Here we show how universal quantum computers based on the quantum circuit model can handle mathematical analysis calculations for functions with continuous domains, without any digitalization, and with remarkably few qubits. The basic building block of our approach is a variational quantum circuit where each qubit encodes up to three continuous variables (two angles and one radious in the Bloch sphere). By combining this encoding with quantum state tomography, a variational quantum circuit of $n$ qubits can optimize functions of up to $3n$ continuous variables in an analog way. We then explain how this quantum algorithm for continuous optimization is at the basis of a whole toolbox for mathematical analysis on quantum computers. For instance, we show how to use it to compute arbitrary series expansions such as, e.g., Fourier (harmonic) decompositions. In turn, Fourier analysis allows us to implement essentially any task related to function calculus, including the evaluation of multidimensional definite integrals, solving (systems of) differential equations, and more. To prove the validity of our approach, we provide benchmarking calculations for many of these use-cases implemented on a quantum computer simulator. The advantages with respect to classical algorithms for mathematical analysis, as well as perspectives and possible extensions, are also discussed.
Entanglement Detection with Variational Quantum Interference: Theory and Experiment
Hefei National Research Center for Physical Sciences at the Microscale and School of Physical Sciences, University of Science and Technology of China（中国科学技术大学物理科学学院）；Shanghai Research Center for Quantum Science and CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China（中国科学技术大学上海量子科学研究中心及中国科学院量子信息重点实验室）；Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University（清华大学交叉信息研究院量子信息中心）；Hefei National Laboratory, University of Science and Technology of China（中国科学技术大学合肥国家实验室）
A hybrid variational quantum circuit approach for stabilizer states classifiers
Laboratoire Interdisciplinaire Carnot de Bourgogne, UMR 6303 CNRS, University of Technology of Belfort-Montbéliard（贝尔福-蒙贝利亚尔技术大学）；Mathematics and Statistics Department, Auburn University（奥本大学）
Efficient Operator Selection and Warm-Start Strategy for Excitations in Variational Quantum Eigensolvers
ICAMS, Ruhr University Bochum（鲁尔大学波鸿校区计算物理与材料科学研究所）；German Aerospace Center (DLR), Institute for Frontier Materials on Earth and in Space（德国航空航天中心前沿地球与空间材料研究所）
