Shadow tomography of quantum states with prediction

Shadow tomography serves as a cheap alternative of state tomography in some situations that only certain linear observables of the target state ρ are cared about. This technique has reduced the dependence of the number of qubits on sample complexity from exponential to linear, but the order of the desired accuracy ε is −4.

To address this challenge, a research team led by Jialin ZHANG published their new research on 15 July 2025 in Frontiers of Computer Science co-published by Higher Education Press and Springer Nature.

The team proposes a new quantum algorithm that can leverage prior knowledge about the unknown state, i.e., a potentially inaccurate prediction ϱ. This algorithm is based on the FTRL framework for learning quantum states, but some adjustments have been made to the previous one, like adding a linear term to the regularizer to make the initial state almost close to ϱ. This linear term should be checked to avoid overflow in the gradient.

In the research, they analyze the number of turns in the FTRL and establish a new upper bound of sample complexity by scaling the conventional one with the trace distance between ρ and ϱ. When this distance is Θ(ε), the order of ε⁻¹ in the upper bound of the sample complexity can be reduced to 3, as demonstrated in their experiments. Their experiments also show the superiority of their algorithm in the shadow tomography task in some state preparation circuits.

Future directions can focus on finding a more effective way to update the hypothesis in the FTRL. Another is to study the possible effects of the noise, since it is still in the NISQ era.