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arXiv:2210.03136

Variational Quantum Continuous Optimization: a Cornerstone of Quantum Mathematical Analysis

Bermejo, Pablo, Román Orús

Published Oct 6, 2022• 4 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.2210.03136

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Variational quantum algorithms (VQAs) are the leading algorithm for achieving quantum advantage using near-term quantum computers. VQAs use parameterized quantum circuits for inference, and the variational parameters in quantum circuits can be trained using a classical optimizer. The parameters are trained to guide how the quantum bits evolve and make the final measurements closely match the ground truth. However, this way of learning from raw data makes it difficult to capture the underlying dynamic information in the data, especially for time series data. To address this limitation, we proposed continuous variational quantum algorithms (CVQAs) for time series in this paper. CVQAs use quantum variational circuits to parameterize the dynamics of time series, thus they can learn the dynamic information behind the data. After the dynamics are trained, the prediction results will be obtained by a differential equation solver working on the dynamics. Since we aim to model the dynamics of data instead of the data itself, the quantum circuit in our approach will need fewer qubits and variational gates. To evaluate our proposed approach, we compare our model with baseline models on several weather time series. Experimental results prove that our approach has better or equivalent results but with fewer qubits and variational gates compared to baseline models.