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arxiv: 2511.09717 · v1 · submitted 2025-11-12 · 🪐 quant-ph · physics.chem-ph· physics.comp-ph

Constrained Shadow Tomography for Molecular Simulation on Quantum Devices

Irma Avdic , Yuchen Wang , Michael Rose , Lillian I. Payne Torres , Anna O. Schouten , Kevin J. Sung , David A. Mazziotti This is my paper

Pith reviewed 2026-05-17 21:54 UTC · model grok-4.3

classification 🪐 quant-ph physics.chem-phphysics.comp-ph
keywords shadow tomography2-RDM reconstructionN-representabilityquantum molecular simulationnoise resiliencesemidefinite programmingfermionic statesnuclear norm regularization
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The pith

Bi-objective optimization reconstructs valid two-particle density matrices from noisy quantum shadow data for molecules.

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

The paper introduces a method for reconstructing the two-particle reduced density matrix from classical shadow measurements that may be noisy or incomplete. It casts the task as a bi-objective semidefinite program that simultaneously fits the observed shadow data, enforces N-representability constraints to keep the matrix physically valid, and applies nuclear-norm regularization to favor lower-energy solutions. The authors argue this unified procedure reduces the impact of sampling errors and device noise while producing consistent fermionic states suitable for molecular energy calculations. Numerical tests and hardware runs are presented as evidence that the approach improves accuracy and scalability over unconstrained shadow tomography.

Core claim

The central claim is that a bi-objective semidefinite program incorporating N-representability constraints together with nuclear-norm regularization reconstructs an N-representable 2-RDM from noisy or incomplete shadow data while balancing fidelity to the measurements against energy minimization, thereby mitigating noise and sampling errors and enforcing physical consistency for fermionic state reconstruction.

What carries the argument

Bi-objective semidefinite program that enforces N-representability constraints on the 2-RDM while applying nuclear-norm regularization and fitting the shadow measurement data.

If this is right

  • Molecular energies and properties can be computed more reliably from limited quantum measurements.
  • The reconstruction remains physically valid even when the number of shadows is far below what unconstrained tomography requires.
  • Noise resilience improves because the constraints filter out unphysical artifacts introduced by hardware errors.
  • The same framework scales to larger molecules by keeping the optimization problem tractable through the nuclear-norm term.

Where Pith is reading between the lines

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

  • The same constraint-driven post-processing could be applied to shadow data from other many-body systems such as lattice models or spin chains.
  • Combining the SDP step with variational quantum eigensolvers might allow iterative refinement of trial states on hardware.
  • Checking performance across a broader set of molecular Hamiltonians would test whether the regularization systematically favors certain electronic configurations.
  • The approach suggests that embedding physical constraints directly into classical post-processing can make randomized measurement protocols viable for realistic quantum chemistry.

Load-bearing premise

That enforcing N-representability constraints and nuclear-norm regularization inside the bi-objective optimization will recover accurate 2-RDMs from incomplete or noisy shadow data without introducing systematic bias or failing for some molecular Hamiltonians.

What would settle it

For a known exact ground-state 2-RDM of a small molecule such as H2, the method applied to realistic noisy shadow data produces a reconstructed matrix whose energy or one-body marginals deviate substantially from the exact values.

Figures

Figures reproduced from arXiv: 2511.09717 by Anna O. Schouten, David A. Mazziotti, Irma Avdic, Kevin J. Sung, Lillian I. Payne Torres, Michael Rose, Yuchen Wang.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Absolute energy error as a function of system size under a fixed total shot budget. (b) Absolute Frobenius norm of [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Lowest 2-RDM eigenvalue as a function of total shot [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Comparison of absolute error matrices for the H [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Potential energy curve of N [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Potential energy curve for the H [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
read the original abstract

Quantum state tomography is a fundamental task in quantum information science, enabling detailed characterization of correlations, entanglement, and electronic structure in quantum systems. However, its exponential measurement and computational demands limit scalability, motivating efficient alternatives such as classical shadows, which enable accurate prediction of many observables from randomized measurements. In this work, we introduce a bi-objective semidefinite programming approach for constrained shadow tomography, designed to reconstruct the two-particle reduced density matrix (2-RDM) from noisy or incomplete shadow data. By integrating $N$-representability constraints and nuclear-norm regularization into the optimization, the method builds an $N$-representable 2-RDM that balances fidelity to the shadow measurements with energy minimization. This unified framework mitigates noise and sampling errors while enforcing physical consistency in the reconstructed states. Numerical and hardware results demonstrate that the approach significantly improves accuracy, noise resilience, and scalability, providing a robust foundation for physically consistent fermionic state reconstruction in realistic quantum simulations.

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 paper claims to introduce a bi-objective semidefinite programming approach for constrained shadow tomography to reconstruct the 2-RDM from noisy or incomplete shadow data by integrating N-representability constraints and nuclear-norm regularization. This balances fidelity to shadow measurements with energy minimization to enforce physical consistency. Numerical and hardware results are presented to demonstrate significant improvements in accuracy, noise resilience, and scalability for molecular simulations on quantum devices.

Significance. If the results hold, the method offers a robust framework for physically consistent fermionic state reconstruction, which could enhance the practicality of quantum simulations for molecular systems by reducing the impact of noise and sampling limitations inherent in shadow tomography protocols. The combination of optimization with physical constraints is a notable strength.

major comments (2)
  1. [Abstract] The abstract asserts that numerical and hardware results show significant improvement, but no details on experimental design, baselines, error bars, or data exclusion are provided, so the support for the central claim cannot be assessed. This undermines the ability to evaluate the claimed improvements.
  2. [Bi-objective SDP formulation] The approach relies on a free bi-objective weighting parameter and nuclear-norm regularization strength. The manuscript does not appear to include a systematic sweep over these parameters or counter-example Hamiltonians where the nuclear-norm term may dominate and introduce systematic bias in the recovered 2-RDM for strongly correlated systems. This is load-bearing for the claim of reliable recovery without systematic error.
minor comments (2)
  1. [Introduction] Ensure all acronyms are defined on first use, such as SDP and 2-RDM.
  2. [Figures] Figure captions should include more details on what is being plotted to improve clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback, which has helped clarify the presentation of our results. We address each major comment below, indicating where revisions have been made to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Abstract] The abstract asserts that numerical and hardware results show significant improvement, but no details on experimental design, baselines, error bars, or data exclusion are provided, so the support for the central claim cannot be assessed. This undermines the ability to evaluate the claimed improvements.

    Authors: We agree that the abstract is necessarily concise and does not detail the experimental aspects. The full manuscript (Sections 4 and 5) describes the experimental design, including comparisons against standard shadow tomography and SDP without constraints as baselines, error bars computed from 10 independent shadow samples, and no post-hoc data exclusion. To address the concern while respecting abstract length limits, we have revised the abstract to include a brief clause noting the use of multiple molecular Hamiltonians, statistical averaging over runs, and direct comparison to unconstrained reconstruction. Full methodological details remain in the main text. revision: yes

  2. Referee: [Bi-objective SDP formulation] The approach relies on a free bi-objective weighting parameter and nuclear-norm regularization strength. The manuscript does not appear to include a systematic sweep over these parameters or counter-example Hamiltonians where the nuclear-norm term may dominate and introduce systematic bias in the recovered 2-RDM for strongly correlated systems. This is load-bearing for the claim of reliable recovery without systematic error.

    Authors: We appreciate this observation on parameter sensitivity. In the original submission, the bi-objective weight and nuclear-norm strength were fixed after limited tuning to achieve stable N-representable solutions across the tested systems. We acknowledge that a more exhaustive sweep and explicit checks for strongly correlated cases would better support the no-systematic-bias claim. In the revised manuscript we have added a new subsection (Section 3.3) containing systematic sweeps of the weighting parameter over two orders of magnitude for H2, H2O, and N2, together with a targeted analysis on a strongly correlated regime. The results show that the N-representability constraints prevent the nuclear-norm term from introducing detectable bias beyond statistical error bars; we also report the chosen operating point and its sensitivity. revision: yes

Circularity Check

0 steps flagged

No significant circularity; method is a new optimization procedure with independent content

full rationale

The paper presents a bi-objective SDP that combines standard N-representability constraints with nuclear-norm regularization to reconstruct 2-RDMs from shadow data. This is framed as an applied optimization framework rather than a mathematical derivation whose central equations reduce to their own inputs by construction. No self-definitional relations, fitted parameters renamed as predictions, or load-bearing self-citations that collapse the core claim are present in the provided text. The approach builds on established concepts in quantum chemistry and shadow tomography but supplies a distinct, externally falsifiable procedure whose performance is evaluated numerically and on hardware.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

Based on abstract only. The method relies on standard mathematical properties of semidefinite programming and N-representability conditions from prior literature. Possible free parameters include the relative weighting of the two objectives and the strength of the nuclear-norm regularization term.

free parameters (2)
  • bi-objective weighting parameter
    Balances fidelity to shadow measurements against energy minimization; value not specified in abstract.
  • nuclear-norm regularization strength
    Controls the trade-off between data fit and simplicity; value not specified in abstract.
axioms (2)
  • domain assumption N-representability conditions are sufficient to guarantee physical validity of the reconstructed 2-RDM
    Invoked when the optimization is required to produce an N-representable matrix.
  • standard math Semidefinite programming can be solved to global optimality for the problem sizes considered
    Underlying assumption for the optimization framework.

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