Beating the natural Grover bound for low-energy estimation and state preparation

TL;DR

This paper introduces quantum algorithms capable of estimating ground state energies and preparing corresponding states for general many-body Hamiltonians, surpassing traditional Grover speedup limits by exploiting the structure of low energy subspaces.

Contribution

The authors develop the first quantum algorithms for low-energy estimation that outperform the standard Grover speedup, applicable to non-local Hamiltonians without geometric constraints.

Findings

Algorithms work for long-range and all-to-all interactions.

Runtime scales as 2^{cn/2} with c<1, beating Grover's bound.

Low energy subspace has exponential dimension, enabling efficient estimation.

Abstract

Estimating ground state energies of many-body Hamiltonians is a central task in many areas of quantum physics. In this work, we give quantum algorithms which, given any $k$-body Hamiltonian $H$, compute an estimate for the ground state energy and prepare a quantum state achieving said energy, respectively. Specifically, for any $\varepsilon>0$, our algorithms return, with high probability, an estimate of the ground state energy of $H$ within additive error $\varepsilon M$, or a quantum state with the corresponding energy. Here, $M$ is the total strength of all interaction terms, which in general is extensive in the system size. Our approach makes no assumptions about the geometry or spatial locality of interaction terms of the input Hamiltonian and thus handles even long-range or all-to-all interactions, such as in quantum chemistry, where lattice-based techniques break down. In this fully general setting, the runtime of our algorithms scales as $2^{cn/2}$ for $c<1$, yielding the first quantum algorithms for low-energy estimation breaking a standard square root Grover speedup for unstructured search. The core of our approach is remarkably simple, and relies on showing that an extensive fraction of the interactions can be neglected with a controlled error. What this ultimately implies is that even arbitrary $k$-local Hamiltonians have structure in their low energy space, in the form of an exponential-dimensional low energy subspace.