Efficient quantum linear solver algorithm with detailed running costs

David Jennings, Matteo Lostaglio, Sam Pallister, Andrew T Sornborger, Yiğit Subaşı

As we progress towards physical implementation of quantum algorithms it is vital to determine the explicit resource costs needed to run them. Solving linear systems of equations is a fundamental problem with a wide variety of applications across many fields of science, and there is increasing effort to develop quantum linear solver algorithms. Here we introduce a quantum linear solver algorithm combining ideas from adiabatic quantum computing with filtering techniques based on quantum signal processing. We give a closed formula for the non-asymptotic query complexity $Q$ -- the exact number of calls to a block-encoding of the linear system matrix -- as a function of condition number $\kappa$, error tolerance $\epsilon$ and block-encoding scaling factor $\alpha$. Our protocol reduces the cost of quantum linear solvers over state-of-the-art close to an order of magnitude for early implementations. The asymptotic scaling is $O(\kappa \log(\kappa/\epsilon))$, slightly looser than the $O(\kappa \log(1/\epsilon))$ scaling of the asymptotically optimal algorithm of Costa et al. However, our algorithm outperforms the latter for all condition numbers up to $\kappa \approx 10^{32}$, at which point $Q$ is comparably large, and both algorithms are anyway practically unfeasible. The present optimized analysis is both problem-agnostic and architecture-agnostic, and hence can be deployed in any quantum algorithm that uses linear solvers as a subroutine.

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