Analysis of quantum Krylov algorithms with errors
IBM Quantum, IBM Research Cambridge, Cambridge, MA 02142, USA
| Published: | 2024-08-29, volume 8, page 1457 |
| Eprint: | arXiv:2401.01246v7 |
| Doi: | https://doi.org/10.22331/q-2024-08-29-1457 |
| Citation: | Quantum 8, 1457 (2024). |
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Abstract
This work provides a nonasymptotic error analysis of quantum Krylov algorithms based on real-time evolutions, subject to generic errors in the outputs of the quantum circuits. We prove upper and lower bounds on the resulting ground state energy estimates, and the error associated to the upper bound is linear in the input error rates. This resolves a misalignment between known numerics, which exhibit approximately linear error scaling, and prior theoretical analysis, which only provably obtained scaling with the error rate to the power $\frac{2}{3}$. Our main technique is to express generic errors in terms of an effective target Hamiltonian studied in an effective Krylov space. These results provide a theoretical framework for understanding the main features of quantum Krylov errors.
Featured image: Converged energy errors from the quantum Krylov algorithm versus noise rate, for a classical simulation of a Heisenberg model on a 3×3 lattice. $\sigma$ is the standard deviation of Gaussian noise added to the Krylov matrix elements (i.e., the simulated outputs of the quantum computer). Each datapoint is a median of 10000 runs. We separately evaluate these for the signed errors that are positive and negative. The dashed line is the best monomial fit to the positive error data. The solid curve shows the analytic bound proven in this work: note that it captures the correct scaling with the noise rate, but is far from tight.
Popular summary
For quantum computers, there are two primary challenges in this task: finding sufficiently good initial guess states, and obtaining sufficiently accurate results in the presence of noise in the quantum circuits. The former is known (up to complexity-theoretic assumptions) to be unavoidable, but for some specific instances of problems it is hoped to be tractable. The latter is due to the fact that, to date, quantum computers have not reached the threshold for quantum error correction, which will permit arbitrarily good suppression of errors.
The quantum Krylov algorithm is appealing in the near-term because it possesses provable convergence to the ground state energy, up to some error depending on the noise rate in the quantum computer as well as the problem parameters. The dependence of that converged energy error on the noise rate is our primary object of study. Prior to this work, the best provable dependence of the energy error on the noise rate $\eta$ was $O(\eta^{2/3})$, but numerical studies instead have shown the preferable dependence $O(\eta)$, which is optimal.
In this work, we solve half of the problem of improving the theoretical analysis to match the numerical performance: we achieve an upper bound of $O(\eta)$ on the $signed$ energy error. Unfortunately, a matching lower bound proved elusive using the techniques introduced in this work, but we provide a numerical example illustrating that the negative energy errors can have nearly identical performance to the positive energy errors that we prove to behave as $O(\eta)$, with an appropriate choice of the regularization threshold in the method. The main proof technique is based on expressing an arbitrary error in the quantum algorithm as an error in the Krylov space together with an error in the Hamiltonian being studied.
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