Assessment of practical quantum information processing (QIP) remains partial without understanding limits imposed by noise. Unfortunately, mere description of noise grows exponentially with system size, becoming cumbersome even for modest sized systems of imminent practical interest. We fulfill the need for estimates on performing noisy quantum state preparation, verification, and observation. To do the estimation we propose fast numerical algorithms to maximize the expectation value of any $d$-dimensional observable over states of bounded purity. This bound on purity factors in noise in a measurable way. Our fastest algorithm takes $O(d)$ steps if the eigendecomposition of the observable is known, otherwise takes $O(d^3)$ steps at worst. The algorithms also solve maximum likelihood estimation for quantum state tomography with convex and even non-convex purity constraints. Numerics show performance of our key sub-routine (it finds in linear time a probability vector with bounded norm that most overlaps with a fixed vector) can be several orders of magnitude faster than a common state-of-the-art convex optimization solver. Our work fosters a practical way forward to asses limitations on QIP imposed by quantum noise. Along the way, we also give a simple but fundamental insight, noisy systems (equivalently noisy Hamiltonians) always give higher ground-state energy than their noiseless counterparts.

中文翻译：

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具有受限纯度状态的可观测量的最大期望


在不了解噪声所施加的限制的情况下，对实用量子信息处理（QIP）的评估仍然是片面的。不幸的是，仅仅对噪声的描述随着系统尺寸的增加而呈指数增长，即使对于具有迫在眉睫的实际意义的中等尺寸的系统也变得很麻烦。我们满足了对执行噪声量子态准备、验证和观察进行估计的需求。为了进行估计，我们提出了快速数值算法，以最大化任何 $d$ 维可观察到的有限纯度状态的期望值。这种对噪声纯度因素的限制以可测量的方式进行。如果可观察量的特征分解已知，我们最快的算法需要 $O(d)$ 步，否则最坏的情况下也需要 $O(d^3)$ 步。该算法还解决了具有凸甚至非凸纯度约束的量子态断层扫描的最大似然估计。数值显示我们的关键子例程的性能（它在线性时间内找到具有与固定向量大部分重叠的有界范数的概率向量）可以比常见的最先进的凸优化求解器快几个数量级。我们的工作提出了一种实用的方法来评估量子噪声对 QIP 的限制。在此过程中，我们还给出了一个简单但基本的见解，噪声系统（相当于噪声哈密顿量）总是比无噪声系统给出更高的基态能量。