Solving approximate hidden subgroup problems: quantum heuristics to detect weak entanglement

TL;DR

This paper develops heuristics to detect weak entanglement in quantum states using quantum algorithms for hidden subgroup problems, broadening their applicability beyond cryptography.

Contribution

It introduces a rigorous link between the hidden cut algorithm's output and entanglement quality, enabling detection with fewer state copies.

Findings

Reduced state copies still reveal weak entanglement patterns

Established a connection between output distribution and entanglement measurement

Broadened the potential applications of hidden subgroup quantum algorithms

Abstract

How can we use a quantum computer to detect the entanglement structure of a quantum state? Bouland et al. (2024) recently provided an algorithm that, given multiple input copies of the state, finds the "hidden cuts"-partitions into fully unentangled qubit registers. Their solution is based on turning cuts into a symmetry which can be detected with a Shor-type quantum algorithm for hidden subgroup problems, the hidden cut algorithm. In this paper we derive heuristics that can find "approximate symmetries", or weakly entangled qubit registers, to unlock this powerful idea for a much broader range of problems. Our core contribution is a rigorous link between the output distribution of the hidden cut algorithm and the reward function that measures the quality of a cut. This implies that reducing the number of state copies in the original hidden cut algorithm leads to measurement samples from which patterns of weak entanglement can be extracted. We believe that these insights are an important step in making quantum algorithms for hidden subgroup problems useful for applications beyond cryptography.