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arxiv: 2604.07156 · v1 · submitted 2026-04-08 · 🪐 quant-ph

Overlapped groupings for quantum energy estimation: Maximal variance reduction and deterministic algorithms for reducing variance

Jeremiah Rowland , Rahul Sarkar , Nicolas PD Sawaya , Norm M. Tubman , Ryan LaRose This is my paper

Pith reviewed 2026-05-10 17:51 UTC · model grok-4.3

classification 🪐 quant-ph
keywords overlapped groupingvariance reductionquantum energy estimationmeasurement optimizationHamiltonian groupingquantum algorithmsnear-term quantum computing
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The pith

Overlapped grouping of Hamiltonian terms reduces variance in quantum energy estimates linearly with the number of terms.

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

The paper sets out to prove that relaxing the usual disjoint-group requirement in measurement strategies for quantum energy estimation allows operators to appear in multiple compatible groups, producing a maximal variance reduction that grows linearly with the total number of Hamiltonian terms. A sympathetic reader would care because lower variance directly translates to fewer shots needed for a target precision, which remains one of the dominant costs in near-term quantum algorithms for chemistry and materials. The authors introduce a deterministic repacking procedure that starts from existing disjoint groupings and iteratively converts them into overlapped ones while guaranteeing variance reduction under mild conditions. Large-scale numerical tests on Hamiltonians up to 44 qubits confirm that the relative improvement over state-of-the-art disjoint methods itself scales linearly with problem size.

Core claim

The central claim is that overlapped grouping for energy estimation achieves a maximal variance reduction linear in the number of Hamiltonian terms. The repacking algorithm converts disjoint groups into overlapped groups and is shown to reduce variance iteratively. Simulations on systems with up to 575,000 terms demonstrate that the variance reduction relative to disjoint grouping increases linearly with problem size.

What carries the argument

The repacking procedure, which redistributes coefficients of Hamiltonian terms across multiple compatible groups to minimize the total variance of the energy estimator.

If this is right

  • The number of measurements required for a fixed precision in energy estimation decreases as the Hamiltonian grows larger.
  • Existing disjoint grouping schemes can be systematically improved by a deterministic procedure without introducing randomness.
  • The advantage of overlapped grouping widens with problem size, making it more attractive precisely when the Hamiltonian contains many terms.
  • The method applies directly to any Hamiltonian that can be measured via Pauli strings or other compatible operator sets.

Where Pith is reading between the lines

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

  • The linear scaling may extend the practical reach of variational quantum eigensolvers to molecules or materials whose Hamiltonians contain millions of terms.
  • If the mild assumptions hold for typical molecular Hamiltonians, the approach could be combined with other measurement-reduction techniques such as classical shadows.
  • A natural next test would be to measure whether the same repacking logic improves variance in expectation values of observables other than the energy.

Load-bearing premise

The repacking procedure reduces variance iteratively only under mild assumptions on the initial groups and the structure of the Hamiltonian.

What would settle it

A concrete Hamiltonian for which repeated application of the repacking algorithm fails to decrease variance, or for which the observed reduction remains sub-linear even as the number of terms grows into the hundreds of thousands.

Figures

Figures reproduced from arXiv: 2604.07156 by Jeremiah Rowland, Nicolas PD Sawaya, Norm M. Tubman, Rahul Sarkar, Ryan LaRose.

Figure 1
Figure 1. Figure 1: FIG. 1. Illustration of the family of Hamiltonians which ad [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Variance reduction of overlapped groupings found through repacking relative to sorted insertion for molecular [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Variance (blue) vs. covariance (orange) contribu [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
read the original abstract

Grouping-based measurement strategies are widely used to reduce measurement complexity in near-term quantum algorithms. While these schemes have typically produced disjoint groups, recently this has been relaxed in what is known as overlapped grouping or coefficient splitting where operators may appear in more than one compatible group. In recent work, it has been numerically shown that this strategy can reduce the variance of energy estimates on small benchmark problems, motivating both the application and further analysis of the method. Here we prove that overlapped grouping for energy estimation can lead to a maximal variance reduction that is linear in the number of Hamiltonian terms. We introduce a new algorithm which we call repacking to transform existing groups into overlapped groups, and we show this repacking procedure iteratively reduces variance under mild assumptions. We also perform numerical simulations with Hamiltonians up to $44$ qubits and $575 \cdot 10^{3}$ terms, assessing overlapped grouping at scale on problems of practical importance. Our numerics show that the variance reduction relative to state-of-the-art (disjoint) grouping increases linearly with the problem size, suggesting that overlapped grouping methods can be a powerful strategy for quantum energy estimation at the scale of Megaquop computers and beyond.

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

2 major / 2 minor

Summary. The manuscript proves that overlapped grouping (via coefficient splitting) for quantum energy estimation can achieve a maximal variance reduction that scales linearly with the number of Hamiltonian terms. It introduces a deterministic 'repacking' algorithm to convert disjoint groups into overlapped ones and shows that this procedure iteratively reduces variance under mild assumptions. Large-scale numerics on Hamiltonians up to 44 qubits and 575k terms demonstrate that the variance reduction relative to state-of-the-art disjoint grouping grows linearly with problem size.

Significance. If the proof and its assumptions hold generally, the linear scaling offers a substantial improvement in measurement overhead for near-term variational quantum algorithms, with clear relevance to scaling toward larger systems. The deterministic algorithm and extensive numerics on practically relevant instances are strengths that enhance the result's applicability.

major comments (2)
  1. [§3] §3 (repacking algorithm and variance-reduction proof): The central claim of linear (in number of terms) maximal variance reduction is established by showing iterative improvement under 'mild assumptions,' but these assumptions are not explicitly enumerated or their robustness tested against common Hamiltonian features such as mixed-sign coefficients or non-trivial commuting structures; this is load-bearing because early termination of the iteration would prevent the claimed linear factor.
  2. [§5] §5 (numerical results, e.g., the scaling plots): The observed linear growth is reported for specific instances, but the manuscript does not compare the achieved variance reduction against the theoretical upper bound derived in the proof, leaving open whether the repacking procedure attains the claimed maximum or saturates earlier on the tested Hamiltonians.
minor comments (2)
  1. [§2-3] Notation for the overlapped groups and the variance expression could be unified between the proof and the algorithm pseudocode to avoid reader confusion.
  2. [§5] Figure captions in the numerics section should explicitly state the number of terms and qubits for each data point to facilitate direct comparison with the linear scaling claim.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading, positive assessment of the significance of our results, and constructive comments. We address each major comment below and outline revisions that will strengthen the manuscript.

read point-by-point responses

  1. Referee: [§3] §3 (repacking algorithm and variance-reduction proof): The central claim of linear (in number of terms) maximal variance reduction is established by showing iterative improvement under 'mild assumptions,' but these assumptions are not explicitly enumerated or their robustness tested against common Hamiltonian features such as mixed-sign coefficients or non-trivial commuting structures; this is load-bearing because early termination of the iteration would prevent the claimed linear factor.

    Authors: We agree that the assumptions should be stated explicitly for clarity and to allow readers to assess their generality. In the revised manuscript we will add an enumerated list of the mild assumptions directly in §3, including the precise conditions under which the iterative repacking is guaranteed to continue. The proof itself is formulated for general coefficients (including mixed signs) and does not rely on all-positive coefficients or trivial commutation relations; we will add a short paragraph clarifying this generality and noting that the assumptions are satisfied by the molecular and spin Hamiltonians used in our numerics. We will also include a brief robustness discussion, supported by additional small-scale tests on Hamiltonians with varied sign patterns and commuting structures, to confirm that early termination does not occur for the problem classes of interest. revision: yes

  2. Referee: [§5] §5 (numerical results, e.g., the scaling plots): The observed linear growth is reported for specific instances, but the manuscript does not compare the achieved variance reduction against the theoretical upper bound derived in the proof, leaving open whether the repacking procedure attains the claimed maximum or saturates earlier on the tested Hamiltonians.

    Authors: We acknowledge that an explicit comparison to the theoretical upper bound would be valuable. For the largest instances (hundreds of thousands of terms) an exact computation of the bound is intractable, as it requires solving a combinatorial optimization problem over all possible overlapped groupings. In the revision we will add a dedicated paragraph in §5 that (i) recalls the theoretical linear upper bound, (ii) explains the computational intractability for large systems, and (iii) presents new comparisons on smaller Hamiltonians (where the bound is computable via exhaustive or integer-programming methods) showing that repacking reaches reductions within a few percent of the maximum. These smaller-scale results, together with the observed linear scaling on the large instances, support that the algorithm attains the claimed maximal reduction in practice. revision: partial

Circularity Check

0 steps flagged

No circularity: proof and algorithm are independent of inputs

full rationale

The paper derives the linear variance reduction claim via a mathematical proof and introduces a deterministic repacking algorithm whose iterative improvement is shown under explicitly stated mild assumptions. These steps are first-principles constructions that do not reduce to self-definitions, fitted parameters renamed as predictions, or load-bearing self-citations. Numerical experiments on large Hamiltonians serve as validation rather than the derivation itself. The central result remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no identifiable free parameters, axioms, or invented entities. The work rests on standard quantum measurement variance formulas and introduces a new algorithmic procedure without new physical postulates.

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