TITLE: Optimal quantum dataset for learning a unitary transformation 

ABSTRACT: 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 3-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.
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 3-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.
Optimal quantum dataset for learning a unitary transformation
http://arxiv.org/abs/2203.00546v3