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Variational Quantum Linear Solver
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abstract
Previously proposed quantum algorithms for solving linear systems of equations cannot be implemented in the near term due to the required circuit depth. Here, we propose a hybrid quantum-classical algorithm, called Variational Quantum Linear Solver (VQLS), for solving linear systems on near-term quantum computers. VQLS seeks to variationally prepare $|x\rangle$ such that $A|x\rangle\propto|b\rangle$. We derive an operationally meaningful termination condition for VQLS that allows one to guarantee that a desired solution precision $\epsilon$ is achieved. Specifically, we prove that $C \geq \epsilon^2 / \kappa^2$, where $C$ is the VQLS cost function and $\kappa$ is the condition number of $A$. We present efficient quantum circuits to estimate $C$, while providing evidence for the classical hardness of its estimation. Using Rigetti's quantum computer, we successfully implement VQLS up to a problem size of $1024\times1024$. Finally, we numerically solve non-trivial problems of size up to $2^{50}\times2^{50}$. For the specific examples that we consider, we heuristically find that the time complexity of VQLS scales efficiently in $\epsilon$, $\kappa$, and the system size $N$.
Forward citations
Cited by 6 Pith papers
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Analog photonic simulator for large-scale transport
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Measurement-Efficient Variational Quantum Linear Solver for Carleman-Linearized Nonlinear Dynamics
Hybrid VQLS pipeline with Carleman linearization recovers high-fidelity solutions to the weakly nonlinear Duffing equation on IBM and Xanadu hardware using symmetry-grouped measurements and optimized ansatzes.
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Variational Quantum Solutions to the Advection-Diffusion Equation for Applications in Fluid Dynamics
Hybrid variational quantum algorithm solves the advection-diffusion equation on small systems using current noisy IBM quantum hardware, with claimed logarithmic scaling in vector space dimension.
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