The aim of this paper is to develop novel quantum algorithms for Gaussian process quadrature methods. Gaussian process quadratures are numerical integration methods where Gaussian processes are used as functional priors for the integrands to capture the uncertainty arising from the sparse function evaluations. Quantum computers have emerged as potential replacements for classical computers, offering exponential reductions in the computational complexity of machine learning tasks. In this paper, we combine Gaussian process quadratures and quantum computing by proposing a quantum low-rank Gaussian process quadrature method based on a Hilbert space approximation of the Gaussian process kernel and enhancing the quadrature using a quantum circuit. The method combines the quantum phase estimation algorithm with the quantum principal component analysis technique to extract information up to a desired rank. Then, Hadamard and SWAP tests are implemented to find the expected value and variance that determines the quadrature. We use numerical simulations of a quantum computer to demonstrate the effectiveness of the method. Furthermore, we provide a theoretical complexity analysis that shows a polynomial advantage over classical Gaussian process quadrature methods. The code is available at https://github.com/cagalvisf/Quantum_HSGPQ.The aim of this paper is to develop novel quantum algorithms for Gaussian process quadrature methods. Gaussian process quadratures are numerical integration methods where Gaussian processes are used as functional priors for the integrands to capture the uncertainty arising from the sparse function evaluations. Quantum computers have emerged as potential replacements for classical computers, offering exponential reductions in the computational complexity of machine learning tasks. In this paper, we combine Gaussian process quadratures and quantum computing by proposing a quantum low-rank Gaussian process quadrature method based on a Hilbert space approximation of the Gaussian process kernel and enhancing the quadrature using a quantum circuit. The method combines the quantum phase estimation algorithm with the quantum principal component analysis technique to extract information up to a desired rank. Then, Hadamard and SWAP tests are implemented to find the expected value and variance that determines the quadrature. We use numerical simulations of a quantum computer to demonstrate the effectiveness of the method. Furthermore, we provide a theoretical complexity analysis that shows a polynomial advantage over classical Gaussian process quadrature methods. The code is available at https://github.com/cagalvisf/Quantum_HSGPQ.

arXiv:2502.14467Feb 20, 20250 citations

Quantum stochastic processes

C. Barnett, R. F. Streater, Ivan F. Wilde

Lecture notes in mathematicsJan 1, 19847 citations

Quantum stochastic processes

E. B. Davies

Communications in Mathematical PhysicsDec 1, 1969237 citations

Characterizing Gaussian quantum processes with Gaussian resources

Logan W. Grove, Pratik J. Barge, Kevin Valson Jacob

Characterizing quantum processes is indispensable for the implementation of any task in quantum information processing. In this paper, we develop an efficient method to fully characterize arbitrary Gaussian processes in continuous-variable quantum systems. This is done by directly obtaining all elements of the symplectic matrix that describes the process. Only Gaussian resources such as coherent probes and quadrature measurements are needed for this task. The method is efficient, involving only $O({N}^{2})$ steps to characterize an $N$ -mode system. Further, the method is resilient to uniform loss. We simulate this procedure using the Python package Strawberry Fields. We observe that heterodyne measurements outperform homodyne measurements for reconstructing Gaussian processes.Characterizing quantum processes is indispensable for the implementation of any task in quantum information processing. In this paper, we develop an efficient method to fully characterize arbitrary Gaussian processes in continuous-variable quantum systems. This is done by directly obtaining all elements of the symplectic matrix that describes the process. Only Gaussian resources such as coherent probes and quadrature measurements are needed for this task. The method is efficient, involving only $O({N}^{2})$ steps to characterize an $N$-mode system. Further, the method is resilient to uniform loss. We simulate this procedure using the Python package Strawberry Fields. We observe that heterodyne measurements outperform homodyne measurements for reconstructing Gaussian processes.

arXiv:2503.13698Physical Review AAug 13, 20250 citations

Quantum Stochastic Processes

Anatolij Dvurečenskij, Sylvia Pulmannová

Springer eBooksJan 1, 19830 citations

O({N}^{2})

steps to characterize an

N

-mode system. Further, the method is resilient to uniform loss. We simulate this procedure using the Python package Strawberry Fields. We observe that heterodyne measurements outperform homodyne measurements for reconstructing Gaussian processes.