基于混合先验的贝叶斯相位估计
Efficient Bayesian phase estimation using mixed priors
摘要：
我们描述了一个有效的实现贝叶斯量子相位估计中存在的噪声和多重本征态。这项工作的主要贡献是不同的相位分布，即截断傅立叶级数和正态分布的表示之间的动态切换。傅里叶级数表示法的优点是在许多情况下是精确的，但随着先验知识的每次更新，复杂性会增加。这就需要截断序列，最终导致分布变得不稳定。我们推导出截断傅立叶级数表示正态分布的误差界限，并使用这些来决定何时切换到正态分布表示。这种表示方法要简单得多，并且是结合近似贝叶斯更新的拒绝过滤提出的。我们表明，在许多情况下，更新可以完全使用解析表达式，从而大大降低了更新的时间复杂度。最后，当处理几个本征态的叠加时，我们需要估计相对权重。这可以用公式表示为凸优化问题，我们使用梯度投影算法来解决。通过在指数缩放的迭代中更新权重，我们在不影响整体精度的情况下大大降低了计算复杂度。
英文摘要：
We describe an efficient implementation of Bayesian quantum phase estimation in the presence of noise and multiple eigenstates. The main contribution of this work is the dynamic switching between different representations of the phase distributions, namely truncated Fourier series and normal distributions. The Fourier-series representation has the advantage of being exact in many cases, but suffers from increasing complexity with each update of the prior. This necessitates truncation of the series, which eventually causes the distribution to become unstable. We derive bounds on the error in representing normal distributions with a truncated Fourier series, and use these to decide when to switch to the normal-distribution representation. This representation is much simpler, and was proposed in conjunction with rejection filtering for approximate Bayesian updates. We show that, in many cases, the update can be done exactly using analytic expressions, thereby greatly reducing the time complexity of the updates. Finally, when dealing with a superposition of several eigenstates, we need to estimate the relative weights. This can be formulated as a convex optimization problem, which we solve using a gradient-projection algorithm. By updating the weights at exponentially scaled iterations we greatly reduce the computational complexity without affecting the overall accuracy.
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