A reconstruction algorithm for compressive quantum tomography using various measurement sets

Quantum State Tomography via Nonconvex Riemannian Gradient Descent

The recovery of an unknown density matrix of large size requires huge computational resources. State-of-the-art performance has recently been achieved with the factored gradient descent (FGD) algorithm and its variants since they are able to mitigate the dimensionality barrier by utilizing some of the underlying structures of the density matrix. Despite the theoretical guarantee of a linear convergence rate, convergence in practical scenarios is still slow because the contracting factor of the FGD algorithms depends on the condition number κ of the ground truth state. Consequently, the total number of iterations needed to achieve the estimation error ϵ can be as large as O(sqrt[κ]ln(1/ϵ)). In this Letter, we derive a quantum state tomography scheme that improves the dependence on κ to the logarithmic scale. Thus, our algorithm can achieve the approximation error ϵ in O(ln(1/κϵ)) steps. The improvement comes from the application of nonconvex Riemannian gradient descent (RGD). The contracting factor in our approach is thus a universal constant that is independent of the given state. Our theoretical results of extremely fast convergence and nearly optimal error bounds are corroborated by the numerical results.

Seasonal Changes in Locomotor Activity Patterns of Wild Social Natal Mole-Rats (Cryptomys hottentotus natalensis)

Differences in individual locomotor activity patterns may be linked to a number of ecological factors, such as changes in ambient temperature or photoperiod. Observations on subterranean mammals suggest that they exhibit diel rhythms despite the lack of visual cues in their underground burrows, but it is unknown how seasonality and individual characteristics affect their activity. In this study we use RFID technology to monitor daily activity patterns of wild, social Natal mole-rats (Cryptomys hottentotus natalensis) during the summer and winter to investigate how their activity varies with season and whether their activity depends on individual characteristics such as body mass, sex and reproductive status. We found that in winter, individuals were more active during the time with the highest soil temperatures, whereas in summer, they showed a bimodal activity pattern during early morning and late afternoon coinciding with cooler soil temperatures. Individual characteristics, including reproductive status, did not affect general activity indicating that reproductive and non-reproductive individuals contribute equally to cooperative behaviors. We suggest that the activity patterns may be a behavioral adaptation to avoid extreme burrow temperatures and a mechanism to maintain a stable core body temperature. We highlight the advantages of RFID technology to study wild small mammal movements.

An Efficient and Fast Quantum State Estimator With Sparse Disturbance

A pure or nearly pure quantum state can be described as a low-rank density matrix, which is a positive semidefinite and unit-trace Hermitian. We consider the problem of recovering such a low-rank density matrix contaminated by sparse components, from a small set of linear measurements. This quantum state estimation task can be formulated as a robust principal component analysis (RPCA) problem subject to positive semidefinite and unit-trace Hermitian constraints. We propose an efficient and fast inexact alternating direction method of multipliers (I-ADMM), in which the subproblems are solved inexactly and hence have closed-form solutions. We prove global convergence of the proposed I-ADMM, and the theoretical result provides a guideline for parameter setting. Numerical experiments show that the proposed I-ADMM can recover state density matrices of 5 qubits on a laptop in 0.69 s, with 6 ×10^-4 accuracy (99.38% fidelity) using 30% compressive sensing measurements, which outperforms existing algorithms.