Matrix product operators allow efficient descriptions (or realizations) of states on a 1D lattice. We consider the task of learning a realization of minimal dimension from copies of an unknown state, such that the resulting operator is close to the density matrix in trace norm. For finitely correlated translation-invariant states on an infinite chain, a realization of minimal dimension can be exactly reconstructed via linear algebra operations from the marginals of a size depending on the representation dimension. We establish a bound on the trace norm error for an algorithm that estimates a candidate realization from estimates of these marginals and outputs a matrix product operator, estimating the state of a chain of arbitrary length $t$. This bound allows us to establish an $O(t^2)$ upper bound on the sample complexity of the learning task, with an explicit dependence on the site dimension, realization dimension and spectral properties of a certain map constructed from the state. A refined error bound can be proven for $C^*$-finitely correlated states, which have an operational interpretation in terms of sequential quantum channels applied to the memory system. We can also obtain an analogous error bound for a class of matrix product density operators on a finite chain reconstructible by local marginals. In this case, a linear number of marginals must be estimated, obtaining a sample complexity of $\tilde{O}(t^3)$. The learning algorithm also works for states that are sufficiently close to a finitely correlated state, with the potential of providing competitive algorithms for other interesting families of states.

翻译：矩阵乘积算子允许对一维晶格上的态进行高效描述（或实现）。我们考虑从未知态的多个副本中学习最小维数实现的任务，使得所得算子在迹范数意义下接近密度矩阵。对于无限链上的平移不变有限关联态，可通过依赖于表示维数的大小区际的线性代数运算精确重构出最小维数实现。我们为一种算法建立了迹范数误差界，该算法从这些边际的估计中计算候选实现，并输出一个矩阵乘积算子，用于估计任意长度 $t$ 链的态。该误差界使我们能够建立学习任务样本复杂度的 $O(t^2)$ 上界，其显式依赖于格点维数、实现维数以及由该态构造的特定映射的谱性质。对于 $C^*$-有限关联态（其在操作上可解释为对记忆系统施加的序列量子信道），可以证明更精细的误差界。对于可通过局部边际重构的有限链上的一类矩阵乘积密度算子，我们也能获得类似的误差界。此时必须估计线性数量的边际，得到 $\tilde{O}(t^3)$ 的样本复杂度。该学习算法也适用于充分接近有限关联态的态，有望为其他有趣态族提供有竞争力的算法。