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 the 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 a 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 practical quantum dataset consisting of $n+1$ elementary tensor product states that are sufficient for exact training. The main idea is to simplify the structure utilizing decoupling, which leads to an exponential improvement in the size of the datasets with pure states. Furthermore, we show that the size of the 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 three-qubit one-dimensional nearest-neighbor Heisenberg model, our circuit only uses $96$ elementary quantum gates, which is significantly less than 4080 gates in the circuit constructed by the Trotter-Suzuki product formula.

Figure 1

The main scheme of using QML to learn an unknown unitary transformation. For the target unitary transformation, the first step is to prepare the training dataset consisting of input and output quantum states. Next, we train a QML model with the data using quantum processing, machine learning algorithms, and optimization methods. After the training process, the built QML model is able to emulate the action of the target unitary transformation.

Figure 2

The parameterized quantum circuit, one layer of which has $n$ generic one-qubit rotation gates with three Euler angles $\mathit{\theta }=(\theta ,\varphi ,\lambda )$ and $n$ cnot gates. The cnot gates in the circuit are only between neighboring qubits on a 1D chain with periodic boundary condition.

Figure 3

Panel (a) displays the number of required gates to simulate Hamiltonian of different quantum system sizes by our method and Trotter-Suzuki method. Note that each generic $U_3$ gates in the PQC is counted as three single-qubit rotation gates. Panel (b) shows the quantum risk in training with the dataset ${\mathcal{D}}_1$. The blue, orange, and green lines correspond to the results on the quantum system with size $n=3,4$, and $5$, respectively. Panel (c) shows the quantum risk in training with randomly generated dataset ${\mathcal{D}}_2$. The blue, orange, and green lines are averaged over ten independent training instances for the three system sizes. The corresponding shaded region is the min-max threshold in different runs.