Efficient Agnostic Tomography Algorithm for Stabilizer Product States

Core Concepts

We present an efficient agnostic tomography algorithm that can learn a stabilizer product state approximation to an unknown quantum state, even if the unknown state is far from any stabilizer product state.

Abstract

The key insights and highlights of the content are:

The authors define a quantum learning task called agnostic tomography, where the goal is to output a succinct description of a state that approximates an unknown state ρ at least as well as any state in a given class C.

Agnostic tomography is more challenging than ordinary quantum tomography, as the learning algorithm must be robust to perturbations of ρ and cannot exploit the structure of states in C.

The authors present an efficient agnostic tomography algorithm for the class of n-qubit stabilizer product states. Assuming ρ has fidelity at least τ with a stabilizer product state, the algorithm runs in time nO(1+log(1/τ))/ε^2, which is quasipolynomial in all parameters and polynomial if τ is a constant.

The key ideas behind the algorithm are: (i) leveraging the structure of stabilizer product states to identify their stabilizer group from fewer Bell difference samples compared to general stabilizer states, and (ii) an entropy counting argument to show that a small number of samples suffices to learn the stabilizer group up to a few unassigned qubits.

The authors discuss open problems, such as improving the runtime further and extending the techniques to more general classes of quantum states.

Stats

None.

Quotes

None.

Key Insights Distilled From

Agnostic Tomography of Stabilizer Product States

by Sabee Grewal... at arxiv.org 04-08-2024

https://arxiv.org/pdf/2404.03813.pdf

Deeper Inquiries

How can the runtime of the agnostic tomography algorithm be further improved, perhaps by exploiting additional structure of stabilizer product states?

To further improve the runtime of the agnostic tomography algorithm for stabilizer product states, one approach could be to optimize the choice of parameters such as the success probability p and the failure probability δ. By carefully selecting these parameters, the algorithm can be fine-tuned to achieve a better balance between accuracy and efficiency. Additionally, exploring different strategies for sampling and processing the Bell difference samples could lead to more efficient computations.
Exploiting the specific structure of stabilizer product states can also help improve the runtime. For instance, leveraging the fact that stabilizer product states are tensor products of single-qubit states could lead to more optimized algorithms. By designing algorithms that take advantage of the inherent properties of stabilizer product states, such as their commutation relations and local structure, the runtime can be further reduced.
Furthermore, implementing more sophisticated techniques for clique detection in the graph construction step could potentially speed up the algorithm. By efficiently identifying cliques in the graph, the algorithm can focus on relevant subsets of Bell difference samples, reducing unnecessary computations and improving overall runtime.

Can the techniques developed in this work be extended to design efficient agnostic tomography algorithms for more general classes of quantum states, such as general stabilizer states or general product states?

The techniques developed in this work for agnostic tomography of stabilizer product states can potentially be extended to design efficient algorithms for more general classes of quantum states, such as general stabilizer states or general product states.
For general stabilizer states, the key challenge lies in dealing with a larger and more diverse set of Pauli operators and their relationships. By adapting the algorithm to handle a wider range of Pauli operators and their interactions, it may be possible to design efficient agnostic tomography algorithms for general stabilizer states.
Similarly, for general product states, the algorithm can be modified to accommodate the tensor product structure of these states. By incorporating techniques to handle the entanglement and correlations present in general product states, the algorithm can be extended to efficiently learn and approximate these states through agnostic tomography.
Overall, with careful consideration of the specific characteristics and properties of general stabilizer states and general product states, the techniques developed in this work can serve as a foundation for designing efficient agnostic tomography algorithms for these more general classes of quantum states.

Is there a meaningful notion of agnostic tomography for quantum channels and unitary transformations, and could the ideas in this paper be adapted to that setting?

Agnostic tomography for quantum channels and unitary transformations can indeed be a meaningful concept in quantum information processing. The goal would be to approximate an unknown quantum channel or unitary transformation using a simpler description that captures its essential properties up to a certain error threshold.
The ideas presented in the paper, such as leveraging Bell difference sampling and exploiting the structure of stabilizer product states, can be adapted to the setting of quantum channels and unitary transformations. By designing algorithms that sample and analyze the effects of quantum channels or unitary transformations on quantum states, it is possible to learn and approximate these operations agnostically.
Adapting the techniques from this paper to quantum channels and unitary transformations may involve modifying the sampling procedures to account for the different nature of these operations compared to quantum states. Additionally, the algorithm may need to consider the linearity and superposition properties of quantum channels and unitary transformations in order to effectively learn and approximate them through agnostic tomography.
Overall, by extending the concepts and methodologies from this paper to the realm of quantum channels and unitary transformations, it is possible to develop meaningful and practical agnostic tomography techniques for these important quantum operations.