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

2024-07-03

AI Summary

This work addresses the problem of estimating the ground-state energy and preparing low-energy eigenstates of arbitrary $k$-local Hamiltonians without geometric or locality assumptions—particularly relevant for long-range or fully connected systems where conventional lattice-based methods fail. The proposed method combines interaction truncation with controlled-error approximation, integrated with quantum phase estimation and randomized sampling, to efficiently probe low-energy subspaces without relying on spatial structure. Crucially, it breaks the quadratic speedup barrier of Grover search under no locality assumptions, revealing that the low-energy subspace of $k$-local Hamiltonians exhibits exponential-dimensional structure. The algorithm achieves energy estimation and corresponding state preparation with precision $varepsilon M$ in time complexity $2^{c n/2}$, where $c < 1$ and $n$ is the number of qubits. This result significantly broadens the scope of quantum Hamiltonian learning beyond geometric constraints.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Learning to Search
• Intelligent Robots: State Estimation

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements

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.

Problem

• Estimate ground state energies
• Prepare quantum states efficiently
• Handle long-range interactions

Innovation

• Quantum algorithms for energy estimation
• Handles long-range interactions efficiently
• Breaks Grover's square root bound