Efficient quantum interpolation of natural data

Abstract

We present efficient methods to interpolate data with a quantum computer that complement uploading techniques and quantum post-processing. The quantum algorithms are supported by the efficient Quantum Fourier Transform (QFT) and classical signal and imaging processing techniques, and open the door of quantum advantage to relevant families of data. We showcase a QFT interpolation method, a Quantum Cosine Transform (QCT) interpolation geared towards natural data, and we improve upon them by utilizing a quantum circuit's capabilities of processing data in superposition. A novel circuit for the QCT is presented. We demonstrate the methods on probability distributions and quantum encoded images, and discuss the precision of the resulting interpolations.

Main: 6 pages, 4 figures. Appendix: 3 pages, 2 figures. Code available online

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• Quantum encoder for fixed Hamming-weight subspaces
• Quantum circuits for partial differential equations in Fourier space
• Approximate real-time evolution operator for potential with one ancillary qubit and application to first-quantized Hamiltonian simulation
• Quantum frequency resampling