[Paper Review] Guaranteed efficient energy estimation of quantum many-body Hamiltonians using ShadowGrouping

Alexander Gresch, Martin Kliesch|arXiv (Cornell University)|Jan 9, 2023

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper introduces ShadowGrouping, a novel method for efficient and provably accurate energy estimation of quantum many-body Hamiltonians using single-qubit measurements. By combining shadow estimation with optimized Pauli string grouping, it reduces measurement overhead and outperforms state-of-the-art methods in both theoretical guarantees and numerical benchmarks on molecular ground-state energies.

ABSTRACT

Estimation of the energy of quantum many-body systems is a paradigmatic task in various research fields. In particular, efficient energy estimation may be crucial in achieving a quantum advantage for a practically relevant problem. For instance, the measurement effort poses a critical bottleneck for variational quantum algorithms. We aim to find the optimal strategy with single-qubit measurements that yields the highest provable accuracy given a total measurement budget. As a central tool, we establish new tail bounds for empirical estimators of the energy. They are helpful for identifying measurement settings that improve the energy estimate the most. This task constitutes an NP-hard problem. However, we are able to circumvent this bottleneck and use the tail bounds to develop a practical, efficient estimation strategy, which we call ShadowGrouping. As the name indicates, it combines shadow estimation methods with grouping strategies for Pauli strings. In numerical experiments, we demonstrate that ShadowGrouping improves upon state-of-the-art methods in estimating the electronic ground-state energies of various small molecules, both in provable and practical accuracy benchmarks. Hence, this work provides a promising way, e.g., to tackle the measurement bottleneck associated with quantum many-body Hamiltonians.

Motivation & Objective

• Address the measurement bottleneck in variational quantum algorithms and quantum simulation, where estimating Hamiltonian expectation values is a critical performance limiter.
• Develop a strategy that minimizes estimation error under a fixed total measurement budget, focusing on single-qubit measurements.
• Provide rigorous tail bounds for empirical energy estimators to guide optimal measurement setting selection.
• Design a practical, efficient estimation protocol that achieves high accuracy while maintaining provable error bounds.
• Demonstrate superior performance over existing methods in estimating electronic ground-state energies of small molecules.

Proposed method

• Introduce new tail bounds for empirical energy estimators to quantify the accuracy of energy estimates under finite sampling.
• Use these tail bounds to guide the selection of optimal measurement settings (unitaries) that maximize estimation accuracy for a given measurement budget.
• Develop the ShadowGrouping framework, which combines randomized measurement protocols (shadow estimation) with grouping strategies for Pauli strings to reduce measurement overhead.
• Formulate the measurement setting optimization as a minimization of an upper bound on estimation error, leveraging the derived tail bounds.
• Implement a preprocessing step to generate a list of unitaries tailored to the Hamiltonian structure and measurement budget.
• Postprocess measurement outcomes from single-qubit measurements to compute a final energy estimator with a provable accuracy guarantee.

Experimental results

Research questions

• RQ1 Can we design a measurement strategy for quantum many-body Hamiltonians that guarantees high accuracy with minimal measurement resources?
• RQ2 How can we leverage tail bounds on empirical estimators to guide the selection of optimal measurement settings in a computationally efficient way?
• RQ3 To what extent does combining shadow estimation with Pauli string grouping improve energy estimation accuracy compared to existing methods?
• RQ4 Can the proposed method outperform random or heuristic measurement strategies in practical benchmarks on molecular Hamiltonians?
• RQ5 Is there a provable advantage in estimation accuracy when using structured measurement settings derived from tail bounds?

Key findings

• ShadowGrouping achieves superior performance in estimating electronic ground-state energies of small molecules compared to state-of-the-art methods, including random Pauli settings and existing grouping strategies.
• Numerical benchmarks show that ShadowGrouping reduces the root mean squared error (RMSE) by up to a factor of 4 compared to random Pauli settings, with improvements observed across all tested molecules.
• For the H₂ molecule in the STO-3G basis, ShadowGrouping achieved an RMSE of 27 ± 3 mHa with Pauli settings, outperforming the 50 ± 6 mHa of the single-shot estimator.
• In larger systems like H₂O and NH₃, ShadowGrouping reduced RMSE by over a factor of 3 compared to random Pauli settings, with errors of 320 ± 40 mHa and 430 ± 50 mHa respectively.
• The method provides provable error bounds that scale favorably with the measurement budget, with theoretical guarantees matching empirical performance up to logarithmic factors.
• The single-shot estimator derived from the tail bound did not perform competitively in practice, suggesting that recycling measurement outcomes via grouped estimation is more effective.