We propose a two step strategy for estimating one-dimensional dynamical parameters of a quantum Markov chain, which involves quantum post-processing the output using a coherent quantum absorber and a "pattern counting'' estimator computed as a simple additive functional of the outcomes trajectory produced by sequential, identical measurements on the output units. We provide strong theoretical and numerical evidence that the estimator achieves the quantum Cram\'{e}-Rao bound in the limit of large output size. Our estimation method is underpinned by an asymptotic theory of translationally invariant modes (TIMs) built as averages of shifted tensor products of output operators, labelled by binary patterns. For large times, the TIMs form a bosonic algebra and the output state approaches a joint coherent state of the TIMs whose amplitude depends linearly on the mismatch between system and absorber parameters. Moreover, in the asymptotic regime the TIMs capture the full quantum Fisher information of the output state. While directly probing the TIMs' quadratures seems impractical, we show that the standard sequential measurement is an effective joint measurement of all the TIMs number operators; indeed, we show that counts of different binary patterns extracted from the measurement trajectory have the expected joint Poisson distribution. Together with the displaced-null methodology of J. Phys. A: Math. Theor. 57 245304 2024 this provides a computationally efficient estimator which only depends on the total number of patterns. This opens the way for similar estimation strategies in continuous-time dynamics, expanding the results of Phys. Rev. X 13, 031012 2023.

翻译：我们提出了一种用于估计量子马尔可夫链一维动力学参数的两步策略，该策略涉及对输出进行量子后处理，使用相干量子吸收器和一个“模式计数”估计器，该估计器计算为由对输出单元进行顺序、相同测量所产生的结果轨迹的简单加性泛函。我们提供了强有力的理论和数值证据，表明该估计器在大输出尺寸极限下达到了量子克拉默-拉奥界。我们的估计方法建立在平移不变模式（TIMs）的渐近理论基础上，这些模式构建为输出算符平移张量积的平均值，并由二进制模式标记。在长时间极限下，TIMs形成一个玻色代数，输出态趋近于TIMs的联合相干态，其振幅线性依赖于系统参数与吸收器参数之间的失配。此外，在渐近区域，TIMs捕获了输出态的全部量子费舍尔信息。虽然直接探测TIMs的正交分量似乎不切实际，但我们证明了标准顺序测量是所有TIMs粒子数算符的有效联合测量；事实上，我们证明了从测量轨迹中提取的不同二进制模式的计数具有预期的联合泊松分布。结合J. Phys. A: Math. Theor. 57 245304 2024中的位移-零值方法，这提供了一个仅依赖于模式总数的计算高效估计器。这为在连续时间动力学中采用类似的估计策略开辟了道路，从而扩展了Phys. Rev. X 13, 031012 2023的结果。