Efficient quantum amplitude encoding of polynomial functions

Javier Gonzalez-Conde1,2, Thomas W. Watts3, Pablo Rodriguez-Grasa1,2,4, and Mikel Sanz1,2,5,6

1Department of Physical Chemistry, University of the Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain
2EHU Quantum Center, University of the Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain
3School of Applied and Engineering Physics, Cornell University, Ithaca, NY 14853, USA
4TECNALIA, Basque Research and Technology Alliance (BRTA), 48160 Derio, Spain
5IKERBASQUE, Basque Foundation for Science, Plaza Euskadi 5, 48009, Bilbao, Spain
6Basque Center for Applied Mathematics (BCAM), Alameda de Mazarredo, 14, 48009 Bilbao, Spain

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Loading functions into quantum computers represents an essential step in several quantum algorithms, such as quantum partial differential equation solvers. Therefore, the inefficiency of this process leads to a major bottleneck for the application of these algorithms. Here, we present and compare two efficient methods for the amplitude encoding of real polynomial functions on $n$ qubits. This case holds special relevance, as any continuous function on a closed interval can be uniformly approximated with arbitrary precision by a polynomial function. The first approach relies on the matrix product state representation (MPS). We study and benchmark the approximations of the target state when the bond dimension is assumed to be small. The second algorithm combines two subroutines. Initially we encode the linear function into the quantum registers either via its MPS or with a shallow sequence of multi-controlled gates that loads the linear function's Hadamard-Walsh series, and we explore how truncating the Hadamard-Walsh series of the linear function affects the final fidelity. Applying the inverse discrete Hadamard-Walsh transform converts the state encoding the series coefficients into an amplitude encoding of the linear function. Thus, we use this construction as a building block to achieve an exact block encoding of the amplitudes corresponding to the linear function on $k_0$ qubits and apply the quantum singular value transformation that implements a polynomial transformation to the block encoding of the amplitudes. This unitary together with the Amplitude Amplification algorithm will enable us to prepare the quantum state that encodes the polynomial function on $k_0$ qubits. Finally we pad $n-k_0$ qubits to generate an approximated encoding of the polynomial on $n$ qubits, analyzing the error depending on $k_0$. In this regard, our methodology proposes a method to improve the state-of-the-art complexity by introducing controllable errors.

Quantum computers offer immense potential for tackling complex problems, yet efficiently loading an arbitrary function onto them remains a critical challenge. This is a bottleneck for many quantum algorithms, particularly in the fields of partial differential equations and linear systems solvers. To partially tackle this issue, we introduce two methods for efficiently encode discretized polynomials into the amplitudes of a quantum state within gate-based quantum computers. Our approach introduces controllable errors while enhancing the complexity of current quantum function loading algorithms, presenting promising advancements with respect to the current state of the art.

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Cited by

[1] Arthur G. Rattew and Patrick Rebentrost, "Non-Linear Transformations of Quantum Amplitudes: Exponential Improvement, Generalization, and Applications", arXiv:2309.09839, (2023).

[2] Javier Gonzalez-Conde, Ángel Rodríguez-Rozas, Enrique Solano, and Mikel Sanz, "Efficient Hamiltonian simulation for solving option price dynamics", Physical Review Research 5 4, 043220 (2023).

[3] Felix Tennie, Sylvain Laizet, Seth Lloyd, and Luca Magri, "Quantum Computing for nonlinear differential equations and turbulence", arXiv:2406.04826, (2024).

[4] A. Termanova, Ar. Melnikov, E. Mamenchikov, N. Belokonev, S. Dolgov, A. Berezutskii, R. Ellerbrock, C. Mansell, and M. Perelshtein, "Tensor Quantum Programming", arXiv:2403.13486, (2024).

[5] Paul Over, Sergio Bengoechea, Thomas Rung, Francesco Clerici, Leonardo Scandurra, Eugene de Villiers, and Dieter Jaksch, "Boundary Treatment for Variational Quantum Simulations of Partial Differential Equations on Quantum Computers", arXiv:2402.18619, (2024).

[6] Pablo Rodriguez-Grasa, Ruben Ibarrondo, Javier Gonzalez-Conde, Yue Ban, Patrick Rebentrost, and Mikel Sanz, "Quantum approximated cloning-assisted density matrix exponentiation", arXiv:2311.11751, (2023).

[7] Xi-Ning Zhuang, Zhao-Yun Chen, Cheng Xue, Xiao-Fan Xu, Chao Wang, Huan-Yu Liu, Tai-Ping Sun, Yun-Jie Wang, Yu-Chun Wu, and Guo-Ping Guo, "Statistics-Informed Parameterized Quantum Circuit via Maximum Entropy Principle for Data Science and Finance", arXiv:2406.01335, (2024).

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