用压缩感知学量子哈密顿量，不依赖局部性假设。
Learning $k$-body Hamiltonians via compressed sensing
- 基于压缩感知与GHZ态，非自适应学习哈密顿量。
- 总演化时间仅需 $\mathcal{O}(M^{1/2+1/p}/ε)$，精度 $ε$。
- 对噪声鲁棒，适合未知稀疏度或非局域哈密顿量。
我们研究了学习包含 $M$ 个未知泡利项的 $k$-体哈密顿量的问题，这些项不一定具有几何局部性。提出一种协议，在总演化时间 ${\mathcal{O}}(M^{1/2+1/p}/ε)$（忽略对数因子）内将哈密顿量学习至精度 $ε$，误差以泡利系数的 $\ell^p$ 距离衡量。该协议仅需单比特控制操作和GHZ态初始态，是非自适应的，对状态制备与测量（SPAM）误差鲁棒，且在 $M$、$k$ 事先未知或哈密顿量不精确 $M$-稀疏时仍表现良好。利用经典压缩感知理论，从所有可能的 $k$-体泡利算符中高效识别出 $M$ 个关键项。同时给出了该学习任务所需总演化时间的下界，并讨论了 $\ell^1$ 与 $\ell^2$ 误差度量的操作意义。与多数先前工作不同，本协议无需几何局部性或其他弱局部性条件。
原文摘要 · Abstract (English)
We study the problem of learning a $k$-body Hamiltonian with $M$ unknown Pauli terms that are not necessarily geometrically local. We propose a protocol that learns the Hamiltonian to precision $ε$ with total evolution time ${\mathcal{O}}(M^{1/2+1/p}/ε)$ up to logarithmic factors, where the error is quantified by the $\ell^p$-distance between Pauli coefficients. Our learning protocol uses only single-qubit control operations and a GHZ state initial state, is non-adaptive, is robust against SPAM errors, and performs well even if $M$ and $k$ are not precisely known in advance or if the Hamiltonian is not exactly $M$-sparse. Methods from the classical theory of compressed sensing are used for efficiently identifying the $M$ terms in the Hamiltonian from among all possible $k$-body Pauli operators. We also provide a lower bound on the total evolution time needed in this learning task, and we discuss the operational interpretations of the $\ell^1$ and $\ell^2$ error metrics. In contrast to most previous works, our learning protocol requires neither geometric locality nor any other relaxed locality conditions.
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