Near-optimal ground state preparation

Abstract

Preparing the ground state of a given Hamiltonian and estimating its ground energy are important but computationally hard tasks. However, given some additional information, these problems can be solved efficiently on a quantum computer. We assume that an initial state with non-trivial overlap with the ground state can be efficiently prepared, and the spectral gap between the ground energy and the first excited energy is bounded from below. With these assumptions we design an algorithm that prepares the ground state when an upper bound of the ground energy is known, whose runtime has a logarithmic dependence on the inverse error. When such an upper bound is not known, we propose a hybrid quantum-classical algorithm to estimate the ground energy, where the dependence of the number of queries to the initial state on the desired precision is exponentially improved compared to the current state-of-the-art algorithm proposed in [Ge et al. 2019]. These two algorithms can then be combined to prepare a ground state without knowing an upper bound of the ground energy. We also prove that our algorithms reach the complexity lower bounds by applying it to the unstructured search problem and the quantum approximate counting problem.

Lecture Notes on Quantum Algorithms for Scientific Computation

“…Using amplitude amplification, the number of repetitions can be reduced to O(γ −1 ), and the total number of queries to to U A and U † A becomes O(∆ −1 p − 1 2 0 log(1/ )). This also matches the lower bound [LT20a].…”

“…This is because the algorithm first prepares the ground state and then estimates the ground state energy. If we are only interested in estimating E 0 to precision , the gap dependence is not necessary (see Section 4.1 as well as [LT20a]).…”

Quadratic speedup for spatial search by continuous-time quantum walk

“…Now we can make use of the Hubbard-Stratonovich transformation which states that, for any x ∈ R, [21], is ε-close to |v 0 with probability scaling as | ψ 0 |v 0 | 2 (See Appendix for details). The performance matches the best circuit model algorithms for this problem but is conceptually simpler as unlike in the discrete-time setting, does not require implementing linear combination of unitaries or quantum phase estimation [22][23][24]. We believe that our approach can also lead to potentially simpler analog quantum algorithms for preparing Gibbs states, as in [25].…”

Fast inversion, preconditioned quantum linear system solvers, fast Green's-function computation, and fast evaluation of matrix functions

“…Nonetheless, it is reasonable to make these assumptions for systems where an ansatz can be constructed that has polynomial overlap with the ground state, and where the spectral gap of the Hamiltonian can be bounded from below. These are the assumptions made in, e.g.. [53,54], and is believed to occur for a wide range of realistic systems in physics and chemistry.…”