We study the performance of classical and quantum machine learning (ML)
models in predicting outcomes of physical experiments. The experiments depend
on an input parameter $x$ and involve execution of a (possibly unknown) quantum
process $\mathcal{E}$. Our figure of merit is the number of runs of
$\mathcal{E}$ required to achieve a desired prediction performance. We consider
classical ML models that perform a measurement and record the classical outcome
after each run of $\mathcal{E}$, and quantum ML models that can access
$\mathcal{E}$ coherently to acquire quantum data; the classical or quantum data
is then used to predict outcomes of future experiments. We prove that for any
input distribution $\mathcal{D}(x)$, a classical ML model can provide accurate
predictions on average by accessing $\mathcal{E}$ a number of times comparable
to the optimal quantum ML model. In contrast, for achieving accurate prediction
on all inputs, we prove that exponential quantum advantage is possible. For
example, to predict expectations of all Pauli observables in an $n$-qubit
system $\rho$, classical ML models require $2^{\Omega(n)}$ copies of $\rho$,
but we present a quantum ML model using only $\mathcal{O}(n)$ copies. Our
results clarify where quantum advantage is possible and highlight the potential
for classical ML models to address challenging quantum problems in physics and
chemistry.

本研究比较了传统机器学习和量子机器学习在物理实验结果预测方面的表现，发现虽然对于任何输入分布，使用经典机器学习模型可以实现平均准确预测，但是利用量子机器学习模型，可以在所有输入分布上达到指数级别的量子优势，此外，我们提出了一种仅需要 O (n) 个副本来预测 n 量子位系统的所有 Pauli 可观察量期望值的量子机器学习模型，为解决物理和化学领域的难题提供了新思路。