Efficient MPS representations and quantum circuits from the Fourier modes of classical image data

Machine learning tasks are an exciting application for quantum computers, as it has been proven that they can learn certain problems more efficiently than classical ones. Applying quantum machine learning algorithms to classical data can have many important applications, as qubits allow for dealing with exponentially more data than classical bits. However, preparing the corresponding quantum states usually requires an exponential number of gates and therefore may ruin any potential quantum speedups. Here, we show that classical data with a sufficiently quickly decaying Fourier spectrum after being mapped to a quantum state can be well-approximated by states with a small Schmidt rank (i.e., matrix-product states) and we derive explicit error bounds. These approximated states can, in turn, be prepared on a quantum computer with a linear number of nearest-neighbor two-qubit gates. We confirm our results numerically on a set of $1024\times1024$-pixel images taken from the `Imagenette' and DIV2K datasets. Additionally, we consider different variational circuit ans\"atze and demonstrate numerically that one-dimensional sequential circuits achieve the same compression quality as more powerful ans\"atze.