Tensor-Programmable Variational Quantum Algorithms for Flow Simulations

Y. Y. Liu, Zhen Chen, Chang Shu +5 more

Marco Cerezo, Andrew Arrasmith, Ryan Babbush +8 more

Marco Baioletti, Fabrizio Fagiolo, Angelo Oddi +1 more

In this paper we present a Variational Quantum Algorithm (VQA) for the Permutation Flow Shop Scheduling Problem (PFSSP). This scheduling problem is challenging to be solved with a quantum algorithm because the objective function is quite difficult to be expressed by means of a Hamiltonian, and a large number of qubits is generally required. For this reason, we propose to use a variational approach where the solution is represented with a limited number of qubits and the objective function can be computed without explicitly synthesizing a Hamiltonian operator. The algorithm has been implemented and tested on a simulator; the preliminary experimental results demonstrate that our solution could represent a viable alternative to solve the PFSSP on real quantum devices.

Mario Guillaume Cecile, Nis-Luca van Hülst, Tomohiro Hashizume +19 more

We present a comprehensive review of quantum approaches for solving partial differential equations (PDEs) arising in computational fluid dynamics (CFD). We examine fully quantum approaches, including quantum linear system algorithms (QLSAs), ranging from the Harrow--Hassidim--Lloyd (HHL) algorithm to quantum singular value transformation (QSVT), Hamiltonian simulation, and quantum lattice Boltzmann methods (QLBMs), while emphasizing hybrid quantum--classical approaches, including quantum physics-informed neural networks (QPINNs) and amplitude-encoded variational PDE solvers. We focus on hardware-agnostic algorithms compatible with present noisy processors and emerging fault-tolerant architectures. For each framework, we analyze the mathematical formulation, algorithmic structure, and principal limitations. We also examine tensor-network (TN) representations, since CFD fields, differential operators, and geometrical information can often be encoded efficiently in low-rank form. The TN formalism bridges CFD discretizations and quantum states, operators, and circuits, enabling compact representations to be translated into tensor-programmable variational quantum algorithms (TP-VQAs). We further review benchmark problems, including Poisson, reaction, diffusion, and nonlinear model equations, and assess how well quantum algorithms capture key features of fluid dynamics. Our analysis highlights that potential quantum advantage is highly problem dependent and governed by condition number, representational complexity, state preparation, and measurement constraints. We outline capabilities, limitations, and challenges toward scalable quantum algorithms for CFD.

Lejia Gu

V. Yu. Yushankhaï, L. Siurakshina

The variational quantum algorithm (VQA) is designed to simulate an $$N$$ -particle wave function of an electronic system of molecules using programmable quantum computers. This brief review, presented at the International Workshop on Mathematical Problems of Quantum Information Technologies (May 2024, Dubna), traces the evolution of the algorithm from a version originating from the unitary coupled cluster method, UCCSD, in computational quantum chemistry to a new version, ADAPT-VQA, adapted for practical implementation on noisy intermediate-scale quantum (NISQ) computing devices.