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  • Open Access

Shot-Noise Reduction for Lattice Hamiltonians

Timo Eckstein1,2,*, Refik Mansuroglu1,3, Stefan Wolf1, Ludwig Nützel1, Stephan Tasler1, Martin Kliesch4, and Michael J. Hartmann1,2

  • *Contact author: Timo.Eckstein@fau.de
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PRX Quantum 7, 020303 – Published 3 April, 2026

DOI: https://doi.org/10.1103/xy36-drb3

Abstract

Efficiently estimating energy expectation values of quantum lattice systems on quantum computers is a crucial subroutine for various quantum algorithms, which can lead to significant overhead due to the high measurement shot numbers required. We introduce a measurement strategy tailored to quantum lattice systems and (noisy) energy eigenstates. It is based on a geometric partitioning of the Hamiltonian into local patches and performing the measurements in the eigenbases of those patches. The resulting energy estimator has a smaller variance than the ones of Pauli grouping schemes, which leads to a reduction of the total number of shots. We provide rigorous guarantees for this variance improvement for energy eigenstates, also in the presence of depolarizing noise. As one can choose the subsystem size, one can ensure that measurement circuits remain within implementable depths. In numerical experiments, we demonstrate the shot count reduction for various 2D lattice models, including the transverse field XY and Ising models, as well as the Fermi-Hubbard model. We find sampling improvements of several orders of magnitude already for plaquettes of two by two qubits, where the required readout circuits remain very moderate in depth.

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Popular Summary

In quantum computing, in contrast to its classical counterpart, the result of our computation is typically not a single deterministic bitstring but a probability distribution encoded into a quantum state. Hence, a central task after state preparation is to efficiently extract information from this probability distribution, which, on a real-world quantum device, corresponds to measuring the generated quantum state. Particularly relevant near-term measurement tasks are those for energy estimation of low-energy stats, as they are considered an enabler for material science research via quantum computing.

Here, we present a measurement scheme for estimating the energy of quantum lattice models from measurements of a prepared quantum state. We prove that it always requires fewer measurements for the same precision as “naive” sampling in mutually commuting groups. Our method is based on partitioning the lattice into local patches and then measuring those in the eigenbases of their local Hamiltonians via a transformation of local eigenstates into the measurement basis, typically reducing the required number of measurements by several orders of magnitude.

We expect our results to have a noticeable impact on near-term eigenstate preparation algorithms, as those typically need to estimate the state’s energy numerous times. Importantly, the patch size can be freely chosen to fit the available gate budget, starting from as little as n/2 additional 2-qubit gates, which are simply appended to the end of the existing quantum circuit.

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PRX Quantum 7, 020303– Published 3 April, 2026
  • Received 14 November 2024
  • Revised 15 July 2025
  • Accepted 14 January 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Open access publication funded by Max Planck Society.

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