基下稠密但具克利福德结构的情形。
AI 中文摘要
结构化量子幺正算子与哈密顿量的学习算法主要针对在泡利基下局域或稀疏的过程类展开。本文聚焦于学习n量子比特的量子幺正算子U和哈密顿量H，给定对U的查询访问权限或H的幺正演化，这些算子可能在泡利基下是稠密的，但仍具有简洁的克利福德分解形式。具体而言，我们考虑幺正算子（或哈密顿量）的形式为U=Σ_i α_i C_i，其中C_i为克利福德算子，且具有有界克利福德范围Σ_i |α_i|。为提取该克利福德结构，我们提出一种针对克利福德幺正算子的不可知层析成像协议：给定对未知幺正算子U的查询访问权限，该协议在时间poly(n,(1/ε)^log(1/ε))内输出一个克利福德幺正算子，其见证保真度≥opt−ε，其中opt为最优克利福德保真度，ε>0为误差参数。随后，我们将该协议应用于获得具有有界克利福德范围的幺正算子和哈密顿量的层析成像协议。这将哈密顿量的可学习性从具有稀疏泡利分解的哈密顿量扩展到在泡利基下稠密（即稀疏度为Ω(2^n)）但具有克利福德结构的哈密顿量。
英文摘要
Learning algorithms for structured quantum unitaries and Hamiltonians have primarily considered classes of processes that are local or sparse in the Pauli basis. We turn our attention to learning $n$-qubit quantum unitaries $U$ and Hamiltonians $H$, given query access to $U$ or the unitary evolution of $H$, that may be dense in the Pauli basis but still admit concise Clifford decompositions. Specifically, we consider unitaries (or Hamiltonians) of the form $U = \sum_i α_i C_i$ over Cliffords $C_i$ with bounded Clifford extent $\sum_i |α_i|$. To extract this Clifford structure, we introduce an agnostic tomography protocol for Clifford unitaries that given query access to an unknown unitary $U$ with optimal Clifford fidelity $\textsf{opt}$, outputs a Clifford unitary witnessing fidelity $\geq \textsf{opt} - \varepsilon$ for some error $\varepsilon > 0$, in time $\textsf{poly}(n,(1/\varepsilon)^{\log(1/\varepsilon)})$. We then apply this protocol to obtain tomography protocols for unitaries and Hamiltonians that have bounded Clifford extent. This extends learnability of Hamiltonians from those with sparse Pauli decompositions to those that are dense (i.e., has sparsity $Ω(2^n)$) in the Pauli basis but are Clifford structured.
