# Quantum Sequential Universal Hypothesis Testing

## TL;DR

This paper introduces the quantum sequential universal test (QSUT), a new adaptive measurement framework for quantum hypothesis testing that reduces the number of copies needed compared to traditional fixed-copy methods.

## Contribution

The paper proposes QSUT, a novel sequential and adaptive approach to quantum hypothesis testing for composite hypotheses, improving efficiency and error control.

## Key findings

- QSUT reduces copy complexity in quantum hypothesis testing.
- Empirical results show QSUT outperforms fixed-copy strategies.
- Two practical implementations of QSUT demonstrate versatility.

## Abstract

Quantum hypothesis testing (QHT) concerns the statistical inference of unknown quantum states. In the general setting of composite hypotheses, the goal of QHT is to determine whether an unknown quantum state belongs to one or another of two classes of states based on the measurement of a number of copies of the state. Prior art on QHT with composite hypotheses focused on a fixed-copy two-step protocol, with state estimation followed by an optimized joint measurement. However, this fixed-copy approach may be inefficient, using the same number of copies irrespective of the inherent difficulty of the testing task. To address these limitations, we introduce the quantum sequential universal test (QSUT), a novel framework for sequential QHT in the general case of composite hypotheses. QSUT builds on universal inference, and it alternates between adaptive local measurements aimed at exploring the hypothesis space and joint measurements optimized for maximal discrimination. QSUT is proven to rigorously control the type I error under minimal assumptions about the hypothesis structure. We present two practical instantiations of QSUT, one based on the Helstrom-Holevo test and one leveraging shallow variational quantum circuits. Empirical results across a range of composite QHT tasks demonstrate that QSUT consistently reduces copy complexity relative to state-of-the-art fixed-copy strategies.