Limitations of measure-first protocols in quantum machine learning

In recent times, there have been major developments in two distinct yet connected domains of quantum information. On the one hand, substantial progress has been made in so-called randomized measurement protocols. Here, a number of properties of unknown quantum states can be deduced from surprisingly few measurement outcomes, using schemes such as classical shadows. On the other hand, significant progress has been made in quantum machine learning. For example, exponential advantages have been proven when the data consists of quantum states and quantum algorithms can coherently measure multiple copies of input states. In this work, we aim to understand the implications and limitations of combining randomized measurement protocols with quantum machine learning, although the implications are broader. Specifically, we investigate quantum machine learning algorithms that, when dealing with quantum data, can either process it entirely using quantum methods or measure the input data through a fixed measurement scheme and utilize the resulting classical information. We prove limitations for the general class of quantum machine learning algorithms that use fixed measurement schemes on the input quantum states.Our results have several implications. From the perspective of randomized measurement procedures, we show limitations of measure-first protocols in the average case, improving on the state-of-the-art which only focuses on worst-case scenarios. Additionally, previous lower bounds were only known for physically unrealizable states. We improve upon this by employing quantum pseudorandom functions to prove that a learning separation also exists when dealing with physically realizable states, which may be encountered in experiments. From a machine learning perspective, our results are crucial for defining a physically meaningful task that shows fully quantum machine learning processing is not only more efficient but also necessary for solving certain problems. The tasks at hand are also realistic, as the algorithms and proven separations hold when working with efficiently preparable states and remain robust in the presence of measurement and preparation errors.

In quantum machine learning, a key question is how to handle input data that come in the form of quantum states. There are two main approaches: one is to apply a fixed measurement scheme to extract classical information from the quantum states, which is then passed to a learning algorithm. The other is to process the input data fully within a quantum computer, allowing the learning algorithm itself to choose and adapt the best measurements during training. We set out to investigate whether these two strategies are equally powerful, or if one has a clear advantage.This paper presents a learning problem that shows a learning separation: when quantum data is processed coherently, the task can be solved using only a polynomial number of samples. In contrast, using fixed measurements followed by classical or quantum processing requires an exponential number of data points. This result is particularly striking given recent advances showing that many properties of quantum states can be estimated efficiently using techniques like classical shadows.Importantly, our task involves quantum states that are realistically preparable in experiments, and the advantage remains even when noise in the state preparation is present. These findings highlight that, in quantum machine learning, quantum computers are essential not just for acquiring data, but also for processing it effectively.