Quantum advantage in variational Bayes inference

• Published in: Proceedings of the National Academy of Sciences - PNAS Vol. 120; no. 31; p. e2212660120
• Main Authors: Miyahara, Hideyuki, Roychowdhury, Vwani
• Format: Journal Article
• Language: English
• Published: United States National Academy of Sciences 01.08.2023
• Subjects: Algorithms; Applied Physical Sciences; Broken symmetry; Free energy; Ground state; Hamiltonian functions; Inference; Iterative methods; Low temperature; Mathematical models; Minima; Optimization; Physical Sciences; Physics; Quantum mechanics; Quantum theory; Qubits (quantum computing); Statistical analysis; Statistical models; Variational methods; deterministic annealing; quantum annealing; quantum machine learning; variational Bayes inference
• ISSN: 0027-8424; 1091-6490; 1091-6490
• DOI: 10.1073/pnas.2212660120

SignificanceQuantum machine learning (QML) is an emerging research field that deals with quantum algorithms for data analysis. It is hoped that QML will yield practical demonstrations of quantum advantage by exploiting the emerging noisy intermediate-scale quantum (NISQ) devices, which cannot yet implement large-scale quantum algorithms. Most of the proposed QML frameworks, such as quantum principal component analysis, quantum circuits, and quantum recommendation systems, provide potential quantum speedups of corresponding classical algorithms. These algorithms, thus, do not improve the quality of the solutions, and QML algorithms that outperform classical ML schemes are rare or nonexistent. This paper shows an example of how quantum mechanics can lead to better solutions for machine learning (ML) problems without incurring increased time complexity overheads. Variational Bayes (VB) inference algorithm is used widely to estimate both the parameters and the unobserved hidden variables in generative statistical models. The algorithm—inspired by variational methods used in computational physics—is iterative and can get easily stuck in local minima, even when classical techniques, such as deterministic annealing (DA), are used. We study a VB inference algorithm based on a nontraditional quantum annealing approach—referred to as quantum annealing variational Bayes (QAVB) inference—and show that there is indeed a quantum advantage to QAVB over its classical counterparts. In particular, we show that such better performance is rooted in key quantum mechanics concepts: i) The ground state of the Hamiltonian of a quantum system—defined from the given data—corresponds to an optimal solution for the minimization problem of the variational free energy at very low temperatures; ii) such a ground state can be achieved by a technique paralleling the quantum annealing process; and iii) starting from this ground state, the optimal solution to the VB problem can be achieved by increasing the heat bath temperature to unity, and thereby avoiding local minima introduced by spontaneous symmetry breaking observed in classical physics based VB algorithms. We also show that the update equations of QAVB can be potentially implemented using ⌈logK⌉ qubits and (K) operations per step, where K is the number of values hidden categorical variables can take. Thus, QAVB can match the time complexity of existing VB algorithms, while delivering higher performance.