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arxiv: 2410.21251 · v3 · pith:RY76BOR3new · submitted 2024-10-28 · 🪐 quant-ph

Shot-noise reduction for lattice Hamiltonians

Timo Eckstein , Refik Mansuroglu , Stefan Wolf , Ludwig N\"utzel , Stephan Tasler , Martin Kliesch , Michael J. Hartmann This is my paper

Pith reviewed 2026-05-25 08:58 UTC · model grok-4.3

classification 🪐 quant-ph
keywords shot-noise reductionlattice Hamiltoniansenergy estimationvariance reductionPauli groupinggeometric partitioningdepolarizing noisequantum measurements
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The pith

A geometric partitioning of lattice Hamiltonians into local patches reduces the variance of energy estimators compared with Pauli grouping and thereby cuts the number of required shots.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a measurement strategy for estimating energies of quantum lattice systems that divides the Hamiltonian into local patches and performs measurements in the eigenbases of those patches. This targets (noisy) energy eigenstates and is meant to produce lower variance than conventional Pauli grouping schemes. The reduction in variance directly lowers the total shots needed while allowing patch size to be chosen so that readout circuits stay shallow. Rigorous bounds on the improvement are given for eigenstates under depolarizing noise, and numerical tests on 2D models confirm large savings even with small patches.

Core claim

By partitioning the Hamiltonian into local patches and measuring in the eigenbases of those patches, the resulting energy estimator exhibits smaller variance than Pauli grouping schemes for energy eigenstates, providing rigorous guarantees even in the presence of depolarizing noise, which reduces the total number of measurement shots required while allowing control over circuit depth through choice of patch size.

What carries the argument

Geometric partitioning of the Hamiltonian into local patches with measurements performed in the eigenbases of those patches.

If this is right

  • Fewer total shots are needed to reach a target precision in energy estimation.
  • The variance reduction holds with rigorous guarantees for energy eigenstates even under depolarizing noise.
  • Patch size can be tuned to keep measurement circuit depth practical.
  • Numerical experiments show shot-count reductions of several orders of magnitude on 2D XY, Ising, and Fermi-Hubbard models.
  • The approach applies to any quantum algorithm subroutine that requires energy expectation values of lattice Hamiltonians.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same partitioning idea could be tested on approximate eigenstates prepared by variational methods if inter-patch correlations remain moderate.
  • Hierarchical or overlapping patch schemes might yield further variance gains on larger lattices.
  • Similar local-eigenbasis measurements could be adapted to estimate other local observables beyond the energy.
  • The method may combine with existing grouping or importance-sampling techniques for additional shot savings.

Load-bearing premise

The prepared states are eigenstates of the full Hamiltonian so that cross-patch correlations do not increase the variance beyond the derived bounds.

What would settle it

A direct numerical comparison of estimator variances on a small 2D lattice for an exact energy eigenstate, where the new method must show strictly lower variance than a Pauli grouping scheme.

Figures

Figures reproduced from arXiv: 2410.21251 by Ludwig N\"utzel, Martin Kliesch, Michael J. Hartmann, Refik Mansuroglu, Stefan Wolf, Stephan Tasler, Timo Eckstein.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
read the original 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 eigentstates, 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript introduces a measurement strategy for estimating expectation values of lattice Hamiltonians on quantum computers. It partitions the Hamiltonian geometrically into local patches, performs measurements in the eigenbases of those patches, and claims that the resulting energy estimator has provably smaller variance than standard Pauli-grouping estimators when the prepared state is a (noisy) energy eigenstate of the full Hamiltonian. This variance reduction is asserted to translate into fewer total shots; rigorous bounds are provided that remain valid under depolarizing noise, and the patch size can be chosen to keep readout circuits shallow. Numerical experiments on 2D transverse-field XY, Ising, and Fermi-Hubbard models report shot-count reductions of several orders of magnitude already for 2×2 plaquettes.

Significance. If the variance bounds hold, the approach offers a concrete route to lowering the dominant measurement overhead in quantum lattice simulations whenever eigenstates (or noisy eigenstates) are prepared. The explicit provision of rigorous guarantees that survive depolarizing noise, together with the tunable patch size that keeps circuit depth practical, constitutes a clear technical strength. The numerical demonstrations across multiple models supply reproducible evidence of the claimed improvement and support the practical relevance of the method within its stated scope.

minor comments (3)
  1. [§2] §2 (or wherever the patch Hamiltonian is defined): the notation for the local patch operators and their eigenbases would benefit from an explicit equation immediately after the geometric-partitioning paragraph to avoid ambiguity when the reader reaches the variance derivation.
  2. [Numerical experiments] Figure 4 and associated table: the reported shot-reduction factors lack error bars or a statement of the number of independent runs; adding this information would make the numerical claims easier to assess.
  3. [Abstract] The abstract states that the method works 'also in the presence of depolarizing noise,' but the precise noise model (local vs. global depolarizing, strength relative to patch size) is only clarified later; a one-sentence reminder in the abstract would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the accurate summary of our geometric partitioning approach, and the recommendation for minor revision. The referee's comments highlight the strengths of the variance bounds under depolarizing noise and the numerical results across models.

Circularity Check

0 steps flagged

No significant circularity; variance bounds derived from eigenstate properties

full rationale

The paper introduces a geometric partitioning strategy and derives variance reduction guarantees directly from the mathematical properties of energy eigenstates (including under depolarizing noise) and the structure of local patch measurements. No steps reduce by construction to fitted parameters, self-citations, or renamed inputs; the central claims rest on explicit bounds that hold for the stated class of states and Hamiltonians without circular redefinition. The derivation is self-contained against external benchmarks such as standard Pauli grouping variances.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, ad-hoc axioms, or invented entities are identifiable from the provided text.

axioms (1)
  • standard math Standard quantum measurement theory and variance formulas for expectation values
    Implicit background for any shot-noise analysis.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Preparation Circuits for Matrix Product States by Classical Variational Disentanglement

    quant-ph 2025-04 unverdicted novelty 7.0

    A layer-by-layer classical variational disentanglement algorithm compiles preparation circuits for matrix product states by minimizing bipartite entanglement to reduce bond dimensions.

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    ”Disordered phase” h>>J We normalize the Hamiltonian to obtain (this does not change the variance ratio): H =− ∑ i Xi− (J h ) ∑ ⟨i,j⟩ ZiZj =− ∑ i Xi−λ ∑ ⟨i,j⟩ ZiZj =H0 +λV (D1) The ground state|E0⟩is given up to second order in λby: |E(0) 0 ⟩=|+⟩⊗n (D2) |E(1) 0 ⟩= ∑ |b⟩̸=|+⟩⊗n ⣨ b ⏐⏐⏐ ∑ ⟨i,j⟩ZiZj ⏐⏐⏐ +⊗n ⟩ −n−(n−2·2) |b⟩= 1 4 ∑ ⟨i,j⟩ ZiZj|+⟩⊗n (D3) |E(2) ...

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    Further note that we defined Hr =T−1HlT1

    ”Ordered phase” h<<J We normalize the Hamiltonian to obtain (this does not change the variance ratio): H =− ∑ ⟨i,j⟩ ZiZj− (h J )∑ i Xi =− ∑ ⟨i,j⟩ ZiZj−λ ∑ i Xi =H0 +λV (D32) Then we can calculate the perturbed state order by order: Var(H2) Var(1 2(H−Hl +Hr)) = 4 Var(H1) Var(Hl−Hr) (D33) We now want to calculate |E(0) 0 ⟩=|0⟩⊗n (D34) |E(1) 0 ⟩= ∑ |b⟩̸=|0⟩⊗...

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