Optimal quantum dataset for learning a unitary transformation

Unitary transformations formulate the time evolution of quantum states. How to learn a unitary transformation efficiently is a fundamental problem in quantum machine learning. The most natural and leading strategy is to train a quantum machine learning model based on a quantum dataset. Although presence of more training data results in better models, using too much data reduces the efficiency of training. In this work, we solve the problem on the minimum size of sufficient quantum datasets for learning a unitary transformation exactly, which reveals the power and limitation of quantum data. First, we prove that the minimum size of dataset with pure states is $2^n$ for learning an $n$-qubit unitary transformation. To fully explore the capability of quantum data, we introduce a quantum dataset consisting of $n+1$ mixed states that are sufficient for exact training. The main idea is to simplify the structure utilizing decoupling, which leads to an exponential improvement on the size over the datasets with pure states. Furthermore, we show that the size of quantum dataset with mixed states can be reduced to a constant, which yields an optimal quantum dataset for learning a unitary. We showcase the applications of our results in oracle compiling and Hamiltonian simulation. Notably, to accurately simulate a 3-qubit one-dimensional nearest-neighbor Heisenberg model, our circuit only uses $48$ elementary quantum gates, which is significantly less than $4320$ gates in the circuit constructed by the Trotter-Suzuki product formula.