Fast digital methods for adiabatic state preparation

Kianna Wan, Isaac Kim

Published 2020-04-08Version 1

We present quantum algorithms for adiabatic state preparation on a gate-based quantum computer, with complexity polylogarithmic in the inverse error. This constitutes an exponential improvement over existing methods, which achieve subpolynomial error dependence. Our first algorithm digitally simulates the adiabatic evolution between two self-adjoint operators $H_0$ and $H_1$, exponentially suppressing the diabatic error by harnessing the theoretical concept of quasi-adiabatic continuation as an algorithmic tool. Given an upper bound $\alpha$ on $\|H_0\|$ and $\|H_1\|$ along with the promise that the $k$th eigenstate $|\psi_k(s)\rangle$ of $H(s) \equiv (1-s)H_0 + sH_1$ is separated from the rest of the spectrum by a gap of at least $\gamma > 0$ for all $s \in [0,1]$, this algorithm implements an operator $\widetilde{U}$ such that $\| |\psi_k(1)\rangle - \widetilde{U}|\psi_k(s)\rangle\| \leq \epsilon$ using $\mathcal{O}(({\alpha^2}/{\gamma^2})\mathrm{polylog}(\alpha/\gamma\epsilon))$ queries to block-encodings of $H_0$ and $H_1$. Our second algorithm is applicable only to ground states and requires multiple queries to an oracle that prepares $|\psi_0(0)\rangle$, but has slightly better scaling in all parameters. We also show that the costs of both algorithms can be further reduced under certain reasonable conditions, such as when $\|H_1 - H_0\|$ is small compared to $\alpha$, or when more information about the gap of $H(s)$ is available. For certain problems, the scaling can even be improved to linear in $\|H_1 - H_0\|/\gamma$ up to polylogarithmic factors.