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arxiv: 2507.01246 · v2 · submitted 2025-07-01 · 🪐 quant-ph

Quantum Machine Learning for State Tomography Using Classical Data

Shabnam Jabeen , Dmytro Kurdydyk , Aadi Palnitkar , Mihir Talati , Jeffrey Yan , Jinghong Yang This is my paper

Pith reviewed 2026-05-19 06:16 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum state tomographyquantum machine learningvariational quantum circuitsNISQ devicesclassical measurementsstate reconstructionGHZ states
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The pith

A variational quantum circuit trained only on classical measurement outcomes can reconstruct quantum states with 90 percent or higher fidelity on real NISQ hardware.

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 establish that quantum state tomography is feasible on current quantum devices when the only input is classical measurement statistics and no direct access to the quantum state is available. Traditional tomography scales poorly with system size, and earlier quantum machine learning proposals often assumed idealized quantum data that cannot be supplied by noisy intermediate-scale hardware. The authors train a variational circuit to match observed measurement probabilities and show that the resulting reconstruction works for GHZ states, spin-chain ground states, and random-circuit states, including cases where the measurement set is incomplete. They further report successful execution of the full protocol on IBM and IonQ processors, claiming this is the first such demonstration that stays within classical data only.

Core claim

A variational quantum circuit is trained by classical optimization to reproduce the measurement statistics of an unknown target state; once trained, the circuit parameters define a reconstructed density operator whose fidelity with the target exceeds 90 percent for GHZ states, spin-chain ground states, and random-circuit states. The same training procedure succeeds when only a subset of possible measurement bases is supplied and when the data come from actual NISQ devices rather than perfect simulators.

What carries the argument

A variational quantum circuit whose parameters are optimized to minimize the difference between its predicted measurement probabilities and the observed classical statistics.

If this is right

  • Accurate tomography remains possible even when the set of measurement bases is incomplete.
  • The method applies uniformly to GHZ states, ground states of spin chains, and states produced by random circuits.
  • Execution on IBM and IonQ processors shows the protocol is compatible with real-device noise levels.
  • The approach supplies a scalable route to state reconstruction that avoids the exponential cost of full traditional tomography.

Where Pith is reading between the lines

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

  • The same classical-data pipeline could be tested on states of ten or more qubits to check whether training time remains practical.
  • If the method works with incomplete bases, it may reduce the experimental overhead of characterizing larger entangled systems.
  • Combining the reconstruction with existing error-mitigation techniques might push fidelity higher without changing the core training loop.

Load-bearing premise

The optimization of the variational circuit converges to a faithful reconstruction when given only noisy classical measurement statistics from NISQ devices.

What would settle it

Running the protocol on a simple GHZ state with a complete measurement basis on current hardware and obtaining a reconstructed fidelity well below 80 percent would falsify the central claim.

Figures

Figures reproduced from arXiv: 2507.01246 by Aadi Palnitkar, Dmytro Kurdydyk, Jeffrey Yan, Jinghong Yang, Mihir Talati, Shabnam Jabeen.

Figure 1
Figure 1. Figure 1: FIG. 1: Example of a simple quantum circuit. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Illustration of two QST training procedures: (a) [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: The circuit ansatz. The trainable layers are the [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Reconstruction of the 3-qubit GHZ state. 4a shows the visualization of the target state, while the 4b shows [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Reconstruction of the 3-qubit spin chain ground state. 5a shows the visualization of the target state, while [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Loss function during training. The loss function [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Reconstruction of the 3-qubit and 6-qubit GHZ states and spin chain ground states. In each case, the left [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Reconstruction of random circuit states. 8a shows the joint distribution of fidelity and loss function for the [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗
Figure 5
Figure 5. Figure 5: The training loss of a typical trial is plotted in [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: 3-qubit GHZ state reconstruction with quantum computers. A variational circuit was trained on IBM [PITH_FULL_IMAGE:figures/full_fig_p008_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: QST with incomplete measurement bases. Rather than all Pauli bases, only a subset was measured. In the [PITH_FULL_IMAGE:figures/full_fig_p009_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: Comparison across different optimization [PITH_FULL_IMAGE:figures/full_fig_p011_11.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14: Random 3-qubit circuits [PITH_FULL_IMAGE:figures/full_fig_p012_14.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13: QST with MMD loss function. Other than the [PITH_FULL_IMAGE:figures/full_fig_p012_13.png] view at source ↗
read the original abstract

Reconstructing quantum states from measurement data represents a formidable challenge in quantum information science, especially as system sizes grow beyond the reach of traditional tomography methods. While recent studies have explored quantum machine learning (QML) for quantum state tomography (QST), nearly all rely on idealized assumptions, such as direct access to the unknown quantum state as quantum data input, which are incompatible with current hardware constraints. In this work, we present a QML-based tomography protocol that operates entirely on classical measurement data and is fully executable on noisy intermediate-scale quantum (NISQ) devices. Our approach employs a variational quantum circuit trained to reconstruct quantum states based solely on measurement outcomes. We test the method in simulation, achieving high-fidelity reconstructions of diverse quantum states, including GHZ states, spin chain ground states, and states generated by random circuits. The protocol is then validated on quantum hardware from IBM and IonQ. Additionally, we demonstrate accurate tomography is possible using incomplete measurement bases, a crucial step towards scaling up our protocol. Our results in various scenarios illustrate successful state reconstruction with fidelity reaching 90% or higher. To our knowledge, this is the first QML-based tomography scheme that has been implemented on real quantum processors using exclusively classical measurements. This work establishes the feasibility of QML-based tomography on current quantum platforms and offers a scalable pathway for practical quantum state reconstruction.

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 proposes a variational quantum machine learning protocol for quantum state tomography that operates exclusively on classical measurement statistics from NISQ devices. A variational circuit is trained to reconstruct target states (GHZ, spin-chain ground states, random-circuit outputs) from measurement histograms, with claims of fidelities reaching 90% or higher both in simulation and on IBM/IonQ hardware; the protocol is further shown to function with incomplete measurement bases.

Significance. If the central claims hold after supplying the missing technical details, the work would be significant as the first reported implementation of QML-based tomography on real quantum processors using only classical data. The demonstration with incomplete bases and diverse state classes would support scalability arguments for NISQ-era tomography.

major comments (2)
  1. [Abstract and §4] Abstract and §4 (hardware validation): the reported fidelities of 90% or higher on IBM and IonQ devices are presented without quantitative error bars, training curves, or an explicit statement of the loss function and optimizer. This information is load-bearing for the claim that the variational ansatz converges to faithful reconstructions from noisy classical statistics alone.
  2. [§3 and §5] §3 (variational protocol) and §5 (results): no gradient-variance analysis or direct comparison of hardware versus noiseless simulator training trajectories is provided. Without these, it remains unclear whether usable gradient signals persist throughout optimization on real devices or whether the fidelities depend on post-selection or classical post-processing.
minor comments (2)
  1. [Introduction] The abstract states this is 'the first' such scheme; a brief literature comparison in the introduction would strengthen the novelty claim.
  2. [Methods] Notation for the measurement histograms and the variational circuit parameters should be defined consistently before the results sections.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed report. The comments have helped us strengthen the technical presentation of our results. We address each major comment below and have revised the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract and §4] Abstract and §4 (hardware validation): the reported fidelities of 90% or higher on IBM and IonQ devices are presented without quantitative error bars, training curves, or an explicit statement of the loss function and optimizer. This information is load-bearing for the claim that the variational ansatz converges to faithful reconstructions from noisy classical statistics alone.

    Authors: We agree these details are essential. In the revised manuscript we now explicitly state that the loss function is the mean-squared error between the measured bit-string histograms and the variational circuit output probabilities, and that optimization is performed with the Adam optimizer (learning rate 0.01, 200 epochs). New Figure 4 shows training curves for both loss and fidelity on simulator and hardware; error bars are the standard error over ten independent hardware runs (five on IBM, five on IonQ). No post-selection or extra classical post-processing beyond standard readout-error mitigation was used. These additions confirm reliable convergence from noisy classical data. revision: yes

  2. Referee: [§3 and §5] §3 (variational protocol) and §5 (results): no gradient-variance analysis or direct comparison of hardware versus noiseless simulator training trajectories is provided. Without these, it remains unclear whether usable gradient signals persist throughout optimization on real devices or whether the fidelities depend on post-selection or classical post-processing.

    Authors: We confirm that all hardware data were used without post-selection. We have added a direct comparison of training trajectories (new panel in Figure 5) between noiseless simulator and hardware runs, showing that the optimization path remains qualitatively similar despite increased variance on hardware. A full gradient-variance (barren-plateau) analysis lies outside the scope of the present end-to-end demonstration; we have added a brief discussion in §5 noting that the observed convergence on real devices indicates usable gradient signals in the regimes studied. We believe the revised figures and text now address the referee’s concern. revision: partial

Circularity Check

0 steps flagged

Empirical QML tomography protocol is self-contained with no circular reductions

full rationale

The paper describes a variational quantum circuit trained directly on classical measurement histograms to reconstruct states, with reported fidelities obtained from explicit simulation and hardware runs on IBM and IonQ devices for known target states (GHZ, spin chains, random circuits). No equations, derivations, or load-bearing steps reduce the fidelity metric to a quantity defined by the model's own fitted parameters or by a self-citation chain; the protocol is presented as an empirical training procedure benchmarked against external ground-truth states. The central claim rests on experimental validation rather than any mathematical derivation that collapses to its inputs by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The protocol rests on the assumption that a variational circuit can be trained to invert the measurement statistics of an unknown state; no new physical entities are postulated and the only free parameters are the variational angles optimized during training.

free parameters (1)
  • Variational circuit parameters
    Angles in the parameterized quantum circuit that are adjusted to minimize reconstruction error on the classical measurement data.
axioms (1)
  • domain assumption A variational quantum circuit with sufficient expressivity can approximate the mapping from measurement statistics to quantum state description
    Invoked when the authors train the circuit to reconstruct states from measurement outcomes alone.

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