$\sigma$-VQE: Excited-state preparation of quantum many-body scars with shallow circuits

Eoin Carolan; Gabriele Cenedese; Giuliano Benenti; Nathan Keenan

arxiv: 2602.20881 · v2 · pith:QPRINSHJnew · submitted 2026-02-24 · 🪐 quant-ph

σ-VQE: Excited-state preparation of quantum many-body scars with shallow circuits

Eoin Carolan , Nathan Keenan , Gabriele Cenedese , Giuliano Benenti This is my paper

Pith reviewed 2026-05-21 12:56 UTC · model grok-4.3

classification 🪐 quant-ph

keywords quantum many-body scarsvariational quantum eigensolverexcited statesshallow circuitsenergy varianceNISQ algorithmsscar statesvariational preparation

0 comments

The pith

A modified VQE with shallow circuits and an energy-variance penalty prepares quantum many-body scar states by favoring their low entanglement.

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

The paper introduces σ-VQE to target mid-spectrum eigenstates such as quantum many-body scars on near-term quantum hardware. It pairs a low-depth variational circuit with a cost function that selects a target energy while explicitly penalizing fluctuations in the measured energy. This works because shallow circuits have restricted expressibility and cannot readily produce generic highly entangled states, so they preferentially locate atypical low-entanglement eigenstates like scars. A sympathetic reader would care because conventional approaches to excited states often require deep circuits or full spectrum methods that exceed current device capabilities.

Core claim

By minimizing a cost that combines a term selecting a chosen target energy with an explicit penalty on energy variance, a shallow variational ansatz converges to quantum many-body scar states rather than typical mid-spectrum eigenstates. The approach is validated on two families of scar-supporting models constructed via the Shiraishi-Mori embedding and via matrix-product-state parent Hamiltonians. An unbiased estimator for the nonlinear cost is provided that remains compatible with qubit-wise commuting grouping and bitstring reuse, and a small-system instance is demonstrated on IBM quantum hardware.

What carries the argument

The σ-VQE cost function that adds an energy-variance penalty to a target-energy term, using the restricted expressivity of shallow circuits to select low-entanglement eigenstates.

If this is right

Scar states become accessible with circuit depths feasible on current hardware.
The same cost function works across different constructions of Hamiltonians that host scars.
An estimation scheme allows the nonlinear objective to be evaluated with standard measurement groupings.
Variational states with appreciable scar overlap can be generated without requiring full diagonalization.

Where Pith is reading between the lines

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

The variance-penalty idea may extend to preparing other atypical low-entanglement excited states in systems that lack scars.
Hardware limitations on entanglement generation can be turned into a feature for targeted state preparation rather than an obstacle.
The method suggests a broader strategy for excited-state search that scales with the natural restrictions of near-term devices.

Load-bearing premise

Penalizing energy variance around a chosen target will cause shallow circuits to select low-entanglement scar states over more typical entangled eigenstates.

What would settle it

Preparing a state whose entanglement entropy matches that of a typical random eigenstate at the target energy, or obtaining negligible overlap with known scar states in the benchmark models.

Figures

Figures reproduced from arXiv: 2602.20881 by Eoin Carolan, Gabriele Cenedese, Giuliano Benenti, Nathan Keenan.

**Figure 2.** Figure 2: FIG. 2. We plot the bipartite entanglement entropy, [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. We simulate the [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗

**Figure 6.** Figure 6: to single-run optimization to characterize baseline behavior, standard variational-algorithm strategies may improve robustness in higher χ regimes. Examples include restart strategies with different initial circuit parameters to mitigate optimizer sensitivity in rugged landscapes combined with the use of cheaper initial optimizers such as SPSA when one wishes to reduce per-iteration measurement overhead to… view at source ↗

**Figure 7.** Figure 7: FIG. 7. Hardware implementation of [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗

**Figure 8.** Figure 8: FIG. 8. The results of our noisy emulation using 535 [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗

read the original abstract

We present and benchmark a type of variational quantum eigensolver (VQE), which we denote $\sigma$-VQE. It is designed to target mid-spectrum eigenstates and prepare quantum many-body scar states. The approach leverages the fact that noisy intermediate-scale quantum devices are limited in their ability to generate generic highly entangled states. This modified VQE pairs a low-depth circuit with an energy-selective objective that explicitly penalizes energy variance around a chosen target energy. The cost function exploits the limited expressibility of the shallow circuit as atypical low-entanglement eigenstates such as scar states are preferentially selected. We validate this mechanism across two complementary families of models that contain many-body scar states: the Shiraishi-Mori embedding approach and a matrix product state parent Hamiltonian construction. We define an unbiased estimation scheme for the nonlinear cost function that is compatible with qubit-wise commuting grouping and bitstring reuse. A proof-of-principle demonstration using a small-system instance was performed on IBM Fez (Heron r2 QPU). These results motivate its use as a practical algorithm for detecting quantum many-body scars and variationally generating states with appreciable scar state overlap.

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.

Desk Editor's Note private letter to a colleague

σ-VQE pairs a variance penalty with shallow circuits to target scars, but lacks controls to show whether limited expressibility is doing real work beyond the cost function.

read the letter

Hey, the one or two things to know about this paper are that they've come up with σ-VQE, which tweaks the standard variational eigensolver by penalizing energy variance around a chosen target energy, and they use shallow circuits to help select quantum many-body scar states because those circuits struggle with high entanglement. What is new here is the specific objective that exploits both the variance penalty and the limited expressibility of shallow ansatzes to favor low-entanglement mid-spectrum states like scars. They test this on two different model families that host scars, the Shiraishi-Mori embedding and the matrix product state parent Hamiltonian approach. They also provide a scheme for estimating the nonlinear cost function without bias, which is compatible with qubit-wise commuting measurements and allows reusing bitstrings. That practical detail is useful. Plus, they include a small-scale demonstration on the IBM Fez Heron r2 QPU. The paper does well in grounding the method in concrete models and showing it's runnable on current hardware. On the soft spots, the central mechanism assumes that the shallow circuit's restricted ability to generate entangled states is what helps select the scars when the variance penalty is applied. But as the stress test notes, there's no control experiment where they use a deeper circuit with the same cost function to check if the variance term by itself would converge to similar states. That leaves open the possibility that the shallowness is not the key selector. The validation is more proof-of-principle, and without detailed benchmarks, error bars, or comparisons to other scar preparation methods, it's difficult to see exactly how effective it is or how it stacks up. This paper is for quantum information researchers and condensed matter physicists working on many-body scars and non-thermalizing dynamics, especially those trying to implement things on NISQ devices. A reader interested in variational quantum algorithms for excited states would find the approach and the estimation scheme valuable. It deserves a serious referee because it introduces a targeted method with some experimental validation and addresses a relevant problem in quantum simulation. I would recommend sending it to peer review, suggesting they add the depth control experiments to clarify the role of the circuit shallowness.

Referee Report

2 major / 1 minor

Summary. The manuscript introduces σ-VQE, a variational quantum eigensolver variant for targeting mid-spectrum eigenstates and quantum many-body scar states. It pairs a shallow-depth circuit ansatz with a cost function that penalizes energy variance around a chosen target energy, claiming that the circuit's limited expressibility preferentially selects atypical low-entanglement scar states. The approach is validated on Shiraishi-Mori embedding models and matrix-product-state parent Hamiltonians, includes an unbiased estimation scheme compatible with qubit-wise commuting grouping, and reports a proof-of-principle hardware run on IBM Fez.

Significance. If the central mechanism is confirmed, the method could provide a practical route to scar-state preparation on NISQ hardware by turning limited circuit depth into an advantage rather than a limitation. The variance-penalized objective is independent of detailed target-state structure, and the measurement scheme supports efficient implementation. These elements, together with the hardware demonstration, position the work as a potentially useful addition to excited-state variational algorithms.

major comments (2)

[Abstract and model-validation sections] Abstract and validation sections: The central claim states that 'the cost function exploits the limited expressibility of the shallow circuit as atypical low-entanglement eigenstates such as scar states are preferentially selected.' This requires a control that increases circuit depth (or expressivity) while holding the σ-cost function and target energy fixed; without such a comparison it remains possible that the variance penalty alone drives convergence to mid-spectrum states of low variance, rendering the shallowness incidental. No such control is reported.
[Abstract] Abstract: The description of the IBM Fez run is presented as a proof-of-principle but supplies no quantitative benchmarks, error bars, overlap values, or comparison against standard VQE or other excited-state methods. This weakens the ability to assess whether the observed states indeed exhibit appreciable scar overlap or outperform alternatives.

minor comments (1)

[Methods] The notation and definition of the nonlinear cost function (denoted σ) should be introduced with an explicit equation early in the methods, including how the target energy is chosen and how the variance term is normalized.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed feedback on our manuscript. We address each major comment below and describe the revisions we will implement to strengthen the validation and presentation of the results.

read point-by-point responses

Referee: [Abstract and model-validation sections] Abstract and validation sections: The central claim states that 'the cost function exploits the limited expressibility of the shallow circuit as atypical low-entanglement eigenstates such as scar states are preferentially selected.' This requires a control that increases circuit depth (or expressivity) while holding the σ-cost function and target energy fixed; without such a comparison it remains possible that the variance penalty alone drives convergence to mid-spectrum states of low variance, rendering the shallowness incidental. No such control is reported.

Authors: We agree that a direct control comparing shallow versus deeper circuits (with the σ-cost function and target energy held fixed) would provide clearer evidence that limited expressibility plays a key role in preferentially selecting scar states. Our current validation demonstrates that σ-VQE successfully targets scar states in the Shiraishi-Mori embedding models and MPS parent Hamiltonians using shallow circuits, but we acknowledge the absence of this explicit comparison. In the revised manuscript we will add numerical results from deeper-circuit controls to address this point directly. revision: yes
Referee: [Abstract] Abstract: The description of the IBM Fez run is presented as a proof-of-principle but supplies no quantitative benchmarks, error bars, overlap values, or comparison against standard VQE or other excited-state methods. This weakens the ability to assess whether the observed states indeed exhibit appreciable scar overlap or outperform alternatives.

Authors: The IBM Fez demonstration is intended as a proof-of-principle showing that the method can be executed on current NISQ hardware. We agree that quantitative metrics would allow better evaluation of scar overlap and relative performance. In the revised manuscript we will augment the hardware section with error bars on the measured energies, available overlap or fidelity estimates, and a brief comparison against standard VQE and other excited-state variational approaches on the same small system. revision: yes

Circularity Check

0 steps flagged

No circularity: novel objective function defined independently of scar states

full rationale

The σ-VQE cost function is introduced as an explicit penalty on energy variance around a chosen target energy, paired with a shallow-circuit ansatz whose limited expressibility is invoked to explain preferential selection of low-entanglement scar states. This selection mechanism is presented as an empirical consequence rather than being encoded by definition into the cost function itself. No equations reduce a claimed prediction to a fitted parameter or prior result by algebraic identity, and no load-bearing uniqueness theorem or ansatz is imported via self-citation. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the modeling assumption that scar states are atypically low-entanglement and that the variance penalty plus circuit shallowness will select them; no free parameters or invented entities are described in the abstract.

axioms (1)

domain assumption Scar states possess atypically low entanglement compared with generic mid-spectrum eigenstates.
Invoked to explain why shallow circuits preferentially select them under the variance penalty.

pith-pipeline@v0.9.0 · 5742 in / 1142 out tokens · 41315 ms · 2026-05-21T12:56:47.311112+00:00 · methodology

discussion (0)

Reference graph

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The second uses a parent-Hamiltonian embedding of random matrix-product states (MPSs) [33], enabling controlled tuning of scar state entanglement

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[1] [1]

The second uses a parent-Hamiltonian embedding of random matrix-product states (MPSs) [33], enabling controlled tuning of scar state entanglement

to embed a product eigenstate at finite energy density into an otherwise chaotic spectrum. The second uses a parent-Hamiltonian embedding of random matrix-product states (MPSs) [33], enabling controlled tuning of scar state entanglement. Finally, in Sec. IV we benchmarkσ-VQE performance in idealized simulation and under realistic finite-shot and hardware ...

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