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The limited capabilities of current quantum hardware significantly constrainthe scale of experimental demonstrations of most quantum algorithmicprimitives. This makes it challenging to perform benchmarking of the currenthardware using useful quantum algorithms, i.e., application-orientedbenchmarking. In particular, the Harrow-Hassidim-Lloyd (HHL) algorithm is acritical quantum linear algebra primitive, but the majority of the componentsof HHL are far out of the reach of noisy intermediate-scale quantum devices,which has led to the proposal of hybrid classical-quantum variants. The goal ofthis work is to further bridge the gap between proposed near-term friendlyimplementations of HHL and the kinds of quantum circuits that can be executedon noisy hardware. Our proposal adds to the existing literature of hybridquantum algorithms for linear algebra that are more compatible with the currentscale of quantum devices. Specifically, we propose two modifications to theHybrid HHL algorithm proposed by Lee etal. leading to our algorithm HybridHHL++: (1) propose a novel algorithm for determining a scaling factor for thelinear system matrix that maximizes the utility of the amount of ancillaryqubits allocated to the phase estimation component of HHL, and (2) introduce aheuristic for compressing the HHL circuit. We demonstrate the efficacy of ourwork by running our modified Hybrid HHL on Quantinuum System Model H-seriestrapped-ion quantum computers to solve different problem instances ofsmall-scale portfolio optimization problems, leading to the largestexperimental demonstrations of HHL for an application to date.

Solving Linear Systems on Quantum Hardware with Hybrid HHL++