Variational Quantum Linear Solver
By Carlos Bravo-Prieto, Ryan Larose, M. Cerezo, Yiğit Subaşı, Lukasz Cincio, Patrick J. Coles. Quantum, 2019
377 citations recorded by the source corpus.
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⟩ such that A|x⟩∝|b⟩. We derive an operationally meaningful termination condition for VQLS that allows one to guarantee that a desired solution precision ϵ is achieved. Specifically, we prove that C⩾ϵ2/κ2, where C is the VQLS cost function and κ 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 1024×1024. Finally, we numerically solve non-trivial problems of size up to 250×250. For the specific examples that we consider, we heuristically find that the time complexity of VQLS scales efficiently in ϵ, κ, and the system size N.
Subfields: Artificial Intelligence
View Paper: https://www.semanticscholar.org/paper/215bc3154ad2150e1814f9fcf9ac8d05d5cf2db9
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In Artificial Intelligence
- Patrick J. Coles
- M. Cerezo
- Lukasz Cincio
- Suguru Endo
- Piotr Czarnik
- Andrew Arrasmith
- Jarrod R. McClean
- Simon C. Benjamin
Citations in the corpus
- Cost function dependent barren plateaus in shallow parametrized quantum circuits (2021, background): “Let us now assume that θν is a parameter inside a given block W, we obtain from (6), (27), and (28)”
- Diagnosing Barren Plateaus with Tools from Quantum Optimal Control (2021, methodology): “[7] Carlos Bravo-Prieto, Ryan LaRose, M.”
- Variational quantum algorithms (2020, background): “This problem can be mitigated by considering a local Hamiltonian with the same ground state [118] or by using a hybrid ansatz strategy [120] where |ψ(θ)〉 = ∑ i αi|ψi(θ1)〉 with αi being variational parameters.”
- Connecting ansatz expressibility to gradient magnitudes and barren plateaus (2021, background): “[12] Carlos Bravo-Prieto, Ryan LaRose, Marco Cerezo, Yigit Subasi, Lukasz Cincio, and Patrick Coles, “Variational quantum linear solver,” arXiv preprint arXiv:1909.”