We analyze the complexity of learning $n$-qubit quantum phase states. A degree-$d$ phase state is defined as a superposition of all $2^n$ basis vectors $x$ with amplitudes proportional to $(-1)^{f(x)}$, where $f$ is a degree-$d$ Boolean polynomial over $n$ variables. We show that the sample complexity of learning an unknown degree-$d$ phase state is $\Theta(n^d)$ if we allow separable measurements and $\Theta(n^{d-1})$ if we allow entangled measurements. Our learning algorithm based on separable measurements has runtime $\textsf{poly}(n)$ (for constant $d$) and is well-suited for near-term demonstrations as it requires only single-qubit measurements in the Pauli $X$ and $Z$ bases. We show similar bounds on the sample complexity for learning generalized phase states with complex-valued amplitudes. We further consider learning phase states when $f$ has sparsity-$s$, degree-$d$ in its $\mathbb{F}_2$ representation (with sample complexity $O(2^d sn)$), $f$ has Fourier-degree-$t$ (with sample complexity $O(2^{2t})$), and learning quadratic phase states with $\varepsilon$-global depolarizing noise (with sample complexity $O(n^{1+\varepsilon})$). These learning algorithms give us a procedure to learn the diagonal unitaries of the Clifford hierarchy and IQP~circuits.

翻译：我们分析了学习$n$比特量子相位态的复杂性。一个$d$次相位态定义为所有$2^n$个基向量$x$的叠加态，其振幅与$(-1)^{f(x)}$成正比，其中$f$是$n$个变量上的$d$次布尔多项式。我们证明，若允许可分离测量，学习未知$d$次相位态的样本复杂度为$\Theta(n^d)$；若允许纠缠测量，则为$\Theta(n^{d-1})$。我们基于可分离测量的学习算法运行时间为$\textsf{poly}(n)$（对于常数$d$），且仅需在泡利$X$和$Z$基上进行单比特测量，因此特别适合近期的演示实验。我们给出了学习具有复值振幅的广义相位态时的类似样本复杂度下界。我们进一步研究了以下情形下的相位态学习：当$f$在$\mathbb{F}_2$表示中具有稀疏度$s$和度数$d$时（样本复杂度为$O(2^d sn)$），$f$具有傅里叶度数$t$时（样本复杂度为$O(2^{2t})$），以及学习存在$\varepsilon$全局退极化噪声的二次相位态时（样本复杂度为$O(n^{1+\varepsilon})$）。这些学习算法为我们提供了一种学习克利福德层级对角酉矩阵和IQP电路的方法。